The Two Cultures: Shared Problems

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The Two Cultures: Shared Problems

Ernesto Carafoli · Gian Antonio Danieli Giuseppe O. Longo Editors

The Two Cultures: Shared Problems

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Ernesto Carafoli Dipartimento di Chimica Biologica, Università degli Studi di Padova Istituto Veneto di Medicina Molecolare Padova, Italy

Giuseppe O. Longo Dipartimento di Elettrotecnica, Elettronica e Informatica Università degli Studi di Trieste Italy

Gian Antonio Danieli Dipartimento di Biologia Università degli Studi di Padova Italy

ISBN 978-88-470-0868-7

e-ISBN 978-88-470-0869-4

DOI 10.1007/978-88-470-0869-4 Library of Congress Control Number: 2008943321 © 2009 Springer-Verlag Italia This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproductiononmicrofilmor in anyotherway, andstorage indatabanks.Duplication of this publication or parts thereof is only permitted under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Italian Copyright Law. Cover concept: Simona Colombo, Milano Cover figure: Arrigo Marcolini Typesetting: le-tex publishing services oHG, Leipzig, Germany Printing and binding: Printer Trento, Trento Printed on acid-free paper Springer-Verlag Italia – Via Decembrio 28 – 20137 Milano springer.com

Preface

The phrase “The Two Cultures” was used by Charles Percy Snow in the title of his 1959 Rede Lecture at the University of Cambridge: although he had already used it in an article he had published three years earlier in The New Statesman, it was the Rede Lecture that attracted to it immediate and very wide attention. In the Lecture, Snow had actually only spoken of scientific as opposed to literary culture and had not discussed artistic culture in general. He had polemically equated the literary culture of his time with the traditional culture, which was totally, and somewhat even proudly, ignorant of science. Predictably, the literary intellectuals reacted furiously, and a few years later in a second publication Snow thus talked optimistically of a Third Culture that would foster the dialogue between scientists and literary intellectuals. The extension of the literary culture to the full panorama of arts, which had not been made by Snow, emerged gradually and, in hindsight, logically. Now, 50 years after the Rede Lecture, the expression “Two Cultures” is normally used to define broadly the scientific and the artistic cultures, not only the scientific and the literary cultures. And it has gradually become acknowledged that the two cultures have basic differences: the scientific culture being objective and in need of verification, the artistic culture, and that would of course include Snow’s original literary culture, being instead subjective and in no need of verification. However, even if intrinsically different, both cultures are concerned with the big problems that have troubled humankind since it started to think: except that they have tackled, and still tackle them, from angles that are sometimes so different that they make the dialogue all but impossible. Yet, in line with Snow’s optimistic concept of a Third Culture, such a dialogue, no matter how difficult, is necessary. This book is the result of a Symposium that had attempted to foster it. It has done so by identifying important problems that are common to the two cultures, and by confronting them with the specific intellectual tools they normally use. They are the largest problems that confront humankind, from that of time and its meaning, to that of infinity, to that of nothingness, to the “number one” problem, which was defined as the “Grundfrage” by German philosophers, synthesized by the famous rhetorical question first clearly asked by Leibniz more than 300 years ago: why is there something instead of

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nothing? The Grundfrage actually defines in a striking manner a fundamental problem the two cultures have tackled in very different ways. The great philosophers of European Christianity have traditionally discussed it with reference to God. Somehow, from Leibniz, to Kant, to Schelling, science was never a significant partner in the discussion. It was only in the last century that Western philosophy became attentive to science when considering this fundamental problem. Perhaps the most notable example of this enlargement of perspective was the famous Opening Lecture by Martin Heidegger in 1929: “Was ist Metaphysik?” After that, answers to the Grundfrage were rationally attempted with the contribution of the scientific culture, without necessarily escaping into metaphysics. One last point could be mentioned, because it emphasizes the striking difference in the ways of thinking between European Christianity, from Augustine all the way to Heidegger, the early Greeks and, especially, Orientals. Three-thousand years before Leibniz, the famously beautiful Hymn of the Creation in the philosophicalmythological text of the Indoarian civilization, the Rig Veda, sung the problem of nothingness and the story of the beginning without claiming the intervention of Divinity: from the dark watery chaos the world emerged by the power of heat: desire, the first seed of thought, linked the nonexistent to the existent. Importantly, the Hymn explicitly declares that the Gods only came later, and ends by saying that nobody, perhaps not even He, who sits up there, knows how it all came about. Essentially, the coming into being of the creation thus remains clouded in mystery Who knows, really? Who will declare, here, Whence it was produced, whence this creation? The Gods came afterwards, with the creation of this universe. Who, then, knows whence it has come into being? Whence this creation has arisen – perhaps it formed itself, or perhaps it did not – He, up there, only He knows. Or perhaps He knows not.

Of course these fundamental problems, of which that sung in the Hymn of the Creation is the most challenging, only have meaning because the human mind is able to think them. Thus, the second part of the book converges into a discussion of the structure of the creativity process, analyzing the mechanism by which the human mind creates, irrespective of whether the product of this creativity ends up in the scientific or artistic culture. Concepts of obvious interest to both cultures will also be discussed in detail, like symmetry and beauty itself. It will appear that the traditional dichotomy between truth and beauty, which are often and so simplistically proclaimed to be the separate goals of the scientific and artistic cultures, is an insufficient approximation. The aim of both cultures indeed is the quest for both truth and beauty. December 2008

The Editors

Acknowledgements We are grateful to Dr. Laura Fedrizzi and Dr. Sebastiano Pedrocco for their precious cooperation. We would also like to thank Arrigo Marcolini for providing the cover figure.

List of Contributors

Luciano Boi École des Hautes Études en Sciences Sociales, Centre de Mathématiques Laboratoire de l’Univers et ses Théories (LUTH), Observatoire de Paris-Meudon, EHESS-CAMS, Paris, France Mario Botta Università della Svizzera Italiana Studio arch. Mario Botta, Lugano, Switzerland Dmitri Bougakov 3/5 West 57th Street, New York, USA Ernesto Carafoli Università degli Studi di Padova, Dipartimento di Chimica Biologica, Istituto Veneto di Medicina Molecolare, Padova, Italy Ruth Durrer Département de Physique Théorique, Université de Genève, Switzerland Maurizio Ferraris Dipartimento di Filosofia, Università degli Studi di Torino, Italy Elkhonon Goldberg Department of Neurology, New York University School of Medicine, New York, NY, USA vii

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Giuseppe O. Longo Dipartimento di Elettrotecnica, Elettronica e Informatica, Università degli Studi di Trieste, Italy Eugenio Mazzarella Dipartimento di Filosofia, Università degli Studi di Napoli “Federico II”, Italy J.J.A. Mooij Rijksuniversitat Groningen, Faculty of Medical Sciences, The Netherlands Piergiorgio Odifreddi Dipartimento di Matematica, Università degli Studi di Torino, Italy Michelangelo Pistoletto Cittadellarte – Fondazione Pistoletto, Biella, Italy Joerg Rasche C.G. Jung Institut Berlin, Germany Marcus du Sautoy Newcollege Oxford, Mathematical Institute, University of Oxford, UK Giancarlo Setti Dipartimento di Astronomia, Università degli Studi di Bologna, Italy Giorgio Vallortigara Centre for Mind/Brain Sciences, Università degli Studi di Trento, Italy Gabriele Veneziano Theory Division, CERN, Geneva, Switzerland Collège de France, Paris, France

List of Contributors

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section 1 Time, Infinity, Cosmology 1

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3

Did Time Have a Beginning? A Meeting Point for Science and Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriele Veneziano 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Time in Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General Relativity and the Beginning-of-Time Myth . . . . . . . . . . 1.4 The Beginning of Time: a Necessity or a Myth? . . . . . . . . . . . . . . 1.5 String Theory: What’s That? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 String-Inspired Cosmologies: a Longer History of Time? . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5 6 7 8 10 11

The Flow and the Map: On the Dynamic and Static Views of Time . J.J.A. Mooij 2.1 Three Key Events in the Modern Philosophy of Time . . . . . . . . . 2.2 The Two Views of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Two Views and the Two Cultures . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Evolution of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ruth Durrer 3.1 From Myths to “Rational Explanations” of Phenomena . . . . . . . . 3.1.1 The Chinese Myth of Phan Ku . . . . . . . . . . . . . . . . . . . . .

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13 15 19 23 24

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3.1.2 The Pythagoreans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Ptolemy’s Epicycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Heliocentric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Galileo’s Principle of Relativity . . . . . . . . . . . . . . . . . . . . 3.2 The Evolution of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 29 30 31 31 35 36 37

And the Eternal Zeno Springs to Mind . . . . . . . . . . . . . . . . . . . . . . . . . Piergiorgio Odifreddi 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Perpetual Race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A Complete Autobiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 End of the Race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 A Perfect Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Kafkaesque Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Borges’ Inventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Creating the Physical World ex nihilo? On the Quantum Vacuum and Its Fluctuations . . . . . . . . . . . . . . . . . . Luciano Boi 5.1 Introduction: The Vacuum as a Philosophical and Scientific Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Dirac Idea of Vacuum as a “Particle Sea”: Wave Function, Dirac Equation and Negative Energy . . . . . . . . . 5.3 The Role of Vacuum in Modern Physics: From the Universe to the Quantum World . . . . . . . . . . . . . . . . . . . 5.4 The Problem of the Vacuum and the Conceptual Conflict Between GRT and QM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Topology and Curvature as a Source of Vacuum Fields . . . . . . . . 5.6 The Dirac “Full-Particles Sea” Idea and the Vacuum in Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Hole Theory, Negative Energy Solutions, and Vacuum Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Further Theoretical Remarks on the Vacuum Fluctuations: Poincaré Conformal Invariance and Spontaneous Symmetry Breaking Symmetry (SSB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Creation of Universes from Nothing . . . . . . . . . . . . . . . . . . . . . . . . 5.10 String Landscape and Vacuum Energy: The Emergence of a Multidimensional World from Geometrical Possibilities . . . 5.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 40 41 42 43 43 44 45

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Worldly Nihilism and Theological Nihilism – A Possible Definition . Eugenio Mazzarella 6.1 What Nihilism Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Submission to Exteriority as the Work of Reason . . . . . . . . . . . . . 6.3 What Overcoming Nihilism Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 104 111 112 112

General Discussion on the Themes of the First Section . . . . . . . . . . . .

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Section 2 Intelligence and Emotions 8

Original Knowledge and the Two Cultures . . . . . . . . . . . . . . . . . . . . . . Giorgio Vallortigara 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Chick and the Baby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Animated and Non-Animated Objects . . . . . . . . . . . . . . . . . . . . . . 8.4 Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Many Faces of Intelligence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elkhonon Goldberg and Dmitri Bougakov References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Cycles of Re-Creation – A Psychoanalytical Approach to Music . . . Joerg Rasche 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Chopin, Fantaisie-Impromptu op. 66 . . . . . . . . . . . . . . . . . . . . . . . 10.3 About the Inner Re-Creation of Our World . . . . . . . . . . . . . . . . . . 10.4 Schumann, Kreisleriana op. 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 The “Invisible Girl” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The Second Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 The Girl’s Melody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Perturbations and Re-Consolidations . . . . . . . . . . . . . . . 10.4.5 Introversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.6 Reappearance of the Girl’s Melody . . . . . . . . . . . . . . . . . 10.4.7 The Heartbreaking Moment . . . . . . . . . . . . . . . . . . . . . . . Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 126 131 137 139 142 143 144

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165 168 170 171 172 173 173 174 175 175 177 178 180 180

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Section 3 Beauty and Creativity 11 Symmetry: A Bridge Between the Two Cultures . . . . . . . . . . . . . . . . . Marcus du Sautoy 11.1 Symmetry: The Language of Nature . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Symmetry: A Blueprint for the Arts . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Music: The Process of Sounding Mathematics . . . . . . . . . . . . . . . 11.4 The Mathematics of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Symmetry in Hyperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Dynamics of Beauty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giuseppe O. Longo 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Spatial Dynamics: Near–Far . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Relational Dynamics: Object and Subject . . . . . . . . . . . . . . . 12.4 Co-Evolutionary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 The Dynamics of Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Remarks on Symmetry and the Golden Ratio . . . . . . . . . . . . . . . . 12.7 The Beauty of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Language and Thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Scientific and Artistic Creativity: In Search of Unifying Analogies Ernesto Carafoli 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Beauty and Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Symmetry and Beauty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 The Quest for Truth in Artistic Creativity . . . . . . . . . . . . . . . . . . . 13.4.1 Music, Mathematics, and the Golden Ratio . . . . . . . . . . 13.4.2 Visual Arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Interplay Between Science and Art: An Example from the Architectural Work of Michelangelo . . . . 13.6 More on Art as a Path to Knowledge: Expressionism and Music 13.7 The Search for Truth in Poetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 The Structure of the Creativity Process. The Essential Role of Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Intuition in Scientific Creativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10 Intuition in the Work of Poets: The “Gift of God” . . . . . . . . . . . . 13.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 187 188 190 194 200 202 203 207 207 208 212 218 223 225 231 233 235 236 239 239 240 244 245 245 248 252 253 254 255 256 258 260 260 261

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14 The Artworld and the World of Works of Art . . . . . . . . . . . . . . . . . . . Maurizio Ferraris 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 From Mimesis to Ready-Mades . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Hermeneutics of Art and the Ontology of the Works of Art . 14.4 The Limits of Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Art as the Class of the Works of Art . . . . . . . . . . . . . . . . . . . . . . . . 14.6 The Paradigm of the Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Diffuse Aesthetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Impossible Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.10 Physical, Ideal, and Social Objects . . . . . . . . . . . . . . . . . . . . . . . . . 14.11 Objects that Pretend to Be Subjects . . . . . . . . . . . . . . . . . . . . . . . . 14.12 The Automatic Sweetheart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Architecture Between Science and Art . . . . . . . . . . . . . . . . . . . . . . . . . . Mario Botta 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Inadequacy of Technical Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Need for Beauty and for the Infinite . . . . . . . . . . . . . . . . . . . . 15.4 Le Corbusier’s Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 A Comparison with History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 The Symbolic Value in the City . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Geometry that Organizes Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Presence in Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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16 The Mirror: An Optical Prosthesis that Multiplies the Reflective Abilities of the Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michelangelo Pistoletto Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section Time, Infinity, Cosmology

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Did Time Have a Beginning? A Meeting Point for Science and Philosophy

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Gabriele Veneziano

1.1 Introduction The question appearing in my title: Did time have a beginning? certainly does look like an obvious “shared question/problem” within a session concerned with time. This does not mean, however, that my task will be a simple one. Why? In his 1959 lecture Charles P. Snow stressed the language barrier between scientists and humanists. He argued that, in response to questions such as What do you mean by mass, or acceleration?, no more than 10% of the highly educated people would have felt that one was speaking their own language. I am myself convinced that this language problem between the two cultures has only gotten worse since that famous lecture. The concepts themselves are perhaps not so difficult to communicate, but the language is an almost unsurmountable barrier. The question of our origins, of how far back in the past we can go, has been a concern since the beginning of mankind. It has been of interest to artists (110 years ago Paul Gauguin painted his famous canvas Where are we coming from? Who are we? Where are we going?), while philosophers have long been arguing about whether time had a beginning or goes back forever. Aristotle, for one, held the latter point of view, whereas St. Augustin said that a question like What was God doing before he created the world? did not make sense since God created time with the world. Giordano Bruno took a different attitude about such matters, and had to pay the consequences. . . Incidentally, one could ask a related (and equally shared?) question: Will time have an end? I will not talk about this question because of lack of time, but it is perhaps interesting to notice that most religions insist that there was a creation – and thus a beginning – while accepting eternity in the future. They appear to take a rather asymmetric attitude towards past and future. 3

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In conclusion, even if we all share these problems, “we do not share the language,” as C.P. Snow himself said. In the following, I will try my best to convey what physicists think today about this issue not without stressing that, unlike in other disciplines, scientists are never 100% sure about their own conclusions: as much emphasized by the late Nobel Prize winner Richard Feynman, always living in doubt is one of the greatest strengths of science. I will start by talking about different attitudes concerning time in “classical” physics, meaning by “classical” (again a matter of language!) “before the advent of quantum mechanics (QM).” I will then recall how classical physics appears to lead, inevitably, to the “beginning-of-time myth” and explain why I regard this as a myth rather than a solid scientific conclusion. Next, I will argue that, once QM is taken into account, our perspective must change drastically. However, the question is only well-posed if one has a consistent way to include quantum mechanics in Einstein’s general relativity, something that has defeated all attempts for many decades. To conclude, I will argue that a new theoretical development, known as String theory, can provide such a consistent framework and allow us to address that question in a serious way. While the final answer to the question is still not known, I will describe some possible new scenarios inspired by string theory and some concrete ways to test them experimentally.

1.2 Time in Classical Physics Even in classical physics the concept of time underwent a few revolutions. For instance, in Galileo’s or Newton’s views, time, or better any time interval, is the same for all observers. In special relativity, a theory introduced by Einstein about 100 years ago, time intervals depend on the relative motion of the observer and the observed. Let me give an example: consider the lifetime of an unstable particle (how long it lives on the average before decaying). For a given particle species, this is a fixed number for any observer who travels together with the particle. However, if we watch the particle from a different observation point, namely if we move with some velocity v with respect to the particle, we discover that the particle’s lifetime is increased by a factor γ = (1 − v2 /c2 )−1/2 > 1, the famous “Lorentz factor” which is always larger than 1 and can be very large if the relative velocity v approaches the speed of light c. This phenomenon is seen everyday in accelerator physics, for instance at CERN, and is used to make beams of an unstable particle last long enough to be useful for doing experiments. When one talks instead about General Relativity (GR), the extension of Special Relativity introduced by Einstein in 1915, then time intervals are even more observer-dependent. They also depend on the gravitational field, on how strong gravity is at the position of the clock. For instance, clocks tick with slower pace if you are here in Venice than if you are on top of Mont Blanc. If we synchronize two clocks on earth and one is taken for a while to a higher altitude, when it comes

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back to its original altitude it shows a slightly (very slightly!) later time than the clock that stayed all the time at the lower altitude. This prediction of GR has been tested and is even of practical interest. Although the differences in time due to the gravitational field of the earth are very small, if one wants to reach the needed precision for the now widely used Global Positioning System (GPS), one needs to take this GR correction into account.

1.3 General Relativity and the Beginning-of-Time Myth General Relativity is a very successful theory for describing gravitational phenomena in many physical situations. The deviations from Newtonian gravity predicted by GR have been tested with great precision, while some of its new implications, like the existence of black holes and of gravitational waves, have by now been confirmed either directly or indirectly. There is mounting evidence, for instance, that at the center of our galaxy (the Milky Way) there is a gigantic black hole whose mass is more than a million solar masses. There is not yet, instead, any direct evidence for gravitational waves. Several detectors are presently looking for them, but do not have, presumably, the required sensitivity. However, we can argue indirectly for their existence through the study of how binary stars change their period of rotation over many years. That fits extremely well with what we would expect on the basis of emission of gravitational waves according to GR. To summarize, GR seems to be very well applicable to isolated systems, to waves in empty space, to the Universe as a whole. For instance, the attractive nature of gravity is responsible for the growth of small initial density fluctuations in the Universe and thus for the emergence of the large-scale structures that we observe in it, galaxies, clusters of galaxies. Ruth Durrer will be talking about this later in this conference. There is no apparent reason for mistrusting GR in yet unexplored regimes, however we have to face some less pleasant consequences of it. That same universal property of gravity, being always attractive, is also responsible for gravitational collapse, the formation of black holes and, unfortunately, of what physicists (or mathematicians) call “singularities.” In their language, a singularity means some place in space, or some instant in time, where some physical quantities become infinitely large. In fact, Steven Hawking and Roger Penrose have shown, under very minor assumptions, that these singularities are almost unavoidable in GR: for instance, there should be a singularity behind the so-called horizon that surrounds a black hole. More important for this talk, GR predicts the existence of a cosmological singularity lying some 13.5 billion years in our past. This is what is now generally known as the Big Bang (BB). Thanks to this powerful theorem, we have to conclude that having a beginning of time is inevitable since once you hit this infinity, it does not make any sense to extend the solution of your equations beyond that point. This general consequence of GR has led us to believe in the beginning-of-time myth.

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1.4 The Beginning of Time: a Necessity or a Myth? Why do I talk about time having a beginning as a myth and not as a consequence of the laws of physics? The point is that the Big Bang singularity is a necessity only if one considers GR as an exact theory at all length and time scales. However, there is a simple argument showing that, once we take Quantum Mechanics into consideration, the whole question has to be reconsidered as one approaches the socalled Planck scale of distances or of time. Planck’s scale was introduced by Max Planck at the very beginning of the last century (actually he did so in 1899). He noticed that a particular combination of the gravitational (so-called Newton) constant GN , of the Planck constant h (appearing in Heisenberg’s uncertainty principle) and the speed of light c, provides a fundamental length scale or, dividing that length by the speed of light, a certain time scale. These length and time scales are incredibly small. The Planck length, for instance, is about 10−35 (meaning zero point followed by 34 zeros and a 1) meters, while the Planck time is about 10−43 seconds. The uncertainty principle implies that, if one would try to perform a measurement in a region which is smaller than a Planck length one would need such a large energy that a black hole will appear and hide the same region where one wanted to carry out the measurement. But when we talk about the Big Bang singularity we talk about time scales which are even smaller than the Planck time and of regions which are smaller than the Planck length. It is therefore very dangerous to neglect quantum effects in these regimes. I hope to have given you an idea of why QM can change our point of view, particularly as we approach the Big Bang singularity. Unfortunately, taking quantum effects into account within Einstein’s General Relativity turned out to be a very difficult – if not impossible – task. This leads us, if you wish, to a modern version of Einstein’s dilemma. Einstein, who, incidentally, contributed a lot to the development of QM (after all he received his Nobel prize for the quantum theory of the photoelectric effect and not for General Relativity!), soon realized that there was a clash between QM and GR. Once faced with this dilemma he decided in favor of his own “baby,” GR. His famous sentence God doesn’t play dice! has often been quoted as meaning that Einstein did not like the lack of determinism implied by QM, a theory that provides only probabilities for certain things to happen rather than others, something like playing dice games. But do we really have to make a choice between GR and QM? Well, perhaps not, perhaps we can have the cake and eat it. Indeed, about 40-years-ago, physicists came up with a new revolutionary theory that appears to avoid Einstein’s dilemma. At the same time, this theory unifies our descriptions of the infinitely large, to which classical physics applies, and of the infinitely small, to which quantum physics applies. It is called String theory and I will try to tell you in as simple terms as I can what it is. I will try to explain that, although the basic assumptions underlying this theory are extremely simple, their consequences are very far reaching and, to a large extent, still widely unexplored.

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1.5 String Theory: What’s That? What are the basic postulates of string theory? In conventional relativistic quantum theory, known as quantum field theory, particles are considered to be point-like. There is, of course, quantum mechanical uncertainty, but the important thing is that a point has a finite number of degrees of freedom: its position, its velocity and possibly a few others. In string theory, instead, all elementary particles, when looked at with sufficient resolution, are actually one-dimensional objects, strings, (called “stringhe,” instead of “corde,” in a very bad Italian translation). Some of them are like a violin string with two ends: they are called open strings. But there are also strings which have no ends: they are like small loops: and are called closed strings. Unlike points, they have infinitely many degrees of freedom. This is the basic assumption of string theory: every truly elementary particle in nature is actually a string and gets its characteristic properties from the way the string vibrates. It’s like a violin, but where different musical notes correspond to different particles, yet all originating from the same basic object. This is why string theory has a magic unifying power. Other than that, there is no new assumption, one is still using Special Relativity and Quantum Mechanics as one does in the more conventional theories. To summarize, Strings, Relativity and Quantum Mechanics, when put together, are the three ingredients of a magic cocktail. Why magic? Because of an amazing series of quantum “miracles” that emerge, automatically. . . once you shake the cocktail well enough. I will limit myself to the miracle directly relevant to our topic today, but there is a list of about a dozen such miracles. Let us go back again by 100 years when QM solved the problem of atomic stability. Take a system like a hydrogen atom which consists of a nucleus, a single proton for hydrogen, and an electron going around it. From the point of view of classical physics this system is quite unstable because the electron going around the proton radiates out energy in the form of electromagnetic waves. You see this phenomenon all the time in particle accelerators: the electrons going around the circular accelerator ring radiate a lot of energy (this is why CERN’s electricity bill is very high: we have to give back this energy to the electrons so that they do not slow down). The same is what happens, classically, to our electron going around a proton: eventually the electron should fall into the proton (like a satellite crashing into the earth’s atmosphere) and form a very small object having the size of the proton itself (about 1 Fermi = 10−15 meters). When, instead, one looks at the same hydrogen atom from the point of view of QM the electron, through the uncertainty principle, finds its optimal orbit, which is not sharply defined, but corresponds to an average distance from the proton of about 10−10 meters i.e. a hundred thousand times bigger. This is what gives the observed size and stability of atoms. What happens with strings? The physics is very similar to that of the atom: classical strings may have zero size, may shrink to a point, and this is how they would reach their minimal possible mass/energy. But quantum mechanics, because of the uncertainty principle, forces strings to have a minimal,

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or rather I should say optimal, size. One can compute what this optimal size of the strings is in terms of the Planck constant, the speed of light and another quantity, the so-called string tension. Like violin strings, also ours have a tension: out of it one can form a length, called the string length, in analogy with how Planck constructed his length. At the end of the day one finds that today the string length is about 10 times bigger than the Planck length (this ratio may have changed over time), hence about 10−34 meters, which is still not a lot; however, it is sufficient to modify physics as one approaches the Plank length (or time), i.e. near the Big Bang, because it is somewhat larger than the Planck length.

1.6 String-Inspired Cosmologies: a Longer History of Time? This finally takes us to string-inspired cosmology and, perhaps, to a longer history of time. The basic question is “Does the existence of this new fundamental length eliminate the singularities of GR?”. This question still occupies the minds of the few thousand string theorists working all over the world. It is a very hard question, but all indications, so far, are in favor of an affirmative answer. For instance: the finite size of strings should imply an upper limit to the density of the Universe because, having a non contractible size, strings cannot be packed beyond a certain limit. String theory is also known to provide a maximal value for temperature. Both features are in striking contrast with what happens at the Big Bang in classical GR where both density and temperature grow without any limit. The next question is then: if the conventional Big Bang singularity is indeed avoided by string theory, what is going to replace it in string cosmology? This is an even harder question, but I emphasize that at least within string theory, it’s a wellposed question. People are working on it and, while the final answer is not known yet, we can discuss a couple of possibilities. The first one is that time, as we perceive it today, is an emergent concept, it will emerge out of what we may call a “stringy epoch” during which neither the concept of a classical distance nor that of a classical time interval make any sense. In this alternative there would be nothing before such a stringy phase. This point of view will be the closest one, in spirit if not in detail, to Steven Hawking’s proposal, in which time, as we approach the Big Bang, becomes “Euclidean,” meaning that it would become like another direction in space. The second alternative, which I personally prefer since it is richer in consequences, is that the Big Bang, rather than representing the beginning of time, is the result of a previous evolution, also a classical one with a well-defined concept of time. In this pre-bang evolution density and temperature grew from some infinitesimally small value up to some maximal, finite value determined by the string length. This pre-bang epoch would be a kind of “mirror image” of what has happened after the Big Bang. Peculiar symmetries of string theory, known as dualities, allow for such a mirror phase. Effects due to the finite string size would take over after the

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pre-bang phase and force the Universe to bounce after going through a high-density, high-temperature string phase. You can say in one word, that the Big Bang becomes a Big Bounce. This scenario offers new ways to solve the problems of standard cosmology through the possibility that our Universe, having such a long pre-history, is much older than in conventional cosmology. But what are these problems of Big Bang cosmology that we would like to solve? One of them can be described as follows. Consider how big the Universe (better: that part of the Universe that we can observe today) was very early on, say a few Planck times after the Big Bang. The answer, in standard hot Big Bang cosmology is that its size was a fraction of a millimeter. One millimeter is very small compared to its present size; yet it is huge if compared with the distance traveled by light during its short “life” after its birth (about 10−43 seconds), namely 10−35 meters. That means that the baby Universe was 30 orders of magnitude larger than the distance traveled by light. The baby Universe was very big for its young age! This puts us in front of a very big puzzle: since the distance traveled by light is the maximal distance over which you can have communication among different parts of the system, it’s very hard to understand how the Universe could become homogeneous enough in such a short time (this homogeneity is reflected in the uniformity of the observed temperature of the cosmic microwave background: 2.7 degrees above absolute zero irrespective of the direction in the sky). It is like if you turn on a heater in a corner of this room: it takes a while before you feel the increase of temperature at the other end of the room, because this room is quite large. The only way to account for present data is to “fine tune” the initial state of the Universe very very precisely. This is what physicists do not like. They have thought a lot about this problem and concluded that, if one insists that the Universe had a beginning at the Big Bang, there is only one way out: the primordial Universe had to be much smaller than in conventional theories. This gave rise to a very successful paradigm, called “inflationary cosmology,” in which we insert in very early cosmology a phase of accelerated expansion of the Universe (inflation) making the baby Universe much smaller than the 1 millimeter mentioned before. However, if we accept that there was a pre-bang epoch, the Universe right after the bounce was already very old and had a lot of time to thermalize. The interesting thing is that these new cosmologies not only have some philosophical interest, they have observable consequences too, so that one can test experimentally what happens near the Big Bang, or even earlier if there was something before. It sounds very bizarre that you may be able to probe today how the Universe was at – or before – the Big Bang, but this is not so. It is related to a phenomenon called “freeze-out” of certain structures when the Universe is expanding very fast (relative to the size of the structure itself). These large-scale structures “defrost” today carrying with them information about the Universe as it was when they froze out immediately after or even before the Big Bang. It is similar to a prehistoric animal caught in the ice millions of years ago, and revealing itself to us now, after defrosting.

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There are several good examples of what we may call pre-bangian relics, relics from before the Big Bang: one is a stochastic background of gravitational waves, very much like the already mentioned electromagnetic background that fills up our Universe at a temperature of 2.7 degrees above absolute zero. Gravitational waves, unfortunately, are much harder to detect but, as I mentioned, there are plans to do so. Other predictions of these unconventional cosmologies concern the galactic and intergalactic magnetic fields whose origin is otherwise mysterious, as well as some peculiar structures in cosmic microwave background anisotropies. For further suggested reading on string cosmolog see [1, 2, 3]. Let me conclude: the belief that time had a beginning is indeed a myth based on extrapolating GR beyond its limits of applicability. If one takes the beginning-oftime point of view, then the inflationary paradigm is the only possible solution to the puzzle of hot Big Bang cosmology. But string theory, thanks to its magic properties, should be able to remove the Big Bang singularity of GR and to allow for the possibility of a pre-bang phase, offering a new solution to those cosmological puzzles. The new symmetries implied by string theory suggest a mirror structure for the pre-bang phase: a cold and almost empty Universe which, through a process similar to gravitational collapse, leads to a high temperature-density phase which replaces the singular Big Bang event of standard cosmology. Most important, these pre-bang or big bounce cosmologies make (better, will make, since we are still working on this difficult problem) distinct predictions on today’s (or near-future) cosmological observables and can thus be discriminated from more conventional scenarios. So, the question of whether time had a beginning may turn out to have an experimental answer one day. . .

Acknowledgements I would like to thank the organizers of this conference, and in particular Professor Ernesto Carafoli, for having provided a unique opportunity for exchanging ideas across so many different cultural areas in such a splendid environment.

References 1. G. Veneziano: The Myth of the Beginning of Time Sc. Am. 290, 5 (2004) 2. M. Gasperini: L’universo prima del Big Bang (Franco Muzzio Editore, Roma 2002) 3. M. Gasperini and G. Veneziano: Beyond the Big Bang, ed. by Rudy Vaas (Springer, Berlin, Heidelberg, in press 2009)

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Discussion Ch. Riedweg: Thank you very much for this very illuminating paper. I, as a classicist, am always struck by the parallels to ancient kinds of thinking and also to mythology and I think that with this pre-bang period we come very close again to traditional myth. For instance, if you look at the genesis with its “tohu va-vohu” this looks like what’s behind the pre-bang phase and we have the same in the Babylonian tradition, or if you turn to Plato:: here again one might bring the two cultures into discussions with each other. For Plato it is with the big bang that time would have started; he also accepts that there is a pre-bang epoch which has left traces in this world – for instance evil is still a trace of that in the world – and these are quite fascinating parallels which I have learned from your highly interesting paper, thank you very much. G. Veneziano: Thank you for the comment. I would like to hear more about this. I knew vaguely that there was something very similar to what I’m saying in some philosophies or religions. I want to emphasize that, actually, a pre-bang chaotic era is precisely how we describe the “beginning” in a paper I wrote a few years ago with Thibault Damour and Alessandra Buonanno, where the initial state of the Universe is indeed taken to be a very chaotic (and generic) one. In that phase there is no clearly defined direction of time. Only in some regions where order comes out of this huge chaos, a time arrow will emerge. So I think that the analogy goes even further than what you have been saying. It’s interesting that such ideas have come up already. However, as I said, the difference is that we physicists try to justify them within a well-defined theoretical framework, which, of course, can be right or wrong. The hope is that, eventually, we shall be able to put this theory to a test through its observable consequences. L. Boi: Could you please tell me the solutions your alternative way offers to the following problems: the first is that of the expansion of the Universe, the second – and I believe is a very important one – is the form of the Universe, whether it is finite or infinite. The third is whether there is only one Universe or a continuous creation of a multiplicity of Universes. I believe a full day would be necessary to address these three questions, but it seems to me that they are the three central problems of cosmology. G. Veneziano: There were actually three questions in one: the first was whether this scenario can explain why the Universe is expanding; the second question was whether it leads to a Universe which is finite or infinite; and the third on whether there is just one Universe or many of them. For lack of time let me limit myself to this last question: the answer is related to my previous one. We start with this very chaotic state similar to a rough sea of waves. As you know waves can collide and interfere and create some bounces. This is what would happen to our pre-bang Universe: occasionally, at some moments and in some places, a critical density forms starting a process of gravitational collapse. It is within such a collapsing region that

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an interesting cosmology may develop. Since the critical density for collapse can occur in different places and at different times, this picture leads actually to a multiverse type scenario. Observationally, our Universe needs not to be the only one. We don’t know what is there beyond the maximum distance we can observe: it could very well be that there are other parallel Universes, possibly with different laws of physics. I didn’t mention that one of the very peculiar predictions of string theory (it’s another miracle, perhaps one we could have done without) is that there are more than three dimensions of space. If this is so, in our Universe the extra dimensions of space must be very small so that we can only see their indirect consequences. But you may well consider the possibility that other Universes have a different number of macroscopic dimensions. So, it’s a long story, but the short answer is yes, there could be many parallel and different Universes. Unidentified: Why should we trust the fact that the constants of physics do not change with time, for example the velocity of light? G. Veneziano: Very good question. The short answer is that, precisely within string theory, we shouldn’t think that the “constants” are necessarily constant, i.e. the same in every point of space and time. However, if you talk about the speed of light, the issue is slightly more subtle, one should always talk about space or time variations of quantities that do not have dimensions. I don’t know whether the audience is familiar with the word “dimension,” in the sense of physical quantities like lengths, masses, time intervals, or combinations thereof. The speed of light has dimensions of space/time, while the Planck length, as its name says, has dimensions of a length. What makes sense, for instance, is to talk about possible variations of the ratio between the string length and the Planck length, which here and now is about 10. However, that number, 10, doesn’t need to be fixed: in string theory that ratio is actually a field and, like the electric or gravitational field, can depend on space and time. Pre-Big Bang cosmology is based very much on the idea that the above ratio was very different (actually much larger) in the far past. So it is true that in string theory there is room for variation of constants that in conventional theories must be fixed once and for all. Another dimensionless number, the so-called fine-structure constant, could very well vary in string theory. There are experimental attempts to detect any such “variation of constants”: it looks like a contradiction in terms, but it isn’t. A. Miller: I very much appreciated your lecture, very enjoyable, and in particular your bringing out a very important point and wonder whether you wish to comment on some of our colleagues in physics who have ventured into philosophy and have criticized string theory on philosophical grounds. G. Veneziano: Thank you. Possibly you are referring to the criticism that string theory cannot be falsified. I certainly disagree: like its old incarnation, also the present version of string theory can be falsified, and not only by experiments that need unrealistically high energies.

The Flow and the Map: On the Dynamic and Static Views of Time

2

J.J.A. Mooij

2.1 Three Key Events in the Modern Philosophy of Time About 50 years ago C.P. Snow delivered his famous lecture on the two cultures. We have to go about a hundred years back, however, to find similar landmarks in the history of the philosophy of time. At that moment three things happened that greatly influenced the philosophy of time in the twentieth century. In the first place, there was the publication of the article “The Unreality of Time” by the British philosopher John McTaggart. It appeared in the philosophical journal MIND in 1908 [1]1 . Here McTaggart distinguished between two series of events and temporal positions, two ways of classifying or ordering them. He called them the A- and the B-series. The A-series is the ordering of events and moments in terms of past, present, and future. This makes the properties of the A-series changeable properties, for any event is first future, then present and ultimately past. Thus, one may call the A-series dynamic. The B-series is the ordering of events and moments in terms of earlier and later. These relations do not change: if it is ever true that a certain event precedes another event, then this is always true. Therefore, the B-series may be called static. According to McTaggart, this implies that the A-series is the basic temporal series because time essentially involves change. Thus, without the A-series there cannot be time and consequently there cannot be another temporal series (in particular, no B-series) either. But McTaggart also argued that the positions of the A-series are contradictory and for that reason the A-series cannot exist. Thus, time does not exist; time is an illusion. Hardly any philosopher has accepted that conclusion. But some discussed his arguments carefully, and many adopted and generalized his terminology, speaking freely about A- and B-notions, A- and B-facts and beliefs, even A- and B-theories and -theorists. Of course, McTaggart did not invent this distinction. But he made 1 The essay also appeared, with slight modifications, in [2], ch. 33. For this version, see also [3], pp. 23–34.

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it a vital instrument in the philosophy of time, especially in the analytical tradition from about the 1960s until the present. Perhaps he is the only Hegelian philosopher who strongly stimulated analytical philosophy, not by serving as a scapegoat but by setting an example. The second event occurred somewhat earlier, in 1905, in Albert Einstein’s annus mirabilis, when he published a number of highly important articles, one of which was the creation of the special theory of relativity. This article, too, would greatly influence the modern philosophy of time. For the theory of relativity implies that time cannot be the absolute entity that Newton had declared it to be, the entity that flows evenly and continuously, independent of everything else in the universe and the same for all observers. Einstein showed that temporal measurements are dependent on movement and distance. Events that are simultaneous within one frame of reference need not be simultaneous within another frame if the two frames move with respect to each other. And if simultaneity becomes unstable, order becomes unstable as well. Indeed, it may happen that if an event X precedes an event Y in one temporal scheme, X may follow Y in another scheme. What is more, the clocks in two such frames go differently, they do not have the same pace. If a system is moving with respect to another, its clocks slow down with respect to the clocks of the other system. This phenomenon is called time dilation. For all these reasons it would be senseless to assume that at any moment (say, now) the whole of the universe is in one well-determined state [4]2 . These are revolutionary implications. However, under normal conditions and in our everyday conceptual framework they are hardly relevant. Velocities and distances must be very large indeed for the relativistic effects to be notable. And so the theory of relativity has not displaced all other perspectives. In particular, there is yet a third event to be remembered. I am referring to a course of lectures given by Edmund Husserl in the same year 1905, in which he tried to find the foundation of time in the absolute flow of consciousness. In the following years he continued his search. Husserl was rather hesitating in his ambition, sometimes stating that objective time is independent of the time of consciousness, sometimes locating the origin of all time in that same consciousness; but certainly his so-called phenomenological approach has highly stimulated the search for authentic time, that is, original time independent of pragmatic and scientific considerations. This search was already started before him by Henri Bergson in 1889 and would be continued by Heidegger in Sein und Zeit (Being and Time) in 1927 and by Sartre in L’Etre et le Néant (Being and Nothingness) in 1943 [5, 6]3 . The three contributions by McTaggart, Einstein, and Husserl have proved to be key events in the three main traditions of twentieth century philosophy of time, viz.: the traditions of analytical philosophy, of scientific empiricism, and of phenomenology. These three key events will suffice as the starting points for my topic, viz.: the 2 In his general theory of relativity, published in 1915, Einstein generalized the principle of relativity from uniform motion to accelerated movements. 3 Husserl’s lectures of 1905 were only published much later. They were, in collaboration with himself, prepared for publication by his assistant Edith Stein about 1916 and on that basis edited by Heidegger in 1928 [5]. In the mean time Husserl had continued to discuss problems of time [6].

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tension between the dynamic and the static views of time, a tension that is already evident in the essay by McTaggart and closely connected with the other two key events [1, 2].

2.2 The Two Views of Time Let me describe the opposition between these two views in some detail. On the one hand, one may regard time as a flowing entity, with an ever-changing present, an ever-increasing past and an ever-decreasing future. This, of course, is nearly everybody’s common experience: more or less urgent, more or less painful, tragic or (sometimes) comic. Here it is taken as the only essential characteristic of time. Indeed, in this perspective past, present, and future will be the central temporal categories. But past and future are experienced as highly asymmetrical for we know a lot about the past and we take the past as fixed and unalterable, whereas we are uncertain about the future and think we are more or less free to act in ways according to our wants and decisions. Time has to pass in order for us to know for sure what will happen, and we know it only when it is no longer future but has become present or past. This view of time as flowing or moving is the dynamic view. On the other hand, the flow of time, like a river, can be mapped. Then we get the static view. Here time is regarded as a fixed, static dimension defined by relations of earlier and later, by temporal distance and simultaneity. Surely, even so time still is the dimension of change, but it is not changing itself. Only because it is a static dimension, time would be capable of delivering a measure of the rate of change. If time would move (or flow) itself, the question “how fast does it move?” would both be unavoidable and unanswerable. Or so the argument on behalf of this static view would run. All events would have their unchanging positions somewhere along the temporal dimension, and any change would simply amount to something having different properties in different temporal positions. Temporal change would appear to be a matter of a space-like difference, a difference between properties at different moments - similar to a difference between properties at different places. This static view might seem rather abstract and far-fetched, but it is connected with our direct experience, too. In particular, it is involved in memory. Memory allows for a kind of panoramic overview, in which time is fixed or crystallized, in a sense. True, memory refers to the past; and even apart from being unreliable it is very partial and selective. Nonetheless, it delivers an overview in terms of earlier and later, in such a way that these notions may take precedence over the notion of the moving present. And what is more, this might apply to history as well. (I’ll come back to this.) These two views have both been developed from McTaggart’s article [1, 2]. As I indicated already, few philosophers accepted his conclusion that time does not exist. Most of them who discussed his work fell into one of two groups. Either they accepted his argument as to the primacy of the A-series of past, present, and future, but rejected his argument that the A-series is fatally contradictory, and so they opted for the dynamic view or A-theory. Or they accepted the unreality of the A-series but

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rejected his claim that this implied the unreality of the B-series as well, and so came to adhere to the static view or B-theory. In both cases, however, the rejected aspect must be, in one way or another, explained or incorporated or (maybe) explained away. It cannot simply be forgotten, for it is a feature of our common experience and belief. Many proponents of the static view of time appeal to physics. There are several reasons for that. For one thing, the laws of physics do not refer to the present. “Now,” in time, is physically as irrelevant as “here,” in space. Secondly, nearly all the fundamental physical laws are symmetrical in time. That is, t and −t are interchangeable, and movements that go in one direction are theoretically possible in the other direction as well. Both reasons make physical time space-like. Indeed, already at the end of the nineteenth century it was sometimes suggested that the dimension of physical time was not essentially different from the dimensions of physical space. Thus, it should be possible to represent the world and its complete history in a four-dimensional continuum. Any momentary state would correspond to a threedimensional cut of a four-dimensional entity; and the history of any particle would crystallize into a line, indicating the position it would have at any moment. This came to be called later its world-line. So a world-line does not change in time, but it represents what happens to a particle in space and time; and the dimension of time is part of a fixed, unchangeable model of the world. This early four-dimensional view of the world has even left some traces in literature. In the first chapter of H.G. Wells’ science-fiction novel The Time Machine (1895) its protagonist, the Timetraveler, states: “There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it”; and he describes man as a “four-dimensional being, which is a fixed and unalterable thing.” Indeed, it is precisely this, according to him, that makes time-traveling possible [7] 4 . Ten years later, that is, in 1905, the connection between space and time became even much more intimate due to the theory of relativity. Motion and distance would be relevant to the determination of all essential characteristics of time such as simultaneity, temporal order, and temporal interval. Surely, there still would be an invariant interval between any two events, the same for all observers, their so-called spacetime interval; but as the term indicates already, its definition involves both space and time. Time and space became theoretically intertwined in a way that was completely new. For quite a long time, say from Aristotle until Descartes, space and time had been conceived as radically different entities, not similar at all. Then, from Newton onwards, space and time came to be seen as cognate quantities, one might say as sibling notions, and accordingly the phrase “space and time” became very common. But in the theory of relativity space and time became, moreover, mutually interconnected; the world came to be seen not only in “space and time,” but in 4

The first quotation is surprisingly similar to a remark made by the mathematician-philosopher Hermann Weyl much later when he argues that events only come to happen because my consciousness moves along the world line of my body [7], p. 116. The idea of something moving in time along a curve in four-dimensional spacetime is of course highly debatable, and Weyl was much criticized for it.

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“spacetime” – a four-dimensional amalgam from which different observers derive different versions of three-dimensional space and one-dimensional time. That was a boost for the four-dimensional view of the world, the presentation of the world as a four-dimensional continuum comprising all events that ever happened or ever will happen. Quite a few natural philosophers inferred from this that the world really is such a four-dimensional whole, or “block-universe” as it was to be called. Time is here block-time, the whole of time timelessly laid out [8]5 . As Einstein put it: happening in three-dimensional space was transformed into existence in a fourdimensional continuum [4]6 . Temporal development and coming-into-being would not be something that occurs but rather something that is implicit in that fixed, unchangeable four-dimensional whole. And the present would simply be a subjective addition to objective reality. However, it is certainly not true that the philosophy of physics as a whole was committed to the idea that the moving present is only a subjective addition to objective reality – that the static view is the only objective view. For instance, Hans Reichenbach argued that, unless determinism is true, the moving present has an undeniably objective reality as the boundary between the fixed past and the still uncertain future [9]7 . And G.J. Whitrow, the author of the excellent and authoritative work The Natural Philosophy of Time, defended even that the moving present has anyhow, whether or not determinism is true, a physical reality of its own [10, 11]8 . But it is true that within the philosophy of physics there has been a strong tendency in favor of the B-view, that is, the static conception of time, denying the objective reality of the present. This was, of course, radically different in the third main tradition in the twentieth century philosophy of time, viz. the phenomenological tradition. Here authentic time is located in some form of consciousness (transcendental or otherwise), and its flowing character is the heart of the matter. Heidegger, for instance, argues that human being (“Dasein”) in its authentic form understands itself as being-to-death (“Sein zum Tode”). In anticipating death and coming forward to it, Dasein creates future and through the future also the past and the present. And this primordial, qualitative time (strictly speaking not a matter of succession at all, but a threefold experience or attitude) is taken to be the source of quantitative time: of the vulgar time used in everyday life as well as of the objective time in the physical world.9 With Sartre, on the other hand, the core of authentic consciousness is not the anticipation of death but the acceptance of absolute freedom and absolute responsibility. In Being and Nothingness we are said to be responsible for everything we do or do not do; it is bad faith (“mauvaise foi”) to want to evade this absolute responsibility. 5

p. 72. The four-dimensional continuum is Euclidean in the special theory of relativity, at least for x, y, z and t  = it, in the general theory it is non-Euclidean (Riemannian). 6 p. 140. Cf. also 171. 7 p. 227. 8 pp. 81–82; pp. 132–134. 9 True, Heidegger did not like the term consciousness. He suggests that in the being of humans (“Dasein”) the distinction between inner and outer loses its edge. However, the whole argument about authentic time supposes consciousness; without it, the argument collapses.

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More in particular, we can never appeal to our past by way of excuse or justification. From moment to moment we have to freely transcend our past, a past that we are but at the same time negate. This makes the present the heart of temporality, rather than the future, let alone the past. In later works of phenomenology still other aspects or perspectives have come to the fore. In the process the degree of separation between primordial time and objective or real or physical time changes as well. This separation may be very incisive (as with Heidegger), or rather smooth (as with Manfred Frank: Zeitbewusstsein, 1990), or surprisingly variable (with Husserl himself). But the fundamental categories remained the dynamic ones, not the static ones.10 So the upshot of the three traditions with respect to the static/dynamic distinction is complex. Whereas the phenomenologists opt for the dynamic perspective as the fundamental one, the analytical philosophers (seen as a collective) hesitate between the two, and the scientific empiricists choose, I think, in majority for the static conception, but with a powerful minority in favor of the dynamic view. These are not arbitrary choices, of course. The several parties have their reasons, and in my survey I have already indicated some of them. But I want to say yet something more about it with reference to one central issue (perhaps the central issue), viz. the mind-dependence or mind-independence of the present. Indeed, it is evident that this issue is of crucial importance for the topic of this lecture. One widespread view (by the way, it is my view too) is that without consciousness, and in particular without mind, there would be no present. The main argument is the following. Without mind in whatever form or version all the moments and periods of time would be equivalent. Time would be the dimension along which the world develops, perhaps from a beginning, perhaps towards an end, but with no specific situation in which it is now. Any now supposes a referential act, that is: some mental act to make the selection of one moment or period from all the moments and periods that make up time. This makes the word “now” analogous to the word “here.” Indeed, it seems evident that without any mental activity no place could be “here,” whether it be this room or Venice or Europe or the earth or our solar system or any solar system or even the universe. “Now” and “here” are demonstrative or indexical words that get their specific reference only from the situation and the context in which they are used, and so they have to be used in one way or another, if not spoken, then at least felt or thought or signaled. This does not make the present an illusion, of course. There are many things that are mind-dependent that would not normally be called illusionary: for instance, memory, feelings of guilt or happiness, and even knowledge. There must be specific reasons why a mind-dependent phenomenon should be an illusion. Let us suppose that what makes it an illusion is a conflict with objective reality, in such a way that something we experience is false. In the case of the present this may be the feeling that the present is part of mind-independent physical reality. For if it is true that our mind constructs the present, then that feeling is false. But only that would be the illusion. Other aspects would remain unproblematical. And so the belief that 10

For further details on the phenomenological philosophy of time see [9], esp. ch. 14 and pp. 239– 243.

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something happens now need not be deemed illusionary, just as it is no illusion to believe that something happens here. There is then no conflict with objective reality. Moreover, the above does not make the temporal Now similar to its spatial analogue (Here) in all respects. It is not even subjective in the same degree. In several respects Now is less subjective than Here. Why should that be so? Very briefly stated: we have less freedom in choosing our Now than in choosing our Here; and the word “now” is less person-bound (that is, more collective) than the word “here” [9]11 . All this would mean that the notion of the present and all that it implies cannot easily be discarded. There is a tension between the static basis of time and its dynamic features. The tension has, for the time being, to be maintained.

2.3 The Two Views and the Two Cultures This suggests that there is no satisfying, objective, scientific theory of time that should be accepted by all rational individuals. One’s view of time may still depend on weighing the arguments, or on weighing one aspect (say, mental experience, or social life) against another aspect (say, physical or cosmological theory). Briefly, one’s view of time still depends on one’s view of the world. But if world-views come into sight, then the two cultures are close by. And so the question arises if there is a certain connection between the two main conceptions of time on the one hand, and the two cultures on the other hand. In particular, it seems probable that most people who opt for the static view belong to the scientific culture. Moreover, it might also be the case that most people who opt for the dynamic view belong to the literary culture. I guess that this correlation exists, but I also guess that it is rather weak. For both static and dynamic features play a role in the natural sciences as well as in the humanities. Let me give some examples. As I said already, the fundamental laws of physics are symmetrical with respect to time and the present plays no role in physical theory at all. However, the temporal symmetry applies only to the fundamental level. Higher up, physical phenomena show striking and massive asymmetries and are not reversible at all. Moreover, the present, the past, and the future are very relevant notions in other natural sciences like cosmology, geology, and (highly topical) climatology. Climatology in particular is focussed nowadays not on general laws and general aspects of climate change, but on what is happening now and on what will happen in the near future, that is, how to explain the present trends. So in natural science as a whole the moving present is far from irrelevant and may sometimes be of central importance. Of course, it may be true that all natural sciences are ultimately reducible to physics. However, this only concerns the general concepts and the general laws. The process of reduction in no way eliminates the relevance of the present context in the sciences as a whole. For 11

More about this in the last chapter of my book [9], esp. pp. 263–269. See also [12], ch. 3. Chapter 5 is a general discussion of the mind-dependency of the present. For a B-theory in which the time-space analogy is wholly rejected, see [13], esp. ch. 5.

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reasons like the above it is quite understandable that many members of the natural scientific community would opt for a dynamic view of time. The more so if they have a keen eye for such things as the thrill of creativity and the mental as well as social pressures in the development of any science. Or, crossing the boundary between the sciences and the humanities, let us take history. Temporal change is essential here, and the dilemmas and uncertainties implicit in any situation that is the present situation for the historical actors would seem to make the dynamic view a matter of course. But some philosophers of history have argued that there is a deep discontinuity between the temporal experience of historical actors on the one hand and the representation of their doings in historical narrative on the other. According to Louis O. Mink this discontinuity derives mainly from the fact that historical narrative connects the several stages of a development that are separated in reality. “To comprehend temporal succession means to think of it in both directions at once, and then time is no longer the river which bears us along but the river in aerial view, upstream and downstream seen in a single survey” [14]12 . This would mean that history largely eliminates time as a moving entity and substitutes B-time for A-time. In its most characteristic products it may even tone down the importance of dates, that are so essential to annals and chronicles. In a very enlightening essay my compatriot Frank Ankersmit has elaborated and supported this point of view, contrasting it with the view of David Carr where the temporality of historical actors is preserved in historical narrative. Ankersmit concludes that time is so to speak the food of history but is, like food, transformed into something different, viz. a narrative interpretation of the temporal events of the past. Surely, without time there would be no history. Paradoxically, however, moving time is necessary in order to be removed [15]13 . And so, according to Mink and Ankersmit, in narrative history time is incorporated in a static, nontemporal representation. This makes narrative history surprisingly similar to the block universe of natural philosophy! As the block universe represents the fortunes of any physical particle through its world line in a fourdimensional continuum, so history would represent the fortunes of its objects in a static narration in which nothing happens but all is only said to happen. History, too, would have, like physics, its own way of integrating what happens in time into a model of reality that is itself static: two different ways, of course, but similar in one important respect. One may also think of the image on the front of Paul Ricœur’s last book, La mémoire, l’histoire, l’oubli (memory, history, forgetting) [17]. It is the image of a baroque sculpture with two figures, Time (Chronos) and History, where Time is tearing out the page of a book and History is hindering him in doing so. This image nicely illustrates that history is a great attempt to save the past from disappearing into a void, to salvage the past from complete destruction by the moving present. Thus, even for historians a static conception of time despite its dynamic features would not be unreasonable. It is evident that this also applies to the specific case of literary history. T.S. Eliot transformed this insight into a striking view of the modern 12 13

p. 57; see also pp. 42–60. esp. p. 123. For Carr’s view see [16].

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poet’s situation and responsibility when he famously declared, in his essay “Tradition and the Individual Talent” (1919), that the historical sense compels a poet to write “not merely with his own generation in his bones, but with a feeling that the literature of Europe from Homer [. . . ] has a simultaneous existence and composes a simultaneous order.” [18]14 In a very different context, Eliot has expressed a similar feeling with regard to personal memory. I am referring to a passage in his own later poetry, viz. in Four Quartets. In the first of them, “Burnt Norton,” he exalts the role of memory as follows: But only in time can the moment of the rose-garden, The moment in the arbour where the rain beat, The moment in the draughty church at smokefall Be remembered; involved with past and future. Only through time time is conquered. [19]15

And this brings us to literature itself. To begin with, something similar to what I said about history is also valid for the novel and cognate literary genres. Surely, story and plot are here mostly fictional and the novelist has many more possibilities to present them than the historian; think of the use of fictional narrators, or the presentation of the inner feelings and secret thoughts of the characters. But here too a period or fragment of time can be presented as a whole: as a medium in which events occur and changes happen, but which does not change itself. Conceived in this way, the fleeting present of the characters is part of a static, nonfleeting representation, and it is only the fleeting present of the reader that re-introduces the dynamic aspect of time. I do not want to suggest, of course, that one must necessarily conceive of fictional narrative in this way. It is enough that it is possible to do so, in order to understand that here too the tension between the dynamic and the static exists. Nor is this a far-fetched possibility. Indeed, in his recent book The images of time Robin Le Poidevin argues that one should explain the time of fictional narrative in such a way. Starting from the role of the future and of future-tensed statements he arrives at the conclusion that fictional time is B-time and does not pass. Fictional A-times would be the result of our imaginative self-projection into the fiction [12]16 . However, the above idea does not apply to art in general. The representation of time in the different arts – in music, literature, theater, film, even in painting and sculpture – is highly variable. Each case differs essentially from the others; moreover, each case is complex in itself and has been the object of quite a lot of analysis and research. Time in literature is very different from time in music or painting; and time in one poem, novel, or painting can even be very different from time in another poem, novel, or painting. 14

p. 23. p. 173 (end of section II). 16 pp. 11–12 and ch. 8, esp. pp. 157–161. This book discusses, in connection with the problem of the reality of the A-series of times, several kinds of temporal representation: in perception, memory, art, and fiction. How strange that there is no discussion of temporal representation in the writing of history! If Le Poidevin had discussed history, would he have arrived at a view similar to the view of Mink and Ankersmit, viz. that historical time (i.e. the time of historical narrative), like fictional time (i.e. the time of fictional narrative), does not pass? 15

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Interestingly, something else now comes to the fore. For despite these differences there has been a common element with respect to time in many artistic ideologies. That common element is the wish to conquer time, to create a work of art that will survive, to approach in a way eternity. Striking examples of a poetic expression of this desire are the poems “Ode on a Grecian Urn” by Keats and “Sailing to Byzantium” by W.B. Yeats. Both poems celebrate an object symbolizing the artistic aspiration to a victory over time, especially over temporal change and decay. In the poem by Keats (1819) this is the scene presented on the marble urn, a scene of eternal youth, love and warmth. The urn itself is called a “sylvan historian” expressing a tale in terms of a scene that involves change but does not change itself – it embodies a moment that is not a fleeting but a staying present, a nunc stans. Let me quote the following lines from the third strophe: Ah, happy, happy boughs! that cannot shed Your leaves, nor ever bid the Spring adieu; And, happy melodist, unwearied, For ever piping songs for ever new; More happy love! more happy, happy love! For ever warm and still to be enjoy’d, For ever panting, and for ever young; [20]17 . . .

In the poem by Yeats (1926), partly inspired by the mosaics in the S. Apollinare in Ravenna, it is the products of Grecian goldsmiths that serve as paradigms of “the artifice of eternity.” The poet elaborates in particular on a golden bird, made To keep a drowsy Emperor awake; Or set upon a golden bough to sing To lords and ladies of Byzantium Of what is past, or passing, or to come. [22]18

Moreover, even a poem that is squarely about the loss of the past or about the inescapable transience of the present does not necessarily refer only to the dynamic view of time. I am thinking now of “Tears, Idle Tears” by Alfred Tennyson, a magnificent elegy on “the days that are no more,” and of an early poem by the Greek poet Kavafis, “The Candles,” in which the sequence of the days is compared to a row of candles, lighted as long as they are future, quenched when they are past. For both these poems may be interpreted as attempts to map the flow of time either in an enduring meditation or in a lasting image [23].19 And so this common element of many artistic ideologies – although surely not universal, let alone always prominent, and even often under attack but nonetheless in one form or another widespread – shows once more that the tension between a dynamic and a static view can be found everywhere, that is: everywhere within the two cultures. The differences are important, of course. For it will be clear that the contrast between the flow of time and its static representation in a four-dimensional 17 For comments and text pp. 151–166 and 290–292. A poem with a similar theme is “The people on the bridge” by the Polish poet Wislawa Szymborska, inspired by a picture by the Japanese painter Hiroshige; see [21], pp. 167–169. 18 It is the first poem of the collection The Tower (1928); pp. 217–218. 19 from The Princess, 1847, p. 265; comments and text in [20], pp. 167–177 and 292. [24], p. 5.

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model is not at all the same as the contrast between the flow of time and its representation in historical or in literary narrative, let alone the contrast between the flow of time and the object of artistic aspiration as something transcending time. For the moment, however, I would like to stress the similarities: For they all take the flow of time for granted, but propose an alternative at the same time. Surely the tension between the static and the dynamic views of time is a shared problem, in terms of the subtitle of our conference. But there is, I am inclined to think, no inkling of a shared solution.

References 1. J.M.E. McTaggart: The unreality of time. Mind 17 (1908) 457–474 2. J.M.E. McTaggart: The nature of existence. Vol. II. (Cambridge University Press, Cambridge 1927) 3. R. Le Poidevin and M. MacBeath (eds.): The philosophy of time (Oxford University Press, Oxford 1993) 4. A. Einstein: Relativity: The special and the general theory. Authorized translation by Robert W. Lawson. (Three Rivers Press, New York 1961 (1916)) 5. E. Husserl: The Phenomenology of internal time-consciousness. Edited by Martin Heidegger, translated by James S. Churchill. Introduction by Calvin A. Schrag. (Nijhoff, The Hague 1964 (1928)) 6. E. Husserl: Zur Phänomenologie des inneren Zeitbewusstseins (1893-1917). Edited by Rudolf Boehm = Husserliana, Vol. X. (Nijhoff, The Hague 1966) 7. H. Weyl: Philosophy of mathematics and natural science. Translated by O. Helmer. (Princeton University Press, Princeton 1949 (1927)) 8. P. Davies: About time. Einstein’s unfinished revolution (Penguin Books, London 1995) 9. J.J.A. Mooij: Time and mind. The history of a philosophical problem. Translated from the Dutch by Peter Mason. (Brill, Leiden 2005) 10. G.J. Whitrow: The natural philosophy of time. 2nd ed. (Oxford University Press, Oxford (1st ed. 1961) 1980) 11. G.J. Whitrow: What is time? With a new introduction by J.T. Fraser, and a new bibliographic essay by J.T. Fraser and M.P. Soulsby. (Oxford University Press, Oxford (1st ed. 1972) 2003) 12. R. Le Poidevin: The images of time. An essay on temporal representation (Oxford University Press, Oxford 2007) 13. D.H. Mellor: Real Time II (Routledge, London and New York 1998) 14. L.O. Mink: Historical understanding. Edited by Brian Fay, Eugene O. Golob, and Richard T. Vann. (Cornell University Press, Ithaca 1987) 15. F.R. Ankersmit: Over geschiedenis en tijd [On history and time]. In: De navel van de geschiedenis: over interpretatie, representatie en historische realiteit [The navel of history: on interpretation, representation and historical reality] (Historische Uitgeverij, Groningen 1990) pp. 108–124 16. D. Carr: Time, narrative, and history. (Indiana University Press, Bloomington, Ind. 1986) 17. P. Ricœur: La mémoire, l’histoire, l’oubli (Seuil, Paris 2000) 18. TS. Eliot: Selected Prose. Edited by John Hayward. (Penguin Books, Harmondsworth, Middlesex 1953) 19. T.S. Eliot: The Complete Poems and Plays (Faber and Faber, London and Boston 1969) 20. C. Brooks: The well wrought urn. Studies in the structure of poetry (Harcourt, Brace & World, New York 1947) 21. W. Szymborska: View with a grain of sand. Selected Poems. Translated from the Polish by Stanislaw Bara´nczak and Clare Cavanagh. (Faber and Faber, London 1996) 22. W.B. Yeats: Collected Poems. 2nd ed. (Macmillan, London 1950, repr. 1987) 23. A. Tennyson: The Poems of Tennyson in three volumes. 2nd ed. edited by Christopher Ricks. Volume Two. (Longman, Harlow 1987) 24. C.P. Cavafy: Poems. Translated, from the Greek, by J.C. Cavafy. With an introduction by Manuel Savidis. (Ikaros, Athens 2003)

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Discussion J. Rasche: Is there a link between number 3 and the dynamic concept of time (pastpresent-future), and number 4 and the static concept (2 + 2)? This would fit to common concept (3 = ∧ , activity, trinitarian world, 4 = ∧ , mandala, static. . . ) J.J.A. Mooij: Perhaps the numbers 3 and 4 have the values of activity and stability, respectively. My own sense of balance is different, however. It is definitely “uneven.” I feel that a group of four poems or an essay in two parts are less well-balanced than a group of five poems or an essay in three parts. Maybe with respect to paintings it is somewhat different, for a symmetrical composition can seem very stable indeed. But with respect to natural landscapes I again feel that two and four are more dynamic than three: three trees in a row can look more firmly rooted to me than two (or four). But I do not see any connection at all with the distinction between static and dynamic views of time, whether on the basis of a Jungian interpretation or on the basis of my own intuitions. G. Veneziano: In General Relativity one talks about two kinds of time: coordinated time and proper time. If two persons at the same point in space synchronize their clocks, and then one of them takes a trip to the moon, when he comes back to earth to meet his friend again the two will be at the same coordinated time, but this clocks will show a different (proper) time. Did philosophers consider the implications of the existence of these two different times? Which one are they talking about when they speak about “time”? J.J.A. Mooij: It is impossible to say in general which notion philosophers have in mind when speaking about “time.” Some of them have discussed the conceptual revolution brought about by the theory of relativity. One of the first to do so in a convincing way was the neo-Kantian philosopher Ernst Cassirer (in 1921). Less successful was Henri Bergson. Several outstanding analyses came from the logical empiricists, in particular from Hans Reichenbach and Adolf Grünbaum. Some (other) analytical philosophers followed suit. Most philosophers, however, either in phenomenological or in analytical traditions, primarily refer to “time” in its common sense meaning. Generally speaking, phenomenologists concentrate on temporal intuitions, analytical philosophers on conceptual features involved in the language of time. Surely, members of both groups may arrive at conclusions far removed from everyday experiences; Heidegger and some Heideggerians even loathe the blindness of everyday life. Nonetheless, these philosophers are interested in time as it occurs in human consciousness and experience, even if they aspire to a metaphysics of time. To them, the physical distinction between proper time and coordinate time is irrelevant.

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A la rigueur, one might say that their notions of time are closer to proper time than to coordinate time, perhaps even that these notions are something like proper time taken in a nonphysical (psychological or metaphysical) sense. It is primarily in the tradition of logical empiricism that the theory of relativity is a subject of discussion or a system in the background. In so far as they studied the philosophy of physics, they certainly considered the implications of the two different times. Reichenbach and Grünbaum differed, however, in their views of the present, the now: the first held that it is based on physical reality, the latter argued that it is dependent on consciousness.

The Evolution of the Universe

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Ruth Durrer

3.1 From Myths to “Rational Explanations” of Phenomena Since its beginning, mankind has been preoccupied with questions of the kind: Where does the Universe end, how big is it? Is there a beginning in time? If yes, how has the Universe been “created”? Will there be an end? What is the role of human beings in all this? Originally, before the advent of scientific reasoning, such questions were addressed with myths and religious beliefs. As Karl Popper puts it: “Science must begin with myths and with the criticism of myths.” We therefore also start this text with a myth.

3.1.1 The Chinese Myth of Phan Ku “In the beginning was a huge egg containing chaos and a mixture of yin–yang (female–male, cold–heat, dark–light, wet–dry, etc). Also within this yin–yang was Phan Ku who broke forth from the egg as a giant who separated the yin–yang into many opposites, including earth and sky. With a great chisel and a huge hammer, Phan Ku carved out the mountains, rivers, valleys, and oceans. He also made the sun, moon, and stars. When he died, after 18,000 years, it is said that the fleas in his hair became the human beings.” (From [1].)

In ancient times there were myths for most natural phenomena like thunderstorms, the seasons, floods etc. For example the sun was a fiery chariot driven across the sky by the god Helios. The Greeks were among the first to turn also to rational explanations. For example, Anaximander of Miletus (610–545 B.C.), a fellow citizen of Thales pictured the sun as a hole in a fire-filled ring which encircles the earth and revolves around it [2]. 27

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Xenophanes of Colophon (570–480 B.C.) explained that the earth exudes combustible gases at night until they reach a critical density and ignite, thereby creating the sun every morning. Night falls when the ball of gas has burned out [3]. Even though these explanations have not been submitted to experimental tests like modern scientific hypotheses, they differ from the myths in that they are rational explanation and no supernatural being, “giant,” is involved. 3.1.2 The Pythagoreans The rationalist movement came into its first bloom with the Pythagorean school (Pythagoras 580–500 B.C.) which included artists and philosophers alike (also women participated). It was known that the earth was round and Eratosthenes (276–194 B.C.) measured its diameter using the famous experiment with the stick via the following reasoning: He had heard that on the longest day of the summer solstice, the midday sun shone to the bottom of a well in the town of Syene (Aswan) and hence the sun was vertically over Syene at this time, see Fig. 3.1. At the same time, he observed that the sun was not directly overhead at Alexandria; instead, it cast a shadow which corresponds to an angle of 1/50 of a circle (7.2°) with the vertical. According to Fig. 3.1, this implies that the angle between Syene and Alessandria subtended from the center of the earth is also 7.2°. To these observations, Eratosthenes applied the known fact that Alexandria and Syene lay on a direct north-south line. Legend has it that he had someone walk from Alexandria to Syene to measure the distance: that came out to be equal to 5000 stadia or (at the usual Hellenic 185

Shadow Cast *

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sun the m o ene s fr t Sy l ray a e l l l a tica Par Ver

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a

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Fig. 3.1 Eratosthenes measurement of the circumference of the earth

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meters per stadion) about 925 kilometers. The linear distance between Alexandria and Syene was hence 1/50 of the circumference of the Earth which thus must be 50 × 5000 stadia = 250,000 stadia or 46,000 km. (The circumference of the Earth is 40,075.16 km). With this it was then possible to measure other cosmological distances (earth–moon, earth–sun, etc) with varying success (see e.g. Aristarchos, 310– 230 B.C. [4]).

3.1.3 Ptolemy’s Epicycles After many hundreds of years the observations of the solar system led to a cosmology which is described fully in Ptolemy’s Almagest [5]. Ptolemy’s model is earth centered, everything falls towards the center of the earth, see Fig. 3.2. To explain the retrograde motions of the planets, one has to introduce epicycles1 and epicycles of epicycles. The model becomes very complicated but is accurate and has significant explanatory power.

Fig. 3.2 The geocentric model of the world

1

The motion on an epicycle is a motion on a circle, the center of which moves on another circle.

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3.1.4 Heliocentric Model The first to propose a heliocentric model were probably the Pythagorean Philolaus of Croton (ca. 480–385 B.C.) and Aristarchos. The heliocentric model, see Fig. 3.3, is simpler, especially, it naturally explains the retrograde motions of planets which come from the fact that the earth itself is not at rest but also revolves around the sun and therefore can overtake other planets (Copernicus). It is also in better agreement with a large sun (Aristarchos). However, it needs a very large Universe, the fixed stars have to be very far away so that no parallaxes are seen by the naked eye (Aristarchos). At the time of Copernicus his heliocentric model was not in good agreement with accurate data, especially of Mars, until Kepler introduced the ellipse [6]. The natural question why do we not “feel” the motion of the earth was only clarified by Galileo’s principle of relativity (see below). Also the question why does not everything fall towards the sun was not resolved until Newton discovered the universality of gravity. The distances to the fixed stars were determined for the first time in 1838 when Bessel measured the parallax to α -Centauri (about 1/4 of an arc second which implies a distance of about 12 light years, the true distance is about 4.3 light years).

Fig. 3.3 The heliocentric model of the world

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3.1.5 Galileo’s Principle of Relativity “Shut yourself up with a friend in the main cabin below deck on some large ship, and have with you some flies, butterflies and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how all the little animals fly with equal speed to all sides of the cabin; how the fish swim indifferently in all directions; how the drops fall into the vessel beneath. And, in throwing something to your friend, you need to throw it no more strongly in one direction than another, the distances being equal; and jumping with your feet together you pass equal spaces in all directions. When you have observed all these things carefully have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship moves or stands still.” (From [7]).

This was probably the most important discovery of Galileo Galilei. Namely that velocity is not absolute but relative and only accelerations (e.g. fluctuations in the ships motion) can influence the outcome of a physical experiment. The physical laws are the same in all inertial systems (i.e. systems which move with respect to each other with a constant velocity).

3.2 The Evolution of the Universe After the earth–sun distance (first measured by Aristarchos) of about 1.5 × 108 km  8 light minutes, and the distance to the nearest star of about 4.3 light years  4.3 × 1013 km, distances to other stars could be measured. The next milestone in the cosmic distance ladder was the distance to the next spiral galaxy, Andromeda, which is about 2.5 × 106 ly  2.5 × 1019 km and was determined first around 1920 by E. Hubble. Until then it was heavily debated whether the “nebulae,” known today as spiral galaxies where truly extra-galactic or simply clouds of gas in our own galaxy. The size of the observable universe as measured today is about 1.4 × 1010 ly  1.3 × 1023 km. Immediately after having discovered General Relativity (in 1915), Einstein realized that, in contrast to Newtonian gravity, his equations of gravitation are capable of describing the Universe as a whole. However, supposing a static, homogeneous, isotropic cosmology, he could obtain a solution to his equations only after introducing the cosmological constant which he later called his “greatest blunder” [8]. The spacetime of Einstein’s cosmological solution is static and finite in size. Furthermore (what Einstein seems not to have realized), it is unstable. A few years later Friedmann and Lemaître [9, 10] developed time-dependent but isotropic and spatially homogeneous world models where space can be finite or infinite and expanding or contracting. After the discovery of the expansion of the Universe by Hubble (1930) also Einstein accepted the model of an expanding universe.

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Lemaître showed that an expanding universe was not only denser but also hotter in the past. The cosmic microwave background (CMB), an isotropic radiation field, an “afterglow” from the hot early phase was first predicted by Gamov (1948) and discovered in 1965 by chance [11] by Penzias and Wilson (Nobel prize 1978). This radiation has a perfect Planck spectrum with temperature T0 = 2.725 ± 0.002 [12] (Mather, Nobel prize 2006). In addition the CMB is very isotropic (the same in all directions). Apart from a dipole anisotropy which is caused by our motion with respect to this radiation (Doppler effect) the temperature fluctuations are of the order of a few ×10−5. After long searches they have been discovered by the DMR experiment aboard the NASA satellite COBE [13] (Smoot, Nobel prize 2006). The latest observations of these fluctuations from the five-year data of NASA’s WMAP satellite, the follow-up to COBE, are shown in Fig. 3.4. The small anisotropies in the CMB can be calculated very accurately within linear perturbation theory, assuming a simple initial spectrum of fluctuations. The resulting spectrum2 of CMB fluctuations depends not only on the initial fluctuations but also on the cosmological parameters describing the curvature and the matter content of the Universe which determine its expansion history. Furthermore, these anisotropies lead to a slight polarization of the CMB radiation which are correlated with the temperature anisotropies. This correlation also depends on cosmological parameters. At present it is mainly these CMB observations which allow us to accurately determine the parameters of the present Universe [15]. For example, the fact that spatial curvature is zero or very small, that the baryonic matter in the universe contributes only about 4% to the expansion of the Universe and the rest must be provided by dark matter (about 22%) and dark energy (about 74%) comes

Fig. 3.4 This is a false color map of the CMB anisotropies after subtraction of the much larger dipole contribution and the emission from the Milky Way. Figure from [14]

2

The spectrum gives the average square amplitude of the fluctuations of a given angular size, see Fig. 3.5.

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Fig. 3.5 The spectrum of the CMB anisotropies and of the temperature-polarization correlation as observed with the WMAP satellite (figure from [14])

mainly (but not only) from the observation of this radiation. In Fig. 3.5 the theoretical curves from the CMB anisotropy spectrum (top panel) and the temperaturepolarization correlation are compared with the data from the WMAP satellite. Of course these determinations of parameters are “indirect” i.e. they come from a best fit for six parameters which also make assumptions about the initial fluctuations. On the other hand, inflationary models (see below) give us a handle on how to parameterize these initial conditions. I think we can only say that the present model, with given parameter values can fit the data very well. However, because of this indirect determination, cosmologists are always very keen on having complementary datasets which determine the same cosmological parameters in independent ways. At present, a simple cosmological model of a spatially flat universe containing baryons (ordinary matter, atoms), nonbaryonic dark matter and dark en-

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ergy (see below) with a nearly scale-invariant spectrum of scalar initial fluctuations is in good agreement with nearly all cosmological data. What is more, it determines the cosmological parameters within a few percent. This precision opens a new era in cosmology. About ten years ago, most cosmological parameters where only known within their order of magnitude. As with all physical theories, observations will never “confirm” this simple model. They can only rule it out or, at best, be compatible with it. In physics we cannot confirm a theory by experiment, but an experiment can be in agreement with the predictions of a given theory and if this happens over and over for different, independent experiments, our confidence in the theory grows. This is of course, where cosmology is still in its infancy. We have only a limited number of independent observations measuring the same cosmological parameters. But within a few years, a couple of new, highly challenging observations will become possible. Gamov also predicted that the universal abundance of Helium had been generated in the “hot big bang.” The resulting Helium and especially also the Deuterium abundance depend on the baryon density of the Universe. The observations are in good agreement with the calculations if the baryon density is taken to be the one required also by CMB observations. This is one of these independent confirmations of a parameter estimated by CMB observations. Actually, this nucleosynthesis result for the baryon density of the Universe is older than the CMB estimate. But what rendered the universe homogeneous and isotropic in the first place? A tentative answer to this is inflation, a phase of very rapid expansion which dilutes inhomogeneities and curvature. What is more interesting, inflation generates also the right spectrum of initial fluctuations to correspond to the fluctuations measured in the CMB. During the rapid inflationary expansion of the Universe, quantum fluctuations which are present in all fields, are stretched to very large scales and then freeze in as classical fluctuations in the space-time metric and in the energy density of the cosmic fluid. According to this inflationary idea, the largest structures in the Universe stem from tiny quantum fluctuations, a breathtaking connection! The calculations of these fluctuations are in perfect agreement with the observed nearly scale-invariant spectrum of CMB anisotropies. Even though the idea of homogenization by inflation looks convincing at first sight, it remains an unproven idea until this day and there are counter examples like “eternal inflation” where it does not work. Therefore, alternative ideas to inflation are still very in demand and need to be studied (see the contribution by Gabriele Veneziano). Also within the framework of the so successful cosmological “standard model” there remain a series of open questions. The most important ones are maybe the following: • What is inflation, the inflaton? • What was before inflation? Did time have a beginning, was there a Big Bang? • Is space finite or infinite? This question cannot be answered in the Big-Bang Universe, since we can see only finitely far, namely 1.4 × 1010 ly  H0−1 (Hubble scale) far. Even if the cur-

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vature of 3d space vanishes, space might be finite in one or several directions, e.g., rolled up like a cylinder. If the sides of the cylinder are several times larger than the Hubble scale, we have no means to ever determine this by an experiment. Apart from the speculative inflationary beginning, there are other open questions in cosmology which concern the bizarre values of the cosmological parameters: • The observed matter in the Universe, the atoms, are not sufficient to explain the formation of the observed large-scale structure (galaxies) and the flat rotation curves of galaxies and the virial motion in clusters of galaxies. Dark matter needs to be added which contributes about 8-times more to the matter density of the Universe than the visible, baryonic matter. Despite intensive searches, dark matter has not been detected so far by any other means than its gravitational influence (on galaxies, clusters and on the CMB). Nevertheless, from well-motivated theories of high-energy physics we have good candidates for such dark matter particles which we actually might discover soon e.g. at the Large Hadron Collider (LHC) which will start operation this year at CERN. • At present, the expansion of the universe seems not to be slowed down as expected by the presence of matter and its gravitational attraction, but it seems to be accelerated. To obtain such a behavior in the framework of general relativity, the energy density of the universe must be dominated by some gravitationally repulsive component which we term dark energy. What is this dark energy? – A cosmological constant? (Einstein’s biggest blunder, is consistent with all present observations.) – A dynamical large-scale coherent field which comes to dominate just now? – Modification of general relativity at large scales? – An effect from our too simplified modeling of cosmology (back-reaction, a large void, under-density around us). Assuming the simplest solution, the cosmological constant presently fits all the relevant cosmological data (for a more detailed discussion see [16]).

3.3 Conclusions With the growth of precise observational data, we have been forced to adjust and complicate the cosmological model more and more. We have introduced an “inflationary phase,” dark matter which contributes about 22% to the energy density of the Universe, “dark energy” which contributes more than 70%, see Fig. 3.6. The only known form of energy, namely ordinary matter and radiation only contributes about 4% to the energy density of the Universe. For the first time in human history, we have such a successful cosmological model which agrees with precise data on the level of a few percent. This can be

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Fig. 3.6 The contribution of the different forms of energy in the Universe. Baryons (atoms), dark matter which does not emit light and is non-baryonic, and dark energy which has strongly negative pressure leading to repulsive gravitational interaction (for a detailed study of cosmological parameters see [17])

considered as a great achievement. Nevertheless, there are several unmotivated coincidences: the contributions from baryons, dark matter, and dark energy are of the same order. If dark energy is a cosmological constant, it was completely subdominant in the past and will entirely dominate the energy density in the future. Why is it of the same order as the matter density just today when we humans are capable of measuring it? A very bizarre coincidence. Because of these “accidents,” the present cosmological model cannot be considered “beautiful” in the scientific sense of beauty. It is too contrived. This reminds me of the epicycles of Ptolemy. Do we need a change of paradigm? Is it possible that the solution to the problem is already out there but nobody takes it seriously.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

From http://www.crystalinks.com/chinacreation.html (2008) C. Sagan: Cosmos (Ballantine Books, New York 1985) E. Heitsch: Xenophanes: Die Fragmente (Artemis Verlag, Zurich 1983) T.L. Heath: Aristarchus of Samos. The ancient Copernicus: Reprint of the 1913 original (Dover, New York 1981) G.J. Toomer: Ptolemy’s “Almagest” (Princeton University Press, Princeton NJ 1998) A. Koestler: Die Nachtwandler. Die Entstehungsgeschichte unserer Welterkenntnis (Vollmer, Wiesbaden 1963) T. Budden: Galileo’s ship thought experiment and relativity principles, Endeavour, 22, 54 (1998) A. Einstein: Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, S.-B. Preuss. Akad. Wiss. 142 (1917) A. Friedmann: Über die Krümmung des Raumes, Z. Phys. 10, 377 (1922) G. Lemaître: L’Univers en expansion, Ann. Soc. Bruxelles 47A, 49 (1927) A. Penzias and R. Wilson: Astrophys. J. 142, 419 (1965) J. Mather et al.: Highlights of Astronomy, 9, 275 (1992) G. Smoot et al.: Astrophys. J. 371, L1 (1991) G. Hinshaw et al., preprint archived as arXiv:0803.0732 (2008)

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15. R. Durrer: The Cosmic Macrowave Background (Cambridge University Press, Cambridge New York 2008) 16. R. Durrer and R. Maartens: Dark energy and dark gravity: theory overview, Gen. Rel Grav. 40, 301 (2008) 17. J. Dunkley et al., preprint archived as arXiv:0803.0586 (2008)

Discussion G. Setti: The direct evidence for an accelerating universe assumes that supernovae type Ia are really standard candles. Up to now, there are no reasons, observational or theoretical, not to believe this. However, who knows in 10–20-years time. . . R. Durrer: This is correct, supernovae type Ia are assumed to be “modified” standard candles. From their light curves we can infer their luminosity, and by comparing it with the observed intensity we can determine the distance. If the inferred intrinsic luminosities are wrong, the distances are also wrong. However, there are also other observations that indicate the presence of a cosmological constant, and the consistency of all these different observations (CMB, galaxy catalogs, lensing, baryon oscillations, supernovae, etc) gives us confidence. But of course one can never be sure. . . L. Boi: What should be changed in general relativity in order to be coherent with the new cosmological problems and observations? R. Durrer: There are possibilities to change the Einstein–Hilbert action from the Riemann scalar R to f (R). Another possibility are scalar-tensor theories or TeVeS (tensor-vector-scalar theories). One always thought that general relativity has to be modified at high energy, due to quantum effects. But to solve the cosmological dark energy problem one has to modify it at low energy. This is most unexpected and difficult to reconcile with solar system constraints which confirm general relativity also at low energy. J.J.A. Mooij: Is it true that Einstein’s problem with the stability of the Universe had a prologue in Newtonian cosmology in that (according to Newton’s mechanics) a finite material Universe is doomed to collapse? And that Newton realized this, and believed that God had to update his creation by supplying energy? R. Durrer: This is correct. Within Newtonian gravity a finite static Universe is not only unstable, but there is no regular solution of the Newtonian gravitational equation where the mass distribution is finite and static. Even more so for an infinite Universe. In Newtonian gravity a so-called “jeans swindle” is needed. A. Miller: There were problems with observations in the Copernican system, too. The image you showed was a very fanciful one. Kepler got rid of them by suggesting a new mathematics. Might this have a lesson for us today.

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R. Durrer: This is true. Copernicus tried to fit the data with spherical orbit, these were already at the time worse fits than the epicycles. Only once Kepler introduced the ellipses especially the Mars data could be well fitted. That we now might be in a similar situation is a very interesting thought. I do not know the answer. Maybe string theory is something in this direction, a new mathematics.

And the Eternal Zeno Springs to Mind

4

Piergiorgio Odifreddi

4.1 Introduction There is a concept which corrupts and upsets all others. I do not refer to Evil, whose limited realm is that of ethics; I refer to the infinite. I once longed to compile its mobile history. The numerous Hydra (the swamp monster which amounts to a prefiguration or emblem of geometric progressions) would lend convenient horror to its portico; it would be crowned by the sordid nightmares of Kafka and its central chapters would not ignore the conjectures of that remote German cardinal – Nicholas of Krebs, Nicholas of Cusa – who saw in the circumference of the circle a polygon with an infinite number of sides and wrote that an infinite line would be a straight line, a triangle, a circle, and a sphere (De docta ignorantia, I, 13, [1]. Five, seven years of metaphysical, theological, and mathematical apprenticeship would allow me (perhaps) to plan decorously such a book. It is useless to add that life forbids me that hope and even that adverb [2].

If such a hope was denied Jorge Luis Borges, an unlimited professional, then we, as limited amateurs certainly cannot nurture it. At most, taking inspiration from those Avatars of the Tortoise of which we have just quoted the admirable incipit, we can try to follow a single thread of the inexhaustible weft woven by the infinite in the warp of literature: a thread that connects the two protagonists of an immortal story we are now going to tell.

4.2 A Perpetual Race Zeno of Elea, who lived in the fourth century B.C., was perhaps the most prolific inventor of paradoxes, and certainly the best known. His aim was to support Parmenides’ theses against motion, and he did this so effectively that he was called the “forked tongue”: a tongue he lost, according to Diogenes Laertius’ Lives and Opinions of the Eminent Philosophers (IX, 25–29) [3], when he bit it off with his teeth and spat it in the face of the tyrant to incite people to revolt. 39

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The most famous of Zeno’s paradoxes, thanks precisely to a captivating literary image, is the one involving fleet-footed Achilles and the slow-moving tortoise. If Achilles allows the tortoise any advantage he will never be able to reach it, because he must first run the distance he has given as a headstart, but at the same time the tortoise has run another stretch, which Achilles must cover, and so on. The oldest version that has come down to us is in Aristotle’s Physics (Z 9, 239b– 240a) [4], but the more recent version given by Borges in his essay Avatars of the Tortoise [2] is perhaps more suitable here: Achilles runs ten times faster than the tortoise and gives the animal a headstart of ten meters. Achilles runs those ten meters, the tortoise one; Achilles runs that meter, the tortoise runs a decimeter; Achilles runs that decimeter, the tortoise runs a centimeter; Achilles runs that centimeter, the tortoise, a millimeter; Fleet-footed Achilles, the millimeter, the tortoise a tenth of a millimeter, and so on to infinity, without the tortoise ever being overtaken. . .

Many of Zeno’s arguments were discovered almost simultaneously by the Chinese sophist of the fourth century B.C. Hui Shi, and included in the last chapter of the Taoist classic Chuang Tzu [5], criticizing them as “words that do not reach the target,” a “desire to run faster than one’s own shadow.” One in particular presents a perfect analogy of Achilles and the tortoise: If a foot-long stick is halved every day, something will always remain of it even after ten thousand generations. Once discovered, infinite regression spread in infinite variants, positive and negative. Plato used it in his Parmenides [6], a dialogue in which Zeno appears as the protagonist. Eudoxus and Archimedes adopted it to approximate the circle by way of regular polygons. Avicenna Ibn Sina and Thomas Aquinas took demonstrations for the existence of God from it. Fermat formalized it in the method of infinite descent. Kant deduced the second antimony of pure reason from it, Schopenhauer the impossibility of knowing oneself. . . After having been the exclusive prerogative of philosophy and mathematics for two millennia, the time finally came for Zeno’s ideas to attract the interest of literature, and for us to begin our brief history.

4.3 A Complete Autobiography The baptism of the art is in this piece from The Life and Opinions of Tristram Shandy, Gentleman (IV, 13) by Lawrence Sterne, of 1761 [7]: I am this month one whole year older than I was this time twelve-month ago; and having got, as you perceive, almost into the middle of my fourth volume – and no farther than to my first day’s life – ‘tis demonstrative that I have three hundred and sixty-four days more life to write just now, than when I first set out; so that instead of advancing, as any common writer, in my work with what I have been doing at it – on the contrary, I am just thrown so many volumes back – was every day of my life to be as busy a day as this – And why not? – and the transactions and opinions of

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it to take up as much description – And for what reason should they be cut short? as at this rate I should just live 364 times faster than I should write – It must follow, an’ please your worships, that the more I write, the more I shall have to write – and consequently, the more your worships read, the more your worships will have to read. As for the proposal of twelve volumes a year, or a volume a month, it no way alters my prospect – write as I will, and rush as I may into the middle of things, as Horace advises, – I shall never overtake myself – whipp’d and driven to the last pinch, at the worst I shall have one day the start of my pen – and one day is enough for two volumes – and two volumes will be enough for one year.

So the protagonist of the story, who had embarked on writing his complete autobiography, had to abandon it because the task proved quite obviously impossible. There is actually nothing paradoxical about this: just a sad recognition of the mortality of human life. The paradox would rather be an infinite life, because it would allow a person not only to write his complete autobiography at the slow pace of Tristram Shandy, but also to spend most of the time doing something else (to make the biography itself interesting). The real paradox would be to postulate not a future but an infinite past just ended: on the basis of the same correspondence between infinite years and infinite days. In this case, too, there would be all the time needed to write one’s own complete autobiography. If it were not for the fact that it would in any case take a year to describe the last day, and so it would have to be started before experiencing it!

4.4 End of the Race In 1895 Lewis Carroll imagined that the two protagonists of Zeno’s paradox had miraculously completed their task, and related What the tortoise said to Achilles [8]: a dialogue that, according to the narrator, went on for months. As a good logician, Carroll used Eleatic methods to argue that syllogisms are not possible. To say that certain premisses are followed by a conclusion means saying that there is a rule that allows one to go from the first to the second; but to be able to apply the rule one needs a meta-rule, which would say that from premises, and by the rule that binds premises and conclusion, it is possible to reach conclusions, and so on. The novelty of Carroll’s argument was that it effectively showed the distinction between linguistic implication and meta-linguistic deduction, now formalized in the so-called theorem of deduction, which proclaims their equivalence. More generally, the argument for the first time explicitly showed the distinction between language and meta-language, which is now a confirmed achievement of modern logic. In a similar way and for similar purposes, though rather belatedly, Ludwig Wittgenstein was to show in his Philosophical Investigations [9] that rules are not learnt by way of meta-rules. Otherwise, in order to learn the meta-rules, there would need to be meta-meta-rules, and so on.

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4.5 A Perfect Map The entire philosophy of Francis Herbert Bradley is an explicit return to the Eleatics: he believed in the absolute and the one, as the only possible reconciliation of the ubiquitous contradictions of multiplicity in the appearance. In particular, in Appearance and Reality (1893) [10] Bradley made a variation of Carroll’s variation of Zeno’s paradox to show that no kind of properties or relations is possible. For example, to say that an object has a unary property means saying that a binary relation exists between the object and the property, and this means saying that there is a ternary relation between the object, the property and the relation, and so on. Bradley’s ideas were discussed in detail in 1899 by Josiah Royce in The World and the Individual [11], in which an effective (and now well known) image of the infinite regression in them is presented: Let us imagine that a portion of the soil of England has been leveled off perfectly and that on it a cartographer traces a map of England. The job is perfect; there is no detail of the soil of England, no matter how minute, that is not registered on the map; everything has there its correspondence. This map, in such a case, should contain a map of the map, which should contain a map of the map of the map, and so on to infinity.

In a piece in The Artifice, of 1960 [12], apocryphally attributed to the nonexistent J.A. Suárez Miranda (From Travels of Praiseworthy Men, 1658), Borges reformulates the argument literally: . . . In that Empire, the craft of Cartography attained such Perfection that the Map of a Single province covered the space of an entire City, and the Map of the Empire itself an entire province. In the course of Time, these Extensive maps were found somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point for point. Less attentive to the Study of Cartography, succeeding generations that followed came to judge a map of such Magnitude cumbersome, and, not without Irreverence, they abandoned it to the Rigours of sun and Rain. In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography.

The problem raised by the infinite regression of Royce’s map is more implicitly found in all the works containing a part that should coincide with the work itself: Homer’s Iliad, in which Helen embroiders a purple garment that illustrates the story of the poem; Valmiki’s Ramayana, at the end of which Rama’s sons seek refuge in a forest, where an ascetic teaches them to read a book that is, precisely, the Ramayana; Vyasa’s Mahabarata, which begins with a narrator who meets a friend and tells him the Mahabarata, a story about the poet Vyasa who told the god Ganesh the Mahabarata, a story that tells of a king who meets the poet Vyasa and has him recount the Mahabarata; the Dream of the Red Chamber, in which the protagonist sees the events of the story at the beginning of a dream; Shakespeare’s Hamlet, in which a tragedy is represented that is more or less the same as Hamlet. It is today perhaps no less interesting to consider the subject mathematically, in the light of the Banach Fixed Point Theorem, according to which a contraction on a complete metric space has a single fixed point: in the case of the map, this means there should be a point of land that coincides with its image on the map.

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4.6 Kafkaesque Situations The entire works of Franz Kafka are a literary reworking of Zeno’s paradoxes, according to the Argentinean writer Carlos Mastronardi. The pathos of his stories arises precisely out of the infinite number of obstacles that block their identical heroes. And the incompleteness of his works, the lack of intermediate chapters, are a consequence of this. Indeed, it is not necessary to number all the points of a segment or all the possible vicissitudes, to suggest infinite regression. As an example of the process, it is worth quoting the Zeno-like Emperor’s Message entirely, because of its brevity, dating from 1917 (later inserted in the contemporary and equally Zeno-inspired The Great Wall of China) [13]: The Emperor, so it runs, has sent a message to you, the humble subject, the insignificant shadow cowering in the remotest distance before the imperial sun; the Emperor from his deathbed has sent a message to you alone. He has commanded the messenger to kneel down by the bed, and has whispered the message to him; so much store did he lay on it that he ordered the messenger to whisper it back into his ear again. Then by a nod of the head he has confirmed that it was right. Yes, before the assembled spectators of his death – all the obstructing walls have been broken down, and on the spacious and loftily mounting open staircases stand in a ring the great princes of his Empire – before all these he has delivered his message. The messenger immediately sets out on his journey; a powerful, an indefatigable man; now pushing with his right arm, now with his left, he cleaves a way for himself through the throng; if he encounters resistance he points to his breast, where the symbol of the sun glitters; the way is made easier for him than it would be for any other man. But the multitudes are so vast; their numbers have no end. If he could reach the open fields how fast he would fly, and soon doubtless you would hear the welcome hammering of his fists on your door. But instead how vainly does he wear out his strength; still he is only making his way through the chambers of the innermost palace; never will he get to the end of them; and if he succeeded in that nothing would be gained; he must next fight his way down the stair; and if he succeeded in that nothing would be gained; the courts would still have to be crossed; and after the courts the second outer palace; and once more stairs and courts; and once more another palace; and so on for thousands of years; and if at last he should burst through the outermost gate – but never, never can that happen – the imperial capital would lie before him, the center of the world, crammed to bursting with its own sediment. Nobody could fight his way through here even with a message from a dead man. But you sit at your window when evening falls and dream it to yourself.

4.7 Borges’ Inventions Borges has taken from Zeno’s paradoxes the basis of his thinking on the infinite, time and reality (ubiquitous in all his writings) on one hand, and on the other the cue for constructing his disturbing situations at the limit. He thinks they are the definitive proof that will reveal the hallucinatory nature of the world: without them our dreams would be so precise and consistent that we would end up believing them, but they introduce fissures of absurdity that end up revealing their unreality. Borges then uses them repeatedly, and for similar purposes, in his literary constructions, which I have already examined in detail for Le Scienze on another occasion [14]. So here it will be enough to quote two examples.

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The first is taken from Death and the Compass, of 1944 [15]. A detective manages to foresee the last crime in a series, but on going to the place to prevent it, finds that he has been lured there to be killed. This is the last conversation between the victim and his murderer: ‘In your labyrinth there are three lines too many,’ he said at last. ‘I know of one Greek labyrinth which is a single straight line. Along that line so many philosophers have lost themselves that a mere detective might well do so, too. Scharlach, when in some other incarnation you hunt me, pretend to commit (or do commit) a crime at A, then a second crime at B, eight kilometers from A, then a third crime at C, four kilometers from A and B, halfway between the two. Wait for me afterwards at D, two kilometers from A and C, again halfway between both. Kill me at D, as you are now going to kill me at Triste-le-Roy.’ ‘The next time I kill you,’ replied Scharlach, ‘I promise you that labyrinth, consisting of a single line which is invisible and unceasing.’

He moved back a few steps. Then, very carefully, he fired. The second example comes from The God’s Script, of 1949 [16]: One day or one night – what difference between my days and nights can there be? – I dreamt there was a grain of sand on the floor of the prison. Indifferent, I slept again; I dreamt I awoke and that on the floor there were two grains of sand. I slept again; I dreamt that the grains of sand were three. They went on multiplying in this way until they filled the prison and I lay dying beneath that hemisphere of sand. I realized that I was dreaming: with a vast effort I roused myself and awoke. It was useless to awake; the innumerable sand was suffocating me. Someone said to me: ‘You have not awakened to wakefulness, but to a previous dream. This dream is enclosed within another, and so on to infinity, which is the number of grains of sand. The path you must retrace is interminable and you will die before you ever really awake.’

Indefinitely multiplying the examples of literary Eleaticism would not only be useless, but impossible. So we may stop here, as we could in any other place, leaving the reader an alternative regarding the incessantly nagging doubt set off by Zeno: should we renounce that scrap of Greek light to maintain our concept of the universe or, as Borges suggested, alter our concept of the universe to adapt it to that scrap of Greek darkness?

References 1. J. Hopkins: Nicholas of Cusa on Learned Ignorance: A Translation and an Appreaisal of De Docta Ignorantia (The Arthur J. Banning Press, Minneapolis 1985) 2. J.L. Borges: Avatars of the Tortoise. In: Labyrinths (Penguin Books, Harmondsworth 1970) 3. L. Diogenes: Lives and Opinions of Eminent Philosophers (Kessinger Publishing Co., Whitefish 2007) 4. Aristotle: Physics (Oxford University Press, Oxford 1996) 5. Hui Shi, in The complete works of Chuang Tzu (Columbia University Press, New York and London 1968) 6. Plato: Parmenides (Focus Philosophical Library, Newburyport 1996) 7. L. Sterne: The life and opinions of Tristam Shandy, Gentleman (Penguin, London 2003) 8. L. Carroll: What the Tortoise said to Achilles, Mind, 4 (1895) 9. L. Wittgenstein: Philosophical Investigations (Prentice Hall, Englewood Cliffs 1987) 10. F.H. Bradley: Appearance and Reality: a Metaphysical Essay (Adamant Media Corporation, Boston 2005)

4 And the Eternal Zeno Springs to Mind 11. 12. 13. 14. 15. 16.

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J. Royce: The World and the Individual (Peter Smith Inc., Gloucester 1976) J.L. Borges: El Hacedor, Dreamtigers (University of Texas Press, Austin 1964) F. Kafka: The great wall of China. In: The Complete Short Stories (Vintage, United Kingdom 1992) P. Odifreddi: Un matematico legge Borges, Le Scienze 372 (1999) 76–81 J.L. Borges: Death and the Compass. In: Labyrinths (Penguin Books, Harmondsworth 1970) J.L. Borges: The God’s Script. In: Labyrinths, (Penguin Books, Harmondsworth 1970)

Discussion G.O. Longo: I think it unnecessary to remark that Borges is a mine of ideas concerning the subjects you have touched upon. It is Borges that reports the anecdote of the man that dreams he is a butterfly. In Otras Inquisiciones (Other Inquisitions), he writes (approximately): “About twenty-four centuries ago, Chuang Tzu dreamt he was a butterfly. When he woke up, he did not know whether he was a man that had dreamt he was a butterfly or a butterfly that was dreaming it was a man.” And on a similar subject Borges quotes a few lines, I think by Coleridge: “If a man could pass through Paradise in a dream, and have a flower presented to him as a pledge that his soul had really been there, and if he found that flower in his hand when he awoke. . . And what then?” But the dream of the butterfly is more interesting, as it hints at an unsolvable circularity, an unending oscillation between A and not-A. And, to mention dreams and circularity, another extraordinary story by Borges, Las Ruinas Circulares (The Circular Ruins) tells about a wizard that tries to create another human being from his own dreams, night after night. At the end, with relief, with humiliation, with terror he understands that he too is a mere appearance, dreamt by another. Again a recursus ad infinitum? As to Kafka, who was one of the most remarkable visionary writers, besides Eine kaiserliche Botschaft (An Imperial Message), that you mentioned, he wrote another paradoxical story, Vor dem Gesetz (Before the Law), that in fact is contained in the novel Der Prozess (The Trial). The story goes like this: a man from the country comes before the law and asks the gatekeeper to gain entry into the law. The gatekeeper, however, does not grant entry. For days and years the man from the country waits to gain entry, to no end. When the gatekeeper sees that the man is going to die, he shouts to him: here no one else can gain entry, since this entrance was assigned only to you. I’m going now to close it. It is a mysterious story, and Kafka aptly devotes several pages to various explanatory hypotheses, without reaching any conclusion. It is good that no conclusion is reached: conclusions are lethal and, as it were, anti-Gödelian. So there are paradoxical narratives not necessarily stemming from Zeno, in fact more fluid and more interesting as they leave interstices where active imagination can slip in. . . Last but not least, I am anxious not to leave the audience under the misleading impression that map and territory coincide or might coincide. . . P. Odifreddi: In one point. . .

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G.O. Longo: . . . in one point, sure, they do, but if they coincided everywhere, that would be a problem for cartographers, those who work in Geographical Institutes, not those working for the Emperor in Borges’ story. Alfred Korzybski coined a simple but mighty aphorism, that Gregory Bateson often quoted: “The map is not the territory.” The territory contains a potentially infinite amount of information, and we all know that when we mention infinity we have to be extremely careful, as you pointed out speaking of Borges. Infinity is a concept that pollutes and corrupts all other concepts, not only from a rational viewpoint, but also from an ethical viewpoint, since perfection, which is so important for mathematicians, is not to be found in human matters, alas, if not as an asymptotic aspiration. As a consequence, in constructing a map from a territory it is not possible to reproduce the territory perfectly, just by reducing its scale. Actually, if the map were a perfect reproduction of the territory, the concept of scale would be meaningless: any reduction would be insignificant. Hence we have to give up perfection, we have to omit many (infinite) details contained in the territory. This is why the map is not the territory. And what details are to be omitted? Those that are not of interest for the people who construct or use the map. Each of us has different interests and goals, and consequently many different maps can be obtained from the same territory. I realize that I did not ask questions, rather I made remarks and digressions. P. Odifreddi: But I will answer, anyway. Concerning the story you have just mentioned of the Emperor who dreams of being a butterfly, and then wakes up asking himself whether he perhaps is a butterfly who in his sleep dreams of being an Emperor, it comes indeed from the Chuang-zu, the famous Taoist book I had quoted. Naturally, Borges quoted it because it adhered to his idealistic concept of life, in which one cannot distinguish between dreams and reality. The Greeks had already thought this, and this indeed is one of the tenets of idealism: that dreams are so vivid that they appear like reality whereas sometimes reality is so confused that it appears like a dream. As for the metaphor of the map and the territory, naturally Borges has used it masterfully. But there are many other examples that one could quote, for instance that of the literary critic who had set up to summarize the Divine Comedy, and began by writing that it was a poem of three Canticas, the Inferno, the Purgatory, and the Paradise. But then he realized that the abridgment had been too drastic, and so he wrote that each Cantica contains 33 Canti, and kept adding details. But he still sensed that this wasn’t enough, and felt that some verses had to be added, and he started doing so. In the end, the summary of the Divine Comedy became exactly the Divine Comedy that Dante had written. He published it, and Borges said that the critics who felt that this was Dante’s Comedy had not understood that this was instead a completely different thing, at which Dante had arrived at from above, writing Canto after canto, whereas the critic had arrived at it from below, trying to approximate it. Borges derived from this metaphor the literary theory that he developed in other short stories for instance that of the famous Pierre Menar, who wanted to write the Quixote without being Cervantes, i.e., a Spaniard of the 16th century. Thus, he could only write it forgetting all the history he had learned, and Borges reads the two pieces, that by Cervantes and that by this guy Menar

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side-by-side. They are absolutely identical, except that the two texts can be read differently depending on the way one interprets them. In reality then these literary plays hide something that is much deeper and the thing that is most interesting from the stand point of the theme of this Symposium is that Borges indeed was a man on the hinge: he was of course a literary man, but also had deep love for, and knowledge of scientific literature. For instance he has read Russell and quoted his works. Another writer of this type was Calvino. Born as a conventional writer, he began to feel a deep interest in science: he even read the Scientific American, which was his source of inspiration for the Cosmicomiche and Ti con zero. Actually, Borges and Calvino are related to one another by an amusing anecdote. As is well known, Calvino stuttered and thus did not like to speak, at least in public. As for Borges he was blind, and anybody can see that a dialogue between one who is blind and one who is mute is bound to be difficult. As the story goes Borges arrived one day in Milan to be greeted by a number of literates. One of them told him that Calvino was there as well. To which Borges replied that, yes, he had recognized him by his silence. It seems marvelous to me that one could distinguish different silences. M. Bresciani: On Calvino, I wanted to mention The non existing rider, the novel in which the rider does not exist but at the same time also exists. The story ends but goes ahead, since in the end Bradamante enters the convent where she is Sister Theodora who tells the story of the nonexisting rider, and then leaves the Sister’s clothes and rides away, and the story goes on. The same thing occurs in Ti con zero, in which the chased and the chaser eventually switch roles. P. Odifreddi: Well, Calvino’s case is a classic, as he has used practically every scientific, logical, deductive procedure: Take for instance the famous short story, The Count of Montecristo, which is part of a series of short stories that develop along lines of mathematical deductions. I have had the chance to ask José Saramago whether he recognized himself as belonging to Calvino’s definition of deductive literature. I did it because in his novels there always is some non-sense assumption at the start, but then everything else proceeds along lucid lines. And he replied that writers always believe that what they do is novel and original, but then they discover that somebody even theorized the novelty they only had in mind, like in the case of Calvino’s deductive literature I was confronting Saramago with. Actually, the case of deductive literature seems to me to be a typical example of the intermingling of the two cultures, of the mixing of the “magmatic” warmth of the literature with the cool rigor of logic. Y. Oudai-Celso: I wonder whether one can detect two different attitudes in the way the two cultures look at these paradoxes. That is, whether one can identify as a common denominator in all the paradoxes you have mentioned, not so much the wish to demonstrate, as Borges had wanted to do, that reality does not exist, but instead the gap between reality and the models we use to describe it. This concept is nicely exemplified by the metaphor of the map and the territory. It seems to me that writers like Kafka or Borges rejoice in these paradoxes, while the mathematicians, the logicians, and the philosophers who try to work seriously, take for instance Plato

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or Aristotle, consider them as something to be avoided. “We do not reason in this way,” Plato and Aristotle seem to be saying. In other words, at variance with writers, philosophers, mathematicians, and scientists, tend to use the paradox to verify their model of reality, i.e., to say that, within limits, the map could efficiently describe the territory. P. Odifreddi: Yes, the two cultures have opposing attitudes: logic abhors the paradoxes, art pampers them. However, I would like to say that I see things a little differently from you and the previous questioner. That is, reality is very poor, literature, and I would include in it mathematics as well, is much richer. I would actually include in the realm of literature theology as well: Borges had already said so much when he stated that theology is a branch of fantasy-literature, which seems a marvelous definition to me. He said so in a positive sense, as he meant that theology is a lot more than what is only related to reality, and on this I believe that mathematicians are like the writers, since they are not confined to the restricted borders of reality. Which is undoubtedly complex, but to a lesser degree than literature or mathematics. Naturally, I say all this in a provocative vein. But one can argue in a way which is opposite to yours and to that of the previous questioner, as we do not only have realistic literature, but fantasy-literature as well. Literature can invent its own worlds, just as mathematics can. Physics, of course, cannot. E. Carafoli: Zeno’s paradox may be mathematically impeccable. Yet we know that Achilles, somehow, catches the turtle in no time. I suspect many in the audience will want to know how one goes about accommodating these paradoxes in reality. I am reminded of the famous quip by Thomas Henry Huxley, who said that “The great tragedy of science is the slaying of a beautiful hypothesis by an ugly fact.” You yourself appear to have shared this preoccupation in one of your recent successful books (Matematico e Impertinente). Could you comment on this? P. Odifreddi: Clearly, something seems absurd, here, and this is why this is a paradox. However, there are many solutions to it. The simplest is one that is central to mathematics, according to which the paradox tells us, intuitively, that one can have an infinite sum of terms, each one different from zero, but the sum of which is finite. Gregorio da San Vincenzo understood that first, he understood that one half plus one quarter, plus one eighth, and so on, represent a geometrical series that has a sum. . . But as one discovers this, one immediately slides into other problems, because if instead of writing a series with a half, a quarter, an eighth, etc., one writes it with a half, plus a third, plus a quarter, plus a fifth, one has a so-called harmonic series, the sum of which is infinite. So, what happens? What happens is that at this point mathematics comes into play, with the theory of converging and diverging series. In my opinion, this was the first intuition of Zeno, which went against the axiom of Archimedes, according to which if one sums up equal quantities, no matter how small, an infinite number of times, he eventually reaches infinity. But if the quantities in question are not fixed, if they do not have a lower limit, but decrease to zero, even if, intuitively, one would think the sum would still reach infinity, this is instead not so. Naturally, we should not forget that Zeno lived about 500 years B.C., and

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200 years later Archimedes and the contemporary Greeks had already understood that Zeno’s was only a paradox, and did not care much about it. Unidentified: I wanted to comment on the relationship between the finite and infinite. In the 19th century Freeman proposed something alternative to the two, by introducing a geometrical space, which is neither finite nor infinite, but is unlimited, and he used the sphere as an example. It seems to me that this concept is very important: on one hand it avoids some of the paradoxes of infinity, at least in mathematics, i.e., the paradox of an object, which is unattainable, undefined, inexhaustible, etc. On the other hand, it expands enormously the border of the finite, i.e., it breaks the finite character of mathematical objects, opening new possibilities. The example of the sphere is very telling: we can go around the sphere an infinite number of times without falling into the paradoxes of infinity, of something that is unattainable. The idea has slowly crept into physics as well, and this is a point worth discussing in the framework of the dialogue of the two cultures. P. Odifreddi: Yes, this is why I had chosen to bring in the concept of infinity with Zeno’s paradox and had decided to illustrate it with literary metaphors. Let me underline, as a proof of the confusion Greek thought had on the concept, that the Greeks referred to infinity with the denotation apeiron, which meant unlimited (peiron indeed means limited), and for them apeiron, i.e., unlimited was the same as infinite. Today, mathematicians, and of course Freeman was one of them, have clarified a number of points on this confused issue, separating actual infinity from the unlimited. The most interesting development on this issue was the discovery by Cantor at the end of the 19th century that in fact there were many “infinites.” This was a real step forward, because the Greeks only had the concept of the potential infinite. On this there is an interesting story, which I would like to summarize for you. A manuscript was discovered in Costantinopolis in 1906 in which some of Archimede’s works, including the famous Method that had been lost, had been erased to make room for some prayers – a good example of the dialogue of the two cultures, I should say. Using modern methods, the manuscript could nevertheless be read, and it was a revolution in mathematics, because nobody until then had known how Archimedes had arrived at his famous results on the quadrature, for example of the parabola or of the sphere. Interestingly, the manuscript disappeared, to surface again about ten years later at the public auction in New York where it was sold for very good money. It showed that, at least for one of the demonstrations of the method, Archimedes had used the actual infinite, which is surprising, as the Greeks did not have this concept. But he apparently had it. Anyway, potential or actual, there was only one infinite. But then, the idea crept in that diverse infinites could exist, and I believe the first one who saw them was Giordano Bruno in his Cena delle Ceneri (Dinner of Ashes), in the episode in which the portion of the spherical hearth that we can see on the horizon increases as we, the observers, go progressively higher. As our height reaches infinity, we will see all earth as a semi-sphere. But where is the other half of the sphere? To be able to see it, Bruno says, we should go past the infinite, we should go to a second infinite that would permit us to see the inferior half of the sphere. Naturally, on this Bruno developed a theology, but

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one could see that, in a sense, he had anticipated the result of Cantor, i.e., that the infinite of the real numbers is different from that of the rational numbers or of the integer numbers. Naturally, one could say that the infinites are of one type only, but that we see them as different because of our inability to relate them to one another because of our mental limitations. But it is important to underline that the concept of infinity in mathematics has come a long way from the uncertain approximation of it that the Greeks had expressed with a concept of apeiron. Not recorded question. It focused on the credibility of paradoxes P. Odifreddi: Somebody said, I believe it was Twain, or maybe Chesterton, that paradoxes are truths with the head down and the legs up, which they move to attract attention. But he also said that paradoxes are born as such, and only after a while they become truths, ceasing to be outrageous, and becoming obvious, trivial. In a book on paradoxes I wrote a while ago, the final chapter is the history of mathematics reconstructed through the paradoxes. I wanted to show that mathematics has developed through paradoxical discoveries that are then gradually incorporated into it. Take for instance the matter of the root of 2, which indeed was quite a paradox, a scandal for which in the old days proponents could even be put to death. Even Jupiter took position on the issue. But what is the root of 2 today? Quite simply, we say that it is a real and irrational number, and that’s it. Something that was perceived as a scandalous thing when it was first proposed gradually became a truth, and eventually something obvious. Not recorded question, also focused on paradoxes P. Odifreddi Why do paradoxes look like paradoxes at the outset, but then gradually become trivial? This is so because at the time of their formulation they are not understood, as they go against common sense. Take for instance heliocentrism, which was perceived as an outrageous concept that even brought people who upheld it to their deaths. Today nobody, not even the Church, would dream of questioning it. Concepts that go against public consensus appear scandalous, but then consensus changes, and they become commonly accepted. It is not a change of logic, it is simply that we understand them differently. Or better, I should say. For instance, take Kepler and the elliptical orbits, which Galileo, even if he was the great scientist we all honor, refused to accept throughout his entire life. He simply did not like them. If you allow me to close in a joking way, I could say that scientists who are jailed, or even burned in the public square, may sometimes have deserved it. . .

Creating the Physical World ex nihilo? On the Quantum Vacuum and Its Fluctuations

5

Luciano Boi

It is only through refined measurements and careful experimentation that we can have a wider vision of things: we see things that are far from what we would guess – far from what we could have imagined. Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there. Richard Feynman (1967) [1]

5.1 Introduction: The Vacuum as a Philosophical and Scientific Concept A vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than standard atmospheric pressure. The root of the word vacuum is the Latin adjective vacuus which means “empty,” but space can never be perfectly empty [1, 2]. A perfect vacuum with a gaseous pressure of absolute zero is a philosophical concept that is never observed in practice, not least because quantum theory predicts that no volume of space can be perfectly empty in this way. The words “nothing,” “void,” and “vacuum” usually suggest uninteresting empty space. To modern quantum physicists, however, the vacuum has turned out to be rich with complex and unexpected behavior [3, 4]. They envisage it as a state of minimum energy where quantum fluctuations, consistent with the uncertainty principle of Heisenberg, can lead to the temporary formation of particle–antiparticle pairs. As we will explain in more detail in a while, according to quantum electrodynamics (QED), the vacuum is filled with electron-positron fields. Real electron-positron pairs are created when energetic photons, represented by the electromagnetic field, interact with these fields. Virtual electron-positron pairs, however, can also exist for minute durations (less than a Planck time h¯ /4π ), as dictated by Heisenberg’s uncertainty principle [5, 6]. Vacuum has been a frequent topic of philosophical debate since ancient Greek times, but was not studied empirically until the 17th century [7]. Experimental techniques were developed as a result of Evangelista Torricelli’s theories of atmospheric pressure. Recall that Greek philosophers did not like to admit the existence of a vacuum, asking themselves “how can ‘nothing’ be something?” Plato found the idea 51

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of a vacuum inconceivable. He believed that all physical things were instantiations of an abstract ideal, and he could not conceive of an “ideal” form of a vacuum; similarly, Aristotle considered the creation of a vacuum impossible – nothing could not be something. Later Greek philosophers thought that a vacuum could exist outside the cosmos, but not within it. We would like to notice here that in Aristotelian physics [8], the medium (such as water or air, etc.) plays a dual role: (1) to provide a cause for motion (toward a proper or natural place) and (2) to resist the motion (lest it to be infinite). In Fig. 5.1 we have a graphic representation of Aristotle’s argument. Body A moves (in natural motion) equal distances, once through water (B) in a longer time C and once through air (D) in a shorter time E. Then Aristotle assumes that the speeds of the bodies are in the (inverse) ratio of the density of the two media. He goes on to conclude that a body would move infinitely rapidly through a void since it would encounter no resistance. Because he considers this impossible, he deduces that a void or vacuum cannot exist. Then, he attempts to strengthen his conclusion that there can be no void by arguing that, since a body moves more rapidly in direct proportion to its weight, bodies would move at different rates through a void without any apparent cause for this difference in speed. Rejecting this alternative, he states that all bodies would move at equal rates in a void, an equally unacceptable result for him. Hence, Aristotle rejects the possibility of the existence of a void. He holds that the medium is necessary for the motion of the body. The Islamic philosopher Al-Farabi (850–970) appears to have carried out the first recorded experiments concerning the existence of vacuum, in which he investigated handheld plungers in water. He concluded that air’s volume can expand to fill available space, and he suggested that the concept of the perfect vacuum was incoherent. In the Middle Age, Christians held the idea of a vacuum to be immoral or even heretical. Evangelista Torricelli argued in 1643 that there was a vacuum at the top of a mercury barometer. Robert Boyle later conducted experiments on the properties of the vacuum. Some people believe that, although Torricelli produced the first sustained vacuum in a laboratory, it was Blaise Pascal who recognized it for what it was. Their barometer consisted of a tube partially submerged, upside down in a bowl of mercury. What keeps the mercury suspended in the tube? Is there an unnatural vacuum that causes the surrounding glass to pull the liquid up? Or is there no vacuum at all

A

B(water) Time: C

A

D (air) Time: E

Fig. 5.1 Motion through a medium

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but rather some rarefied and invisible matter in the “empty space”? Pascal answered that there really was nothing holding up the mercury. The mercury rises and falls due to the variations in the weight of the atmosphere. The mercury is being pushed up the tube, not pulled up by anything [9]. When Pascal offered this explanation to Descartes, Descartes wrote to Christian Huygens that Pascal had too much vacuum in head. Descartes identifies bodies with extension and so had no room for vacuums. If there were nothing between two objects, then they would be touching each other. And if they are touching each other, there is no gap between them. Galileo was disappointed by Johannes Kepler’s hypothesis that the moon influences the tidal waves because the hypothesis seems to require causal chains in empty space. How could the great Kepler believe something so silly? When Isaac Newton resurrected Kepler’s hypothesis he was careful to suggest that the space between the moon and the Earth was filled with ether. Indeed, the universality of Newton’s laws of gravitation seems to require that the whole universe be filled with a subtle substance. Many doubts were addressed against Newton’s theory. It is here worth noticing that, in 1887, the Michelson–Morley experiment, using an interferometer to attempt to detect the change in the speed of light caused by the Earth moving with respect to the ether, was a famous null result, showing that there really was no static, pervasive medium throughout space. While there is therefore no ether, and no such entity is required for the propagation of light, space between the stars is not completely empty. Besides the various particles which comprise cosmic radiation, there is a cosmic background of photonic radiation (light), including the thermal background at about 2.7 K, seen as a relic of the Big Bang. None of these findings affect the outcome of the Michelson–Morley experiment to any significant degree. These doubts about the existence of ether were intensified by the emergence of Einstein’s theory of relativity [10]. Einstein argued that physical objects are not located in space, but rather have a spatial extent. Seen this way, the concept of empty space loses its meaning. Rather, space is an abstraction, based on the relationship between local objects. Nevertheless, the general theory of relativity admits a pervasive gravitational field, which, in Einstein’s words, may be regarded as an “ether,” with properties varying from one location to another. One must take care, however, to not ascribe to it material properties such as velocity and so on.

5.2 The Dirac Idea of Vacuum as a “Particle Sea”: Wave Function, Dirac Equation and Negative Energy In 1930, Paul Dirac proposed a model of vacuum as an infinite ocean of particles possessing negative energy, called the “Dirac sea.” [11] This theory helped refine the predictions of his earlier formulated Dirac equation, and successfully predicted the existence of the positron, discovered two years later in 1932. The idea of Dirac was further developed in the context of Quantum Field Theory (QFT). Let us say

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a few words more about this striking idea. The Dirac electron is a four-component entity, rather than just having the two components of a “Pauli spinor,” describing the two independent states of spin that a nonrelativistic particle of spin 1/2 possesses. In fact, there are only two components of spin for a particle described by Dirac’s equation, despite there being four components for the wave-function. Mathematically, the reason for this is closely related to the fact that the Dirac equation ∂ ψ = iM ψ is a first-order equation, and its space of solutions is spanned by only half as many solutions as in the case of the second-order wave equation (χ + M 2 )ψ = 0. Physically, this “counting” of solutions of the Dirac equation must take into account the fact that the degrees of freedom of the electron’s antiparticle, namely the positron, are also hiding in the solutions of the Dirac equation. It turns out that the solutions of the Dirac equation are not restricted to being of positive frequency, despite all Dirac’s cleverness and hard work to eliminate the square root in the Hamiltonian. In fact, the presence of interactions, such as a background electromagnetic field, will cause an initially positive-frequency wave to pick up negative-frequency parts [12]. When Dirac finally became convinced that the negative-frequency solutions could not be mathematically eliminated, he made the astounding suggestion that all the negative-energy states should already be filled up by the electrons (electrons satisfy the Pauli principle, and it is not allowed for such a particle to occupy a state if that state is already occupied). This ocean of occupied negative-energy states is now referred to as the “Dirac sea.” Then, the core of Dirac’s reasoning is the following [13]. He proposed that almost all negative energy states of the electron are filled, the Pauli principle preventing an electron from falling into such a state. The occasional unoccupied such state – a “hole” in this negative-energy sea – would appear as an antielectron (positron), thereby having positive energy. (a) An electron falling into such a hole would be interpreted as the annihilation of the electron and positron, with the release of energy – the sum of the positive contributions from the electron and positron. (b) In reverse, the supplying of sufficient energy to the Dirac sea could produce an electron-positron pair; in other words, if a hole were not present initially, but a sufficient amount of energy (normally in the form of photons) enters the system, then an electron can be kicked out of one of the negative-energy states to leave a hole. So, in this model, Dirac’s “hole” is indeed the electron’s antiparticle, now referred to as the positron. The development of quantum mechanics has made more complex the modern interpretation of vacuum by requiring indeterminacy [14, 15]. Heisenberg’s uncertainty principle predicts a fundamental uncertainty in the instantaneous measurability of the position and momentum of any particle, and which, not unlike the gravitational field, questions the emptiness of space between particles. In the late 20th century, with especially the development of QFT, the principle was understood to also predict a fundamental uncertainty in the number of particles in a region of space, leading to predictions of virtual particles arising spontaneously out of the void [16]. In other words, there is a lower bound on the vacuum, dictated by the lowest possible energy state of the quantized fields in any region of space. So, in quantum mechanics, the vacuum is defined as the state (i.e. solution of the equations of the theory) with the lowest energy [17].

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However, even an ideal vacuum, thought of as the complete absence of anything, will not in practice be empty. One reason is that the walls of a vacuum chamber emit light in the form of black-body radiation: visible light if they are at the temperature of thousands of degrees, infrared light if they are cooler. If this soup of photons is in thermodynamic equilibrium with the walls, it can be said to have a particular temperature, as well as a pressure. Another reason that perfect vacuum is impossible is the Heisenberg uncertainty principle which states that no particles can ever have an exact position (see the next section for a full account). Each atom exists as a probability function of space, which has a certain nonzero value everywhere in a given volume. Even the space between molecules is not a perfect vacuum. More fundamentally, quantum mechanics predicts that the vacuum energy will be different than its naïve, classical value. The quantum correction to the energy is called the zero-point energy and consists of energies of virtual particles that have a brief existence. This is called vacuum fluctuation [18]. Vacuum fluctuations may also be related to the so-called cosmological constant in cosmology (see Sect. 5.9 for further comments) [19]. The best evidence of vacuum fluctuations is the Casimir effect and the Lamb shift [20]. In QFT and string theory, the term “vacuum” is used to represent the ground state in the Hilbert space, that is, the state with the lowest possible energy. In free (noninteracting) quantum field theories, this state is analogous to the ground state of a quantum harmonic oscillator. If the theory is obtained by quantization of a classical theory, each stationary point of the energy in the configuration space gives rise to a single vacuum. String theory is believed to have a huge number of vacua – the so-called string theory landscape (see Sect. 5.10 for a more thorough account of this topic) [21].

5.3 The Role of Vacuum in Modern Physics: From the Universe to the Quantum World The vacuum is fast emerging as the central structure of modern physics. This issue is especially important in the context of classical gravity, quantum electrodynamics, and the grand unification program. The vacuum emerges as the synthesis of concepts of space, time, and matter; in the context of relativity and the quantum world this new synthesis represents a structure of the most intricate and novel complexity. This synthesis raises genuine philosophical issues, and has also connections with modern metaphysics, in which the concepts of substance and space interweave in the most intangible of forms, the background and context of our experiences: vacuum, void, or nothingness. There is sufficient evidence at present to justify the belief that the universe began to exist without being caused to do so. This evidence includes the Hawking–Penrose singularity theorems that are based on Einstein’s general theory of relativity, and the recently introduced quantum cosmological models of the early universe. The singularity theorem led to an explication of the beginning of the universe that involves the notion of a Big Bang singularity [22], and the quantum cosmological models

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represent the beginning mainly in terms of the notion of a vacuum fluctuation [23]. Theories that represent the universe as infinitely old or as caused to begin are shown to be at odds or at least unsupported by these and other current cosmological notions. It is today recognized by mostly physicists that, in fact, the General Theory of Relativity (GTR) fails to apply when quantum mechanical interactions predominate, and these predominate when the temperature is at or above 1032 K, when the density is at or above 1094 gm cm−3 , and when the radius of curvature becomes of the order of 10−33 cm. Since these conditions are obtained during the Planck era at the first 10−43 s after the singularity, the GTR-based big bang theory cannot be used as a reliable guide in reconstructing the physical processes that occurred during this time and a fortiori cannot be used as a reliable basis for predicting that the density, temperature, and curvature reached infinite values prior to this time. Accordingly it seems that the foregoing probabilistic argument to an uncaused beginning of the universe is in difficulty. There are, however, three sound reasons for a continued support of the idea that the universe spontaneously began to exist [24]. To comprehend the reasons, we must first observe that the reason why GTR is inapplicable during the Planck era is that the theory of gravity in GTR is unable to account for the quantum mechanical behavior of gravity during this era. A new quantum theory of gravity is needed [25]. Although such a theory has not yet been developed, there are some general indications of what it may predict. It is in terms of these indications that our three reasons are to be understood. Firstly, it is thought that a quantum theory of gravity may show gravity to be repulsive rather than attractive under conditions that attain during the Planck era. During this time regions of negative energy density may be crated by the force and particles present, and these regions lead to a gravitational repulsion. This suggests that any given finite set of past-directed time-like or null geodesics will not converge in a single point but will be pushed apart, as it were, by the repulsive gravitational force. This possibility is consistent with an oscillating universe, for as each contracting phase ends gravity becomes repulsive and prevents converging geodesics from terminating in a point; gravity repels them so that they enter a new expanding phase. But this way of avoiding the singularity predicted by the Hawking-Penrose theorems does not give us a universe that is infinitely old. For – and this is the first of the three reasons I want to mention – this oscillating quantum-gravitational universe would still be subject to some serious difficulties. One, namely, increase in radius, length of cycle, radiation and entropy with each new cycle. Consequently, this theory does no more than others push the cosmological singularity further into the past, at a time just before (or at) the beginning of the first cycle when the radius of the universe is zero (or near zero). The second reason is that there is a way in which the Hawking-Penrose theorems’ prediction of a singularity at the beginning of the present expansion can be made consistent with a quantum theory of repulsive gravity. These theorems do not define a singularity as that wherein curvature, density, and temperature are infinite and the radius is zero. A singularity is defined as a point or series of points beyond which the spacetime manifold cannot be extended. Consequently, if the effects of quantum gravity prevent a build up of temperature, density, and curvature to infinite

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values, and a decrease of radius to zero, this need not mean there is no singularity at the beginning of the present expansion. The singularity could occur with finite and nonzero values. The third reason is that most theoretically developed attempts to account for the past of the universe on the basis of specifically quantum mechanical principles have represented the universe as spontaneously beginning at the onset of the present expansion. These theories are collectively known as the “vacuum fluctuation models of the universe.” The models developed by Tryon (1973), Brout, Englert and Gunzig (1978), Grishchak and Zel’dovich (1982), Atkatz and Pagels (1982), and Gott (1982) picture the universe as emerging spontaneously from an empty background space, and the model of Vilenkin (1982), which we shall discuss thoroughly in Sect. 5.9, depicts it as emerging without cause from nothing at all. In the first vacuum fluctuation model developed by E. Tryon, a vacuum fluctuation is an uncaused emergence of energy out of the empty space that is governed by the uncertainty relations. Wave packets can be described just as well in the momentum-space representation as in the position representation. One can introduce a precise notion of the “spread” – or lack of localization – of a wave packet in either the position description or the momentum description. Let us denote these spread measures, respectively, by Δx and by Δp; Heisenberg’s uncertainty relation tells us that the product of these spreads cannot be smaller than the order of Planck’s constant. Let’s also mention that, consistent with relativity, there is a similar Heisenberg uncertainty relation between energy and time; the usual interpretation of the energy/time uncertainty is that if energy of a quantum system is ascertained in some measurement which is performed in a time Δt, then there is an ΔE uncertainty in this energy measurement which must satisfy the relation below, h¯ 2 h¯ ΔEΔt > 2 and which thus has zero net value for conserved quantities. (Recall that Heisenberg’s Principle of Uncertainty states that the position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision. There is a minimum for the product of the uncertainties of these two measurements. There is likewise a minimum for the product of the uncertainties of the energy and time1 ). Tryon argues that the universe is able to be a fluctuation from a vacuum in a larger space in which the universe is embedded since it does have a zero net value for its conserved quantities [26]. Observational evidence (Tryon claims) supports or is consistent with the fact that the positive mass-energy of the universe is canceled by its negative gravitational potential energy, and that the amount of matter created is equal to the amount of antimatter. ΔxΔp >

1 This is not a statement about the inaccuracy of measurement instruments, nor a reflection on the quality of experimental methods; it arises from the wave properties inherent in the quantum mechanical description of nature. Even with perfect instruments and technique, the uncertainty is inherent in the nature of things.

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A disadvantage of Tryon’s theory, and of other theories that postulate a background space from which the universe fluctuates, is that they explain the existence of the universe but only at the price of introducing another unexplained given, namely, the background space. This problem is absent from Vilenkin’s theory, which represents the universe as emerging without a cause “from literally nothing” (see Sect. 5.9 for a more detailed account of this theory.) The universe appears in a quantum tunneling from nothing at all to de Sitter space [27, 28]. Quantum tunneling is normally understood in terms of processes within spacetime; an electron, for example, tunnels through some barrier if the electron lacks sufficient energy to cross it but nevertheless still does cross. This is possible because the abovementioned uncertainty relations allows the electron to spontaneously acquire additional energy for the short period of time required for it to tunnel through the barrier. Vilenkin applies this concept to spacetime itself; in this case, there is not a state of the system before the tunneling, for the state of tunneling is the first state that exists. The state of tunneling thus is an analogue of the Big Bang, according to the definition that the universe began neither at nor after the singularity, for it is the first state of the universe and there is no time before this state. The equation describing this state is a quantum tunneling equation, specifically the bounce solution of the Euclidean version of the evolutionary equation of a universe with a closed Robertson–Walker metric. The universe emerged from the tunneling with a finite size (a = H −1 ) and with a zero rate of expansion or contraction ( da/dt = 0). It emerged in a symmetric vacuum state, which then decays and the inflationary era begins; after this era ends, the universe evolves according to the standard Big Bang model. These quantum mechanical models of the beginning of the universe are explanatorily superior in one respect to the standard GTR-based Big Bang models; they do not postulate initial states at which the laws of physics break down but explain the beginning of the universe in accordance with the laws of physics. The GTR-based theory predicts a beginning of the universe by predicting initial states at which the laws of the theory that are used to predict these states break down. The singularity and the explosion of 4-dimensional spacetime from the singularity obey none of the laws of GTR that are obeyed by states within the universe or subsequent states of the universe. In contrast, the quantum mechanical theories represent the universe as coming into existence via the same laws that processes within the universe obey. Instead of an exploding singularity, there is a quantum fluctuation or tunneling that is analogous to the fluctuations or tunnelings within the universe and that obeys the same a-causal laws as the latter fluctuations or tunnelings.

5.4 The Problem of the Vacuum and the Conceptual Conflict Between GRT and QM According to Einstein and the Special Theory of Relativity (STR), the ether does not exist at all. The electromagnetic fields are not states of a medium, but are independent realities not reducible to anything else. This conception suggests itself the more readily as electromagnetic radiation, like ponderable matter, carries momen-

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tum and energy and, according to STR, both matter and radiation are but special forms of distributed energy. This was the point-of-view subsequently adopted by many theorists. Einstein’s assertion is clear. According to the general theory of relativity spacetime itself is a medium2. “To deny the ether is to assume that empty space has no physical qualities whatever.” General relativity not only restores dynamical properties to empty space but also ascribes to it energy, momentum, and angular momentum. In principle, gravitational radiation could be used as a propellant. Since gravitational waves are merely ripples on the curvature of spacetime, an anti-etherist would have to describe a spaceship using this propellant as getting something for nothing – achieving acceleration simply by ejecting one hard vacuum into another. This example is not as absurd as it sounds. It is not difficult to estimate that a star undergoing asymmetric (octupole) collapse may achieve a net velocity change of the order of 100 to 200 km s−1 by this means. But Einstein’s conception went beyond this. According to him curvature is not the only texture that the ether (that is, spacetime) possesses. It must have other, more subtle textures which, like curvature, are best described in the language of differential geometry and differential topology [29]. In an address delivered in 1920 at the University of Leyden, Einstein laid down the following challenge to his Leyden audience: As to the part the new ether is to play in the physics of the future we are not yet clear. We know that it determines the metrical relations in the spacetime continuum, [. . . ] but we do not know whether it has an essential share in the structure of the elementary particles [. . . ] It would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together as one unified conformation.

Einstein’s attack on the unified field problem, and his failure, are well known. The equal failure of others of high stature – Weyl, Klein, Pauli, to name but three – led to a strong reaction among theorists and to a turning away from such problems for many years. Yet the dream never wholly died. Two features of Einstein’s conception remained permanently compelling: the potential richness of a geometrically based reality, and the predictive power of theories based on local invariance groups. These ideas have survived the explosive arrival in 1926 of the golden age of quantum mechanics, the great era of quantum electrodynamics, the subsequent disillusionment with quantum field theory, and the turning in despair to the applied arts of dispersion theory. Near the end of his Leyden lecture Einstein uttered some words of caution: “In contemplating the [. . . ] future of theoretical physics we ought not to reject the possibility that the facts comprised in the quantum theory may set bounds to field theory beyond which it cannot pass.” These words were spoken eight years before the birth 2 We cite the following homely example: “Think of waves on the surface of water. Here we can describe two entirely different things. Either we may observe how the ondulatory surface forming the boundary between water and air alters in the course of time; or else – with the help of small floats – we can observe how the positions of separate particles of water change. If the existence of such floats for tracking the motion of the particles of a fluid were observable than the shape of the space occupied by the water [. . . ], we should have no ground for assuming that water consists of movable particles. But all the same we could characterize it as a medium.” (A. Einstein, Mein Weltbild, Querido Verlag, Amsterdam 1933).

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of quantum field theory, and therefore Einstein could not know that the quantum theory, far from setting bounds on field theory, would transmute and enrich it. Yet he was right in supposing that it would introduce severe new problems. Let’s look briefly at the ways in which the general theory of relativity and the quantum theory of fields, taken together in a new synthesis, have thus far enriched each other. One of the most striking examples of mutual enrichment is to be found in the impact that the ideas of general relativity have had upon the concept of “the vacuum,” and, conversely, in the reinforcement that quantum field theory has given to the idea that the vacuum may be viewed as a textured ether. It has been known from the earliest days of quantum electrodynamics that field strengths, in the vacuum state, undergo random fluctuations completely analogous to the zero-point oscillations of harmonic oscillators, and that when couplings to the electron field are taken into account these fluctuations are accompanied by pair-creation and annihilation events [30]. The vacuum is thus in a state of constant turmoil. From Einstein’s point-of-view it would be natural to regard field fluctuations as having their seat in the ether (namely, spacetime continuum) and contributing to it by qualities additional to its geometrical properties. A mathematical description of the vacuum that effectively embodied this idea was given years ago by Schwinger (1951). In the presence of an external source, a quantized field initially in the vacuum state need not stay in that state. Schwinger showed that all physical properties of the field can be derived from a knowledge of how the probability amplitude for the field to remain in the vacuum state varies as the source is changed. Functional derivatives of the vacuum-to-vacuum amplitude, with respect to the source, are response functions that describe how the ether reacts to external stimuli. The ether itself thus contains a complete blueprint for the field dynamics [31, 32]. The ether may be probed by other means than sources. One may vary boundary conditions and external fields. For example, the vacuum-to-vacuum matrix element of the stress tensor T μν , of any combination of fields, including the gravitational field, is given by the functional derivative out, vac|T μν |in, vac = −2i(δ /δ gμν )out, vac|in, vac .

(1)

Here |in, vac and |out, vac are the initial and final vacuum state vectors respectively and gμν is an external metric field, frequently referred to as background field, which serves as an arbitrary zero point for the quantum fluctuations of the gravitational field, and which can be used to fix the topology of the spacetime manifold. It is assumed in the foregoing equation that the “vacuum” states are unambiguously (although not necessarily uniquely) defined relative to the background, and that topological transitions can only be described (if indeed they occur at all) by allowing gμν to move into the complex plane. It is also assumed that renormalizations have been carried out to eliminate any divergences that may arise. The analogous equation in quantum electrodynamics is out, vac|J μ |in, vac = −i(δ /δ Aμ )out, vac|in, vac ,

(2)

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where J μ is the current vector, and Aμ is the vector potential of an external or background electromagnetic field. Expression (2) is generally nonvanishing whenever the background field is nonvanishing – a phenomenon known as vacuum polarization (see Sect. 5.8 for a description of this phenomenon). Similarly, expression (1) is generally different from zero whenever the background geometry is curved. But curvature is not the only source of gravitational “vacuum polarization.” Topology also contributes. This means that the properties of the ether depend on the whole manifold [33]! This fact is so striking that it is worth looking at in some detail. The phenomenon was first discovered in the Casimir effect. In the course of computing Van der Waals forces between very close molecules, Casimir (1948) found that the interaction energy could be expressed as a sum of terms involving, in addition to the molecular separation distance and internal molecular parameters, powers of the curvature of the molecular surface. One term, however, depended on neither the curvature nor the molecular details. Its presence implied that an attractive force must exist between any two parallel flat conducting surfaces in a vacuum. The effect was soon verified in the Philips laboratories. Because the force is too tiny, great care had to be taken to insure that the surfaces were absolutely clean, neutral, and micro-flat so that they could be brought nearly into contact without other effects intervening. The relevant field in the Casimir effects is the electromagnetic field, and the manifold involved is the plane-parallel slab between the conducting surfaces. In the mathematical terminology this is an incomplete manifold. Nothing prevents one from doing physics on an incomplete manifold provided one is supplied with appropriate boundary conditions – perfect-conductor boundary conditions in this case. The properties of the ether between the conductors are entirely determined by the field Green’s functions (response functions) appropriate to the slab [34]. Consider first the infinite Minkowski vacuum – the standard vacuum of particle physicists. T μν , the operator describing the energy, momentum, and stresses in the electromagnetic field, is formally a bilinear product of operator-valued distributions (the field operators) and hence is meaningless. It is given meaning by a subtraction process that sets the expectation value T μν  equal to zero in the Minkowski vacuum. This subtraction corresponds to ignoring the zero-point energy of the field oscillators, but it is by no means arbitrary. T μν  must vanish in an empty Minkowski spacetime if quantum field theory is to be ultimately consistent with general relativity. The Minkowski vacuum serves as a standard to which all other vacua, whenever possible, are to be compared. Now suppose a single, infinite plane conductor is introduced into the Minkowski vacuum. One may imagine it to be brought adiabatically from infinity so that the field suffers no excitations but remains in its ground state. The manifold of interest has become an infinite half-space. Introduce Minkowski coordinates xμ , μ = 0, 1, 2, 3, oriented so that x3 -axis is perpendicular to the plane of the conductor. By considerations of symmetry it is clear that T μν  must be diagonal and independent of x0 , x1 and x2 . Moreover, because a perfect conductor remains a perfect conductor in any state of motion parallel to its surface, the vacuum stresses in its vicinity must look the same no matter how rapidly one is skimming over its surface.

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That is to say, the ether always keeps its relativistic properties, and hence T μν  must be invariant under Lorentz transformations that correspond to boosts parallel to the (x1 , x2 )-plane. This means that the first three rows and columns of T μν  must be proportional to the metric tensor of a (2 + 1)-dimensional Minkowski space, μ namely diag (−1, 1, 1). If, to this inference, one adds the observation that Tμ = 0 in μν the case of the electromagnetic field, one concludes that T  has the form T μν  = f (x3 ) × diag(−1, 1, 1, −3) .

(3)

But that is not all. The form of the function f (x3 ) too may be deduced. For this μν one invokes the conservation law T.ν  = T μν .ν = 0. In particular 0 = T 3ν .ν = −3 f  (x3 ) ,

(4)

which implies that f is a constant, independent of x3 . Now T μν  has the dimensions of energy density. The only fundamental constants that enter into the theory are h¯ and c. To get a constant having the dimensions of energy density one needs also a unit of length, mass, or time. No natural units with these dimensions exist in the present problem. Therefore, one can only conclude that f = 0 and hence T μν  = 0 in an infinite half-space. All the above arguments concerning the form of T μν  hold equally well for the slab manifold, except that there is now a natural unit of length – the separation distance, a, between the parallel conductors. In the region between the conductors, therefore, we expect T μν  = f (a) × diag(−1, 1, 1, −3) .

(5)

The form of the function f (a) may be determined by considering the work required to separate the conductors adiabatically. From the infinite half-space analysis one knows that the conductors experience no forces from the outside. There is an internal force, however, of amount 3 f (a) per unit area, tending to pull them together. If the conductors are moved a distance da farther apart an amount of work dW = 3 f (a) da, per unit area must be supplied. This must show up as an increase in the energy per unit area. E = −a f (a). Setting dW = dE and integrating, one immediately obtains f (a) = A/a4 , (6) where A is some universal constant. It will be observed that the energy density in the ether between the conductors is negative. It is a tiny energy, too small by many orders of magnitude to produce a gravitational field that anybody is going to measure. Yet one can easily construct Gedankenexperimente in which the law of conservation of energy is violated unless this energy is included in the source of the gravitational field. It turns out that the energy density in the quantum ether is often negative. The quantum theory therefore violates the hypotheses of the famous Hawking–Penrose theorems concerning the inevitability of singularities in spacetime, which imply the ultimate breakdown of classical general relativity.

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5.5 Topology and Curvature as a Source of Vacuum Fields In order for T μν  to be nonvanishing in a flat empty spacetime it is not necessary for the manifold to be incomplete, as in the Casimir effect. Complete manifolds also exhibit the phenomenon: for example, the manifold R × Σ , where the “slices,” Σ , are flat, space-like Cauchy hypersurfaces having any one of the following topologies: R2 × S1 , R × T 2 , T 3 , R × K 2 , etc., where T n is the n-torus, K 2 is the 2-dimensional Klein bottle, etc. The Σ = R2 × S1 has the closest resemblance to the Casimir example. The only difference is that, instead of imposing conductor boundary conditions on the faces of the slab, one imposes periodic boundary conditions. T μν  again takes the form (5), (6), where a is now the period of the coordinate x3 . Again the renormalized Green’s function is easily computed, and one finds in this case A = π 2 /45. The cases Σ = R × T 2 and Σ = T 3 are more complicated. Although T μν  is coordinate-independent its form is no longer given by (5) and (6) but depends on the ratios of the various coordinate periodicities. In the case Σ = R × K 2 , T μν  is not even coordinate-independent but is itself periodic (and smooth). One advantage in studying quantum field theory on complete manifolds is that any field may be selected. One does not have to worry about the question: what boundary conditions are analogous to perfect-conductor boundary conditions for the electromagnetic field? Scalar and spinor fields, and even the gravitational field, may be introduced. The spinor field is of particular interest because on some manifolds, for example on Σ = R2 × S1, one may introduce spinor fields that are homotopically inequivalent. This means that more than one “vacuum” state can be defined [35]. Situations of this type have been studied end in the 1970s in connections with kinks, solitons, and instantons. General relativity, with the richness of alternative topologies that it allows, increases the variety and complexity of these situations. Moreover, as a model for other (usually simpler) field theory, it has drawn attention to the fact that had often been overlooked earlier, namely that the configuration space of any set of interacting fields is itself a Riemannian manifold or, more generally (if fermions fields are involved), a graded Riemannian manifold possessing a metric that is determined (at least in part) by the field Lagrangian. The topology of this manifold need by no means be trivial. In all the above examples the background manifold is flat; the energy and stresses in the ether are entirely due to topology. Not all topologies admit a flat metric – for example Σ – S3 or Σ = R × S2 . In these cases too, T μν  is nonvanishing. However, the “vacuum polarization” is no longer exclusively produced by topology; curvature plays a role. Curvature also complicates the renormalization algorithm and gives μ rise to the phenomenon of trace anomalies: the formal identity Tμ = 0, valid for conformally invariant classical field theories, fails for quantized fields when curvature is present. We remark here only that trace anomalies are similar in a number of respects to the axial-vector current anomalies of weak-interactions theories in which, again, a formal identity (a divergence identity) fails in the quantum theory.

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5.6 The Dirac “Full-Particles Sea” Idea and the Vacuum in Quantum Field Theory The many interpretations of the wave function in nonrelativistic quantum mechanics has been one of the most important theoretical problems of quantum field theory [36]. Certain difficulties indicated that the classical field was not an appropriate candidate for the ontology of nonrelativistic quantum mechanics. Let us first ask: What is the ontology of quantum theory? In a very schematic way, one can say that De Broglie and Schrödinger held a realistic interpretation of the wave function; they assumed a field ontology and rejected the particle ontology because, they argued, quanta particles obeying quantum statistics showed no identity, and thus were not classically observable individuals. In his probabilistic interpretation, Max Born rejected the reality of the wave function, deprived it of energy and momentum, and assumed a particle ontology. Heisenberg interpreted the wave function as a potential. The introduction of fermion field quantization by Jordan and Wigner initiated further radical changes in the interpretation of quantum field theory. First, a realistic interpretation replaced the probabilistic interpretation of the wave function: the wave function in their formulation had to be interpreted as a kind of substantial field, otherwise the particles as the quanta of the field could not get their substantiality from the field, leaving unsolved some of the original difficulties faced by Schrödinger’s realistic interpretation. Second, the field ontology replaced the particle ontology: the material particle (fermions) was no longer regarded as having an eternally independent existence, but as being in transient excitation of the field, a quantum of the field, thus justifying the claim that QFT started a major variant of the field programme, namely, the quantum field programme. But the reversion of this view of the real field evidently leaves a gap in the logic of models of reality and destroys the traditional conception of substance. A new conception of ontology is introduced. This new ontology cannot be reduced to that of classical particle ontology because the field quanta lack a permanent existence and individuality. It also cannot be reduced to that of the classical field ontology because the quantized field has lost its continuous existence. It seems that 20th century field theories suggest that the quantized field, together with some nonlinear fields, constitute a new kind of ontology (called “ephemeral ontology” by M. Redhead, 1983). The new ontology of QFT was embodied in the Dirac vacuum. As an ontological background underlying the conceptual schema of quantum excitations and renormalization, the Dirac vacuum was crucial for calculations, such as Weisskopf’s calculation of the electron self-energy and Dancoff’s discussion on the relativistic corrections to scatterings [37]. The fluctuations existing in the Dirac vacuum strongly indicate that the vacuum must be something substantial rather than empty. On the other hand, the vacuum, according to the special theory of relativity, must be a Lorentz-invariant state of zero energy and zero momentum. Considering that energy has been loosely thought to be essential to substance in modern physics, it seems that the vacuum could not be taken as a kind of substance. Here we run into

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a profound dilemma, which hints at the necessity of changing our conception of substance and of energy being a substantial property. The mode of conveying interactions in QFT is different from that in the classical field programme in two aspects. First, interactions are realized by local couplings among field quanta, and the exact meaning of coupling here is the creation and annihilation of the quanta. Second, the actions are transmitted, not by a continuous field, but by discrete virtual particles that are locally coupled to real particles and propagate between them. Thus, the description of interactions in QFT is deeply rooted in the concept of localized excitation of operator fields through the concept of local coupling. Yet the local excitation entails, owing to the uncertainty relation, that arbitrary amounts of momentum are available. Then the result of a localized excitation would not only be a single momentum quantum, but must be a superposition of all appropriate combinations of momentum quanta. And this has significant consequences. First, the interaction is transmitted not by a single virtual momentum quantum, represented by an internal line in the Feynman diagram, but by a superposition of an infinite number of appropriate combinations of virtual quanta. This is an entailment of the basic assumption of a field ontology in QFT. Second, the infinite number of virtual quanta with arbitrarily high momentum leads to infinite contributions from their interactions with real quanta. This is the famous divergence difficulty. Thus, QFT cannot be considered a consistent theory without this serious difficulty being resolved. Historically, the difficulty was first circumvented by a renormalization procedure. The essence of the original renormalization procedure is the absorption of infinite quantities into the theoretical parameters of mass and charge. This is equivalent to blurring the exact point model underlying the concept of localized excitations. While quantum electrodynamics meets the requirement of renormalizability, Fermi’s theory of weak interactions and the meson theory of the strong nuclear force fail to do so. Gauge invariance is a general principle for fixing the forms of fundamental interactions, on the basis of which a new programme, the gauge field programme, for fundamental interactions develops within the quantum field programme [38]. Gauge invariance requires the introduction of gauge potentials, whose quanta are responsible for transmitting interactions and for compensating the additional changes of internal degrees of freedom at different spacetime points. The role gauge potentials play in gauge theory is parallel to the role gravitational potentials play in GTR. While the gravitational potentials in GTR are correlated with a geometrical structure (the linear connection in the tangent bundle), the gauge potentials are correlated with a similar type of geometrical structure, that is, the connection on the principal bundle. Deep similarity in theoretical structures between GTR and gauge theory suggests the possibility that the gauge theory may also be geometrical in nature. Recent developments in fundamental physics (supergravity, modern Kaluza-Klein theory, and string theory) have proposed to associate gauge potentials with the geometrical structures in extra dimensions of spacetime (see Sect. 5.10 for further remarks on this topic) [39]. Thus, it seems reasonable to regard the gauge field programme as a synthesis of the geometrical programme and the quantum field programme if we express the gauge field programme in such a way that interactions

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are realized through quantized gauge fields (whose quanta are coupled with material fields and are responsible for the transmission of interactions) that are inseparably correlated with a kind of geometrical structure existing either in internal space or in the extra dimensions of spacetime. We now return to the role of the vacuum in quantum mechanics and QFT. The operator fields and field quantization, especially Jordan-Wigner quantization, had their direct physical interpretation only in terms of the vacuum state. Let’s first say a few words about the physicists’ conception of the vacuum before and after the introduction of field quantization in the late 1920s. After Einstein had put an end to the concept of the ether, the field-free and matter-free vacuum was considered to be truly empty space. The situation, however, had changed with the introduction of quantum mechanics. From then onwards, the vacuum become populated again. In quantum mechanics, the uncertainty relation for the number N of light quanta and the phase θ of the field amplitude, ΔNΔθ ≥ 1, means that if N has a given value zero, then the field will show certain fluctuations about its average value, which is equal to zero. The next step in populating the vacuum was taken by Dirac. In his relativistic theory of electrons (1928) Dirac met with a serious difficulty, namely the existence of states of negative kinetic energy. As a possible solution to that difficulty, he proposed a new conception of the vacuum: all of the states of negative energy have already been occupied, and all of the states of positive energy are not occupied (see Sect. 5.1). Then the transition of an electron from the positive energy state to the negative energy state could not happen owing to Pauli’s exclusion principle. This vacuum state was not an empty state, but rather a filled sea of negative energy electrons. The sea as a universal background was unobservable, yet a hole in the negative energy sea was observable, behaving like a particle of positive energy and positive charge. Dirac took the hole as a new kind of particle with a positive charge and the mass of the electron. It is easy to see that Dirac’s infinite sea of unobservable negative energy electrons of 1930-1931 was analogous to his infinite set of unobservable zero energy photons of 1927. Such a vacuum consisting of unobservable particles allowed for a kind of creation and annihilation of particles. Given enough energy, a negative energy electron can be lifted up into a positive energy state, corresponding to the creation of a positron (the hole in the negative energy sea) and an ordinary electron. And of course the reverse annihilation process can also occur. One might say that all these processes can be accounted for by the concept of the transition between observable and unobservable states, without introducing the ideas of QFT. This is true. But on the other hand, it is precisely the concepts of the vacuum and of the transition that have provided QFT with a deep ontological basis. In fact, the introduction of the negative energy sea, as a solution to the difficulty posed by Dirac’s relativistic theory of the electron, implied that no consistent relativistic theory of a single electron would be possible without involving an infiniteparticle system, and that a QFT description was needed for the relativistic problems. Besides, the concept of transition between observable and unobservable states did indeed provide a prototype of the idea of excitation of QFT, and hence gave a direct physical interpretation to the field operators.

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The concept of creation and destruction (annihilation) of a particle and that of transition from one state (of negative energy) to another state (of positive energy) is maybe the most striking (and strange) feature of Dirac’s visionary theory of the vacuum. In fact, Dirac idea of the vacuum as a kind of substratum shared some of its characteristic features with the ether model, which explains why he returned to the idea of an ether in his later years, although in his original treatment of creation and annihilation processes he did not explicitly appeal to an ether model. One of the striking physical features of Dirac’s vacuum, similar to that of ether, is that it behaves like a polarizable medium. An external electromagnetic field distorts the simple electron wave function of the negative energy sea, and thereby produces a charge-current distribution acting to oppose the inducing field. As a consequence, the charges of particles would appear to be reduced. That is, the vacuum could be polarized by the electromagnetic field. The fluctuating densities of charge and current (which occur even in the electron-free vacuum state) can be seen as the electron-positron field counterpart of the fluctuations in the electromagnetic field. In sum, the situation after the introduction of the filled vacuum is this. Suppose we begin with an electron-positron field Ψ . It will create an accompanying electromagnetic field which reacts on the initial field Ψ and alters it. Similarly, an electromagnetic field will excite the electron-positron field Ψ , and the associated electric current acts and alters the initial electromagnetic field. So the electromagnetic field and the electron-positron field are intimately connected, neither of them has a physical meaning independent of the other. What we have then is a coupled system consisting of electromagnetic field and electron-positron field, and the description of one physical particle is not to be written down a priori but emerges only after solving a complicated dynamical problem. The idea that the vacuum was actually the scene of wild activities (fluctuations), in which infinite negative energy electrons existed, was mitigated later on by eliminating the notion of the actual presence of these electrons. Dirac’s idea of the vacuum entails a revolutionary meaning, and its regarding of the vacuum not as empty but substance-filled remains in our present conception of the vacuum. One may say that the conception of the substance-filled vacuum is strongly supported by the fact that the fluctuations of matter density in the vacuum remain even after the removal of the negative energy electrons, as an additional property of the vacuum besides the electromagnetic vacuum fluctuations. Here we run into the most profound ontological dilemma in QFT. On the one hand, according to special relativity, the vacuum must be a Lorentz invariant state of zero energy, zero momentum, zero angular momentum, zero charge, zero whatever, that is, a state of nothingness [40]. Considering that energy and momentum have been thought to be the essential properties of substance in modern physics and modern metaphysics, the vacuum cannot definitely be regarded as a substance. On the other hand, the fluctuations existing in the vacuum strongly indicated that the vacuum must be something substantial, certainly not empty. A possible way out of the dilemma might be to redefine “substance” and deprive energy and momentum from being the defining properties of a substance. But this would be too ad hoc, and could not find support from other instances. Another possibility is to take the

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vacuum as a kind of pre-substance, an underlying substratum having a potential substantiality. It can be excited to become substance by energy and momentum, and become physical reality if some other properties are also injected into it. The real particles come into existence only when we disturb the vacuum, when we excite it with energy and other properties. What shall we say about the local field operator whose direct physical meaning was thought to be the excitation of a physical particle? First of all, the localized excitation described by a local field operator O(x) acting on the vacuum means the injection of energy, momentum and other special properties into the vacuum at a spacetime point. It also means, owing to the uncertainty relations, that arbitrary amounts of energy and momentum are available for various physical processes. Evidently, the physical realization of these properties symbolized by O(x)|vac will not only be a single particle state, but must be a superposition of all appropriate multi-particle states. For example, Ψel (x)|vac = a|1 electron + ∑ a |1 electron + 1 photon ∑ a |1 electron + 1 photon + 1 electron + . . ., where Ψel (x) is the so-called dressed field operator, and a2 is the relative probability for the formation of a single bare (naked) particle state by the excitation Ψel (x), and so on. As a result, the field operators no longer refer to the physical particles and become abstract dynamical variables, with the aid of which one constructs the physical state. Then how can we pass from the underlying dynamical variables (local field operators), with which the theory begins, to the observable particles? This is a task that is fulfilled temporally only with the aid of the renormalization procedure. Let us roughly describe this procedure [41]. In quantum field theory and the statistical mechanics of fields, renormalization refers to a collection of techniques used to construct mathematical relationships or approximate relationships between observable quantities, when the standard assumption that the parameters of the theory are finite breaks down, giving the result that many observables are infinite. Renormalization arose in quantum electrodynamics as a means of making sense of the infinite results of various calculations and extracting finite answers to properly posed physical questions. When developing QED in the 1940s, S. Tomonaga, J. Schwinger, R. Feynman, and F. Dyson discovered that, in perturbative calculations, problems with divergent integrals abounded, one way of describing the divergence in QED is that they appear as the consequences of calculations involving Feynman diagrams with closed loops of virtual particles in them [42]. These diagrams appear in the perturbative approximation of QFT. Each looped diagram represents a perturbation, or small correction, to a diagram without loops. Intuitively, diagrams with more and more loops should give smaller and smaller corrections to the values of diagrams which do not contain any loops. However, when the contributions of these loop diagrams are naively calculated they become infinitely large. One type of loop would be a situation in which a virtual electron-positron pair appear out of the vacuum, interact with various photons, and then annihilate. There is another electron-photon interaction in which the electron’s charge at one renormalization point is revealed to consist of more complicated interactions at another point. While virtual particles obey conservation of energy and momentum, they can possess combinations of energies and momenta not allowed by the classical laws

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of motion; generally, physicists are not really comfortable with that. Furthermore, whenever a loop appears, the particles involved in the loop are not individually constrained by the energies and momenta of incoming and outgoing particles, since a variation in, say, the energy of one particle in the loop can be balanced by an equal and opposite variation in the energy of another particle in the loop. Therefore, in order to calculate the contribution of a probability amplitude, one must integrate over all possible combinations of energy and momentum in the loop – and these integrals are often divergent, that is, they give infinite answers (infinite mathematical terms). The most theoretically troublesome divergences are the “ultraviolet” (UV) ones associated with large energies and momenta of the virtual particles in the loop, or, equivalently, very short wavelengths and high frequencies of the fields for which these particles are the quanta. These divergences are, therefore, fundamentally shortdistance, short-time phenomena [43]. The solution was to realize that the quantities initially appearing in the theory’s formulae (such as the formula for the Lagrangian), representing such things as the electron’s electric charge and mass, as well as normalizations of the quantum fields themselves, did not actually correspond to the physical constants measured in the laboratory. As we already said, they were bare quantities that did not take into account the contribution of virtual-particle loop effects to the physical constants themselves. Among other things, these effects would include the quantum counterpart of the electromagnetic back-reaction that so displeased classical theorists of electromagnetism. In general, these effects would be just as divergent as the amplitudes under study in the first place; so finite measured quantities would in general imply divergent bare quantities. In order to make contact with reality, then the formulae would have to be rewritten in terms of measurable, renormalized quantities. The charge of the electron, say, would be defined in terms of a quantity measured at a specific kinematic renormalization point or subtraction point (which will generally have a characteristic energy, called the renormalization scale or simply the energy scale). The parts of the Lagrangian left over, involving the remaining portions of the bare quantities, could then be reinterpreted as counterterms, involved in divergent diagrams exactly canceling out the troublesome divergences for other diagrams. For example, in the Lagrangian of QED   1 μ μν L = ψ B iγμ (∂ μ + ieB AB ) − mB ψB − FBμν FB 4

(7)

the fields and coupling constant are really bare quantities, hence the subscript B above. Conventionally the bare quantities are written so that the corresponding Lagrangian terms are multiples of the renormalized one: (ψ mψ )B = Z0 ψ mψ μ

(ψ (∂ + ieAμ )ψ )B = Z1 ψ (∂ μ + ieAμ )ψ (Fμν F μν )B = Z3 Fμν F μν .

(8)

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A term in this Lagrangian, for example, the electron-photon interaction mentioned above, can be then written as follows: LI = −eψγμ Aμ ψ − (Z1 − 1)eψγμ Aμ ψ .

(9)

The physical constant e, the electron’s charge, can then be defined in terms of some specific experiment; we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory (up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the magnetic moment). The rest is the counterterm. If we are lucky, the divergent parts of loop diagrams can all be decomposed into pieces with three or fewer legs, with an algebraic form that can be canceled out by the second term (or by the similar counterterms that come from Z0 and Z3 ). As a result we have that QED theory is indeed renormalizable. The diagram with some Z1 counterterms interaction vertex may cancel out the divergence from the loop in some other diagram.

5.7 Hole Theory, Negative Energy Solutions, and Vacuum Fluctuations In spite of the successes of the Dirac equation, we must face the interpretation of negative energy solutions. Their presence is difficult to accept, since they make all positive energy states unstable in the final analysis. A solution was proposed by Dirac as early as 1930 in terms of a many-particle theory. Although this shall not be the final standpoint, as it does not apply to scalar particles, for instance, it is instructive to retrace his reasoning. It provides an intuitive physical picture useful in practical instances, and permits fruitful analogies with different situations such as electrons in a metal. Its major assumption is that all the negative levels are filled up in the vacuum state. According to the Pauli exclusion principle, this prevents any electron from falling into the negative energy states, and thereby insures the stability of positive energy physical states. In turn, an electron of the negative energy sea may be excited to a positive energy state. It then leaves a hole in the sea. This hole in the negative energy, negatively charged states appears as a positive energy, positively charged particle – the positron. Besides the properties of the positron, its charge |e| = −e and its rest mass me , this theory also predicts new observable phenomena: 1. The annihilation of an electron-positron pair. An (positive energy) electron falls into a hole in the negative energy sea with the emission of radiation. From energy momentum conservation at least two photons are emitted, unless a nucleus is present to absorb energy and momentum. 2. Conversely, an electron-positron pair may be crafted from the vacuum by an incident photon beam in the presence of a target to balance energy and momentum. This is the process mentioned above; a hole is created while the excited electron acquires a positive energy.

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Thus, the theory predicts the existence of positrons which were in fact observed in 1932. Since positrons and electrons may annihilate, we must abandon the interpretation of the Dirac equation as a wave equation. Also, the reason for discarding the Klein-Gordon equation no longer holds. It actually describes spinless particles, such as pions. However, the hole interpretation is not satisfactory for bosons, since Fermi statistics play a crucial role in Dirac’s argument. Even for fermions, the concept of an infinitely charged unobservable sea looks rather queer. We have instead to construct a true many-body theory to accommodate particles and antiparticles in a consistent way. This will be achieved by the “second quantization,” i.e., the introduction of quantized fields capable of creating and annihilating particles. Hole theory implies the existence of electrons and positrons with the same mass and opposite charges which obey the same equation. The Dirac equation must therefore admit a new symmetry corresponding to the interchange particle ↔ antiparticle [44]. We thus seek a transformation ψ → ψ c reversing the charge, i.e., such that (i∂ − eA − m)ψ = 0

(10)

(i∂ + eA − m)ψ c = 0 .

(11)

and The vacuum fluctuations are beautifully evidenced by the Casimir effect. For Dirac fields, the stability of the vacuum and the exclusion principle lead to quantization according to anti-commutation rules [45]. The original observation of Casimir (1948) is that, in the vacuum, the electromagnetic field does not really vanish but rather fluctuates. If we introduce macroscopic bodies – even uncharged – some work will be necessary to enforce appropriate boundary conditions. Intuition on the sign of this effect is lacking, so that work here is meant in some algebraic sense, meaning the difference in zero point energies between the two configurations. We may disregard the (infinite) contribution ∑ 12 hωα to the Hamiltonian by arguing that it is observable. However, its variation can be measured. Let us illustrate this point for the simple configuration of two large parallel perfectly conducting plates as considered first by Casimir. Of course, we can study different geometries and different materials with similar results (except perhaps for crucial signs). We idealize the plates by two large parallel squares of size L at a distance a with a L. Consider the energy per unit surface of the conductor with respect to the vacuum. Its derivatives will be a force per unit surface with dimension ML−1 T −2 (where M is mass, L length, and T time). The only quantities entering the problem are h¯ , c, and the separation a (the boundary conditions E perpendicular and B parallel to the plate at the interface do not introduce any dimensional quantity). Of course, the effect is proportional to h¯ , as is the zero point energy. The force per unit surface is therefore proportional to h¯ c/a4 , the only quantity with the required dimension. We shall see that it is attractive. Consider the modes inside the volume L2 a, where L  a and we ignore the edge contributions. As we know, only transverse modes contribute to the energy. If the component kz perpendicular to the plates is different from zero it can only take discrete values kz = nπ /a (n = 1, 2, . . .) to allow for the nodes on the plates and

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there are two polarization states. If, however, kz vanishes only one mode survives so that the zero point energy of the configuration is E=

1

∑ 2 h¯ ωα =

h¯ c = 2



h¯ c |kα | 2 ∑ α 

1/2  ∞  L2 d2 k n2 π 2 2 |k | + 2 ∑ k + 2 . (2π )2 a n=1

(12)

As it stands this expression is, of course, meaningless, being infinite but we must subtract the free value which contributes in this same volume a quantity E0 =

h¯ c 2

=

h¯ c 2

 

L2 d2 k  +∞ a dkz 2 2 k + kz2 (2π )2 −∞ 2π

L2 d2 k  ∞ √ n2 π 2 dn2 k 2 + 2 . 2 (2π ) 0 a

(13)

Therefore, the energy per unit surface is E = E− h¯ c = 2π

E0 L2



 ∞

k dk 0

∞ √ n2 π 2 k + ∑ k2 + 2 − 2 n=1 a

 ∞ 0

√ n2 π 2 dn k2 + 2 a

 .

(14)

This quantity is apparently still not defined due to ultraviolet (large k) divergences. However, for wavelengths shorter than the atomic size it is unrealistic to use a perfect conductor approximation. Let us therefore introduce in the above integral a smooth cut-off function f (k) equal to unity for k < km and vanishing for k  km where km is of the order of the inverse atomic size. Set u = a2 k2 /π ; then √  ∞ π2 π√  u f du E = h¯ c a3 u 4 2 a 0   ∞ √ 

π √

π √ ∞ √ 2 2 2 2 dn u + n f u+n − u+n = ∑ u+n f a a 0 1   ∞ h¯ cπ 3 1 a F(0) + F(1) + F(2) + . . . − dnF(n) . (15) E = 4 2 0 Here we have defined F(n) =

 ∞ 0



π √ √ du u + n2 f u + n2 . a

(16)

The interchange of sums and integrals was justified due to the absolute convergence in the presence of the cut-off function. As n → ∞, F(n) → 0. We can use the Euler-MacLaurin formula to compute the difference between the sum and integral

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occurring in the above bracket: 1 F(0) + F(1) + F(2) + . . . − 2

 ∞ 0

1 1 dnF(n) = − !B2 F  (0) − !B4 F  (0) + . . . (17) 2 4

The Bernoulli numbers Bν are defined through the series ∞ yν y − 1 = B ν ∑ ν! ey ν =0

(18)

and B2 = 1/6, B4 = −1/30, . . . We have F(n) =

∞

√ du u f

 √ 

πn π u F  (n) = −2n2 f . a a

(19)

n2

We assume that f (0) = 1, while all its derivatives vanish at the origin, so that F(0) = 0, F  (0) = −4, and higher derivatives of F are equal to zero. All reference to the cut-off has therefore disappeared from the final result E =

π 2 hc h¯ cπ 2 B4 =− . 3 a 4! 720 a3

(20)

The force per unit area F reads F =−

π 2 h¯ c 0.013 dyn =− 4 240 a4 aμ m cm2

(21)

and its sign corresponds to attraction. This very tiny force has been demonstrated experimentally by Sparnay (1958), who was able to observe both its magnitude and dependence on the interplate distance! The above derivation may be criticized on account of the fact that we have seemingly disregarded the effects outside the plates. In the present case they turn out to cancel exactly. The lesson is that vacuum fluctuations manifest themselves under very different circumstances than those encountered in particle creation or absorption. By considering various types of bodies influencing the vacuum configuration we may give an interesting interpretation of the forces acting on them.

5.8 Further Theoretical Remarks on the Vacuum Fluctuations: Poincaré Conformal Invariance and Spontaneous Symmetry Breaking Symmetry (SSB) In modern physics, the classical vacuum of still nothingness has been replaced by a quantum vacuum with fluctuations of measurable consequence. In fact, when we think about the vacuum in classical physics (say, in classical mechanics), we think of empty space unoccupied by any matter, through which particles can move unhin-

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dered and in which fields are free from any of the nonlinear interaction effects which make e.g. electrodynamics in media so much more difficult. In Quantum Field Theory, the vacuum turns out to be quite different from this inert stage on which things happen; in fact the vacuum itself is a nonlinear medium, a foamy bubble bath of virtual particles popping into and out of existence at every moment, a very active participant in the strange dance of elementary particles that we call the universe [46]. A metaphor which may make this idea a little clearer could be to think of the vacuum as a sheet of paper on which you write with your pen; looked at on a large scale, the paper is merely a perfectly flat surface on which the pen moves unhindered. On a smaller scale, the paper is actually a tangle of individual fibers going in all directions and against which the pen keeps hitting all the time, thus finding the necessary friction to allow efficient writing. In the case where the paper is the vacuum, the analogue of the paper fibers are the bubbles of virtual particle pairs that are constantly being created and annihilated in the quantum vacuum, the analogue of the pen is a particle moving through the vacuum, and the analogue of friction is the modification of the particle’s behavior as compared with the classical theory which happens as a result of the particle interacting with virtual particle pairs. At first sight, this description of the vacuum may appear like wild speculation, but it has in fact very observable consequences. In QED, the famous Lamb shift is a consequence of the interactions of the electron in a hydrogen atom with virtual photons, as are the anomalous magnetic moment of the electron and the scattering of light by light in the vacuum. In fact, none of the amazingly accurate predictions of QED would work without taking into account the effects of the quantum vacuum. At the quantum level, we get Feynman diagrams with loops in them that describe how particles traveling through the quantum vacuum interact with virtual particles; the problem with these is that the virtual particles exist at very short distances and hence can have very large momenta by virtue of Heisenberg’s uncertainty relation. At very large momenta, the deviation of the lattice theory from the continuum become very evident, and hence the loops on the lattice contribute terms that differ a lot from what the same loops would contribute in the continuum. And then we find that this difference reintroduces the a-dependence that we got rid of classically by tuning our theory! Quantum fluctuation is the temporary appearance of energetic particles out of nothing, as allowed by the Uncertainty Principle. It is synonymous with vacuum fluctuation. As mentioned before (see Sect. 5.3), the Uncertainty Principle states that for a pair of conjugate variables such as position/momentum and energy/time, it is impossible to have a precisely determined value of each member of the pair at the same time. For example, a particle pair can pop out of the vacuum during a very short time interval. An extension is applicable to the “uncertainty in time” and “uncertainty in energy” (including the rest mass energy mc2 ). When the mass is very large (such as a macroscopic object), the uncertainties and thus the quantum effect become very small, classical physics is applicable one more. In classical physics (applicable to macroscopic phenomena), empty spacetime is called the vacuum; the classical vacuum is utterly featureless. However, in quantum mechanics, the vacuum is a much more complex entity. It is far from featureless

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and far from empty. The quantum vacuum is just one particular state of a quantum field (corresponding to some particles). It is the quantum mechanical state in which no field quanta are excited, that is, no particles are present. Hence, it is the “ground state” of the quantum field, the state of minimum energy. A certain kind of processes go on in a quantum vacuum: particle pairs appear, live a brief existence, and then annihilate one another in accordance with the uncertainty principle [47]. Heisenberg’s uncertainty principle applied to energy E and time t states that ΔEΔt ≥ h¯ , then, if Δt is sufficiently small ΔE can become large enough to allow the appearance of virtual particle-antiparticle pairs. In the quantum vacuum virtual particle–antiparticle pairs are continuously created and annihilated: the quantum vacuum is full of activity. The classical vacuum on the other hand is defined as the absence of matter; it is empty. 1. The quantum vacuum can be thought of as a medium much like an ordinary gas. Its structure, however, is normally invisible, much like the water of a transparent pond. To “see” the structure one must perturb the medium: in the case of the transparent water one could evidence the water by throwing a stone (a “disturbance”) in the pond. 2. When studying the quantum vacuum the idea is to set up a disturbance with an external electromagnetic field and to use a “probe” of some sort to investigate possible changes in the structure. A light beam could be such a probe. 3. In the PVLAS experiment, the disturbance is introduced by a high intensity (up to 6.6 T) magnetic field and the probe is a linearly polarized laser beam: the properties of the quantum vacuum are recorded in the polarization state of the probe light, which has changed from linear to elliptical. This phenomenon is also called vacuum magnetic birefringence. In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. By definition, it contains no physical particles. The term “zero-point field” is sometimes used as a synonym for the vacuum state of an individual quantized field. If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator (or more accurately, the ground state of a QM problem). In this case the vacuum expectation value (VEV) of any field operator vanishes – the VEV of a field φ , which should be written as 0|φ |0, is usually condensed to φ . For quantum field theories in which perturbation theory breaks down at low energies (for example, Quantum chromodynamics (QCD) or the BCS theory of superconductivity), field operators may have nonvanishing vacuum expectation values called condensates. In the Standard Model, the nonzero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking (SSB), is the mechanism by which the other fields in the theory acquire mass [48]. In many situations, the vacuum state can be defined to have zero energy, although the actual situation is considerably subtler. The vacuum state – written as |0 or | – is associated with a zero-point energy, and this zero point energy has measurable effects. In the laboratory, it can be detected as the Casimir effect. In physical cosmol-

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ogy, the energy of the vacuum state appears as the cosmological constant [49]. An outstanding requirement imposed on a potential Grand Unification Theory (GUT) is that the vacuum energy of the vacuum state must explain the physically observed cosmological constant [50]. For a relativistic field theory, the vacuum is Poincaré invariant. Poincaré invariance implies that only scalar combinations of field operators have nonvanishing VEVs. The VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum (that is, the ground state of the QFT) has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred. This means that, when the Hamiltonian of a system (or the Lagrangian) has a particular symmetry, but the ground state (i.e., the vacuum) does not, then one says that spontaneous symmetry breaking has taken place. When a continuous symmetry is spontaneously broken, massless gauge bosons (messenger particles which mediate the interactions) appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called Goldstone bosons. In other words, spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. At this point the system no longer appears to behave in a symmetric manner. It is a phenomenon that naturally occurs in many situations. The symmetry group can be discrete, such as the space group of a crystal, or continuous (i.e., a Lie group), such as the rotational symmetry of space. A common example to help explain this phenomenon is a ball sitting on top of a hill. This ball is in a completely symmetric state. However, it is not a stable one: the ball can easily roll down the hill. At some point, the ball will spontaneously roll down the hill in one direction or another. The symmetry has been broken because the direction the ball rolled down in has now been singled out from other directions. Consider now a well-known very nice mathematical example: the Mexican hat potential. In the simplest example, the spontaneously broken field is described by a scalar field. In physics, one way of seeing spontaneous symmetry breaking is through the use of Lagrangians. Lagrangians, which essentially dictate how a system will behave (evolve in time), can be split up into kinetic and potential terms L = ∂ μ φ ∂μ φ − V (φ ) .

(22)

It is in this potential term (V (φ )) that the action of symmetry breaking occurs. An example of a potential is illustrated in the graph which has the form V (φ ) = −10|φ |2 + |φ |4 . This potential has many possible minima (vacuum states) given by √ φ = 5eiθ

(23)

(24)

for any real θ between 0 and 2π . The system also has an unstable vacuum state corresponding to ϕ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (corresponding to a choice of θ ) this

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symmetry will be lost or spontaneously broken. If a vacuum state obeys the initial symmetry then the system is said to be in the Wigner mode, otherwise it is in the Goldstone mode. In the Standard Model, spontaneous symmetry breaking is accomplished by using the Higgs boson and is responsible for the masses of the W and Z bosons. Vacuum energy is an underlying background energy that exists in space even when devoid of matter (known as free space). The vacuum energy results in the existence of most (if not all) of the fundamental forces – and thus in all effects involving these forces, too. It is observed in various experiments, and it is thought (but not yet demonstrated) to have consequences for the behavior of the Universe on cosmological scales, where the vacuum energy is expected to contribute to the cosmological constant3, which affects the expansion of the Universe. Vacuum energy has a number of consequences. For one, vacuum fluctuations are always created as particle–antiparticle pairs. The creation of these “virtual particles” near the event horizon of a black hole has been hypothesized by physicist S. Hawking to be a mechanism for the eventual “evaporation” of black holes. The net energy of the Universe remains zero so long as the particle pairs annihilate each other within Planck time. If one of the pair is pulled into the black hole before this, then the other particle becomes “real” and energy/mass is essentially radiated into space from the black hole. This loss is cumulative and could result in the black hole’s disappearance over time. The time required is dependent on the mass of the black hole, but could be on the order of 10100 years for large solar-mass black holes. The Grand unification theory predicts a nonzero cosmological constant from the energy of vacuum fluctuations.

5.9 Creation of Universes from Nothing The standard hot cosmological model gives a successful description of many features of the evolution of the Universe. However, it is not totally satisfactory, since it requires rather unnatural initial conditions at the big bang. One has to postulate that the Universe has started in a homogenous and isotropic state with tiny density fluctuations which are to evolve into galaxies. Homogeneity and isotropy must extend to scales far exceeding the causal horizon at the Planck time. In addition, the energy density of the universe must be tuned to be near the critical density with an incredible accuracy of ∼ 10−55. In the last few years there has been a growing hope of explaining these initial conditions as resulting from physical processes in the very early Universe. Guth (1981) has suggested that the homogeneity, isotropy and flatness puzzles can be solved if the Universe passed through a de Sitter phase of exponential expansion (inflation) in its early history. [a(t) ∝ exp(Ht), where a(t) is the scale factor.] Such a phase 3 It should be recalled that, as early as 1931, the physicist and astronomer Georges Lemaître had the insight to interpret the cosmological constant as originating in the behavior of matter at very high energies.

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can arise in a first-order phase transition with strong super-cooling. It has been suggested (Witten 1981; Albrecht and Steinhardt 1982) that extreme super-cooling can occur in grand unified models with Coleman–Weinberg type of symmetry breaking. Initially it was not clear how to end the exponential expansion and get back to a radiation-dominated Universe. A plausible answer has emerged recently. At some temperature T0 the false vacuum becomes unstable due to the thermal or gravitational effects [51]. The Higgs field φ starts rolling down the effective potential towards the absolute minimum, φ = σ . The Coleman-Weinberg potential is very flat for small values of φ (φ σ ), and the typical rollover time, τ , can be much greater than the expansion time, H −1 . Until φ becomes of the order σ , exponential expansion continues, and the scale of the Universe grows by a factor ∼ exp(H τ ) ≫ 1. To solve the homogeneity and flatness problems we need exp(H τ ) ≥ 1028. Most of this growth takes place after the destabilization of the false vacuum. When φ becomes ∼ σ , the vacuum energy thermalizes, and the Universe enters a radiationdominated period. The baryon number can be generated during the thermalization or shortly afterwards. Density fluctuations can be generated by vacuum strings produced at a later phase transition. Another attractive feature of this scenario is that the problem of superabundance of heavy magnetic monopoles does not arise: the Higgs expectation value is uniform over the whole visible universe. Now that we have a plausible ending to the inflationary scenario, we can start wondering about its beginning, where the situation is still rather depressing. There is a cosmological singularity at t = 0 and the origin of the initial thermal state is mysterious. Besides, there is another problem if we assume that the universe is closed (which seems to be a more aesthetically appealing choice). It is natural to assume that at Planck time (t ∼ tP ) the size and the energy density of the universe are O(1) in Planck units. But then the universe will expand and recollapse in about one Planck time, its size will never much exceed the Planck length, and the phase of exponential expansion will never be reached (assuming that the grand unification mass scale is much smaller than the Planck mass, σ mP ). In order to cool down to temperatures ∼ 1014 GeV, the energy density at t ∼ tP must be tuned to be near the critical density with an accuracy of ∼ 10−10. This is just a milder version of the same flatness problem that we have faced before. A new cosmological scenario was suggested in the 1980s by A. Vilenkin, A.H. Guth and P.J. Steinhardt in which the universe is spontaneously created from literally nothing, and which is free from the difficulties I mentioned in the preceding paragraph. This scenario does not require any changes in the fundamental equations of physics; it only gives a new interpretation to a well-known cosmological solution. It must be, however, recalled that E.P. Tryon (1973) already proposed a big bang model in which our universe is a fluctuation of the vacuum, in the sense of quantum field theory. The model predicts a Universe which is homogeneous, isotropic and closed, and consists equally of matter and anti-matter. All these predictions seem to be supported by, or be consistent with, present observations. Following previous works of Guth, Steinhardt and Linde, Vilenkin consider a model of interacting gravitational and matter fields. The matter content of the model can be taken to be that of some grand unified theory. The absolute mini-

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mum of the effective potential is reached when the Higgs field φ responsible for the GUT symmetry breaking acquires a vacuum expectation value, φ  = σ m p . The symmetric vacuum state, φ  = 0, has a nonzero energy density, ρν . For a ColemanWeinberg potential, ρν ∼ g4 σ 4 , (25) where g is the gauge coupling. Suppose that the universe starts in the symmetric vacuum state and is described by a closed Robertson–Walker metric  dr2 2 2 2 2 2 ds = dt − a (t) (26) + r dΩ . 1 − r2 The scale factor a(t) can be found from the evolution equation 8 a 2 + 1 = π Gρ ν a 2 , 3

(27)

where a = da/ dt. Note that the Euclidean version of Eq. (27) is −a2 + 1 = H 2 a2 , and the solution is a(t) = H −1 cos(Ht) . (28) The solution of the Eq. (27) is the de Sitter space, a(t) = H −1 cos h(Ht) ,

(29)

where H = (8π Gρν /3)1/2 . It describes a universe which is contracting at t < 0, reaches its minimum size (amin = H −1 ) at t = 0, and is expanding at t > 0. This behavior is analogous to that of a particle bouncing off a potential barrier at a = H −1 . (Here a plays the role of the particle coordinate.) We have seen in the previous section that particles can tunnel through potential barriers. This suggests that the birth of the universe might be a quantum tunneling effect. Then the universe has emerged having a finite size (a = H −1 ) and zero “velocity” (a = 0); its following evolution is described by Eq. (29) with t > 0. Equations (26) and (28) describe a four-sphere, S4 . This is the well-known de Sitter instanton. The solution of (28) does bounce at the classical turning point (a = H −1 ); however, it does not approach any initial state at t → ±∞. In fact, S4 is a compact space, and the solution (28) is defined only for |t| < π /2H. The instanton (28) can be interpreted as describing the tunneling to de Sitter space (29) from nothing [52, 53]. The concept of the universe being created from nothing is an astonishing one (we already introduced this issue in Sect. 5.3). It is possible to show how the instanton solution describing the creation of a pair is obtained from the equation 1/2  x − x0 = ± κ 2 + (t − t0 )2

(30)

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(where κ = |m/eE| and x0 , t0 = const.; the classical turning points are at x = x0 ± κ ) by changing t to −it: (x − x0 )2 + (t − t0)2 = κ 2 . (31) (It describes a circular trajectory, that is, again we have a compact instanton.) The instanton solution (32) can be used to estimate the semi-classical probability, P, of pair creation per unit length per unit time: P ∝ exp(−SE ), where SE is the Euclidean action,    m(1 + x2)1/2 − eEx dt . SE = (32) Obviously, the evaluation of the probability P is possible because the pair creation takes place in a background flat space. The instanton solution contributes to the imaginary part of the vacuum energy. Such a calculation does not make sense for our de Sitter instanton: it is silly to evaluate the imaginary part of the energy of nothing. The only relevant question seems to be whether or not the spontaneous creation of universes is possible. The existence of the instanton (28) suggests that it is. One can assume, as usual, that instanstons, being stationary points of the Euclidean action, give a dominant contribution to the path integral of the theory. There may be several relevant instanton solutions. For example, we can have a de Sitter instanton with broken grand unified symmetry, but unbroken Weinberg–Salam symmetry. 4 ρ , where σ Then the vacuum energy is ∼ ρν ∼ σWS ν WS ∼ 100 GeV is the energy scale of the SU(2) × U(1) symmetry breaking. The Euclidean action of a de Sitter instanton is negative, SE = −3m4p /8ρν . If one assumes that instantons with the smallest value of SE correspond, in some sense, to most probable universes, then most of the universes never heat up to temperatures greater than 100 GeV and have practically vanishing baryon numbers. Obviously, we must live in one of the rare universes which tunneled to the symmetric vacuum state. What remains is to talk about what happens to the universe after the tunneling. The symmetric vacuum state is not absolutely stable. It can decay by quantum tunneling or can be destabilized by quantum fluctuations of the Higgs field. The Higgs field starts rolling down the effective potential towards the celebrated ending of the inflationary scenario. When the vacuum energy thermalizes, the universe heats up 1/4 to a temperature T∗ ∼ ρν . In Vilenkin’s model this is the maximum temperature the universe has ever had. The only verifiable (in principle) prediction of the model is that the universe must be closed. However, Guth has argued that the inflationary scenario almost certainly overshoots, so that ρ = ρcrit with a very high accuracy even at the present time [54]. This means that we shall have to wait for a long time until the sign of (ρ = ρcrit ) can be determined experimentally. The advantages of the scenario proposed by A. Vilenkin (1982, 1984, 1998) are also of aesthetic nature. It gives a cosmological model which does not have a singularity at the big bang (there still may be a final singularity) and does not require any initial or boundary conditions. The structure and evolution of the universe(s) are completely determined by the laws of physics.

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5.10 String Landscape and Vacuum Energy: The Emergence of a Multidimensional World from Geometrical Possibilities One possible starting point of the string picture is that the solution to Einstein’s field equations (roughly speaking, they say that matter tells spacetime how to curve, and spacetime tells matter how to move) is not unique, so many different geometries are allowed. The case of five-dimensional Kaluza-Klein geometry provides a simple example of this nonuniqueness (the central theme of Kaluza-Klein theory is: the physical laws we see depend mainly on the geometry of hidden extra dimensions). The circumference of the small dimension can take any size: in the absence of matter, four large flat dimensions, plus a circle of any size, solve Einstein’s equations. (Similar multiple solutions also exist when matter is present.) In string theory we have several extra dimensions, which results in many more adjustable parameters. One extra dimension can be wrapped up only in a circle. When more than one extra dimension exists, the bundle of extra dimensions can have many different shapes – i.e. topologies –, such as a sphere, a doughnut, two doughnuts joined together and so on. Each doughnut loop (a “handle”) has a length and circumference, resulting in a huge assortment of possible geometries for the small dimensions. In addition to the handles, further parameters correspond to the locations of branes and the different amounts of flux wound around each loop. Any given solution to the equations of string theory represents a specific configuration of space and time. In particular, it specifies the arrangement of the small dimensions, along with their associated brane and lines of force known as flux lines. Our world has six extra dimensions, so every point of our familiar three-dimensional space hides an associated tiny six-dimensional space, or manifold – a six-dimensional analogue of the circle. The physics that is observed in the three large dimensions depends on the size and the structure of the manifold: how many doughnut-like “handles” it has, the length and circumference of each handle, the number and locations of its branes, and the number of flux lines wrapped around each doughnut. Yet the solutions composing this vast collection are not all equal: each configuration has a potential energy, contributed by fluxes, branes and the curvature itself of the curled-up dimensions. This energy is called the vacuum energy, because it is the energy of the spacetime when the large four dimensions are completely devoid of matter or fields. The geometry of the small dimensions will try to adjust to minimize this energy, just as a ball placed on a slope will start to roll downhill to a lower position. To understand what consequences follow from this minimization, focus first on a single parameter: the overall size of the hidden space. We can plot a curve showing how the vacuum energy changes as this parameter varies. At very small sizes, the energy is high, so the curve starts out high at the left. Then, from left to right, it dips down into three valleys, each one lower that the previous one. Finally, at the right, after climbing out of the last valley, the curve trails off down a shallow slope to a constant value. The bottom of the leftmost valley is above zero energy; the middle one is at exactly zero; and the right-hand one is below zero. These three

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local minima differ by virtue of whether the resulting vacuum energy is positive, negative, or zero. In our universe the size of the hidden dimensions is not changing with time: if it were, we would see the constants of nature changing. Thus, we must be sitting at a minimum. In particular, we seem to be sitting at a minimum with a slightly positive vacuum energy. Because there is more than one parameter, we should actually think of this vacuum energy curve as one slice through a complex, multidimensional mountain range: the landscape of string theory. The minima of this multidimensional landscape – the bottoms of depressions where a ball comes to rest – correspond to the stable configurations of spacetime (including branes and fluxes), which are called stable vacua. The landscape of string theory is very complicated (much more than the usual one counting only two independent directions: north-south and east-west) with hundreds of independent directions. The landscape dimensions should not be confused with the actual spatial dimensions of the world; each axis measures not some position in physical space but some aspect of the geometry, such as the size of a handle or the position of a brane. The landscape of string theory is far from being fully mapped out. Calculating the energy of a vacuum state is a difficult problem and usually depends on finding suitable approximations. We cannot be sure how many stable vacua there are – that is, how many points where a ball could rest. But the number could very well be enormous. Some research suggests that there are solutions with up to about 500 handles, but not many more. We can wrap different numbers of flux lines around each handle, but not too many, because they would make the space unstable. If we suppose that each handle can have from zero to nine flux lines (ten possible values), then there would be 10500 possible configurations. Even if each handle could have only zero or one flux unit, there are 2500 , or about 10150 , possibilities. As well as affecting the vacuum energy, each of the many solutions will conjure up different phenomena in the four-dimensional macroscopic world by defining which kinds of particles and forces are present and what masses and interaction strengths they have. String theory may provide us with a unique set of fundamental laws, but the laws of physics that we see in the macroscopic world will depend on the geometry of the extra dimensions. However, many profound questions about the string theory landscape must be answered. Which stable vacuum describes the physical world we experience? Why has nature adopted this particular vacuum and not any other? Have all other solutions been demoted to mere mathematical possibilities, never to come true? Instead of reducing the landscape to a single chosen vacuum, recently some physicists proposed a very different picture based on two important ideas. The first is that the world need not be stuck with one configuration on the small dimensions for good, because a rare quantum process allows the small dimensions to jump from one configuration to another. The second is that Einstein’s general relativity theory, which is a part of string theory, implies that the universe can grow so rapidly that different configurations will coexist side-by-side in different sub-universes, each large enough to be unaware of the others. As mentioned before, each stable vacuum is characterized by its numbers of handles, branes, and flux quanta. But one can also take into account that each of these

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elements can be created and destroyed, so that after periods of stability, the world can snap into a different configuration. In the landscape picture, the disappearance of a flux line or other change of topology is a quantum jump over a mountain ridge into a lower valley. Consequently, as time goes on, different vacua come into existence. Suppose that each of the 500 handles in our earlier example starts out with nine units of flux. One by one, the 4500 flux units will decay in some sequence governed by the probabilistic predictions of quantum theory until all the energy stored in fluxes is used up. We start in a high mountain valley and leap randomly over the adjoining ridges, visiting 4500 successively lower valleys. We are led through some varied scenery, but we pass by only a minuscule fraction of the 10500 possible solutions. A key part of the picture is the effect of the vacuum energy on how the universe evolves. Ordinary objects such as stars and galaxies tend to slow down an expanding universe and can even cause it to recollapse. Positive vacuum energy, however, acts like antigravity: according to Einstein’s equation, it causes the three dimensions that we see to grow more and more rapidly. This rapid expansion has an important and surprising effect when the hidden dimensions tunnel to a new configuration. Remember that at every point in our three-dimensional space there sits a small six-dimensional space, which lives at some point on the landscape. When this small space jumps to a new configuration, the jump does not happen at the same instant everywhere. The tunneling first happens at one place in the three-dimensional universe, and then a bubble of the new low-energy configuration expands rapidly. If the three large dimensions were not expanding, this growing bubble would eventually overrun every point in the universe. But the old region is also expanding, and this expansion can easily be faster than that of the new bubble. Because the original configuration keeps growing, eventually it will decay again at another location, to another nearby minimum in the landscape. The process will continue infinitely many times, decays happening in all possible ways, with far separated regions losing fluxes from different handles. In this manner, every bubble will be host to many new solutions. Instead of a single sequence of flux decay, the universe thus experiences all possible sequences, resulting in a hierarchy of nested bubbles, or sub-universes. The result is very similar to the eternal inflation scenario proposed by A. Guth, A. Vilenkin, and A. Linde [55, 56]. The picture we have described explains how all the different stable vacua of the string landscape come into existence at various locations in the universe, thus forming innumerable sub-universes. This result may solve one of the most important and long-standing problems in theoretical physics – one related to the vacuum energy [57]. To Einstein, what we now think of as vacuum energy was an arbitrary mathematical term – a “cosmological constant” – that could be added to his equation of general relativity to make it consistent with his conviction that the universe was static. To obtain a static universe, he proposed that this constant takes a positive value, but he abandoned the idea after observations proved the universe to be expanding. With the advent of quantum field theory, empty space – the vacuum – became a busy place, full of virtual particles and fields popping in and out of existence, and each particle and field carries some positive or negative energy. Concerning the nonzero value of the vacuum energy, the general idea of string theory

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is that the complicated geometries of hidden dimensions might produce a spectrum for vacuum energy that includes values in the experimental window. If the landscape picture is right, a nonzero vacuum energy should be observed, most likely not much smaller than 10−118Λ p (the value of Λ p is about 1094 grams per cubic centimeter, or one Planck mass per cubic Planck length).

5.11 Concluding Remarks In this paper we tried to stress the point that relativistic quantum field theory makes a very clear distinction between what we would intuitively understand to be an absolute void and what we experience as the vacuum of space. Of course, even in the deepest reaches of space we will find an atom or molecule here and there, and photons of energy are flying through at the speed of light continuously, but there is also a quantum potential which exists at every point in the vacuum of our threedimensional physical space. Under the proper conditions, matter and energy can literally be made to materialize out of what we used to think of as nothing. Of course, this does not mean that any conceivable fantasy can be “popped” out of space with ease. The conditions to create anything more than a scattering of subatomic particles would require extremely high energies and a control of the process far beyond any practical ability, at least with our present means. Nevertheless, according to the present mathematical models of physical theories, it is scientifically possible to create anything right out of the vacuum. The classical notion of empty space has obviously been updated. According to the physicist John A. Wheeler, “No point is more central than this, that empty space is not empty. It is the seat of the most rich and surprising physics.” The quantum vacuum is thought of as a seething froth of real particle-virtual particle pairs going in and out of existence continuously and very rapidly. As we have seen, each of these strange pairs consists of a particle and its antiparticle, one of which has a negative energy and is thus called “virtual.” Out of a singularity in space, which by definition really is nothing, the pair simply comes into existence. Why? Because the probability exists, because the universe is open to many (maybe infinite) possibilities which had not been predicted before, and which yet could occur in the future [58]. The future, in other words, seems to be intrinsically unpredictable, at least according to quantum mechanics but also to chaos theory and other system-dynamics theories. It is impossible to predict in any way, from any information ahead of time, which direction or trajectory a certain physical entity or event will follow (go along), or which it will not follow. That means that physics has, in a way, given up, if the original purpose was to know enough so that given the circumstance we can predict (with precision and certainty) what will happen next. This likely intrinsic indeterminate and in some sense fuzzy character of quantum laws of nature (at the Planck scale) represents a strong denial of the philosophical idea associated with the crude deterministic vision of the natural phenomena, according to which “it is necessary for the very existence of science that the same

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conditions always produce the same results.” And it is not merely our ignorance of the internal mechanisms, of the internal complications, that makes nature appear to have uncertainty and many different possibilities in it. It seems to be somehow intrinsic. Nature herself does not even know which way its constituents are going to go. The fact that this view of reality arises from simply the possibility that it can, sounds very contrived and fantastic, but it is nevertheless a very real reality. It is fundamentally rooted, as we already said, in the uncertainty principle. The quantum potential of the vacuum, revealed dramatically by the particle accelerators in the mergence of new and exotic particles, is an excellent example of the fact that the strange implications of quantum theory are more than just a temporary limitation of the mathematical scheme – the “strangeness” is an actual view of a reality that does not (necessarily and always) conform to everyday common sense logic. Virtual particles with negative energy, just like antiparticles with opposite charge, actually can exist, at least for a tiny fraction of a second, as if they were perfect “mirror images” of the normal partners. Inside a black hole they can even become real particles. The idea of a quantum vacuum gives you an idea of what quantum field theory is like and is a good illustration of what modern physics considers quanta to be. Quanta are not “things in space” so much as they are discrete manifestations of underlying fields which pervade all of space, even where no quanta appear to be. From here it is not such a leap to consider one of the strangest revelations of quantum field theory: quanta actually appear to be completely independent of spacetime separation. For example, a quantum can disappear at one point in spacetime only to instantly reappear somewhere else entirely (faster than the speed of light, if thought to have actually gone through spacetime). This very unconventional and logic-boggling behavior, called the tunnel effect, is very real. Let us sum up the most important points we addressed here: 1. In QFT, a quantum fluctuation is the temporary change in the amount of energy in a point in space, arising from Heisenberg’s uncertainty principle. According to one formulation of the principle, energy and time can be related by the relation ΔEΔt ≈ h/2π . That means that conservation of energy can appear to be violated, but only for small times. This allows the creation of particle-antiparticle pairs of virtual particles. The effects of these particles are measurable, for example, in the effective charge of the electron, different from its “naked” charge. Quantum fluctuations may have been very important in the origin of the structure of the universe [59, 60]: according to the model of inflation the ones that existed when inflation began were amplified and formed the seed of all current observed structure. 2. In QFT, and specifically QED, vacuum polarization describes a process in which the background electromagnetic field produces virtual electron-positron pairs that change the distribution of charges and currents that generated the original electromagnetic field [61]. It is also sometimes referred to as self-energy of the gauge boson (photon). In the true vacuum, i.e., the ground state of the interacting theory (in which electrons, positrons, and photons do interact) con-

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tains short-lived “virtual” particle-antiparticle pairs which are created in pairs out of the Fock vacuum and then annihilate each other. Some of these particleantiparticle pairs turn out to be charged; e.g., the virtual electron-positron pairs. Such charged pairs act as an electric dipole. In the presence of an electric field, e.g., the electromagnetic field around an electron, these particle-antiparticle pairs reposition themselves, thus partially counteracting the field (a partial screening effect, a dielectric effect). The field therefore will be weaker than would be expected when the vacuum would be completely empty. The reorientation of the short-lived particle-antiparticle pairs is referred to as vacuum polarization. 3. In the everyday world, energy is always unalterably fixed; the law of energy conservation is a cornerstone of classical physics. But in the quantum microscopic world, energy can appear and disappear out of nowhere in a spontaneous and unpredictable fashion. The uncertainty principle implies that particles can come into existence for short periods of time even when there is not enough energy to create them. In effect, they are created from uncertainties in energy. Since these particles do not have a permanent existence, they are called virtual particles. Even though we can’t see them, we know that these virtual particles are “really there” in empty space because they leave a detectable trace of their activities. One effect of virtual photons, for example, is to produce a tiny shift in the energy levels of atoms. They also cause an equal change in the magnetic moment of electrons. 4. In modern physics, there is no such thing as “nothing.” Even in a perfect vacuum, pairs of virtual particles are constantly being created and destroyed. The existence of these particles is no mathematical fiction. Though they cannot be directly observed, the effects they create are quite real. There is a still more remarkable possibility, which is the creation of matter from a state of zero energy. The possibility arises because energy can be both positive and negative. The energy of motion or the energy of mass is always positive, but the energy of attraction, such as that due to certain types of gravitational or electromagnetic field, is negative. Consider just the gravitational case, where the situation is especially interesting, for the gravitational field is only a warp-curved space. The energy locked up in a space-warp can be concerted into particles of matter and antimatter. This occurs, for example, near a black hole. Thus, matter appears spontaneously out of empty space. Maybe the universe itself sprang into existence out of nothingness – a gigantic vacuum fluctuation. At least to some significant extent, the laws of contemporary physics allow for this possibility. 5. Let’s conclude by stressing that many researchers see the vacuum as a central ingredient of 21st century physics [62]. We know now that the vacuum can have all sorts of wonderful effects over an enormous range of scales, from the microscopic to the cosmic. The vacuum’s amazing properties all stem from a combination of quantum theory and relativity. The uncertainty (and unstability) of the subatomic world manifests itself in random, causeless fluctuations in energy: the larger the fluctuations, the shorter the time it survives. Thanks to the famous equation E = mc2 , Heisenberg’s uncertainty principle also implies that particles can flit into and out of existence, their duration is dictated only by their mass.

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This leads to the astonishing realization that all around us “virtual” subatomic particles are perpetually popping up out of nothing, and then disappearing again within about 10−23 s. “Empty space” is thus not really empty at all, but a seething sea of activity and dynamic processes that pervades the entire universe.

Acknowledgments I am grateful to Ernesto Carafoli and Giuseppe O. Longo for giving me the opportunity to participate in the International Symposium on “The Two Cultures: Shared Problems” (Venice, 24-26 October 2007). I express my gratitude to the Istituto Veneto di Scienze, Lettere ed Arti for kind hospitality during the symposium. I wish also to thank Salvatore Califano, Gabriele Veneziano, Giorgio Vallortigara, and Arthur Miller for helpful comments and stimulating conversations.

References 1. J.D. Barrow: The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe (Vintage, London 2002) 2. H. Genz: Die Entdeckung des Nichts. Leere and Fülle im Universum (Hanser, München 1994) (English edition: Nothingness: The Science of Empty Space. New York 1998) 3. I.J.R. Aitchison: ‘Nothing’s plenty’. The vacuum in modern quantum field theory. Contemp. Phys., 26 (1985) 333–391 4. P.W. Milonni: The Quantum Vacuum. An Introduction to Quantum Electrodynamics (Academic Press, New York 1994) 5. P.A.M. Dirac: Théorie du positron. In: Rapport du 7e Conseil Solvay de Physique, Structure et Propriétés des Noyaux Atomiques (Gauthier-Villars, Paris 1933) pp. 203–212 6. W. Heisenberg: Bemerkungen zur Diracschen Theorie des Positrons. Zeit. Phys., 90 (1934) 209–231 7. S. Saunders, H.R. Brown (Eds.): The Philosophy of Vacuum (Oxford University Press, Oxford 2002) 8. J.T. Cushing: Philosophical Concepts in Physics (Cambridge University Press, Cambridge 1998) 9. B. Pascal: Préface du Traité du vide (1651). In: Oeuvres complètes, t. 2. (GM Flammarion, Paris 1985) 10. L. Boi: Theories of space-time in modern physics. Synthese, 139 (2004) 429–489 11. P.A.M. Dirac: Quantized Singularities in the Electromagnetic Field. Proc. Roy. Soc. A 133 (1931) 60–72 12. R. Penrose: The Road to Reality. A Complete Guide to the Laws of the Universe (Vintage, London 2004) 13. C. Itzykson and J-B. Zuber: Quantum Field Theory (McGraw-Hill, Singapore 1988) 14. P.A.M. Dirac: The Principles of Quantum Mechanics (Clarendon, Oxford 1930) 15. W. Heisenberg: The Physical Principles of the Quantum Theory (University of Chicago Press, Chicago 1930) 16. R. Feynman: The character of physical laws (The MIT Press, Cambridge 1967) 17. H.E. Puthoff: Source of Vacuum Electromagnetic Zero-Point Energy. Phys. Rev. A 40(9) (1989) 4857–4862 18. C. Beck: Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields. Advanced Series in Nonlinear Dynamics, Vol. 21. (World Scientific, Singapore 2002) 19. S.W. Hawking: Particle creation by black holes. Commun. Math. Phys., 43 (1975) 199–220 20. H.B.G. Casimir: On the attraction between two perfectly conducting plates. Proc. Koninkl. Ned. Akad. Wetenschap B 51(7) (1948) 793–796

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21. S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi: de Sitter vacua in string theory. Phys. Rev. D 68(4) (2003) 046005 22. J.B. Hartle and S.W. Hawking: Wave function of the universe. Phys. Rev. D 28 (1983) 2960–2975 23. A. Vilenkin: Quantum creation of universes. Phys. Rev. D 30(2) (1984) 509–511 24. A. Vilenkin: Creation of universes from nothing. Phys. Lett. B 117(1–2) (1982) 25–28 25. B.S. DeWitt: Quantum gravity: the new synthesis. In: General relativity. An Einstein centenary survey. S.W. Hawking and W. Israel (eds.) (Cambridge University Press, Cambridge 1979) pp. 680–745 26. E.P. Tryon: Is the universe a vacuum fluctuation? Nature, 246 (1973) 396–397 27. B. Allen: Quantum states in de Sitter space. Phys. Rev. D 32(12) (1985) 3136–3152 28. D. Atkatz and H. Pagels: Origin of the Universe as a Quantum Tunneling Event. Phys. Rev. D 25 (1982) 2065–2073 29. L. Boi: Geometrization, Classification, and Unification in Mathematics and Theoretical Physics. In: Proceedings of the Albert Einstein Century International Conference. J-M. Alimi & A. Füzfa (eds.) (American Institute of Physics Publishers, 2006) 314–326 30. A.D. Sakharov: Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation. Dokl. Akad. Nauk. SSSR, 12 (1968) 1040–1057 31. J. Schwinger: On gauge invariance and vacuum polarization. Phys. Rev., 82 (1951) 664–679 32. J. Schwinger: Selected Papers on Quantum Electrodynamics (Dover, New York 1958) 33. J.A. Wheeler: Superspace and the Nature of Quantum Geometrodynamics. In: Battelle Rencontres. 1967 Lectures in Mathematics and Physics, C.M. DeWitt and J.A. Wheeler (eds.) (Benjamin, New York 1968) 242–307 34. H.B.G. Casimir and D. Polder: The Influence of Retardation on the London-van der Waals Forces. Phys. Rev., 73(4) (1948) 360–372 35. P. Candelas, G.T. Horowitz, A. Strominger and E. Witten: Vacuum configurations for superstrings. Nucl. Phys. B 258 (1985) 46–74 36. T.Yu. Cao: Conceptual Developments of 20th Century Physics (Cambridge University Press, Cambridge 1997) 37. V. Weisskopf: Über die Selbstenergie des Elektrons. Zeit. Phys., 89 (1934) 27–39 38. L. Boi: Geometrical and topological foundations of theoretical physics: from gauge theories to string program. Inter. J. Math. Mathem. Sci., 34 (2004) 1777–1836 39. G. Veneziano: Quantum Geometric Origins of All Forces in String Theory. In: The Geometric Universe: Science, Geometry and the Work of Roger Penrose. S.A. Huggett et al. (eds.) (Oxford University Press, Oxford 1998) 40. R. Sorensen: Nothingness. In: Stanford Encyclopedia of Philosophy (Stanford 2006) 1–22 41. T.D. Lee: Particle Physics and Introduction to Field Theory (Harwood Academic Publishers, New York 1990) 42. R. Feynman: Space-time approach to quantum electrodynamics. Phys. Rev., 76 (1949) 769–789 43. G. ’t Hooft: In search of the ultimate building blocks (Cambridge University Press, Cambridge 1997) 44. S. Coleman: Aspects of Symmetry. Selected Erice Lectures (Cambridge University Press, Cambridge 1985) 45. P. Hut and M.J. Rees: How stable is our vacuum. Nature, 302 (1983) 508–602 46. E. Gunzig and S. Diner : Le Vide. Univers du Tout et du Rien (Editions Complexe, Bruxelles 1998) 47. M. Cassé : Du vide et de la création (Odile Jacob, Paris 1993) 48. P.W. Higgs: Broken Symmetries and the Mass of Gauge Bosons. Phys. Rev. Lett., 13 (1964) 508– 509 49. S. Weinberg: The cosmological constant. Rev. Mod. Phys., 61 (1989) 1–23 50. S. Coleman: Why there is nothing rather than something: a theory of the cosmological constant. Nucl. Phys. B 310 (1988) 643–658 51. S.W. Hawking and N. Turok: Open inflation without false vacua. Phys. Lett., 1998 [hep-th/9802030] 52. W. de Sitter: On the curvature of space. Proc. Kon. Ned. Acad. Wet., 20 (1917) 229–243 53. A.A. Starobinsky, Ya.B. Zel’Dovich: The spontaneous creation of the Universe. Sov. Sci. Rev. Sect. E 6(2) (1988) 103–144 54. A. Guth: The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (AddisonWesley, Reading 1997)

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55. A. Linde: A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Phys. Lett. B 108 (1982) 389–412 56. A.H. Guth and D.I. Kaiser: Inflationary Cosmology: Exploring the Universe from the Smallest to the Largest Scales. Science, 307(5711) (2005) 884–890 57. G. Scharf: Vacuum stability in quantum field theory. Il Nuovo Cimento A 109(11) (1996) 1605– 1607 58. J.R. Gott: Creation of Open Universes from de Sitter Space. Nature, 295 (1982) 304–307 59. E. Streeruwitz,: Vacuum fluctuations of a quantized scalar field in a Robertson-Walker universe. Phys. Rev. D 11(12) (1975) 3378–3383 60. Ya.B. Zel’Dovich: Cosmology and the early universe. In: General relativity. An Einstein centenary survey. S.W. Hawking & W. Israel (eds.) (CUP, Cambridge, 1979) 518–532 61. R. Feynman: QED: The strange theory of light and matter. (Princeton University Press, Princeton 1985) 62. L.B. Crowell: Quantum Fluctuations of Spacetime (World Scientific, Singapore 2005)

Suggested readings A. Anderson and B. DeWitt: Does the Topology of Space Fluctuate?. In: Between Quantum and Cosmos. Studies and Essays in Honor of John Archibald Wheeler. W.H. Zurek, A. van der Merwe, W.A. Miller (eds.) (Princeton University Press, Princeton 1988) 74–88 L. Boi: Topological structures in classical and quantum physics. JP J. Geom. Topol., 37 (2008) (to appear) R. Brandenberger: On the Spectrum of Fluctuations in an Effective Field Theory of Ekpyrotic Universe. Journal of High Energy Physics, 11 (2001) 1088–1126 R. Brout, F. Englert and E. Gunzig: The Creation of the Universe as a Quantum Phenomenon. Annals of Physics (N.Y.), 115 (1978) 78–106 C. Callan and S. Coleman: Fate of the false vacuum. II. First quantum corrections. Phys. Rev. D 16 (1977) 1762–68 S. Coleman: The Invariance of the Vacuum is the Invariance of the World. J. Math. Phys., 7 (1966) 787–812 V.A. Fock: Konfigurationsraum und zweite Quantelung. Zeit. Phys., 75 (1932) 622–647 B. Haisch, A. Rueda and Y. Dobyns. Inertial mass and the quantum vacuum fields. Ann. Phys., 10(5) (2001) 393–414 J. Hong, A. Vilenkin and S. Winitzki: Particle creation in a tunneling universe. Phys. Rev. D 68(2) (2003) 1103–1124 S.K. Lamoreaux: Demonstration of the Casimir force in the 0.6 to 6 ìm Range. Phys. Rev. Lett., 78(1) (1997) 5–8 T.D. Lee and G.C. Wick: Vacuum stability and vacuum excitation in a spin-0 field theory. Phys. Rev. D 9 (1974) 2291–2316 T. Levi-Civita: Realtà fisica di alcuni spazi normali del Bianchi. Rend. R. Acad. Lincei, 26 (1917) 519– 531 J. Maldacena: The illusion of gravity. Sci. Amer. Rep. (April 2007) 75–81 T. Padmanabhan: Quantum conformal fluctuations and stationary states. Inter. J. Theor. Phys., 22(11) (1983) 1023–1036 R. Penrose: On gravity’s role in quantum state reduction. Gen. Rel. Grav., 28 (1996) 581–600 M. Redhead: Quantum field theory for philosophers. In: Proceedings of the Biennial Meeting of the Philosophy of Science Association. P.D. Asquith and T. Nickles (eds.), vol. 2 (1983) 57–99 A. Rueda and B. Haisch: Gravity and the Quantum Vacuum Inertia Hypothesis. Ann. Phys., 14(8) (2005) 479–498 B. Russel: A critical exposition of the Philosophy of Leibniz. With a new introduction by J. G. Slater. (Routledge, London 1951) S. Sarangi and S-H.H. Tye: A Note on the Quantum Creation of Universes. (2003) hep-th/0603237 Q. Smith: The Uncaused Beginning of the Universe. Philosophy of Science. 55(1) (1988) 39–57 D. Solomon: Gauge invariance and the vacuum state. Can. J. Phys., 76(2) (1998) 111–127 M.S. Turner and F. Wilczek: Is our vacuum metastable. Nature, 298 (1982) 633–637 A. Vilenkin: The quantum cosmology debate (1998) gr-qc/9812027 F. Wilczek: La musica del vuoto. Indagine sulla natura della materia (Di Renzo Editore, Roma 2007)

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Discussion A.I. Miller: Can you comment further on the difference between causality/determinism in classical physics and quantum physics? In quantum physics these concepts are treated very differently. L. Boi: You raise a very important point. I think the Heisenberg’s “uncertainty relation” has had profound implications for such fundamental notions as causality and the determination of the future behavior of a subatomic particle, as well as philosophical implications for the conceptions of physical reality. Indeed the Uncertainty Principle deeply questions the basic principles of classical physics. It states that for a pair of conjugate variables such as position/momentum and energy/time, it is impossible to have a precisely determined value of each member of the pair at the same time. In particular, when the mass is very small (such as in subatomic objects), the uncertainties and thus the quantum effect become very important, and classical physics is no more applicable. I would like to mention another important aspect related to your question and which clearly shows the difference between the classical and the quantum vision of the physical world. By contrast with classical mechanics, the quantum vacuum is thought of as a seething froth of real particle-virtual particle pairs going in and out of existence continuously and very rapidly. Each of these strange pairs consists of a particle and its antiparticle, one of which has a negative energy and is thus called “virtual.” Out of a singularity in space, which by definition really is nothing, the pair simply comes into existence. Why? This is because the probability exists, because the universe is open to many (maybe infinite) possibilities which had not been predicted before, and which yet could occur in the future. The future, in other words, seems to be intrinsically unpredictable, at least according to quantum mechanics but also to chaos theory and other system-dynamics theories. It is impossible to predict in a deterministic manner, from any information ahead of time, which direction a certain physical entity or event will follow, or which it will not follow. This means that quantum physics has, in a way, given up, if the original purpose was to know enough so that given the circumstances we can predict (with precision and certainty) what will happen next. This likely intrinsic indeterminate and in some sense fuzzy character of quantum laws of nature (at the Planck scale) represent a strong denial of the philosophical idea associated with the deterministic vision of the natural phenomena, according to which “it is necessary for the very existence of science that the same conditions always produce the same results.” And it is not merely our ignorance of the internal mechanisms, of the internal complications, that makes nature appear to have uncertainty and many different possibilities in it. It seems to be somehow intrinsic. Nature herself does not even know which way its constituents are going to go. Ch. Riedweg: I should like to point to similarities of “quantum vacuum” with what Plato in his Timaeus labels chora, space (later interpreted by Aristotle to be the material substrate). A space which, according to Plato, is very hard to perceive: it is

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a continuous fluctuation, and the origin of all potentialities as well. The analogies are very surprising. L. Boi: Thank you for your interesting question. I essentially agree with your remark. There is indeed some striking resemblance between Plato’s notion of space and the concept of “vacuum” in quantum physics. Somewhat, one might say that the conception developed by Quantum electrodynamics (a relativistic Quantum Field Theory (QFT) of classical electrodynamics, which describes how electrically charged particles – electrons, positrons, photons – interact by means of exchange of photons; more precisely, QED can be described as a perturbative theory of the electromagnetic quantum vacuum) constitute a theoretical and physical realization of Plato’s idea. First, I wish to draw attention to the chora’s quality of being in motion (a potentiality, as you noted); as a cause of motion, it is also the source of time and change for the world of Becoming. Moreover, there is some kind of endless generativity of the chora, which is not restricted to its actual content, but is also a site of, say, virtual productivity. The chora in Plato’s Timaeus is conceived as a space existing and acting between Being and Becoming. Timaeus dialog indicates a return to a picture of the cosmic space that gives rise spontaneously to a world of becoming and change. While the cosmos is always moving and changing, any specific change remains, strictly speaking, unaccomplished, unrealizable, and unhypostatizable. Of course, I should recall that, in despite of what I just said, Greek philosophers did not like to admit the existence of a vacuum, asking themselves “how can ‘nothing’ be something?”, which means that they point forwards to a sort of logical (linguistic) contradiction in the hypothesis of the physical realty of the vacuum. Plato found the idea of a vacuum inconceivable, essentially because he believed that all physical things were instantiations of an abstract ideal, and he could not conceive of an “ideal” form of a vacuum. Similarly, Aristotle considered the creation of a vacuum impossible – nothing could not be something. Later Greek philosophers thought that a vacuum could exist outside the cosmos, but not within it. However, this logical (linguistic) objection fails when one considers the following point, and this is the second aspect of my answer to your question. According to quantum mechanics and especially to quantum electrodynamics, one has to make a very clear distinction between what we would intuitively understand to be an absolute void and what we experience as the vacuum of space. Of course, even in the deepest reaches of space we will find an atom or molecule here and there, and photons of energy are flying through at the speed of light continuously, but there is also a quantum potential which exists at every point in the vacuum of our three-dimensional physical space. Under the proper conditions, matter and energy can literally be made to materialize out of what we used to think of as nothing. So, according to the present mathematical models of quantum physical theories, it is scientifically possible (under certain physical conditions, energy, temperature, and so on) to create forms of matter and energy out of the vacuum. The classical notion of empty space has obviously been updated. To quote the physicist John A. Wheeler, “No point is more central than this, that empty space is not empty. It is perhaps the seat of the most rich and surprising physics.” There is, for instance, a remarkable possibility, which is the creation of matter from a state

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of zero energy. The possibility arises because energy can be both positive and negative. The energy of motion or the energy of mass is always positive, but the energy of attraction, such as that due to certain types of gravitational or electromagnetic field, is negative. Consider just the gravitational case, where the situation is especially interesting, for the gravitational field is only a warp-curved space. The energy locked up in a space-warp can be converted into particles of matter and antimatter. This occurs, for example, near a black hole. Thus, matter appears spontaneously out of empty space – and here we actually join again Plato’s philosophical idea of the chora and the cosmic space. Maybe the universe itself sprang into existence out of nothingness – a gigantic vacuum fluctuation. At least to some significant extent, the laws of contemporary physics allow for this possibility. G. Veneziano: Beside the words that may be confusing, the truth is that there are important quantum effects which are essential for explaining experimental data. Some of these quantum effects come out to be infinite and are disposed of through “renormalization.” However, there are other QFT, called supersymmetric, where these dramatic quantum effects are not present. Supersymmetry, even if true, must be broken and the quantum effects may become again very large. This is an active area of research in theoretical physical at present. L. Boi: I thank very much Prof. Veneziano for his specification and illuminating comment. The questions of quantum effects, renormalization and broken symmetry are indeed among the most fundamental issues in theoretical physics today. Quantum effects were for example considered in the development of cosmology. The properties of matter, radiation, spectral lines, light scattering, statistics of Bose or Fermi – all these topics were taken into account in the calculation of pressure, energy density, spectrum transport coefficients, etc. Therefore, the right-hand side of Einstein’s general relativity equations already included quantum effects. Speaking about these effects now we emphasize the influence of spacetime curvature on particles and fields, as opposed to the usual physics of (flat) Minkowskian space. The most interesting effect is the creation of particles by the gravitational field in vacuum. The reactions of the type e+ + e− = g + g, with g being gravitons, were considered and calculated in the 1930s and 1940s. Taking many coherent gravitons, one obtains a classical gravitational wave. The creation of e+ e− pairs obviously occurs in colliding beams of (classical) gravitational waves, i.e., in a vacuum with time-dependent metric. In a cosmological context the excitation of fields, i.e., the creation of field quanta (for example photons) in an expanding Universe, was considered notably by Utiyama and DeWitt in 1962. The general principle is that particle creation is due to the nonadiabatic behavior of the corresponding field in a changing metric. Maybe, the most important result of the marriage between quantum theory (quantum effects) and general relativity (gravity) is Hawking’s theory (1975) of black hole evaporation. Concerning its physical content, in a simplified manner one can say that collapse to a black hole is an event which does not end in a static situation (unlike the collapse of a white dwarf to a neutron star for example). Another important aspect concerns the relationship between conformal invariance and spontaneous symmetry breaking, a topic so beautifully elucidated by Sidney Coleman in the 1980s. We first

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recall that for a relativistic field theory, the vacuum is Poincaré invariant. Poincaré invariance implies that only scalar combinations of field operators have nonvanishing Vacuum Expectation Values (VEVs). The VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case, the vacuum (that is, the ground state of the QFT) has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred. This means that, when the Hamiltonian of a system (or the Lagrangian) has some symmetry, but the ground state (i.e., the vacuum) does not, then one says that Spontaneous Symmetry Breaking (SSB) has taken place. When a continuous symmetry is spontaneously broken, massless gauge bosons (messenger particles which mediate the interactions) appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called Goldstone bosons. In the electroweak model, the Higgs field acts as the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. In empty space, the Higgs field has an amplitude different from zero. This is also known as a “nonzero vacuum expectation value,” the existence of which plays a fundamental role: it gives mass to every elementary particle which has mass, including the Higgs boson itself. In particular, the acquisition of a nonzero vacuum expectation value spontaneously breaks electroweak gauge symmetry (often referred to as the Higgs mechanism). In quantum field theory and the statistical mechanics of fields, renormalization refers to a collection of techniques used to construct mathematical relationships or approximate relationships between observable quantities, when the standard assumption that the parameters of the theory are finite breaks down, giving the result that many observables are infinite. Renormalization arose in quantum electrodynamics as a means of making sense of the infinite results of various calculations and extracting finite answers to properly posed physical questions. As pointed out by Gabriele Veneziano himself (see “Quantum Geometric Origins of All Forces in String Theory,” in The Geometric Universe. Science, Geometry, and the Work of Roger Penrose, edited by S.A. Huggett et al., Oxford, 1998, 235–243), in Kaluza-Klein theory, in particular, both gauge theory and gravity diverge in a similar way in the ultraviolet, another expected consequence of Kaluza-Klein unification; in other words, they become nonrenormalizable. “On the one hand quantum mechanics is essential to the success of the Kaluza-Klein idea [to unify electromagnetic and gravitational interactions by adding a new dimensional space, a circle, which allows for the quantization of momentum hence of electric charge]. At the same time, quantum field theory gives meaningless infinities and spoils the nice semiclassical results. If the beautiful Kaluza-Klein idea is to be saved we need better quantum theory than quantum field theory. Now such theory already exists: it is called superstring theory.” Superstring theory relies crucially on the two ideas of supersymmetry and a spacetime structure of 11 dimensions. Supersymmetry is a conjectured symmetry between fermions (particles having an integer spin, measured in quantum units) and bosons (particles with the same mass but half-integer spin). It is an inherently quantum mechanics symmetry, since the very concept of fermions is quantum mechanical. Bosonic quantities can be described by ordinary (commuting) numbers or by operators obeying commutation relations. Fermionic quantities involve anticommuting numbers or operators.

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Supersymmetry is an updating of special relativity to include fermionic as well as bosonic symmetries of spacetime. In developing relativity, Einstein assumed that the spacetime coordinates were bosonic; fermions had not yet been discovered! In supersymmetry the structure of spacetime is enriched by the presence of fermionic as well as bosonic coordinates. If true, supersymmetry explains why fermions exist in nature. Supersymmetry demands their existence. From experiments, we have some hints that nature may be supersymmetric. However, as we know, one of the most remarkable discoveries of the last decades is the theory worked out by Wess and Zunino in the 1970s that predicts a boson-fermion mass degeneracy which is not observed in nature and thus the supersymmetry must be broken. The Goldstone fermions associated with spontaneous symmetry breaking have the wrong property to be neutrinos and hence the symmetry needs to be implemented as a local gauge invariance with the Higgs-Kibble mechanism in action. In the original version of the standard model, the key to electroweak-symmetry breaking is an entity called the Higgs particle. At high temperatures, Higgs particles, like other particles move at random. But as the Universe cools, Higgs particles combine into a “Bose condensate,” an ordered state in which many particles share the same quantum wave function, leading – in the case of helium – to superfluidity. The electroweak symmetry is broken by the preferred “direction” of the Bose condensate. Although this proposal is simple and fits the known facts, it is unlikely to be the whole story. A seemingly artificial adjustment of parameters is needed to make the Higgs particle mass small enough for the model to work. Numerous proposed alternatives solve this particular problem, although they introduce puzzles of their own. One idea, motivated by a phenomenon that occurs in superconductors, is that the Higgs particle arises as a bound state. This would solve the problem of getting its mass right, but also requires a host of new particles and forces, which have not yet been observed. As we have seen above, a more radical and convincing idea is “supersymmetry,” a new symmetry structure of elementary particles in which quantum variables are incorporated in the structure of spacetime. Other ideas about electroweak symmetry breaking go even further afield. One line of thought links this problem to extra dimensions of spacetime, subnuclear in size, but observable in accelerators. Finally, another line of thought links electroweak-symmetry breaking to the dark energy of the Universe, which astronomers have discovered in the past few years by observing that the expansion of the Universe is accelerating. Although these theoretical proposals are all conceptually incomplete and experimentally unverified so far, the diversity and scope of ideas on electroweak-symmetry breaking suggests that the solution to this riddle will provide new and significant insights into the particle physics search for the understanding of the laws of nature. S. Califano: I wonder whether it is not better to use the Fock space to define all possible degrees of freedom needed to represent the vacuum fluctuations in Quantum Field Theory. L. Boi: Thank you. Your remark points to a very important aspect of (axiomatic) quantum field theory. As you certainly know, the origin of Fock space concept lies in physics. A construction made by the Russian physicist V.A. Fock in 1932 suggested

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the way of passing from states of single objects to states of collections of these objects. Actually, it gave an abstract formulation of Hermite expansion in L2 (R) also for n = ∞. It was then finally realized that Fock space is rather an algebra generated by a given Hilbert space and unity and provided with a scalar product fulfilling certain natural requirements. The Fock space F is an algebraic system or Hilbert space used in quantum mechanics to describe quantum states with a variable or unknown number of particles. More precisely, the Hilbert space describes the quantum states for a single particle, and to describe the quantum states with n particles, or superpositions of such states, one must use a larger Hilbert space, the Fock space, which contains states for unlimited and variable numbers of particles. Fock states are the natural basis of this space. If |Ψi  is a basis of H, then we can agree to denote the state with n0 particles in state |Ψ0 , n1 particles in state |Ψ1 ,. . . , nk particles in state |Ψk  by |n0 , n1 , . . . , nk υ , with each ni taking the value 0 or 1 for fermionic particles and 0, 1, 2,. . . for bosonic particles. Such a state is called a Fock state. Since |Ψi  are understood as the steady states of the free field, i.e., a definite number of particles, a Fock state describes a collection of noninteracting particles in definite numbers. The most general pure state is the linear superposition of Fock states. Two operators of paramount importance are the creation and annihilation operators, which upon acting on a Fock state respectively add and remove a particle in the ascribed quantum state. These operators serve as a basis for more general operators acting on the Fock space. One important mathematical point is that Fock space can be examined in several categories of vector spaces. For example, Fock space has a natural interpretation as a space of holomorphic functions. This suggests that these “nonlinear” functions, obtained via Fock space, are not merely continuous but analytic. Let us be a little more precise on the relationship between what vacuum state means and the corresponding algebraic-geometric construction. There appear to be alternative choices for the “vacuum state,” and this issue of “alternative vacua” seems to have some considerable importance in modern QFT. In fact, the choice of vacuum state is a matter of importance comparable with (and complementary to) the choice of the algebra A generated by creation and annihilation operators (see above), later defining, in a sense, the dynamics of QFT. In the case of free electrons, the two vacua |0 (containing no particles and no antiparticles) and |Σ  (in which all the negativeenergy particle states are filled) can be considered as being, in a sense, effectively equivalent despite the fact that |0 and |Σ  give us different Hilbert spaces. As we already said, quantum states in which there are superpositions of different numbers of such particles. These states are obtained by acting on |0, with an arbitrary element of A, i.e. an expression in creation and annihilation operators (polynomials or power series). The space of such states is referred to as Fock space, and it can be thought of as what is called a direct sum of Hilbert spaces with increasing numbers of particles. The number of particles in a state may be unlimited, such as with the coherent states which are, in a certain well-defined sense, the most “classical-like” of the quantum states. These are states of the form eΞ |0, where Ξ is the field operator associated with the particular field configuration F (a free real Maxwell field). F is defined to be the sum of the creation and annihilation operators (not normalized) corresponding to the positive- and negative-frequency parts of F, respectively. In the last years,

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there have been many attempts at interpreting different puzzling aspects of quantized Yang-Mills gauge theories, such as electrodynamics and chromodynamics, in the framework of a (Minkowski) Fock space, and the problem of vacuum fluctuation is one of the most prominent aspects which pertain to these attempts. G. Giacometti: It might be somewhat ambitious but I believe that I could write your paper without ever mentioning the word “vacuum” which is a word of the old times and should not be used in modern physics. L. Boi: Well, if you will be successful in writing such a paper without ever mentioning the concept of vacuum, I will appreciate your endeavor and read your paper with interest. On the other hand, I humbly think you are wrong and somehow unfair towards the concept of “vacuum.” Of course, I am not denying that the use of the word can be confusing, just as can be confusing the use of many philosophical and scientific words; however, that depends on our capacity to give a definition (or statement) as far as possible accurate of these words. Besides, the fact that there still are several notions and entities in modern physics that possess, say, a certain conceptual indetermination and vagueness, doesn’t mean that they are as far confusing and meaningless. The concept of vacuum is a very instructive example of what I am trying to say, for its initial indefiniteness was actually a sign of an unexpected mathematical and physical depth. I would like to remember that, after all, the concept of vacuum was introduced by the great physicist P. Dirac in the 1930s and then largely applied by many other outstanding physicists, such as R. Feynman, J. Schwinger, B.S DeWitt, J.A. Wheeler, T.D. Lee, B. Zumino, S. Coleman, S. Hawking, A. Linde, A. Vilenkin, and many others. Quantum electrodynamics, one of the most important theories of 20th century physics, would have been inconceivable without applying the concepts of vacuum, vacuum polarization, vacuum states, vacuum fluctuations, vacuum energy, which have been all well defined in the framework of the quantum field theory. Furthermore, the Casimir effect and Lamb shift can be explained physically only by means of the concept of vacuum, which is clearly a dynamical and effective entity. More recently, the concept of vacuum plays a very fundamental role in supersymmetric quantum field theories and especially in string theory. In this theory, the concept of vacuum energy is a key ingredient of what is called the “landscape of string theory.” The minima of this multidimensional landscape correspond to the stable configurations of spacetime which are called stable vacua. All these different stable vacua of the string landscape come into existence at various locations in the universe, thus forming innumerable sub-universes. This result might solve one of the most important and long-standing problems in modern theoretical physics – one related to vacuum energy. To Einstein, what we now think of as a vacuum energy was an arbitrary mathematical term – a “cosmological constant” – that could be added to his equation of general relativity to make it consistent with his conviction that the universe was static. To obtain a static universe, he proposed that this constant takes a positive value, but he abandoned the idea after observations proved the universe to be expanding. With the advent of quantum field theory, empty space – the vacuum – became a busy place, full of virtual particles and fields popping in and out of existence, and each particle and field carries some positive

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or negative energy. Concerning the nonzero value of the vacuum energy, the general idea of string theory is that the complicated geometries of hidden dimensions might produce a spectrum for vacuum energy that includes values in the experimental window. If the string landscape picture is right, a nonzero vacuum energy should be observed, most likely not much smaller than 10−118Λ p (the value of Λ p is about 1094 grams per cubic centimeter, or one Planck mass per cubic Planck length). So there are two key features in this new picture: first, the problem of the change of the vacuum configuration is linked with the multidimensional structure of the string landscape; second, the vacuum energy might have an important effect on how the universe evolves. Many researches see the vacuum as a central part of 21st century physics. We know now that the vacuum can have all sorts of wonderful effects over an enormous range of scales, from the microscopic to the cosmic. I wish to conclude by a general remark, namely, by stressing that often in science (in this respect, like in philosophy and in art) it is by imagining new models and inventing new words and concepts that we can reach a wide and deep vision of things. Thus we can “see” unexpected things: we “see” things that are far from what we would guess, far from what we could have conceived. Our imagination (and the concept of vacuum belongs to the scientific creativity) is stretched to the utmost, not, as in fiction, to only imagine things which are not really there, but particularly to comprehend those things which are there or could be there.

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Eugenio Mazzarella

6.1 What Nihilism Is What is nihilism? Nihilism is a feeling. Nihilism is a fact. Both things concern existence, human dasein, or as Sartre says, taking up a fundamental concept from Heidegger1, “Man is the being through whom nothingness comes to the world” [3]. Which, furthermore, means that the “world” in reality has no knowledge of nothingness. Presence as such – “being” – does not know, nor can it know nihilism, a problem regarding nothingness, as its absence having been thought of, in general, as thought, is a presence, and declares that in our “we are,” being as such always 1 “Das Seiende wird doch durch die Angst nicht vernichtet, um so das nichts übrigzulassen. Wie soll es das auch, wo sich doch die Angst gerade in der völligen Ohnmacht gegenüber dem Seienden im Ganzen befindet. Vielmehr bekundet sich das Nichts eigens mit und an dem Seienden als einem entgleitenden im Ganzen [. . . ] wir kämen auch mit einer solchen Verneinung, die das Nichts ergeben sollte, jederzeit zu spät. Das Nichts begegnet vordem schon [. . . ] es begegne “in eins mit” dem entgleitenden Seienden im Ganzen,” [1], pp. 113–114. Bergson in Creative Evolution had already contested the idea of absolute nothingness: “The idea of the absolute nought, in the sense of the annihilation of everything, is a self-destructive idea, a pseudo-idea, a mere word. If suppressing a thing consists of replacing it by another, if thinking an absence of one thing is only possible by the more or less explicit representation of the presence of some other thing, if, in short, annihilation signifies before anything else substitution, the idea of an “annihilation of everything” is as absurd as that of a square circle [. . . ] there is more, and not less, in the idea of an object conceived as “not existing” than in the idea of this same object conceived as “existing”; for the idea of the object “not existing” is necessarily the idea of the object “existing” with, in addition, the representation of an exclusion of this object by the actual reality taken in block” [2], p. 308 and p. 311. Nothingness, as long as there is a reason around which lives and thinks – and if it were the Reason of God as the good will that creates – is always and only a being of the reason. Nothing is more loosened, absolutus, from reason than nothingness, and vice versa: we hang from nothingness like the horizon of nothing, of the nothing-making of our being, and this horizon is the other face – like a source – of the way, as we know and see, of possible ways of being, of our ways of being; and nothingness hangs from, depends on our reason which opens our eyes to life and so it becomes reason.

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goes on, and surpasses us. On the pathway of truth – i.e., of the manifestation that takes form in thought – nothingness does not exist, it cannot be investigated, it cannot be found: Come now, I will tell you – and bring away my story safely when you have heard it – the only ways of inquiry there are to think: the one, that it is and that it is not possible for it not to be, is the path of Persuasion (for it attends upon Truth), the other, that it is not and that it is necessary for it not to be, this I point out to you to be a path completely unlearnable, for neither may you know that which is not (for it is not to be accomplished) nor may you declare it [4]2 .

As for being, or being-nothingness, nihilism is a nonexistent problem. In the ordered cosmos that sees me emerge and that emerges in my eyes, there is no nihilism. In physics, this means, as Heidegger says, that physis, what there is, that is to say, what is manifest, is unavoidable (das Unumgängliche) for physics: Die Natur bleibt so für die Wissenschaft der Physik das Unumgängliche. Das Wort meint hier zweierlei. Einmal ist die Natur nicht zu umgehen, insofern die Theorie nie am Anwesenden vorbeikommt, sondern auf es angewiesen bleibt. Sodann ist die Natur nicht zu umgehen, insofern die Gegenständigkeit als solche es verwehrt, daß das ihr entsprechende Vorstellen und Sicherstellen je die Wesensfülle der Natur umstellen könnte. . . Auch dort, wo die Theorie aus Wesensgründen wie in der modernen Atomphysik notwendig unanschaulich wird, ist sie darauf angewiesen, daß sich die Atome für eine sinnliche Wahrnehmung herausstellen, mag dieses Sichzeigen der Elementarteilchen auch auf einem sehr indirekten und technisch vielfältig vermittelten Wege geschehen [6]3 .

Nothingness in physics is no more than the field of the infinite effort to know, to advance the frontier of the number, to bring within the mathematical, the already known through calculation, the pre-seen of theory, the not yet seen. This nothingness which is anyway the possibility of the evident, of the becoming seen of something that is the false vacuum of quantum physics, where the vacuum is not a static space, and where nothing can happen, but something dynamic, where phenomena can happen in quantity, as there is a probability greater than zero that an energy fluctuation 2

p. 152. As for the Platonic “parricide” of this Parmenidean idea, it is necessary to weaken the meaning of alternative in the sense of an opposition as the principle between the two conceptions of nothingness which have appeared in the history of philosophy – Nothingness as not-being, i.e., nothingness as otherness or negation, one headed by Parmenides, the other by Plato. The Platonic idea expressed in the Sophist where the being of the not-being is admitted, defining nothingness in the relational sense as otherness: “Then it unquestionably follows that not-being is, throughout all our kinds, no less than in the case of motion. Throughout the whole series, the character of otherness makes each of them other than being, and consequently not-being. Hence by parity of reason we may correctly call them one and all, in this sense, nonentity, though we may also speak of all, by participating in being, as being and entities” [5], pp. 163-164. “Parricide” (Soph. 242 d) – whereby with this standpoint Plato would seem to have decided – and upon closer observation, he is far less parricidal than he appears, as Plato does none other than arrive at the logical consequence of the truth expressed by Parmenides. Precisely because not-being cannot be predicated or thought of in the sense, to subtract it from the flatus vocis to which it is condemned, and rightly so, by Parmenides’ thesis, it is necessary to place it within the context of a relationship with otherness. Nothingness is the fruit of the fact that every positive reality produces a negation, i.e., it is really no-thing, but this negativization in being (as a power of the negative which pro-duces the world of appearance in its multiplicity), which opens the world – to conscience – as the kingdom of Opinion, is possible only on the basis of the original “position” of being, as its being always placed, whose truth as an original and originating manifestation, kept in thought by Persuasion. 3 p. 58. The sections quoted have been inverted.

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of the vacuum can lead to the appearance of a particle. The vacuum now comes to have a potential energy which in some cases becomes manifest, on account of quantum fluctuations. In metaphysical terms, this false vacuum could well be understood as the matter defined by St. Augustine which is and is not something [7]4 , or in the nonbeing of Plotinian matter, as “a tendency towards existence.” [8]5 As long as hypotheses regarding the first instants of the life of the universe, on the beginning of the cosmos, the ordered world, the 0-1 instant, zero already one, so to speak, creating by itself its own laws of expansion and/or the organization of space/time, only say something measurable, they can only speak of the “how” of that which is, and never the “why” of “what is” or of its “how.” The how that we have concerning the what is of that which is, is already a discursive why of that which already is, according to its laws, and not a generative why. Our question is already in the actual generation of what is asked: it is the “what-is” that already creates the discourse which regards it concerning the law which governs it and the grasp within us, depending on number, of this law. The why of the what which knowing seeks always and only finds the how – perhaps ever more cognitively advanced – of its happening6. If this is so, ultimately, from a cosmological point-of-view, parodying a lexicon that could belong to a Nietzschean “stellar” view, nihilism is a petit-bourgeois problem. However, we are petits-bourgeois, and thus rather than playing at being supermen, we will be more honest with ourselves, and the true state of things, if we accept our human condition: the anguish of our own loss, fear conscious of disappearing together with the world of the beloved things, given that we come to the “beloved light of day” (Theognis) when it has already reached its sunset: the regret that so typically characterizes so much literary expression of the Greek soul and, in general, the middle-eastern world of thought. It is more honest to accept the groan that Saint Paul understood to run through creation, and pointed to the wooden cross upon which to hang in order to pull up one’s feet from nothing (Rm., 8.19–25) or the long lament of Gilgamesh, the hero of the Babylonian epic on the unattainableness of the “Land of the Living.” And in this humility, to see what can be done to face this given situation. In the end, the true history of nothingness is the history of the 4

p. 297 (XII, 6). p. 241 (III, 6). 6 The more accurate hypotheses on the first moments of the life of the universe, i.e., the Big Bang hypothesis to which the foundations of general relativity lead, and the hypothesis of the prebig bang of string cosmology to which the foundations of string theory lead, whereby the life of the universe extends to the infinite towards past and future, and whereby the Big Bang is only a violent moment of transition (inflationary unleashing of the expansion of the universe) between the cosmological primordial pre-big bang phase and the current post-big bang phase (our Universe may have been created from the collapse of a previous universe, and its origin should perhaps be imagined less as an explosion, a big bang, than as a big bounce, a bounce of matter itself) do no more than re-present the speculative undecidability between an ontology of necessity (with which string cosmology is coherent) and a creationist ontology (with which the Big Bang cosmology is coherent), between the idea that ex nihilo nihil fit or ex nihilo omne fit or ex nihilo omne ens qua ens fit, and this in an absolute sense as the total of that which is. As what existed before the observed beginning is not observable, so is it not decidable whether this “before” is: we who observe the beginning are first of all observed from the beginning. 5

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soul: it is in being naked to itself, to its unrelatedness in being, or in the creation, that nothingness – which is anyway no part of it – emerges in its view. In the quaestio nihili between Jünger and Heidegger, I believe that, in the end, Jünger is right: Nothingness is – and here I paraphrase the other way round the splendid title of Franco Volpi’s Introduction to the Italian edition of Über die Linie, Itinerarium mentis in nihilum – a route into the mind, of the mind. Wenn ich die Augen schließe, erblicke ich zuweilen eine dunkle Landschaft mit Steinen, Klippen und Bergen am Rande der Unendlichkeit. Im Hintergrund, am Ufer eines schwarzen Meeres, erkenne ich mich selbst, ein winziges Figürchen, das wie mit Kreide aufgezeichnet ist. Das ist mein Vorposten, ganz hart am Nichts – dort unten am Abgrund Kämpfe ich für mich. [9]7 Die eigene Brust: das ist, wie einst in der Thebais, das Zentrum der Wüsten- und Trümmerwelt. Hier ist die Höhle, zu der die Dämonen andrängen. Hier steht ein jeder, gleichviel von welchem Stand und Range, im unmittelbaren und souveränen Kampfe, und mit seinem Siege verändert sich die Welt. Ist er hier stärker, so wird das Nichts in sich zurückweichen. Es wird die Schätze, die überflutet waren, auf der Strandlinie zurücklassen. [9]8

A note of 9 July 1942, regarding the position of Anarca, and a passage from the dialogue with Heidegger Über die Linie. They point in the same direction. Resistance against nothingness should be exerted in the place where it arises, in one’s “own bosom.” Where the feeling of one’s own unrelatedness to everything, to which we are related by a relationship that we have to learn once again to inhabit, as they – men of immanence – have our only known dwelling place there. This is not so very different from the position of the Christian soul, for which the transcendence of existence is not only transcendence in the world, but transcendence to the world. Also here, we have a feeling that apprehends a state of fact. In the pain that almost destroys it – this soul– at not being able to relate to anything concrete in the ocean of being, should it lose its grip on the cross of Christ. It is the situation of the believer today, in the “era of nihilism,” as Joseph Ratzinger described it in 1968, taking up an image of Paul Claudel’s, in his Introduction to Christianity [10]. Fastened to the cross – with the cross fastened to nothing, drifting over the abyss. The situation of the contemporary believer could hardly be more accurately and impressively described. Only a loose plank bobbing over the void seems to hold him up, and it looks as if he must eventually sink. Only a loose plank connects him to God, though certainly it connects him inescapably and in the last analysis he knows that this wood is stronger than the void which seethes beneath him and which remains nevertheless the really threatening force in his day-to-day life. [10]9

I am fastened to “God”, to the deus sive natura, ontologically deconcretized by the “taking the floor” of the “I am” of the Old Testament and replaced by him on the creationist scene of Biblical monotheism (to be is a need for me only because this is what the God of Liberty has wanted and continues to want), only by the wood of the Cross, the support with the nature of grace. The fundamental metaphysical question – expressed by Leibniz, “why is there something rather than nothing?” (“pourqoui il y a plutôt quelque chose que le rien?”) – arises from the disconcerted “car le rien est plus simple et plus facile que quelque 7 8 9

Bd. II, p. 344. Bd. VII, p. 279. p. 19.

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chose”: “for nothing is simpler and easier than something” [11]10 ; and why then am I? The rien is certainly easier than the quelque-chose, especially the quelquechose that thinks, i.e., the someone. Nothingness is easier than me. For this reason, a someone is needed, and so nature has need of Grace, to be “existentifying,” God, without whom “non seulement il n’y auroit rien d’existant, mais il n’y auroit rien de possible” [12]11 . And this is because all things, and not only the “spirits,” are in a kind of society with God by virtue of reason and eternal truths, are members of the City of God [. . . ] in virtue of the harmony pre-established from all time between the kingdoms of nature and grace, between God as architect and God as monarch. Consequently, nature itself leads to grace, and grace perfects nature by making use of it. [11]12

It is the breach endured or feared of this “natural” societas with God – as His creature – that creates cognitive dissociation in the Christian: created not believing in his Creator, this relates him to the position of “anarch,” of the “without law,” abandoned “to the margins of the infinite,” between “rock, cliff and mountain,” “at the edge of a black sea,” reduced to a “minute figure, almost traced in chalk,” left alone in the abyss to fight his fight, to see if it is enough for him to be a law unto himself, parodying god by himself. This theological nihilism of the created is the only nihilism possible other than worldly nihilism, the nihilism of the “loss of the world.” Nihilism always regards my being involved or not in the being-nothingness. It is something that concerns me, and nothing else: In this question which in me comes from me and is posed to me, all other things are involved in it through me as one living it; the groan of nature is my groan for the things and the life that go on together with me, and speaks to no-one. That is to say, nihilism is the risk of staying outside the gates of Him that has drawn me out from nothing, of Him to Whose Cross I owe the chance of a grip to pull my feet out of nothingness. It is the risk of remaining outside the House of the Living. For the Christian soul, one does not lose the cosmos, it loses Him. The descent into hell does not really refer to any outer depths of the cosmos; these are quite unnecessary to it. In the basic text, the prayer of the crucified Christ to the God who has abandoned him, there is no trace of any cosmic reference. On the contrary, this article of the Creed turns our gaze to the depths of human existence, which reaches down into the valley of death, into the zone of untouchable loneliness and rejected love, and thus embraces the dimension of hell, carrying it within itself as one of its own possibilities. [10]13

For the Christian Soul, the abyss, which is dwelt alone, is not the dust of Sheol, but it is the nothingness of Him around me, His nothingness that I have denied myself. Hell is become cold. It is this freezing of the heart. You can also spatialize it, project it “outwards,” but it is a world of within, the autism you close yourself in. The other face of Heaven, always crowded, with choirs, virgins, trees, “Edenlike.” Heaven is this fullness of the relationship. Common use maintains this experience of mutual inhabitation of “I” and “world” in language. When we are perfectly at peace 10 11 12 13

p. 210 (§ 7). Vol. VI, pp. 226–227 (§ 184). p. 212 (§ 15). p. 238.

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with everyone and with everything to the eyes of another, we say (in Italian) that “we touch the sky with one finger”: we express, in an external way, a peak in our inner life, in our self-awareness. When life is truly life, meaning also the inner life, i.e., a dynamic relationship, one always drags along, expressedly or unexpressedly, immediately or regained, all the wealth lived or relived of relationships with the external, with others. Nihilism is remaining outside, alienation, from this state of grace: alienation from the “great” grace of the fullness of life or from the “small” grace of relating to the life where we are born into the “world” and where, what’s more, we allow ourselves to be transported by it far from it far from the edges of nothingness, from the loss threatened or feared of the relationship that binds us to the world and/or to God: in the end – and Heidegger should not hear us say this! – the blessings of the inauthentic, of who, never having seen or weighed nothingness, is not born, and not made, for philosophy: “der steht endgültig und ohne Hoffnung außerhalb der Philosophie” [13]14 . Nihilism relies on anticipating “unrelatedness” to the world as its coming to be without us, leading on to the world of “insignificance”; an anticipation which does not regain the local teleology of existence, the original existential “supply” of our being-in-the-world as a single answer to the absence of ultimate purpose to this emerging, in the sea of being, of the Earth of the Self. It is an anticipation which cannot go back to itself, bless and say yes to its own worldly transcendence, remembering that only those who have emerged into “salvation” can be submerged. Nihilism is the inability of knowing to “bless” – the great yes and amen of life in itself. When this grief has become known to us, to our sense of doing and overdoing, making such doing useless, a useless barrier to the done in vain, a task for Sisyphus, overcoming nihilism can be only the ability to re-establish ourselves on “a completely different plane from that of making and ‘makability’, essentially, entrusting oneself to that which has not been made by oneself and never could be made” [10] 15 can only be the great yes and amen of life for itself, whether this be submission to the exteriority of the world, or the exteriority which for us in interior, of a God, and which is not a mask, deus sive nature, but the True Person. The Christian will call it faith, the man of the death of God, “great health” or Gelassenheit.

6.2 Submission to Exteriority as the Work of Reason This submission to exteriority as that from which it receives, first of all itself, is the pure knowledge of reason, of a reason naturally finite – its true abyss like “the unconditioned need, which we need so indispensably as the ultimate sustainer of all things” [14]16 . Before this pure knowledge, without specifications of any kind to 14 15 16

p. 413. p. 40. p. 574.

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cling to, other than the extremely general “being,” and participating in “being,” of being alive, compared with dass originally without the pure existing, as Schelling says, the rational being has three possibilities to choose from. It can remain paralyzed in stupor at a reason which it had entrusted too much to itself: The discovery of its groundlessness is here the shaking beneath the feet of any terrain of being if this, as pure existing always precedes me and this preceding is suffered as the choice of a Liberty (which is necessarily self, but not me) which can also not give me to myself. But here the abyss is not the effective losing of support, that if it happened it would not find me, and would not even find my anguish, but precisely the anguish that that support – that is – could very well not be there, and for me there was a time when it was not there and it will not be – if I remain a mere natural conscience, not saved – with every certainty for the future, and a future I will not be able to control, but not for this any less certain, as my origin was uncontrollable and uncertain. Here, being takes on nothingness not as its nature (since being-nothingness is always being to itself, in itself and to itself it hides no abyss; and does not even know it, nor could it know it, because for it, it simply would not be, if in being there were a coextensive conscience principle), but it takes on our nature. Nothingness as an abyss of being is none other than the discovered crisis of the self-creating illusion of finite reason. Only a reason believing itself fully able to explain itself, even its mere existence, as a coming into existence as it is, with its own modalities, can then discover itself radically unfounded, i.e., balanced on the nothingness of a mere abyss, desperate and trying to reassure itself – and this is the second possibility – telling stories about a free, but logical creation of the world, of a world of logic, i.e., precisely that of reasoning reason, continuing thus to believe it can guarantee to itself the uncovered foundation as insignificant (but in truth it is only insecure and contingent) in the continual narration of gratuitousness – which however, if it begins is necessary – of its own foundation. Nothing more than a way of reasoning to arrive necessarily at itself in the creation, at my freedom starting from the Freedom of Another to whom I submit myself, only more baroque than the a priori Hegelian solution of the speculative co-extensitivity of being and thought: In Hegel one begins from here. In Schelling, ones arrives anyway, at least in the essentials, guaranteeing the me, the reason that thinks: Die positive Philosophie geht von dem aus, was schlechterdings außer der Vernunft, aber die Vernunft unterwirft sich diesem nur, um unmittelbar wieder in ihre Rechte zu treten. [15]17

“In its rights”: in the possibility, that is, of having as its subject matter, firm in its capacity to determine, that “absolutely outside” of pure existence. As if knowing life meant possessing it as life; where the life that one really knows knows for certain only this, that certainly one cannot possess, not even if one knows. The assurance that reason seeks here, not exactly intellectual coolness of analysis, but as life thrown into uncertainty, life of the mind which is aware of itself as life and which needs to continue to be (what is written into its instinct), this assurance can not come to reason from reason – which is precisely the realization in a living being of the 17

Ergänzungsband VI, p. 171.

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opposite, but only from a warm personal faith in a personal Being, not manifest, deduced, from any thought or syllogism on being necessary, but which comes forward by itself without having need of anything – other than to be believed in, like, what is more, all of revelation. Only if reason psychologically abandons its claim to self-foundation, one can advance its third possibility: to be thought that thanks, being, for being. If it accepts itself, that is, for what it is, truly un-founded as founded-in . . . something which surpasses it, but always (that for which it gives thanks, for the right time, which is its time) it admits of it, the pure existing is not an abyss for it any more than the sea is for the waves; the pure existing is its source territory that as a totality always precedes and supersedes it, but which for it is always determinedly what sustains it. And it is not, whatever Kant thinks, the need for this support that frightens it, because it might not be there, but the fact that this support (unless a personal God appears to say the opposite) is not eternal as the being necessary to itself ab eterno, that being that we encounter again every time it starts thinking as reason, whether it exists or not in itself in its necessity in itself and as such. It is an anthropomorphism of reason to have the necessary being thought of “as the highest among all possible beings,” say to the pure existing, “whence then I am?” [14]18 , and see him afraid in this, or frighten us in his alleged vacillation. Here, it is only the small ego of the mind which, at its own apex, rolls on its ignorance of its origin and destination – and the true sublime that frightens, for a rational being, is not the presumed vacillation of its eternal, necessary support, which, in this respect, as long as it exists, always exists (albeit for the space of a contingency, the thought that it exists always touches being, the Aristotelian tigheîn), but the measure of eternity, in whose light the reasoning being sees itself for what it is, and even now it can no longer be seen, in the sea of time, that in the breaking of a wave, it has given it its moment, not a “nonfoundation” but a transient destiny. This means that there are first and last things, to fear only if there is Christ, a personal God of inviting and eternal refusal – otherwise there are only things, in whose presence which does not transcend “there is no need to hope in order to undertake, nor to win to persevere” (William the Silent). Finite reason can be truly astounded only in the face of time, since it knows – because of its original familiarity with being – that being means always being established, and as far as we are concerned, established as rational. Thus, thought is truly astounded only at being present, i.e., at falling within a fixed time: and its only real effort, being life, is to remain. Only a theological nihilism – post Christum, so to speak– is entitled, theoretically speaking, to pose to the cosmos the question of “nothingness,” of its being able not to be, to unite itself to it in its destiny of “nothingness,” to sympathize with, sympathized with, the groan that runs through creation. But also here, nihilism is a relational problem, no longer (only) of our being in relation to the configured all, to the “cosmos,” the ordered world in the biotic turf that allows our survival, and the pro tempore survival of “our” “nihilistic” problem, which can only be answered by 18

p. 574.

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philosophy as a second philosophy, the application by its own existence in physis as an antientropic programme of human transcendence, of human identity as a stationary metaphysical programme (I have offered some considerations on this stationary metaphysical programme elsewhere). For Christian anthropology es gibt keine metaphysischen Ordnungsgefüge, die man aus aller geschichtlichen und heilgeschichtlichen Konkretion herauslösen könnte. Es gibt theologisch nur das eine Bleibende, daß der Mensch der von Gott geschichtlich Aufgerufene und zur Antwort Angerufene ist. Diese geschichtliche Wort- und Antwortbestimmtheit ist seine “Natur,” sie begründet seine je einmalige Würde. Er muß sie in geschichtlichem Gehorsam und in Verantwortung vor Gott je neu verwirklichen. [16]19

It is Christianity which has turned God into a historical question and one of the essence of man. A question of “word and response,” a word heard as emeth, undoubtable truth of God, and considered (or refuted) as such. As Kittel recalls, Emeth is the Old Testament term for “a reality which is to be regarded as amen, “firm,” and therefore “solid,” “valid,” or “binding.” It thus signifies what is “true” [17]20 . The God of this fact, always known by word for the people that recognized Him, is a “God of the amen,” “elohe amen”; and the man who complies with these facts of its expressed truth and holds firm in it, trusting in loyalty, is an i’s emeth, “one whose conduct falls under the norm of truth and therefore a man of integrity”: emeth means a “normal state,” and special serves to express its “certainty and force”; in legal language it is “the ‘actual truth of a process or cause’,” and in common language means “that a revelation has really happened and is incontestable” [17]21 . The word of Yahweh is therefore emeth, or it is truly and indisputably present and therefore operative in the mouth of the prophet”; and “emeth is a witness to the true facts which are to be disclosed by judicial trial, and as such he is a deliverer of soul. [17]22

The Savior of life is a true witness; those who spread lies are impostors (Pr., 14, 25). All the more so will the prophet, God’s witness be a savior of lives. Now, however, it is precisely the fact that emeth goes beyond the legal sphere – a sphere where in any case an assessment of the truth is required, as falsehood is always possible – which creates problems for the purity of a religious experience based on an extremely concrete and historical confrontation between words and response. That is to say, that if emeth takes on the meaning – as Kittel states – “facts which always demand recognition by all men as reality,” [17] what faces crisis is the unshakable possibility of the human amen for the divine emeth in the universe of Christian experience. It is this universalization of the concept of emeth, in other words, that makes faith as confidence into a proof of reason, and its testimony a rea-

19

p. 56. pp. 232–233. I have dealt more widely with the notion of the emeth of God and man in [19], pp. 157–173. 21 pp. 233–234. 22 p. 234. The quoted sections have been inverted. 20

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sonable reason where Christ, as the scandal of mortal weakness which proclaims itself divine has already gone23. This is the reason why the authenticity of the Christian claim of the extra nos of Salvation by means of a Mediator, which is not the transcendent hypostasis of a soteriological capacity of the conscience regarding itself, must always return to the authenticity of the religious significance of the term, not a metaphor of the juridical in its thrust towards a normalizing and universalizing conceptualization of experience. Religious authenticity, where emeth means the substance of the meeting between the righteous man, who obeys the norm accepted by God and thus knows his will, making it his, the “just” man, and the truth as the basis of action and the word of God, divine veracity which demand the veracity of men [17]24 : “Behold, thou desirest truth in the inward parts: and in the hidden part thou shalt make me to know wisdom” (Ps. 51.8). In the face of the legal connotation of the word emeth, which reconsiders from a rationalizing gnoseological angle a truth and a rule of life, lived, however, directly by the religious conscience and are imposed with the force of evidence, the Christian experience of the truth restores the lived experience and experience of a conversation with God that says yes. This yes is the reason Moses and his people became the elect. We come back to the God “abundant in goodness and truth” of Ex 34, 6, which emeth has always preached and “on Sinai He gave torot emeth, laws which establish the truth and are themselves truth” [17]25 : “Thou art that God, and thy words be true” (2 Sam, 28). Connected to this Old Testament religious authenticity of emeth is the aletheia of John, since it “has the meaning of nonconcealment. It thus indicates a matter 23

Paradoxically, this universalization holds more in myth consciousness than in the Old Testament and Christian traditions. Here the divine is fascinating and the tremendous, whose manifestation crushes and convinces a conscience already linked to it, which is already immanent and whose re-ligio is the scruple in the rite of repeating the origin as it has been handed down: relegere, or as Benveniste translates the meaning of the word: ‘recollecter, reprendre pour un nouveau choix, revenir à une synthèse antérieure pour la recomposer’: “la religio ‘scrupule religieux’ est ainsi, à l’origine, une disposition subjective, un mouvement réflexif lié à quelque crainte de caractère religieux.” [18], p. 265. Now this obvious, truthful, normal taking up of the acts behind worship, which are thus normative, is no longer possible with the original Christian historicization. For Christian consciousness, this is no longer situated in illo tempore, outside history and myth but in the here and now of the Lord who comes and speaks to me, in Christ’s time as his actual-now, come to the world which makes even his annunciation in the beginning real: which gives historical reality, as actual history of salvation, even to the time, which is otherwise mythical, of the fall of Adam, which requires and awaits the new Adam. For the Christian consciousness, re-ligio becomes religare, the meaning philologically upheld by Benveniste, as “‘obligation’, lien objectif entre le fidèle et son Dieu” [18]. A notional necessity which has become a religious consciousness lived apart from the divine to the extent by which it transcends it, that divine to which it is now necessary to return by an act of will, awakening in us “un sentiment qui dirige vers une action, ou qui incite à pratiquer le culte” ([18], p. 270); yet again the notion which Schleiermacher sees as decisive in religious consciousness is the ability to listen in feeling, as the place of the “presence of the whole and undivided whole,” the “voice” of the totality from which we are torn while being included within it. 24 p. 235. 25 p. 236.

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or a state to the extent that it has seen, indicated or expressed, and that in such seeing, indication or expression it is disclosed, or discloses itself, as it really is, with the implication, of course, that it might be concealed, falsified, truncated [. . . ] and therefore, denotes the ‘full or real state of affairs’” [17]26 . The opposition between the sphere of the divine, which is revealed in Jesus Christ, and the demonic sphere, of which man is prisoner after the fall – unlike of the cosmological HellenisticGnostic dualism – is assumed as a pure possibility of human existence. In the same way, the revelation is a word which is heard; it too is a possibility offered to man: since rebelion against God, the breach of immanence to Him, happened in knowing, it is only with the humility of faith and not rational or esoteric indoctrination which actually participates in its transcendence, and by its initiative in an extra nos of the salvation mediated for us by Jesus Christ. John’s revelation is not “a complex of statements or ideas [. . . ][and] is not cosmological or soteriological speculation,” [17]27 rather an appeal which occurs in a concrete and personal meeting: it cannot be separated from the person and the work of Jesus: I am the way, the truth and the life (John, 14, 6). The adoration of God in spirit and truth (John, 4, 23) does not mean that true worship takes place in spirituality and pure knowledge on the basis of a concept of God purged of anthropomorphic conceptions, but that it takes place as determined by God’s own essence, i.e. by the pneuma. If aletheia is added, this is an indication that such worship can take place only as determined by the revelation accomplished in Jesus, and consequently as determined by the Revealer who is the only way of access to God. [17]28

The divine truth is nothing more than the salvific essence of revelation for a righteous man who decides on this truth as his foundation. However, this decision of the “just” man before God has already always decided against the world, against truth of the world, the other fundamental decision for the historical existence, for the existence which dissolved due to the unobserved participation of a world time which crosses it, but without touching it. This is because, for the historical “being,” one is always immanent in one or the other decision as regarding the foundation: Either the personal Christian God, or the world as the prevalence of existence over being and all the names of God that speak of Him29 . On the historical-existential plane this name is the radical historical finitude of human life, so the God of life is a promise that consoles but is not the truth. It is history as the occurrence of what happens consigned to its very happening, to the mere beat of its passing, to the changeable time of its existence, the truth for man of the god of the world. Here, the man of truth must decide, and really decide, for another decision regarding the basis: Nothingness revealed in its unprejudiced seeing as the basis 26

p. 238. p. 246. 28 pp. 246–247. 29 For an illustration of this tesi that only in denying credit to the word of the personal god on his super-worldly divinity, is the problem of atheism posed for philosophy, which is as religious as any natural Deus sive Natura theology as a recognition of the inclusive transcendence of the All with regard to all beings which emerge from it, are supported or wrecked in it, I take the liberty of referring to E. Mazzarella, Pensare e credere. Tre scritti cristiani, [19], in particular Filosofia e teologia di fronte a Cristo. 27

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of its own existence, future nothingness promised to his life as aeternitas which is interminabilis vitae tota simulates et perfecta possession (Boethius). It is aletheia which is the manifesting self-liberation of the world for man, implicated in dictating to reason the norm of necessity – that reason which sees what happens in the world: coming to it as always, a going away from it that fights to stay there for the appropriate. The truth which obliges the phenomenological reason of that transient phenomenon which is human existence is this self-perception as a nonfoundation of a nothingness not redeemed by any hope as something definitive and which cannot be renounced in the temporariness of history. Only from this opening up to the world – which nullifies us at the very moment when it gives us to ourselves – the judgment of truth on things present can receive die Weisung, sich nach dem Gegenstande zu richten und gemäß der Richtigkeit zu stimmen [. . . ] in ein Offenes für ein aus diesem waltendes Offenbares, das jegliches Vorstellen bindet. [1] 30

In the world, it is the discovery of our mortality as a free offering to us of a manifestivity which constrains us to a relationship of truth, the conformity of thought to its thing as much to itself. What is truly absolute in history, literally ab-solutus, freed from every thing which is seen in the presence and is handed down in hearing (that is to say the end of the life as life in itself), is the hope of a future which is not, in what is seen and heard in a human way among men. The truth of the world is another rigor, which for all it lives and wants to live them, does not save the phenomena from themselves; nor can it. It is its phenomenological integrity, the fundamental atheism of philosophy. For this integrity of reason not shaken by faith, the Christological claim to emphasize one specific historical event in human life on Earth – Jesus Christ – as a determinant factor for the whole human reality as much as its salvation really remains no more than the advent of the world to the light of an existence which understands itself, that is human life as such. In the historicity of Christ as beginning and end of all things and human life, there is nothing more for it than the advent of historicity itself as a feeling of time as conscious life, which suspends it, to live, in the hope of the without-time of the “blessed” unconsciousness of the origin. The event of man is, for it, simply the event of the historical, which finishes with him if there is no personal God (aware and willing) of whom Christ is the herald. In the absence of this existence of God in my faith, Christ is only the prototype of man, as the historical, his idea as created in the origin of the phylum which is renewed. Of course, if the Christian God existed in the way He is presented in the preaching of Christ, his human historicity would be based on another historicity which would not only understand itself, but would give itself to itself by itself from self and would interact in us and with us through the son, whose kenosis among men is analogous to its kenosis in creation. So the new Adam would no longer be only a living being and, as such, subject to death; rather, his life is permeated by the source of all life and is therefore immortal. He is therefore a life-giving spirit (1 Cor. 15, 45 ff.). In this context, what is first and original is no longer to be regarded as the highest. [21] 31 30 31

p. 185. p. 498.

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As recognized in Pauline anthropology, according to which the first human being was created earthly and mortal, the second and final human being, on the other hand, heavenly and immortal – as he has emerged in the resurrection of Jesus [. . . ] only Jesus Christ, the eschatological human being, is image of God (2 Cor. 4.4), [21]

if Christ dies, life is death.

6.3 What Overcoming Nihilism Is In the truth, emeth, which gives stability in the relationships between man and God, God and man bet on each other. It is the transcripts “in Heaven” of the bet that has always linked man to the world: I have confidence in the world because the world – manifestly and manifestatively – has confidence in me, trusting me and conceding that I am. Also when the world is thought of, as per Nietzsche, as nature which becomes nature, as a continuous self-interpretation of its “being” as “will to power,” this eternal becoming does not diminish the stability of the mutual wager which links the ego and the world. It gives it, on the other hand, its actual stability, which is nothing more than the setting in place of all levels of interpretation of becoming on the level below, which for the level above has the consistency – precisely as a fact, i.e., the result of an interpretation immanent to being as physis – of a given, literally, of a “given fact”: the unquestionable result as the starting point for each successive interpretation. In Nietzsche’s universe of interpretation, in the end, it is physis which was brought about and is brought about in the legality of its immanent self-interpretation, working as a “foundation,” unavoidable consistence “by law” of its “fact” for the “hermeneutic” reality of the life that is measured there. It is only a merely anthropological interpretation of Nietzsche’s will to power as being, in line with the metaphysics of modern subjectivity, not without guilt on the part of Nietzsche himself, which would lead to thinking that the abused expression, “there are no facts, there are only interpretations,” is the herald of nihilism even in epistemology. Philosophically an unsustainable foolery, that adds nothing of nihilism in the true sense to what is already there already of nothingness-being in the great game of the world; and our solitude in it. Only those who are alone can depend. And since only we are alone in nature or in creation, we have to decide whether we want also to depend on what we depend on anyway, on what hung and hangs from, the world, a ball thrown into the universe, a “well-rounded truth” which cools down, or the Cross. Only if we succeed in wanting this, nihilism – when presented to us and it has made us philosophers or desperate – is it “taken away,” “overcome”; nihilism, our solitude no longer the abyss, but the haversack strapped to our shoulders on our journey. And this, without going too far, is the immediate work of life which bends or breaks over its work, or – when “nothingness” has been known to it – the mediated work of reason as submission to exteriority, to the nonfact and nondoable of the “world” which it does. Or to the “done-by-itself” (unattainable for us) of a God who makes us and continually wants us. All the rest is secondary literature.

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References 1. M. Heidegger: Was ist Metaphysik?. In: Wegmarken, Gesamtausgabe Band IX (Vittorio Klostermann, Frankfurt a. M. 1996) 2. H. Bergson: Creative Evolution, eng. tr. by A. Mitchell (The Modern Library, New York 1944) 3. J.-P. Sartre: Being and nothingness, eng. tr. by Hazel E. Barnes (Washington Square Press, New York 1966) 4. R.D. McKirahan Jr.: Philosophy Before Socrates (Hackett Pubblishing Company, Indianapolis/ Cambridge 1994) 5. Plato: The Sophist & The Statesman, ed. by R. Klibansky and E. Anscombe (Thomas Nelson and Sons LTD, London 1961) 6. M. Heidegger: Wissenschaft und Besinnung. In: Vorträge und Aufsätze (Günter Neske Vr., Stuttgart 2000) 7. St. Augustines’s Confessions, eng. tr. by W. Watts (Heinemann LTD, London 1979) 8. Plotinus: Enneads, eng. tr. by A.H. Armstrong (Heinemann LTD, London 1980) 9. E. Jünger: Sämtliche Werke (Klett-Cotta, Stuttgart 1979) 10. J. Ratzinger: Introduction to Christianity, eng. tr. by J.R. Foster (Herder and Herder, New York 1970) 11. G.W. Leibniz: Principles of Nature and Grace Based on Reason. In: Philosophical Essays, edited and translated by R. Ariew and D. Garber (Hackett Publishing Company, Indianapolis/Cambridge 1989) 12. G.W. Leibniz: Essais de Théodicée. In: Die philosophische Schriften von G.W. Leibniz, hrsg. von C.I. Gerhardt (unveränderter Nachdruck der Ausgabe Berlin 1875–90) (Hildesheim 1965) 13. M. Heidegger: Nietzsche, Gesamtausgabe Band 6.1 (Vittorio Klostermann, Frankfurt a. M. 1996) 14. I. Kant: Critique of pure reason, eng. tr. by P. Guyer and A.W. Wood (Cambridge University Press, Cambridge 1998) 15. F.W.J. Schelling: Philosophie der Offenbarung. In: Schellings Werke, Ergänzungsband VI, hrsg. von M. Schröter (Beck und Oldenbourg, München 1954) 16. W. Kasper: Glaube und Geschichte (Matthias-Grünewald-Verlag, Mainz 1970) 17. Theological Dictionary of the New Testament, translated and edited by G. W. Bromiley (Eerdmans Publishing Company, Grand Rapids-Michigan 1964) 18. E. Benveniste: Le vocabulaire des institutions indo-européennes. 2. Pouvoir, droit, religion (Les éditions de minuit, Paris 1969) 19. E. Mazzarella: Pensare e credere. Tre scritti cristiani (Morcelliana, Brescia 1998) 20. E. Mazzarella: Cristianesimo e storia. In: Millenarismi nella cultura contemporanea, pp. 157–173, a cura di E. Rambaldi (Milano, Franco Angeli 2000) 21. W. Pannenberg: Anthropology in Theological Perspective, eng. tr. by M.J. O’Connell (The Westminster Press, Philadelphia 1985)

Discussion G.O. Longo: My question deals with the matter of the interpretation. It is said that everything is interpretation: on one hand, we experience a primary, obscure and uncertain, contact with reality, but when we try to translate into words what we perceive of it, the interpretation is filtered through our body, our spirit, our mind. Thus, is the interpretation merely one of the linguistic, even mathematical, translation, or does a primary interpretation occur already in the “primordial” contact between us and the world?

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E. Mazzarella: A primary interpretation which pre-exists the cognitive process, an “affective mood”, as Heidegger would say, is at work in every contact with the world, through which we orient ourselves “in the world” from which we emerge, but in which – in the emerging of a distinct perceptive identity linked to an auto-perception – we at the same time remain immersed. Any interpretation, even the most sophisticated, stems from an act of sensitivity which is then verified pragmatically in the confrontation with reality. An act which could even become “neutralized”, should this be convenient to our orientation in the world: which is a procedure linked to conflict. Logic, in essence, is sensitive, and sensitivity, in turn, has its logic. The Principle of Indetermination, according to which the tools of our knowledge, our very knowledge in fact, our body, our mind, would “modify” the “thing” as they get in touch with it, transforming it into something different, is not to be understood as something that makes us “miss” the thing, to lose its truth. It is to be understood as a means to actually get in touch with it, to “grab” it. E. Carafoli: The key point in the “Grundfrage” of Leibniz was the “why.” Asking “why” we are forced to escape into metaphysics. Perhaps, in the vision of the scientific culture, if one would still want to attempt an answer, it would be more appropriate to ask “how come.” I don’t think it is only a matter of semantics. E. Mazzarella: Right, but the two formulations are not mutually exclusive. A fitting example that underlines both the proximity and the distance of the two cultures, without privileging one or the other, would be provided by a very famous verse by Leopardi in the Song of a Migrant Sheperd in Asia (Canto di un Pastore Errante dell’Asia): “What do you do, moon, in the sky, tell me, what do you do?” (Che fai tu luna in ciel, dimmi, che fai?) I have already used this poem, in a slightly different context, to answer a question by G. Setti in the General Discussion a while ago. So, the physicist would have the moon answering “how come” that it is in the sky, “how” is it that the law of gravity prevents it from falling on our heads. The philosopher and the poet, instead, having satisfied themselves with the “how,” with the reasons that keep the moon up in the sky, keep asking “what do you do,” wanting to know “why” the mood does what it does. They talk of the wonder of seeing it up in the sky, in the starry sky. They want to know “for whom” the show is displayed, and what its meaning could be, torn between the aesthetics of the indifference of the universe to our human events, and the help to withstand this indifference that the nocturnal glow of the moon gives us. For science, the wonder “disappears” only in the established notions of the laws of physics and astronomy, which deal with the “how come.” But at the root of this scientific reductionism of the “metaphysical” wonder that the moon is up there, of a “why” of it being up there, still lies the marvelous discovery of the laws that permit it to be there, the wonder of realizing that it stays up there rather than falling down as if it were an apple falling from a tree. A fascination that turns into curiosity, and that tries to exit from the “problem” to find satisfaction in an answer that translates the “why” into the “how come.” L. Boi: Don’t you think that the answer of Leibniz to the question “why is something preferable to nothing” is unsatisfactory from the angle of modern physics and mathematics?

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E. Mazzarella: I think Leibniz in his metaphysical “Grundfrage” tries to preserve the wonder of our being in the world, which questions him, and the question is not obvious. That being is preferable to nothingness ought to be obvious to us, as we are nothing else than an accident with a conscience, an accident that is aware of being one, even of being one which is due to disappear. I do not believe modern physics and mathematics can provide satisfactory answers to the question of Leibniz. They do not answer it, at least I do not think they do. They take for granted what the question asks, i.e. that there is something. G. Setti: To go back to one of the questions, and to your answer to it: it seems to me that the concept that the interpretation is a description of reality (i.e., of its modeling), to which it confers meaning, is what is really meaningful. E. Mazzarella: The meaning of reality provided by the interpretation, seen as a description, as modeling, does not answer the question of the meaning of reality, of the network of meanings in which we are immersed. The philosophical question is fundamentally one of sense.

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G. Setti: Since we have been through a long discussion following the last couple of talks, we should start from the topics of the morning. There may be questions that people have not asked directly or comments that have not been made, so that there is now an opportunity for all this. I would like to start with questions or comments about time, the first topic of this morning. We have heard the point of view of physics as expressed by Gabriele Veneziano, who questioned the often assumed coincidence of the beginning of time with the big bang by presenting a model, based on modern theory, of a pre-big bang universe. Then we have heard Hans Mooij discussing different concepts of time – at some point he mentioned mind and consciousness, and then time from the point of view of a static or a dynamical position – so there were questions after his talk, such as historical time and so on. I thus think it is important to come back to this topic. Are there any questions? R. Durrer: I’d like to put a question to Gabriele Veneziano. What about the arrow of time and the growth of entropy? You haven’t said anything about that. Would the string theory help us understand that too, why entropy grows or, to put it in another way, why the universe starts out in a state of very low entropy? G. Veneziano: So, the question is whether in cosmologies of the type I tried to describe this morning there is the origin of an arrow of time. We all have the feeling, as you know, that there is a direction which is the future and a direction which is the past, and we tend not to reverse it easily. So what determines this arrow of time? It is an old question and it seems that the only possible answer is that, somehow, the universe started in a very ordered state, in what we define in thermodynamics as a low entropy state, and the arrow of time simply means that time progresses in the direction of an increase of entropy, of an increase in disorder. When you break a glass you increase the disorder and it is very hard to reverse this, to recompose your glass from the fragments. This is a question that, for instance, Roger Penrose asked some time ago making conjectures about this by stating, for instance, that the initial singularity, the big bang, was a particularly low entropy situation due to the fact that the curvature of space-time was dominated by the Ricci tensor and not

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by the Weyl tensor. Yes, it’s very technical! We asked this question ourselves in the cosmology in which there is something before the big bang and we would like to start with very generic conditions. So, you would say, how does the arrow come about if you do not start with something which is of very low entropy? Well, the only thing I could say is that there seems to be a possibility not to have an arrow of time developed everywhere throughout space, but perhaps in some particular regions. I illustrated this morning that we see this pre-bang, or pre-bounce era, as something like a gravitational collapse, and this gravitational collapse would not have been everywhere, but would have happened in some particular regions of space which are over-dense, that is, more dense than average. There is a possibility that within the horizons of these collapsing regions you do form a low entropy state, a sufficiently low entropy state. There are some calculations which seem to support this idea, and that would lead to an analogue of a big bang with sufficient entropy to have a really hot start, but low enough to establish an arrow of time. There is another intriguing possibility which I also like: theoretical physicists have recently come up with an idea that there is some upper bound to the entropy, based on what is called holography. Now it is a little bit complicated and it would take a long time to explain exactly what it is, but the bottom line is that in an expanding universe, as it has been since the big bang, this entropy bound on disorder goes up. The upper bound goes up, so entropy would grow because it is allowed to grow by this increasing upper bound on entropy. So the universe would not start in a low entropy state, it would start in a state which has the maximum entropy at that point, but since the bound goes up, the entropy is allowed to grow afterwards, and that would be another way to get to the arrow of time. In any case I would say that in all this I don’t think that the string theory plays a very basic role, for the time being at least, meaning that one can make this last reasoning irrespective of whether you have a string theory or not. Basically, this bound on entropy has to do with black holes. A black hole carries a lot of entropy and is considered to be the most entropic object in the world. Perhaps this is why, as we were discussing during the coffee break, there are these huge black holes in the center of galaxies, the result of a long evolution which has increased the entropy and so we are ending up with these giant black holes that contain a lot of entropy. These bounds on entropy are very much based on what the entropy of the black hole is. For those of you who don’t know, the entropy of a black hole is proportional to the area of the horizon surrounding the black hole and so that’s why it is a holographic bound, because it is proportional to an area rather than to a volume as one would usually think. So that’s all I can say. I wish I could say more but that’s all I know. G. Setti: If one considers all supposed black holes in the universe and, because we roughly know what the mass distribution of those black holes is, one calculates the total amount of entropy, how big is it compared to the entropy of the universal radiation? G. Veneziano: That’s a good question. You know, I never bothered doing the calculation, but it’s a meaningful question. However, a black hole satisfies the second law of thermodynamics in the sense that the entropy always increases because it ac-

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cretes things, its horizon grows and the entropy grows because it is proportional to the horizon area. So, even if the entropy in black holes would still be lower than the entropy of the universe, the fact that they keep forming and accreting matter means that there is still a long way to go to increase the entropy, and therefore our arrow of time might still have a long life time. Unidentified: [A question from an unidentified person off of the microphone. Thus, only the answer has been recorded] G. Veneziano: That is a very good question. Suppose you have two black holes. When they are separated, the entropy is the sum of the two entropies, that is the sum of the two horizons’ areas. Suppose they coalesce, then, roughly, their masses adapt. But in general relativity the size of the black hole is proportional to the mass while the entropy is proportional to the area, and therefore the entropy is proportional to the square of the mass and that’s why the entropy of the huge, combined system is higher than the initial entropy. G. Setti: Let’s leave the black holes for the time being and discuss other topics. Unidentified: Just another paradigm from philosophy, from ancient philosophy actually: instead of entropy, low entropy and high entropy, Empedocles would talk in a poetical manner about love and strife. And he similarly says that at the beginning you have a low entropy state which is this complete roundedness of everything, and then you have an entropy start because strife, which is seen as a personalized driving force, takes over from love and then there is also a limit to this entropy when the pendulum strikes back again and goes to the other position. It seems to me again that, in a way, the ways of thinking are quite similar to what you have presented. G. Veneziano: I am not sure I understood you fully. I accept that the second law of thermodynamics would tell you that you cannot swing back and forth. So, for instance, even in the cosmological model that I was proposing, which has a version that goes through a bounce, then into another bounce, then into another bounce. . . you can say that the universe goes back to itself. That would not be really true. If entropy keeps growing you never go back to where you started. Actually a well known scientist once made a blunder: he said that the growth of entropy went with the expansion of the universe and the entropy will decrease again if the universe re-contracts. But then he took it back as he realized that this was not correct. Let me emphasize something. If we look, for instance, at the gas in a room, the molecules of air spread uniformly and everything is at the same temperature, that would be the highest entropy state, so that is where you would expect to go in the future. However, in the presence of gravity it is not true that the most homogeneous state is the most entropic one – because of the attractive nature of gravity things tend to clump into structures. So this is a very basic difference and this is why black holes, which are clumps of matter, tend to increase the entropy relative to something which is very homogeneous. The structures form in the universe very naturally, you start with very small in-homogeneities, as Ruth Durrer has explained, and then slowly you build more and more, because once you form a dense region, that dense region

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accretes more, so that it becomes even denser and so on and so on. So the tendency is to clump and thanks to this we have structures in the universe, we have galaxies. . . G. Setti: I wonder whether there is life as well. . . G. Veneziano: . . . and there is life also, yes. But precisely in life we have an example of a system which locally makes order, of course not in contradiction with the second law because you throw disorder somewhere else. As explained by Penrose, for instance, we get low entropy photons from the Sun, because they are at high frequencies and quite energetic, and then the Earth radiates back into space low energy photons with a lot of entropy and through this entropy gain we can make order on Earth. So my idea is to play a similar game with the whole universe – we have this big, big thing and you create a region of low entropy in it, and that would be what became our universe. G. Setti: At this point, since we have a mixed audience, I think it would be appropriate to consider other aspects of time. On the one side the time as measured and discussed by physicists – we have watches, we can measure what the intervals of time are and so on, the things we have discussed up to now. On the other hand, as we have heard this morning, there is the awareness of time, the flow of time as we talk about it. In the “two cultures” I think it would be important to try to clarify what that means. I wonder, for instance, whether the perception of time is again something that we might have acquired through the evolution, a clear advantage that human beings, and perhaps other beings too, have acquired by experience and learning. Does anybody wish to comment on this? L. Boi: I would just like to say that in psychophysics people have tried to make an exact measure of the perception of time. But what they measure is not the perception itself, but rather its mental aspect; I mean the physiological process which underlies this perception. But this is totally different from the ontological nature of perception, which is not only the physiological process which underlies this perception. What I mean is that the physiological, the neurological process can be measured in some sense, like in physics, but the perception itself cannot. G. Setti: Thank you for this comment. Perhaps questions about conscious perception of time may come back tomorrow, since there will be several talks dedicated to the brain and its functions. So let me switch to another subject that has been discussed this morning: the “nothingness” problem. I believe that the question of the “why” may be perceived in different ways in the two cultures. The “why” as perceived by people studying nature and wishing to understand the basic mechanisms underlying nothingness, like the universe coming out of nothing as we have heard in a couple of talks this morning. And there is the other “why,” i.e., why we are here and so on, and that brings up questions that have been discussed this morning by another speaker. Is there any question or comment on this topic? E. Mazzarella: I put a question to myself, as I would like to make more intuitive than in the discussion this morning, and maybe in my talk, the different approach to the problem of the two cultures, as if they were the two faces of a sheet. Many of

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you probably know that renowned poem by Giacomo Leopardi, Canto di un pastore errante nell’Asia (Nocturnal poem of a shepherd errant in Asia) that begins as follows “What are you doing, Moon, up in the sky? Tell me, what are you doing?” Let’s suppose that the shepherd is a physicist. He would tackle this question by trying to understand the conditions that prevent the Moon from falling on his head, that allows it to move around and so on. Later on the same shepherd, although not yet clear about the how and about what the moon is doing up there, even if he still has no answers, begins to ask himself questions such as “Why do you exist?”, thus moving from physics to philosophy. The question’s angle underlines the passage not only from physics to philosophy, but to poetry and to various other things as well. However, these two aspects of the way in which one addresses the question of the how and the why actually mutually enrich each other because, without the curiosity and the initial astonishment in the question “What are you doing?” addressed to the Moon, one wouldn’t even attempt to come out of the wonder into the safe ground of the “how.” But when the refuge into the safe answer to the question of “how” fails, as for instance when a physical law that meets an exception needs to be reformulated, or if an accident in life shakes strongly held beliefs, then the wonder comes out again. This, I believe, underlines the dialectic of the two cultures. G. Setti: Let’s now move on to another subject that came in the morning, that is the matter content of the universe. If one goes back to history, as Ruth Durrer has pointed out, the Earth was believed to be at the center of the universe, but then Copernicus came. And so the center of the universe has been gradually displaced from the Earth to somewhere else, to find that there is no center of the universe, and that the Earth is far from the center of the Milky Way, which is just a normal galaxy, one of the many billions of galaxies populating the universe. Now, more than 450 years after the Copernican revolution, and four centuries since Galileo’s observations, we have discovered that the normal matter of which we and the stars are composed is just a tiny 4% of the total matter-energy of the universe. In a way the Copernican revolution is approaching completion. Perhaps we may say that the normal matter is the noble part, as the nobles have always been but few, or we might think that, after all, we are only a negligible part in the fabric of the universe. In any case, I feel these findings astonishing, and worth further consideration by the two cultures. G. Berlucchi: If you place the emphasis on energy, obviously it is a very small quantity. But if you think of mental powers, it has been shown for many years that they are conducted by the brain, of course at the expense of an amount of energy, but by an amount which is not very different while one is asleep or doing nothing. In other words, energy is important but information is even more important. And information, as told by the brain, does not correspond to the energy. The total energy spent by the brain while one is resting, completely resting, or while one is asleep is more or less the same as one spends while thinking, and walking, and willing and doing all these things. Thus, energy may not be that important, although you need it for all psychological processes.

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G. Setti: Thank you. Yes, it’s not only a matter of quantity, but of quality. . . the noble part in the metaphor I have made before. However, let me just remind you that according to our present knowledge the normal matter is about 17% of the dark matter and only about 5% of the dark energy’ equivalent mass. Ruth, what’s your view on all this? R. Durrer: Maybe this is the reason why I prefer the creationist myth of the Chinese, where the human beings are the flea in the hair of the creator, than the pseudoChristian myth where the human beings are created according to the creator. I think that the universe makes us more and more modest even though, I agree, it’s of course an enormous miracle that we can reflect about ourselves and that we can reflect about our role in this world. That is probably the biggest miracle, and one which I don’t think science can still address – whether for instance there is any limit, any fundamental limit, to our understanding of the universe. I don’t know whether there is any, or whether there is none. It is true that there are things other than energy, maybe information, or you might call it entropy, which might be very important and the amount of information the human brain can store is amazing. I’m still surprised by this – by comparing, for example, human chess players competing with a computer which can calculate so much faster than we can, that can do a calculation, which took Kepler many years to complete, in a split second. Still, there are chess players who may not beat the best computers any more, but do play roughly as good as the best computers, so. . . I find this very surprising, our mind is capable of amazing things. G. Setti: But this opens up the question of artificial intelligence or goes very close to it, right? I would also agree that we have perhaps a little more power than an automaton, but, you know, this is an open question on which not everybody agrees. But we have a question on this. . . M. Du Sautoy: It’s not so much a question, but I think that the point you have raised is very interesting to mathematics as well. The limit of what we can know when you look at something like the Riemann hypothesis and primes. I think that Hardy and Littlewood gave up and said it’s probably false because we just can’t understand it. You work on a problem for so long, and sometimes you think that it’s just not going to yield, your mind just isn’t going to capture it. But the amazing thing is how often, as you say, we can find ways. . . Take the proof of Fermat’s last theorem, for example, which just seemed so beyond anything that we can conceive. But of course the 20th century has given us mathematics to show that there are limitations to what we can do. For example, Hilbert’s 10th problem which says there is not going to be an algorithm which will tell you, given an equation, whether it has a solution or not, which is kind of an example. Gödel’s incompleteness theory as well, which says that within any axiomatic system there will be things, which are true within first order logic, which are not provable within that system. But then again, it’s rather curious that a mathematical mind can articulate what we can’t understand and what will be out of our limits. It’s just quite striking. G. Setti: Other comments?

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L. Boi: A question for Ruth Durrer and Gabriele Veneziano. Do you think that this new cosmological scenario in which normal matter is only 4% of the matter-energy in the universe can have some effect on the level of the human beings? I mean, this is a problem of scale. There is this macroscopic scale, the cosmological scale: in which way can it interact, have effects at the microscopic level of the human beings? There is also the problem of the relationship between physics and biology. G. Veneziano: I don’t know. At first sight I would say that there is no direct implication except that, as Ruth has explained, in the absence of the dark matter the evolution of the universe would have been much different: not the same objects would form, not the same stars, so life would not have been possible. On the other hand, if dark matter is there, and it seems to be there, it is not very strongly interactive with us anyway. So I don’t think it can really affect. . . I mean biology will not change as a result of the fact that there is dark matter, let’s put it this way, but the evolution of the universe will. That is very sensitive. And the future evolution of the universe will depend very much on this dark energy or cosmological constant, or whatever, because that will tell us whether the universe will expand forever accelerating, separating out everything, or whether it will possibly re-collapse, or do things like this. But on a local basis of physics I don’t think that there is a direct influence. Fortunately physics is there for every given problem. For every given scale one discusses which the relevant and irrelevant things are, because if you have to take everything into account for every single problem you would be really in an impossible situation. But in the same way I don’t think that physics or biology would care whether the electron is a string or whether it is a point particle. In a sense, yes, I think that the dark matter is not relevant to local physics or chemistry or biology. This is my point of view. R. Durrer: I of course agree with what Gabriele has said, but I have maybe just something to add. If dark matter is made of elementary particles, as we think, then we hope of course to be able to discover it with local experiments. By building some detectors we hope to detect it, eventually. With these xenon detectors and things which are now being built, once they are sensitive enough, if dark matter is for example the neutralino, a particle once predicted to be the dark matter, then we hope to be able to discover it within, say, the next ten years or a bit more. So we hope that there will be small-scale experiments which should see these particles. The dark energy is much more complicated. I would guess that there is no way to detect it on the small scale. There has actually been a paper in which the authors wanted to detect the cosmological constant locally, but I think it was completely wrong. I mean, we know that we can measure vacuum energies by the Casimir forces mentioned this morning, but that has nothing to do with the cosmological constant. G. Setti: I would say that on the human scale we can expand, but for other reasons. For instance, tonight at dinner we can clearly try to expand. . . but not because of the cosmological constant! Without joking, however, let me add something to the previous comment. I believe that people normally don’t realize that our bodies are continuously penetrated by a host of particles, cosmic rays and neutrinos – we don’t

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see them but in a way they are familiar, part of our matter. But now we should realize that we are probably plunged in a bath of dark matter particles, too. There are probably millions of them per cubic meter all around us and through our bodies, the precise number depending on their mass, which is not yet known. We don’t see them either, and we cannot interact with them. . . they are of a different nature. This doesn’t add to our knowledge, but I feel that in a meeting on the two cultures this is something that should be perceived. It’s time to close this session which ends today’s programme. Thank you very much!

Section Intelligence and Emotions

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Original Knowledge and the Two Cultures

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Giorgio Vallortigara

8.1 Introduction A common, crucial theme for the two cultures is the issue of the origin of knowledge. In this paper I shall argue, on the basis of evidence from comparative cognitive neuroscience, for widespread common mechanisms among vertebrates underlying basic cognitive processes, and I shall also argue that these mechanisms are largely available at birth, little (if at all) affected by past experience. These are in no way novel ideas. They belong to the tradition initiated by Socrates (more precisely by Plato’s account of Socrates’ thinking) and have their most inspiring source in today’s research in developmental sciences (see in particular [1, 2, 3]). Nonetheless, I hope the type of evidence that I am providing is novel, or at least “personal”. Speaking of the theme of the “two cultures”, I was impressed years ago by a short paper by Nicholas Humphrey [4] on the occasion of the 300th anniversary of Newton’s Principia Mathematica.In contrast to C.P. Snow who, in The Two Cultures, labeled great scientists as “scientific Shakespeare”, Humphrey stressed that the individual persons that make science are replaceable: without the person Newton, sooner or later, the same scientific discovery would be made by someone else, whereas without the person of Shakespeare nothing similar to his specific creation would have appeared. For ultimately Newton (and science in general) uncover pre-existing truths in nature, whereas the same cannot be said for artists and humanities in general. In Nicholas Humphrey’s words: “There are no pre-existing novels out there waiting to be written, nor pre-existing pictures waiting to be painted”. I pondered about Humphrey’s beautiful writing over the years, and I am now uncertain whether I would entirely agree with his position. If Borges is right when arguing that universal history is made of a limited set of metaphors, then there is a sense in which novels do pre-exist to their writing and pictures to their painting, for novelists and artists can only play around on a limited set of very constrained materials. And, very likely, the constraints are imposed on our brains by natural

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selection. Moreover, and more important in the present context, I gained over the years a strong feeling that the “personal touch” is in fact not unique to humanities, but extends to scientific work as well. Thus, no doubt the discovery of an important phenomena such as “blindsight” in neuroscience [5] would have occurred anyway by someone, sooner or later. But the fact remains that Humphrey’s original version of it was quite peculiar and highly personal [6, 7]. I think this is common experience among scientists: when we read a paper, with a beautiful experiment or an ingenious hypothesis, we say to ourselves “Oh. . . that must be Nick’s beautiful work! Nobody else’s!”. In a similar vein, I hope the readers of this chapter will comment on it saying that, yes, that’s an old story, and we already knew very well about all these ideas, though, admittedly, not in this particular version, not with these creatures and these sorts of experiments. . .

8.2 The Chick and the Baby The protagonist of this chapter (Fig. 8.1) is not traditionally considered a champion of mental life, bird brain being in several languages the epitome for poor intellect. Actually, there are good reasons, I believe, to elect such a humble creature as a model for investigation on the nature and origin of knowledge. Recent research in developmental sciences has put forward the so-called “core knowledge” hypothesis, i.e., the idea that human cognition would be composed of a set of core systems – a tool-kit of cognitive mechanisms – for representing significant aspects of the environment [8]. Core knowledge systems would be mental “modules”, which are in place at birth, for building mental representations of objects, persons, spatial relationships, and number. According to the hypothesis, core systems would be shared by other animals. However, apart perhaps from number representation, little evidence is available for this hypothesis on the comparative side. Moreover, even the hypothesis that core knowledge systems are available at birth is difficult to prove in human infants, for the human species is an altricial species, and human newborns are very immature at birth. Developmental studies on human newborns thus suffer from the unavoidable limitation of the lack of any precise control on early experiences. To use an example: Are mechanisms that preferentially orient the attention of neonates to human faces inborn? It is virtually impossible to provide a complete absence of face stimulation even in neonates studied a few hours after birth. Similar limitations are also present with altricial animals. For example, Sugita [9] recently reported that infant Japanese monkeys reared with no exposure to any faces for 6–24 months nevertheless showed a preference for human and monkey faces. However, monkeys still had a few hours of experience with their mother after birth, making any strong claim for an experience-independent ability for face processing untenable. Besides, it could be argued that monkeys maintained for several months without exposure to faces would show an abnormal pattern of psychological development. On the other hand, testing monkeys soon after birth would prove impractical because of their immaturity. Luckily, enough nature seems to have provided biologists with an

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Fig. 8.1 The newly-hatched domestic chick. A highly precocial species providing ideal material for investigations on the origins of knowledge

appropriate system model organism for every sort of scientific problem, a “God’s organism” as it has been dubbed by neuroscientist Steven Rose [10]. There are species, such as the domestic chick, which are extremely precocial with regard to their pattern of motor development, making possible quite sophisticated behavioral analyses at an early age (i.e., soon after hatching) combined with a very precise control on the effects of past (even in ovo) sensory experience, including a complete lack of it. The domestic chick has been largely used as a model system in neurobiology and in the study of early learning, and recently it has become a focus of interest and insight with respect to several classical issues in developmental psychology in my lab, taking advantage in particular of the behavioral techniques made available by filial imprinting (for a review see [11]). I imagine you are familiar with imprinting: it is the learning process by which the young of some animals learn the characteristics of an object – usually a social partner – by simply being exposed to it soon after hatching [12]. Most of what I am going to describe can be in fact characterized as a way to “abuse” imprinting. I will start with some old work I did on a very basic mental ability of humans – that is, the ability to recognize partly occluded objects. Usually, when we see an object which is partly occluded by another, opaque object, we have a strong, compelling impression that the object is continuing in some way behind and beyond the obstacle. Sometimes people tend to dismiss this ability, saying it is just a matter of past experience, or memory. However, there are several nice examples in which you

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Fig. 8.2 An elongated car. Amodal completion exerts its action even in contrast to past experience (see text for a discussion). (After an advertisement of Volkswagen car)

can see that amodal completion, as this phenomenon has been dubbed, could act in contrast to past experience and previous knowledge. Figure 8.2 shows an example: quite an elongated car, which is not very familiar in the environment. Clearly, however, the argument would be much more compelling if we can prove that amodal completion also occurs in absence of past experience of occlusive events. Some years ago we investigated this issue using imprinting as a tool [13]. We imprinted newly-hatched chicks on an artificial object – a cardboard triangle – and then we tested chicks for choice between two different versions of this “mother”, a partly occluded version and an amputated version (as shown in Fig. 8.3). Consider the two stimuli from an anthropomorphic point of view, that is from the point of view of our own perceptual experience. In spite of the fact that the overall

Fig. 8.3 Amodal completion in the newly-hatched chicks. Following imprinting on a complete triangle chicks prefer to associate with the partly occluded rather than with the amputated triangle

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red and black areas are exactly the same in the two stimuli, the two patterns look very different. The one on the left looks like a complete triangle, which is simply by accident partly occluded by a bar. The other is something different: an amputated triangle or, if you favor a more analytical description, a small, irregular triangle plus another geometric figure. It turns out that chicks behave like us in this task – they prefer to approach the partly occluded object. However, several control experiments were needed before we could safely say that chicks really do perceptually “complete” partly occluded objects. I would like to describe just one of these control experiments, which also offers us material to discuss amodal completion phenomena in the newborns of our own species. Some colleagues working in developmental psychology, Stephen Lea, Catriona Ryan, and Allan Slater at Exeter University, UK, were interested in duplicating our results with exactly the same stimuli and procedure that are usually used with human infants. To study amodal completion human infants are presented with a rod moving back and forth, partly occluded by a bar (Fig. 8.4, top), until they are habituated to the stimulus. Then the occluder is removed and they are shown a complete rod (Fig. 8.4, bottom left) or the two pieces of rod that were visible during habituation (Fig. 8.4, bottom right). The idea is that if human newborns do perceive completion in this case, they will not be very surprised to look at the complete-rod stimulus but they will be surprised to look at the two-pieces stimulus – or vice versa if they do not perceive completion. It turns out that human infants need to be about four months of age to exhibit amodal completion.

Fig. 8.4 Habituation-dishabituation technique to investigate amodal completion in the human neonates. Babies are visually habituated to a rod moving back and forth behind an occluder. Then dishabituation is tested following the occluder removal, with the babies presented with either the complete rod or with only the two visible end-parts

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On their first day of life, different groups of chicks were imprinted (i) on a moving rod or (ii) on a partly occluded rod, or (iii) on a two-piece rod [14]. They were then tested for choice between a complete and an amputated rod. As expected, chicks imprinted on the complete preferred at test the complete over the two-piece rod, whereas chicks imprinted on the two-piece rod preferred the two-piece rod to the complete rod. However, in the crucial condition, when imprinted on the partly occluded rod, chicks preferred the complete rod over the two-piece rod, in spite of the fact that, physically, there was exactly the same red area visible in the partly occluded rod as in the two-piece rod (see Fig. 8.5). Chicks are a precocial species, whereas human babies are the most altricial of the species. Human babies do not walk soon after birth and thus they would find of little use possession of amodal completion in everyday behavior. Therefore it could perhaps be argued that a basic difference may exist between the two species, with chicks equipped from the start with the mechanisms for completing partly occluded objects and humans who, in contrast, would learn about amodal completion in a period of about 4 months, taking advantage of experience with occluding events in the world. Recently, however, evidence has been provided for a rationalistic interpretation (i.e. in terms of innate behaviors) of human babies’ data. It seems that human newborns have problems with recognizing motion and that when stroboscopic motion is used instead of continuous motion (the former being processed early in development by sub-cortical structures in the brain, when cortical mechanisms are not yet mature), then neonates of only a few hours of life can show evidence of amodal completion [15]. Thus, maturation of certain other areas of the brain seems to be needed in altricial species in order to exhibit mental competences in behavior that are already available from the start, i.e., competences that are innately predisposed in the brain.

Fig. 8.5 Results of an imprinting test that emulates in the newborn chicks the conditions used to study amodal completion in the human neonates. Chicks were imprinted on either a complete, an occluded, or a two-piece rod and then tested for choice between a complete and a two-piece rod. In the crucial condition, chicks imprinted on the partly occluded rod preferred at test to associate with the complete rather than with the amputated (two-piece) rod (re-drawn after [14])

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8.3 Animated and Non-Animated Objects Recently I became interested in the issue of whether young animals possess mechanisms for recognition of other animated creatures. Consider the pattern shown in Fig. 8.6. This was obtained by simply strategically locating some points of light in correspondence of the joints of a human walking and videorecording it in the dark. It is very difficult to figure out what sort of stimulus this is when you look at a single photogram. But when the stimulus is put into motion you can immediately perceive a human figure, a walking man [16]. This sort of biological motion stimuli has been used recently with chicks in our lab ([17]; see Fig. 8.7). We found that before any imprinting had occurred, chicks preferred the motion of a hen to rigid, scrambled or random motion of the same number of points of light. Interestingly, however, the preference was not specific of the movement of a hen, for chicks were perfectly happy in approaching a cat, i.e. a potential predator. It seems that the “life detector mechanism” (as it has been dubbed, see [18]) is of a quite general type, i.e., that it is the general properties of semi-rigid motion that are encoded in the brain.

Fig. 8.6 Biological motion stimulus

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Fig. 8.7 A newly-hatched chick observing a point-light moving hen

Such an encoding seems also to include a gravity bias. The evidence for this arises from the so-called “inversion effect” in biological motion. The left-hand side of Fig. 8.8 shows the point-light display of a hen, whereas the right-hand side shows the same stimulus upside down. People find extremely difficult to recognize a hen when the upside down stimulus is set into motion. (Of course, the effect occurs for any type of animated object, not just a hen.) We found that, other than having a preference to approach biological motion stimuli, chicks also tend to align their body in the same direction as the apparent movement of the hen. For instance, if during the first four minutes of the test the hen is moving in one direction, the chicks orient their body in the same direction of movement of the hen. And when starting from, say, minute 5, the direction of motion of the hen is reversed, chicks reverse their orientation accordingly, in such a way to remain oriented with their body with the apparent direction of movement of the hen. However, when presented with an upside-down version of the hen, chicks show completely random orientation of their body with respect to the apparent direction of the moving hen. So they do apparently experience the inversion effect, as humans do [19].

Fig. 8.8 A point-light moving hen shown upright (on the left) and upside down (on the right). Chicks showed difficulty to recognize the hen when presented upside down

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Fig. 8.9 Marcel Duchamp’s “Rotoreliefs” (1935) © by SIAE 2009

Note, however, that the results obtained with these point-light displays provide evidence that chicks respond to the kinematics of the stimuli but do not provide evidence that they would perceive the point-light displays in the same ways as we did, i.e., as three-dimensional objects. An example of our ability to extract three-dimensional information from twodimensional displays is clearly shown in the so-called stereokinetic phenomenon. This is also a nice case of convergence of arts and science, because stereokinetic phenomena were discovered independently by the visual artist Marcel Duchamp, in 1930, and more or less in the same years, by Vittorio Benussi, the founder of the tradition of experimental perceptual psychology in North-east Italy. Figure 8.9 shows the Rotoreliefs painted by Marcel Duchamp. If you look at these stimuli when they are set into slow motion in the frontoparallel plane, even better if you cover one eye in such a way as to avoid the use of any stereoscopic cue to depth, after a few seconds of visual inspection you can perceive them as 3D objects tilting in three-dimensional space. With the chicks we used a very simple version of the stereokinetic stimuli, the one originally devised by Benussi, as shown in Fig. 8.10. When set into slow ro-

Fig. 8.10 Vittorio Benussi’s stereokinetic stimuli

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Fig. 8.11 The set up for imprinting chicks using the stereokinetic stimuli

tation, after a while you can see in Fig. 8.10 (top) a cone, with an end point either towards the observer or farther away (the stimulus is multistable). The image shown in Fig. 8.10 at the bottom, on the other hand, produces after some inspection the impression of a cylinder when set into slow motion. We exposed chicks to this very strange type of “mother” [20]. Soon after hatching, chicks were presented with these moving stimuli for just 90 minutes (as shown in Fig. 8.11). The idea being that if they do perceive something solid, threedimensional, after looking at this 2D stimulus, they would probably prefer to associate with a real 3D cone rather than a real 3D cylinder, or vice versa. Chicks were tested for choice in the apparatus schematically shown in Fig. 8.12. The results showed that, when imprinted on the cone, chicks preferred the cone to the cylinder and when imprinted on the apparent cylinder – the stereokinetic one – they preferred the cylinder to the cone. (Note that we were careful to check that no information was available on the 2D frontal plane to perform the discrimination between the two stimuli.) You should be probably not terribly surprised by these findings, because they seem to be examples of perceptual rather than truly cognitive abilities. So let us now turn to something more complicated. Sometimes objects are not simply partly occluded by other objects; they may be completely hidden by other objects. Yet, we usually maintain this compelling impression that an object that is no longer available to direct sensory experience has nonetheless not gone out of existence. Such an object permanence concept has been widely studied by developmental psychologists, and it has been shown that it takes some time for young infants to be able to exhibit the object permanence concept. Is this so because of the fact that maturation of the nervous system occurs (with different timing in different species) or because learning occurs? We studied the problem with an imprinting procedure in the newly-hatched chicks. A schematic representation of the testing set up is shown in Fig. 8.13. The chick is located in a sort of a maze, with a window with a grid through which it can look and see its “mother” (an imprinting red ball). In order to rejoin the mother

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Fig. 8.12 The set up for testing chicks’ preferences for 3D objects of different shapes following imprinting on 2D stereokinetic stimuli

Fig. 8.13 The chick facing a detour problem that requires maintenance of an object representation

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the chick should perform a detour, moving back, entering one or other of the two apertures along the runway, and then turning right or left to arrive in one of the four compartments labeled A, B, C, and D. Obviously, we knew that chicks are able to learn a detour task, but our question was different, i.e., what would happen at the very first presentation, when the chick is faced with the problem for the first time? If out of sight really means out of mind, then turning to the right or to the left after entering one or other of the two apertures along the corridor would be exactly the same. However, if chicks have some sort of representation of the disappeared object, together with an idea of its spatial location, we can expect it to turn in to the D or C compartments rather than to the B or A compartments. This is exactly what we observed [21]. One problem with this test is that the amount of time during which the ball, the imprinting object, is no longer visible is very short, and it is out of control by the experimenter (it merely depends on chicks moving). I shall describe a better experimental condition that we developed more recently in a moment. But before that I would like to discuss some evidence for object permanence in another species of birds. We have a group of European jays in the lab, and we tested them with the experimental set up shown in Fig. 8.14. We used and adapted for birds a battery of tests for object permanence that was developed for human infants. We found that jays were able to reach Stage 6 in object permanence tasks, with performances that fully paralleled those of our species at the top end of cognitive development. For instance, jays showed the ability to deal with the so-called invisible displacements tasks. A mealworm was placed into a cylinder and then the mealworm was displaced, invisibly, from the cylinder behind a towel. The bird could then look at the cylinder, noticing that is was empty, and then it searched behind the towel. (Videorecordings of the behavior of these birds can be seen as supporting material in [22].)

Fig. 8.14 The experimental set up to investigate object permanence in jays (after [22])

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What sorts of brain mechanisms are available to animals to perform object permanence tasks? We know that in our species the prefrontal cortex is very important in this regard. Birds have a non-laminated cortex, in which a (presumed) equivalent of the prefrontal cortex, the NCL or nidopallium caudolaterale, has been identified. Note that NCL is not a homologous structure (in fact, it is located posteriorly in the brain), but is likely the result of convergent evolution. We did some lesion experiments implicating this area. The chick was confined within a transparent partition, and a “mother”, an imprinting object, disappeared behind one or other of two identical screens. Then an opaque partition was placed in front of the cage and there was a delay the duration of which could be precisely controlled by the experimenter. After the delay, the chick was allowed to search for the “mother”. In each trial the chick had to remember behind which screen the “mother” had disappeared, then it had to erase the memory and look for the subsequent trial in which the mother was hidden in the same or in a different screen. Thus, the task qualified as a sort of working memory task. Chicks behaved quite well with delays of up to 60 seconds. When chicks were tested after NCL lesion, if the delay was zero – that is, if they were allowed to search for the “mother” immediately after its disappearance – there was no effect at all. But a 10-second delay was enough to observe the animal showing completely random behavior in searching for the “mother”. Thus, the ability to keep in existence the disappeared object depended by the integrity of NCL.

8.4 Number Now, I would like to move from object cognition to another mental ability which is encoded precociously in the vertebrate brain and is part of the core domain of knowledge, number. To investigate the ability of young chicks to use number we used again filial imprinting as an experimental tool. We imprinted chicks on three or one imprinting objects and then we tested them for choice between three and one imprinting objects, using both an absolute or a relative discrimination – in the latter case, the overall number of objects was exactly the same (Fig. 8.15). Chicks were very good at the task. When they were reared with three objects, they preferred three objects at test; when they were reared with one object, they still preferred three objects at test. Apparently they go for more. When tested with two versus three, a more difficult task, the results were similar: reared with three, they preferred three; reared with two, they preferred three again. The choice was clearly related to imprinting, because spontaneous choice in a group of chicks tested without any imprinting did not yield any preference for the larger number of objects. However, it was quite clear that chicks were unlikely to use number in this task, they were probably estimating the overall amount of “stuff” – e.g., the volume or the overall area or something like that. Interestingly, this has been shown to be true also for human infants in some circumstances, for when the number of stimulus elements is contrasted with their overall area or contour length

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Fig. 8.15 Chicks approaching imprinting objects of different numerosities (see text for explanation)

infants sometimes preferentially relied on the continuous physical extent [23]. However, this is not always the rule, for there seems to be circumstances in which infants rely on numerosity disregarding the continuous physical extent [24]. In general, it seems that when objects have similar properties or are from a domain in which physical extent can be supposed to be particularly important (e.g., food, see [23]), then infants seem to favor extent over number. When, however, a task requires reaching for individual objects [25] or objects contrasting in color, pattern, or texture [24] then infants seem to respond to number and not to extent. To check whether something similar occurs even in animals we used stimuli such as those shown in Fig. 8.16 (top), in which different shape, color, area and volume can be obtained for each individual object (taken from a communication by Rugani, Regolin, and Vallortigara). Chicks were imprinted on three objects or on two objects. Then they were tested with completely different novel objects – different in size, color, and shape with respect to the exposure imprinting phase (Fig. 8.16 bottom), and in this case either the volume or the overall surface area was computed in such a way to be the same in the three-element and in the two-element stimulus display. We found that in this case chicks chose the familiar numerosity. If reared with three objects, they chose three objects; if reared with two objects, they chose two objects. So this time they were really using number. These findings were confirmed using conditioning experiments [26]. We employed a traditional simultaneous learning experiment, but with an unusual procedure. We trained chicks to discriminate between two and three small identical dots. Obviously, in this case there were several differences between the two stimuli, due to the continuous physical variables that co-varied with number, i.e. differences in area, overall contour length, spatial position and so on. We thus checked for this by systematically changing or equalizing these variables. Chicks proved able to maintain the two vs. three elements discrimination when they were tested with a change in spatial position of the stimuli and when the overall surface and the overall contour length was made identical in the two stimuli. However, when chicks were tested with a four vs. five elements task, they failed to discriminate. Interestingly, they proved capable of performing the task providing it was turned into a perceptual rather than a numerical task, for instance if the dots were arranged to form a square-shaped (four dots) vs. a pentagonal-shaped (five dots) figure. These findings thus add to a host of evidence that has been accumulating in these last years for the existence of a primitive, pre-linguistic numerical representation

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Fig. 8.16 After imprinting on objects of different aspect and numerosities, chicks were tested for choice between novel objects of different numbers but similar continuous physical properties such as volume and area (see text for details)

system, that allows neonates and non-human animals to count precisely up to a set limit of about three-four elements [27].

8.5 Space Animals show very sophisticated knowledge of their spatial layout, with some rudimentary understanding of geometric properties [28]. A simple example is shown in Fig. 8.17. Imagine that you are located in a rectangular cage, completely uniform, with identically colored walls and in the absence of any intra- or extra-cage cues. In a corner there is something very interesting to you. Then you are displaced from

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Fig. 8.17 Schematic representation of the geometrical information available in a rectangular-shaped environment. The target (filled dot) stands in the same geometric relations to the shape of the environment as its rotational equivalent (empty dot). Metric information (i.e., distinction between a short and a long wall) together with sense (i.e., distinction between left and right) suffices to distinguish between locations A–C and locations B–D, but not to distinguish between A and C (or between B and D)

the room and, with your eyes closed, turned around passively a few times. Finally, you are re-introduced in the room. The goal object is not longer visible. Your task is to determine the corner where the goal object was previously located. Apparently, there is no solution for the problem and perhaps you can predict random choices, 25% of search associated with each corner. However, if you ponder over the problem a little bit you may realize that there is a possibility for a partial – a partial not a complete one – disambiguation of the task. This is because you can make use of the geometric shape of the enclosure – a rectangular shape, not a square. For instance, when you are facing corner A, on the right there is a short wall and on the left a long wall. And there is only another location that stands in the same geometric relationships with respect to the shape of the room, namely corner C, its rotational equivalent. Thus, corners A and C cannot be distinguished from each other, they are geometrically equivalent, but they can be distinguished from corners B and D. In order to perform the task two very simply forms of geometric knowledge are needed, the ability to discriminate metric properties of surfaces as surfaces (short wall vs. long wall) and the ability to discriminate sense (left vs. right). Several species have been proved able to solve these geometrical tasks. For instance, we demonstrated this in both domestic chicks [29] and fish (redtail splitfins, Xenotoca eiseni, [30]). Fascinating enough for those who believe that geometry is not learned by experience but, as followers of Socrates and Plato, incline to the idea that this knowledge is inborn, we recently found that little, if any, experience is needed for young chicks to deal with geometry [31]. We tested the navigational abilities of newborn domestic chicks reared in cages of different shapes and metrical properties (Fig. 8.18), without noticing any difference in their ability to learn to deal with a task such as the one shown in Fig. 8.17. In a particularly revealing experiment, chicks were trained in a rectangularshaped enclosure with panels at the corners providing salient featural cues (as shown in Fig. 8.19). Rectangular-reared and circular-reared chicks proved identically able to learn the task. When tested after removal of the featural cues, both rectangular-

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Fig. 8.18 Chicks reared soon after hatching into environments of different shape, even with little or no explicit metric information provided by surfaces of different extent and by right angles, proved to be capable nonetheless to learn about geometry as well as animals accustomed to these environmental geometric properties (from [31])

Fig. 8.19 After training in a rectangular environment with conspicuous landmarks at the corners (top), chicks proved able to re-orient using the shape of the environment after removal of the panels (bottom), irrespective of their being reared in a circular or rectangular environment. The data in the bottom figure represent percentages of choices for the four corners; group means (in bold) with standard errors (below) are shown (from [31])

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reared and circular-reared chicks showed evidence that they had spontaneously encoded geometric information. These results strongly suggest that effective use of geometric information for spatial reorientation does not require experience in environments with right angles and metrically-distinct surfaces. I suspect that the foundation of natural geometry, at least in its most basic aspects, is far removed from any strictly linguistic and cultural constraint (see also [32]) and deeply rooted in phylogenetic history.

8.6 Conclusions Evidence from developmental and comparative cognitive neuroscience seems to suggest that the vertebrate brain is equipped from the start with a sort of tool-kit of very basic cognitive abilities to deal with physical and social objects as enumerable entities that persist in space and in time. Apparently, cognition of objects, number, and space requires very little experience and interaction with the environment, being available at birth in newly-hatched chicks as well as (I believe) in human newborns. This is, of course, not to deny the powerful effects that learning in a social and cultural environment may exert on initial knowledge. But the point that I would like to stress is that the knowledge that is available to biological organisms at birth is in a sense the necessary knowledge – the sine qua non for learning and subsequent growth of knowledge. On the other hand, the basic similarity of core knowledge of objects, number, space (not to mention time, which was not discussed here) between humans and other animals should not make us oblivious of the fact that humans alone engage in a series of activities that have no obvious equivalents in other species. These are, in fact, the activities that constitute the proper domain of the humanistic culture – arts, religion, law, politics, architecture, theater. . . Science, too, as a uniquely human activity fully belongs in this regard to the humanistic culture. The difference with respect to other animal species is striking, for all animal species need some sort of knowledge to deal with the world of material objects and, some species at least, with artifacts and tools. Yet, only humans organize this knowledge in a systematic way, which is known as science. And, all social animals need to develop some sort of knowledge of the social relationships between individuals. Yet, only humans organize this knowledge in the form of juridical and political institutions. And, again, all animals deal with the need to find and treat food in some ways. Yet, only humans develop cooking as art and culture. What is peculiar of human cognition that make us capable of all this? What is the extra of “necessary knowledge” that makes us build up and discuss two cultures? It seems likely that language and the cognitive abilities associated with it should be in some way crucial. Data from cognitive anthropology concur with this view. For instance, Pica et al. [33] studied numerical cognition among Mundurukú, an Amazonian tribe with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their

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naming range. However, exactly as our chicks, they fail in exact arithmetic with numbers larger than 4 or 5. These findings suggest a distinction between nonverbal systems allowing either number approximation or exact arithmetic up to a set limit of about 3-4 elements, and a language-based counting system for exact number and arithmetic without any upper limit. Mundurukù, of course, do possess (differently from my chicks) the cognitive capabilities to learn words for numbers beyond 5 and to learn such a precise arithmetic for numbers beyond 5. Trying to explain how and why humans alone have developed these capacities is an issue that is likely to occupy the minds of cognitive scientists and neuroscientists for the next decades.

References 1. E.S. Spelke: What makes us smart? Core knowledge and natural language. In D. Gentner and S. Goldin-Meadow (eds.): Language in mind: advances in the investigation of language and thought (MIT Press, Cambridge 2003a) 2. E.S. Spelke: Core knowledge. In N. Kanwisher and J. Duncan (eds.): Attention and performance, vol. 20: Functional neuroimaging of visual cognition (Oxford University Press, Oxford 2003b) 3. S. Carey: Bootstrapping and the origins of concepts. Daedalus, (2004), 59–68 4. N. Humphrey: The mind made flesh (Oxford University Press, Oxford 2002) p. 336 5. L. Weiskrantz: Blindsight: A case study and its implications (Oxford University Press, Oxford 1986) 6. N.K. Humphrey and L. Weiskrantz: Vision in monkeys after removal of the striate cortex. Nature, 215 (1967) 595–597 7. N.K. Humphrey: Vision in a monkey without striate cortex: a case study. Perception 3 (1974), 241– 255 8. E.S. Spelke: Core knowledge. American Psychologist 55 (2000), 1233–1243 9. Y. Sugita: Face perception in monkeys reared with no exposure to faces. PNAS, 105 (2008), 394– 398 10. S.P.R. Rose: God’s organism? The chick as a model system for memory studies. Learning and Memory, 7(1) (2000), 1–17 11. G. Vallortigara: The cognitive chicken: Visual and spatial cognition in a non-mammalian brain. In E.A. Wasserman and T.R. Zentall (eds.): Comparative cognition: Experimental explorations of animal intelligence (Oxford University Press, Oxford 2006) pp. 41–58 12. P.P.G. Bateson: What must be known in order to understand imprinting? In: C. Heyes and L. Huber (eds.), The evolution of cognition (MIT Press, Cambridge 2004) pp. 85–102 13. L. Regolin, G. Vallortigara: Perception of partly occluded objects by young chicks. Perception and Psychophysics, 57 (1995), 971–976 14. S.E.G. Lea, A.M. Slater, C.M.E. Ryan: Perception of object unity in chicks: a comparison with the human infant. Infant Behaviour and Development, 19 (1996), 501–504 15. E. Valenza, I. Leo, L. Gava and F. Simion: Perceptual completion in newborn human infants. Child Development, 77 (2006), 1810–1821 16. G. Johansson: Visual perception of biological motion and a model for its analysis. Perception and Psychophysics, 14 (1973), 201–211 17. G. Vallortigara, L. Regolin, F. Marconato: Visually inexperienced chicks exhibit a spontaneous preference for biological motion patterns. PLoS Biology, 3(7) (2005), 1312–1316 18. N.F. Troje, C. Westhoff: Inversion effect in biological motion perception: Evidence for a “life detector”? Current Biology 16 (2006), 821–824 19. G. Vallortigara, L. Regolin: Gravity bias in the interpretation of biological motion by inexperienced chicks. Current Biology, 16 (2006), 279–280 20. E. Clara, L. Regolin, G. Vallortigara, M. Zanforlin: Domestic chicks perceive stereokinetic illusions. Perception, 35 (2006), 983–992

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21. L. Regolin, G. Vallortigara, M. Zanforlin: Object and spatial representations in detour problems by chicks. Animal Behaviour, 49 (1994), 195–199 22. P. Zucca, N. Milos, G. Vallortigara: Piagetian object permanence and its development in Eurasian Jays (Garrulus glandarius). Animal Cognition, 10 (2007), 243–258 23. L. Feigenson, S. Carey, M.D. Hauser: The representations underlying infants’ choice of more: object file versus analog magnitudes. Psychological Science, 13 (2002), 150–156 24. L. Feigenson: A double dissociation in infants’ representation of object arrays. Cognition, 95 (2005), B37–B48 25. L. Feigenson, S. Carey: On the limits of infants’ quantification of small object arrays. Cognition, 97 (2005), 295–313 26. R. Rugani, L. Regolin, G. Vallortigara: Discrimination of small numerosities in young chicks. Journal of Experimental Psychology: Animal Behavior Processes, 34 (2008), 388–399 27. M.D. Hauser, E.S. Spelke: Evolutionary and developmental foundations of human knowledge: A case study of mathematics. In M. Gazzaniga (ed.), The cognitive neurosciences, Vol. 3. (MIT Press, Cambridge 2004) 28. G. Vallortigara: Animals as natural geometers. In L. Tommasi, L. Nadel, and M. Peterson (eds.): Cognitive biology: Evolutionary and developmental perspectives on mind, brain and behavior (MIT Press, Cambridge 2008) 29. G. Vallortigara, M. Feruglio, V.A. Sovrano: Reorientation by geometric and landmark information in environments of different spatial size. Developmental Science, 8 (2004), 393–401 30. V.A. Sovrano, A. Bisazza, G. Vallortigara: Modularity and spatial reorientation in a simple mind: Encoding of geometric and nongeometric properties of a spatial environment by fish. Cognition, 85 (2002), 51–59 31. C. Chiandetti, G. Vallortigara: Is there an innate geometric module? Effects of experience with angular geometric cues on spatial reorientation based on the shape of the environment. Animal Cognition, 11 (2008), 139–146 32. S. Dehaene, V. Izard, P. Pica & E.S. Spelke: Core knowledge of geometry in an Amazonian indigene group. Science, 311 (2006), 381–384 33. P. Pica, C. Lemer, V. Izard, S. Dehaene: Exact and Approximate Arithmetic in an Amazonian Indigene Group. Science, 306 (2004), 499–503

Discussion C. Montecucco: To what extent does the outcome of your experiments depend on binocular vision? G. Vallortigara: In the chick the binocular field is mainly used during control of feeding, to target small objects at short distance. Thus, most of the data I discussed are likely to involve the use of the lateral field of vision, which in birds is mainly served by the tectofugal pathway. This suggests a difference with respect to mammals: it could be that similar cognitive mechanisms are implemented in different neural pathways in the two classes. J. Rasche: Are the chicks “better” in learning when they are not only “one” in the box but are with other companions? Is there a sort of “collective brain” in these very social animals? G. Vallortigara: Actually, the mere presence of an artificial goal object suffices for the chick to show quite normal social and affiliative behavior. The lack of any adequate imprinting object deeply affects behavior and cognitive functions. For instance, one striking outcome of rearing chicks in isolation is a change in the duration of some

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part of memory (intermediate memory) during the process of consolidation. Interestingly, the same effects can be obtained pharmacologically by injecting sociallyreared chicks with those substances that are produced as a result of stress (e.g., noradrenaline, ACTH, vasopressine). As to problem solving requiring some form of social collaboration and cooperation among different individuals, we did not test the issue in this species. But there is some evidence from other laboratories that this could be important in some species of birds, i.e., in corvids. G. Setti: You showed that chicks, like infants, have difficulties in considering more than a few objects for discrimination. On the other hand, you showed chicks to be able to deal with a series of 9–10 objects in a series. How can the results of these experiments be reconciled? G. Vallortigara: The distinction between the exact numerosity discrimination system (which has a upper limit of 3–4 items) and the approximate number system (which deals with large numerosities in an approximate way, i.e., with the only limitations imposed by Weber’s law) clearly apply to the cardinal aspect of number, i.e., that involving the magnitude of the sets. The experiments in which chicks learned the correct serial position in a series (i.e., the fourth, the sixth, . . . , the nth item of a series) did not require any representation of the magnitude, just the serial procedure of counting. In other words, chicks used only the ordinal position of the stimuli that imply, for instance, that food container 3 comes before food container 4, but without any representation that food container 4 is larger in magnitude with respect to food container 3. J.J.A. Mooij: Memory is involved in knowledge about objects, number, and space. But what about autobiographical knowledge, so essential for the feeling of human identity? Is the preference for certain shapes and numbers already a kind of autobiographical memory? Or does autobiographical memory suppose self-consciousness – and can self-consciousness only arise in higher animals, perhaps only humans? G. Vallortigara: That is a difficult question. Evidence for autobiographical, episodic memory in our species relies deeply on verbal reports, which is something we cannot obtain in nonhuman species. From a purely behavioral point of view, however, it has been argued that episodic memory is equivalent to the ability to combine together information concerning a “what”, a “where”, and a “when”, and there is some evidence that both birds and mammals are able to encode these sorts of episodic-like memories. I do not know whether self-consciousness is unique to our species; certainly it can be expected only in species in which there is a very sophisticated social life, as I am inclined to think that self-consciousness in itself is basically a social, more that an individual, phenomenon.

The Many Faces of Intelligence

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Elkhonon Goldberg and Dmitri Bougakov

Lofty notions such as intelligence are often concerned with concepts like rationality versus emotion, and scientific reasoning versus more artistic thought. In this chapter we discuss how these lofty notions are mediated by the brain, and even more importantly, how they are not. When people talk about higher order cognition they usually implicate the neocortex, and the neocortex is a very complex organ. Just by looking at Broadman’s cytoarchitectonic map [1] (see Fig. 9.1) one can readily appreciate its heterogeneity and structural complexity. However, one should realize that different parts of this quilt do not necessarily correspond in a one-to-one fashion to diverse cognitive faculties, but they do give us an order of magnitude approximation of the complexity of this organ. There is a certain “neuro-mythology” that has evolved in the scientific community, and also in the general media and among the general public, which needs to be addressed. The first of these “neuro-myths” pertains to the notion of intelligence. Of course, the concept of intelligence can be looked upon in at least three different ways. It may be used to denote in a colloquial sense a certain cognitive sophistication. It may denote something very specific and very limited, such as a number, which is generated by the Wechsler Adult Intelligence Test [2]. It may also mean a concept accepted and shared by many in the scientific community and probably by most among the general public that a certain variable exists, a certain aspect of the mind, which defines the mind, which is critical to the mind, which is constant in any given individual, for that individual lifespan, but which is highly variable across individuals. This concept is very often referred to as the “G-factor.” However the notion of the “G-factor,” thus defined, is probably a consequence of a linguistic accident, and a consequence of the way David Wechsler named his test. Essentially, what is WAIS IQ? WAIS IQ is the number generated by Wechsler Adult Intelligence Scale, an instrument designed by David Wechsler many years ago, which has since undergone several revisions. His focus was to sample cognition in a variety of ways, which is

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Fig. 9.1 Broadman’s cytoarchitectonic map [1]

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pertinent to your life, which would enable you to rank individuals on a certain scale in an actuarial sense. Because Wechsler named it Wechsler Intelligence Scale, as opposed to something like the “Wechsler Performance Scale,” it immediately invited the conjecture that there is something behind the number and beyond the number, that this number really reflects some kind of invariant property of the mind, which defines the mind, in a way that was just described. This gave rise to the notion of the “G-factor,” and there are scores of scientific papers and volumes of research have been conducted trying to identify this “G-factor.” From a neurobiological standpoint, the existence of such a “G-factor” would imply the presence of some kind of a variable, some kind of feature in the brain that satisfies these requirements, namely, which predates all of the brain in a likewise fashion in a given individual, which is constant throughout a given individual’s lifespan and which exhibits high variability across individuals. And in a theoretical sense it is not inconceivable that such a variable exists, but so far it has not been found. Just as a theoretical proposition, it is not inconceivable that the rate of protein synthesis necessary for long term memory formation would satisfy this requirement, or that some features of the action potential would satisfy these requirements, or that some features of long pathway myelinization would satisfy these requirements, but in reality, no such constant biological property of the brain has been found, and so at this point, we have absolutely no reason to believe, no neurobiological reason, that the “G-factor” exists. So, intelligence in the sense of a single cardinal property of the mind which informs and defines all aspects of cognition in a given individual in a constant fashion and is highly variable across individuals remains a “neuro-myth” which so far has not found support. Furthermore, the more we learn about how dynamic the brain is, the more we appreciate that fact that the function of cortical neuroanatomy is not constant, it changes in the course of ontogeny in rather dramatic ways. To the extent that we recognize brain plasticity in response to diverse neurological assaults on the brain, and today there is growing evidence that the brain is extremely plastic (for an in depth review see [3]) at least at the neocortical level, we have to conclude that the brain is relatively amodular. In other words, the way function is mapped into the neocortex, at least in the association cortices, cannot be described through the discrete kind of boundaries and discontinuities between regions. The function appears to be mapped into the brain in rather continuous ways devoid of discrete boundaries [4], so there are less and less reasons to believe that the “G-factor” exists in the sense in which it was defined earlier. Another common “neuro-myth” is the notion that in the neocortex, rational thought and emotional, affective thought are somehow segregated and separated in the brain, and are mediated by different neuro-machineries. It is not uncommon to hear allusions that rational thought is mediated by the cortex and emotions are mediated by subcortical structures, such as, for instance, amygdala. That is a “neuromyth” too, because we now have plenty of evidence that emotional processes are motivated and controlled to a very large extent by the neocortex as well (for a review see [3]).

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We know that patients with depression, for example, exhibit a pattern of hypofrontality, in other words, the level of activation of their prefrontal cortex is reduced [3]. We know that strokes, particularly frontal strokes, produce various extreme affective alterations, so that a stroke affecting the left prefrontal cortex would produce a depression-like condition, while a stroke affecting the right prefrontal cortex, the dorsolateral cortex, is likely to produce something akin to euphoria [3], so, clearly, the neocortex plays an important role in the generation and control of emotions. So, it appears that this distinction between the rational, affectless neocortex and the most known subcortical structures subserving emotions is also a fictitious distinction. Another “neuro-myth”, which has gained quite a bit of currency, not only in general media, but also among scientists and neuropsychologists, is the notion of the analytic left hemisphere and artistic right hemisphere. It is a “neuro-myth” because it has been demonstrated that, in fact, professional musicians utilize predominantly the left hemisphere in processing, appreciating, and generating music [3]. So, this division of processing into analytic and artistic, and allocating the two to the two hemispheres is yet another “neuro-myth.” It does not appear that the way the brain parses cognition can be captured by any of these dichotomies. Artistic versus analytic, or emotional versus rational, this is how we, for a variety of reasons, talk about cognition and parse it out in our colloquial parlance – this is not how the brain parses out cognition. But how does the brain parse cognition? Let us draw our attention to two interesting dichotomies, which have consumed much of our research, time, and scientific interest. One of these issues is on the convergence of functional cortical neuroanatomy and the issue that has befuddled computational neuroscientists for quite some time, namely, how does the brain reconcile two needs: to acquire new information and to preserve previously accumulated information? The brain needs to store previously acquired information, and at the same time to acquire new information without disrupting old, previously formed representations. Researchers from the “neuromodeling community,” who try to capture certain formal properties of the brain with devices such as neural nets, have been basically confounded trying to figure out how it can be done, and, currently, this is a matter of much research and much controversy in computational neuroscience. The consideration of this issue involves hemispheric specialization. The following discussion is on the intercept of these two theories. To make a biographic detour, one of the author’s (Elkhonon Goldberg, (EG)) interest in this subject basically arose from a discussion that has a certain Oedipal beginning. It arose from a discussion he had with his father when he was 18 or 19 years old. During this particular incident, EG was very glibly talking about what he had learned in psychology classes at the time, namely that the left hemisphere is the language hemisphere and the right hemisphere is the non-linguistic visual-spatial hemisphere. To which his father, who was an engineer and an applied mathematician, and like some mathematicians, but mercifully not all, had quite a bit of disdain for everybody who made their living in ways other than manipulating mathematical abstractions, sort of sneered and said that one has to be a psychologist to say some-

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thing this nonsensical. He proceeded to note quite cogently, that this dichotomy is limited in its application only to humans, it cannot be extended to other species, because other species do not have language, at least not in its narrow definition. Therefore hemispheric specialization based on the distinction between language and visuospatial processing would imply some kind of drastic evolutionary discontinuity, which should never be a null hypothesis in a scientific explanation. So without any particular knowledge of biology, but with a lot of common sense, EG’s father pointed out this fundamental flaw of this distinction, namely, that it did not allow evolutionary continuities. This early discussion had a major impact on EG’s thinking, and in much of his own later work he tried to identify and formulate some properties, certain contrasting properties of the two hemispheres, which would be applicable across species, which would be meaningful outside of human species. Some of EG’s clinical work led him to conclude that one interesting distinction, one functional distinction between the two hemispheres, is captured by the following notion that the left hemisphere is basically the repository of previously formed knowledge in some sense, where the right hemisphere is uniquely well-suited for dealing with cognitive novelty [5]. Subsequently, the contrast that has guided much of his work has been the contrast between cognitive novelty and cognitive routines. A graphic illustration of this concept is that when an individual is exposed to a cognitively novel task, the right hemisphere is the leading hemisphere, and then with increased familiarity with the task, somehow, the left hemisphere takes over. This is drawn schematically in Fig. 9.2. An interesting question, which is beyond the scope of this chapter, is how this functional distinction between the two hemispheres arises as an emergent property, as a consequence of certain basic architectural differences between the two hemispheres, differences in pathway length, and the allocation of cortical space to different types of neuronal populations. Indeed, how these fundamental features of neural architecture lead to higher levels of distinctions as an emergent property is a very interesting and intriguing issue, but it remains the topic of future discussion.

PERFORMANCE LEVEL

LEFT HEMISPHERE RIGHT HEMISPHERE

TIME

Fig. 9.2 Right to left shift of hemispheric control as a function of task acquisition (adapted from [3])

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Similarity Similarity

Dissimilarity Dissimilarity

Preference Preference •Similarity left fronto-parietal •Dissimilarity right fronto-parietal fixed effects model height threshold p = 0.001, corr. extent threshold 30 voxels

Fig. 9.3 Hemispheres, novelty and cognitive bias task (adapted from [6])

Nevertheless, we can examine a kind of an experimental demonstration of this principle in action. Take a look at Fig. 9.3. This figure summarizes an fMRI study conducted with our German colleague, Kai Vogeley [6], who at the time was at the University of Bonn and is now at the University of Cologne. In this study the subject was given a choice of either engaging in a matching task on the basis of similarity (in other words, they could choose on the basis of familiarity) or on the basis of dissimilarity (in other words, they could choose on the basis of novelty). One can readily see that these different tasks produce differential activation in the two hemispheres, in a complementary fashion. A green pattern of activation is one which corresponds to choices on the basis of similarity, a blue pattern of activation is one which corresponds to making choice on the basis of novelty. Blue “resides” in the right hemisphere, and green “resides” in the left hemisphere. This is direct neuroimaging support for the novelty-routinization concept, which goes to show that to the extent that we are so adamant about finding some kind of binary characteristics of brain function and some critical dichotomies, defining dichotomies, cognitive novelty vs. cognitive routinization may indeed capture, and is a relevant and pertinent way of talking about the way the brain parses cognition. A study conducted by Japanese scientist Masau Aihara and his colleagues [7] further supports this notion (see Fig. 9.4). This study used a high frequency EEG signal, a so-called Gamma frequency range, 30-40 Hertz, which many scientists use as a marker of cognitive effort. This marker has been accepted by many people in the field and it appears to be a good general marker of cognitive effort. In this study the subjects were yet again engaged in a kind of a perceptual discrimination task, perceptual choice task, which is a novel task, but as in most human experiments or animal experiments, an experiment consists of a number of trials, so

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RIGHT TO LEFT SHIFT DURING TASK FAMILIARIZATION (EEG spectral power in the gamma frequency band (30-40 Hz) (Kamiya et al, 2004))

2sec

Target Card

Choice Cards

V2 2.00

1.00

0.00 Fig. 9.4 Right to left shift during task familiarization (adapted from [7])

that if the sequence is sufficiently long, the task which starts out being novel becomes familiar. One can see from the figure that at the very early stages, the task engages the right hemisphere, specifically the right frontal regions, the dorsolateral regions, and then gradually migrates from the right hemisphere to the left hemisphere with task familiarity. There has been a growing number of studies using various functional neuroimaging devices, using PET, positron emission tomography, fMRI and others, which basically lent support to this notion [8, 9, 10, 11], and they all show that if you introduce a subject to a task which is initially novel, and then in the fullness of time it becomes familiar, it corresponds to this kind of a gradual shift from the preponderance of right activation to the preponderance of left hemisphere activation, and this is true for verbal tasks, for nonverbal tasks, memory tasks, for all kinds of tasks, which means that the novelty-routinization distinction overrides other variables, and there appears to be indeed a kind of cardinal distinction in defining which hemisphere is critical for controlling a cognitive task. There is more evidence of this kind using various approaches, longitudinal studies, and cross-sectional studies. Of particular interest is one relatively old study (not everything that is old is bad – old and false is outdated, old and true is enduring). This study was conducted by Thomas Bever and his associates at Columbia Univer-

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sity at the time [12]. The group created a task of musical discrimination. According to the traditional neurological notions, musical discrimination is mediated by the right hemisphere, and agnosia is presumed to be a symptom of right hemispheric damage. This task was given to two kinds of people: musically naive, like most of us are, people untrained in music, and to a group of graduate students from Juilliard School, which is a conservatory school in New York City. Their results showed that in musically naive individuals the right hemisphere was the one that contributed the most to processing music. In professional musicians, it was the other way around – the left hemisphere was the one which carried the cognitive load. Which again goes to show, as alluded to earlier when we talked about the notion of ascribing rational thought, analytic thought, to the left hemisphere and artistic pursuits to the right hemisphere, in fact it is incorrect. It turns out that people engaged in artistic pursuits for a living do it to a much greater extent with reliance on the left hemisphere and not the right hemisphere. With respect to the kinds of things which we commonly leave to the left hemisphere, such as language, it turns out that at very early stages of development, language is mediated to a large extent by the right hemisphere, so this notion of a kind of an exclusive association of language with the left hemisphere may capture functional properties of an adult brain, but not the brain of a child. Which again points to the essentially unidirectional right to left shift of the center of cognitive gravity, if you will, as a function of increasing familiarity. When a certain cognitive domain is novel, as language is for little children, for infants, it appears that the right hemisphere plays a very important role in its mediation, and it’s only in adults that it becomes a more or less exclusive monopoly of the left hemisphere. Lesion effects involving radical hemispherectomy, something that unfortunately sometimes has to be done to treat intractable epilepsy, point in the same direction. It turns out that radical hemispherectomy affecting the right hemisphere in children interferes with their language development, pointing to the important role played by the right hemisphere in language at early stages (for a review see [5]). Neurochemical evidence basically points in the same direction. Studies in rats show that by stimulating two major catecholamine neurotransmitters, one attains opposite effects. Stimulating noradrenergic transmission increases exploratory behavior, so-called orienting response, stimulating dopaminergic transmission stimulates stereotypic behavior [13]. In the mammalian brain, these two types of catecholamines are somewhat lateralized. The work by Stanley Glick [14], at New York Mount Sinai Medical Center, and colleagues has shown that noradrenergic pathways tend to slightly favor the right hemisphere, whereas dopaminergic pathways tend to slightly favor the left hemisphere. Thus, it appears that the right hemisphere, where noradrenergic modulation appears to be more prominent than it is in the left hemisphere, is the one that mediates exploratory behaviors and novelty. The left hemisphere, where dopaminergic

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NOREPINEPHRINE

-RIGHT HEMISPHERIC PREPONDERANCE -MEDIATES SENSORY STIMULATION AND RESPONSE TO PERCEPTUAL NOVELTY

DOPAMINE

-LEFT-HEMISPHERE PREPONDERANCE -MEDIATES MOTOR ACTIVITY AND REDUNDANCY OF RESPONSREPERTOIRE

Fig. 9.5 Neurochemical asymmetry in the brain (after [15])

modulation appears to be more prominent than in the right hemisphere, mediates exploitation of cognitive routines, stereotypic behavior (for a summary of neurochemical data see Fig. 9.5 (after [15]). Let us bring our attention to neural net modeling, the quasi-mathematical models of the brain that are then implemented as computer programs so that one can actually conduct experiments with these models. It has been demonstrated by Steven Grossberg [16, 17] that the computational efficiency of a neural net is increased by separating these two aspects of cognition within the net. It is always reassuring when there is convergent evidence, or at least convergent argument coming from different areas of neuroscience pointing in the same direction: namely, that the distinction between cognitive novelty and cognitive routines is a very salient distinction in the brain. There are good reasons to believe that these two aspects of cognition are mediated separately to some degree by the two hemispheres. Initially, we spent some time talking about “neuro-myths,” false dichotomies that have gained a lot of currency in popular science, but do not seem to reflect the way cognition is actually mediated in the brain. Our discussion is concerned with various dichotomies that do seem to capture some essential aspect of the way cognition is mediated in the brain. The distinction between cognitive novelty and cognitive routines appears to be one of them. The two cerebral hemispheres seem to have a “preferential relationship” with one or the other. Let us now talk about something else – the human lifespan. Clearly, as we move throughout the lifespan, the domain of the novelty shrinks in a kind of a broad sense, and the domain of the familiar expands. As it turns out, the two hemispheres of the brain age at an unequal rate. A number of studies using quantitative brain morphometry have shown that the left hemisphere somehow ages more slowly than the right hemisphere. It turns out that the rates of this shrinkage, the rates of atrophy, are different in the two hemispheres. The right hemisphere seems to age more rapidly than the left hemisphere, and the left hemisphere ages more slowly, in other words the effects of this atrophy are delayed in the left hemisphere (for a review see [3]). One way of explaining this is through the effects of neuroplasticity. We utilize the left hemisphere increasingly more with age, because of accumulated cognitive abilities, and the domain of the familiar expands, and the domain of the novel shrinks as we age.

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Because we rely on the left hemisphere more, it benefits more from lifelong morphogenesis. This is what may account for these unequal rates of atrophy. The left hemisphere seems to benefit somewhat more than the right hemisphere from neuroprotective effects of lifelong morphogenesis, which in turn is a function of the degree to which different neural structures are used. So far, we have talked about the functional differences between the two hemispheres. Let us now shift our attention to the other axis – the front-to-back axis. The prefrontal cortex and the studies of the frontal lobe function, and the prefrontal cortex, has been another enduring theme of EG’s career (see [18]). The frontal lobe is a large entity, but we will be talking about a certain subdivision of the frontal lobe, the prefrontal cortex, (see Fig. 9.6). It is a distinct set of cortical regions, which can be identified cytoarchetectonically, or projection-wise, and in a variety of other ways. The prefrontal cortex is a very interesting part of the brain. It is phylogenetically very young, (see Fig. 9.7, a graph depicting the ratio of the frontal lobe region volume to the total volume of the cortex in various simians). One can easily see from the figure that the frontal lobes are relatively small in the primitive monkeys, and become progressively larger in lesser apes, the great apes, and finally in humans, the ultimate great apes. The frontal lobes enjoy what looks like exponential expansion. Phylogenetically it is a very young part of the brain. It has been implicated in all kinds of cognitive functions such as consciousness and theory of mind, which we humans tend to ap-

Fig. 9.6 The frontal lobe (from [3])

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Fig. 9.7 Evolution of the frontal cortex (from [3])

propriate to ourselves. However, it is doubtful that these are uniquely human, and like most traits, they exhibit evolutionary continuities. The prefrontal cortex is engaged in a range of functions, which we call executive functions (see Fig. 9.8). An interesting distinction which is not often invoked, in fact it is quite often ignored in cognitive science and certainly in clinical neuropsychology and behavioral neurology, is the distinction between “descriptive” or “veridical” decision making, and “prescriptive” or “actor-centered” decision making (see Fig. 9.9).

Prefrontal cortex is critical for:

Fig. 9.8 Function of the frontal lobe

Fig. 9.9 Descriptive and prescriptive knowledge

- goal-directed behavior - planning and temporal organization of cognitive processes - critical judgment - ability to project into the future and anticipate the consequences of one’s behavior - the capacity for insight into other people’s mind - impulse control

• DESCRIPTIVE DECISION-MAKING: WHAT IS TRUE AND WHAT IS FALSE? • PRESCRIPTIVE DECISION MAKING: WHAT SHOULD I DO AND WHAT SHOULD I REFRAIN FROM DOING? • THEIR NEUROANATOMICAL TERRITORIES ARE DIFFERENT AND ONE MAY SUFFER WITHOUT THE OTHER IN BRAIN DAMAGE

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To understand the difference between the two let us consider an example, where the brain could be involved in solving two kinds of problems. One task, one type of problem, is a veridical task. For instance: “How much is ten plus ten” is a veridical task. There is one correct response, other responses being incorrect. The correct response is inherent in the task, it does not depend on the actor – ten plus ten is twenty regardless of who does the math. Now, imagine a different kind of problem. Imagine you graduate from high school and you have to decide whether to study medicine or study engineering. In this particular scenario, saying that medicine is an intrinsically correct choice, and engineering is an intrinsically false choice, is an oxymoron. These types of tasks are not about finding the objective truth, but they are about deciding what is best for you. They require a different type of decision making. One directed at finding the correct response, which is intrinsic in the task, veridical, the other one actor-centered, deciding what is best for me, a very different type of decision making. Most of our cognitive paradigms, used in cognitive neuroscience research, and virtually all of the paradigms used for clinical purposes used to diagnose patients are of the former variety. They are designed to look at veridical decision making. There is precious little, bordering on nothing, in a kind of neuropsychological, neurological arsenal, to look at veridical, actor-centered decision making. Based on everything that we know about the brain, it is clear that these two types of decision making are somewhat compartmentalized in the brain (see Fig. 9.10).

Fig. 9.10 Descriptive and prescriptive knowledge “territories” (from [3])

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The veridical decision making is mediated mostly by posterior association cortices, and non-veridical, actor-centered decision making is mediated mostly by the prefrontal cortex. This is another very salient distinction, a salient parceling of cognition, implemented in the brain, so that the posterior cortex is mostly engaged in veridical decision making, the prefrontal cortex is mostly engaged in actor-centered decision making. Consider the following experiment. The subject sees a geometric form (the target), followed by two other forms below. Consider, further, two tasks that can be superimposed on these stimuli (see Fig. 9.11). In one task, you ask the subject: “Look at the target and choose one of the two forms below which you like the most.” It is a non-veridical preference task. In another situation you ask the subject: “Please look at the target and choose one of the two forms below which is more similar to the target.” Damage to the frontal lobes severely impairs performance on the first, but not on the second task. This contrast highlights the particularly important role played by the prefrontal cortex in actor-centered, nonveridical decision making. Going back to the previously mentioned study by Kamiya et al. [7], which examines Gamma frequency EEG, one can see that in the course of learning the task, the activation shifts is two dimensional (see Fig. 9.4). It proceeds from right to left but it also shifts from front to back. As the subject becomes familiar with a certain way of making the preference, the subject decides: “OK, I’ll choose on the basis of color, or on the basis of something else”, the role of the frontal lobe decreases so the task becomes increasingly veridical, and the frontal lobe plays less and less of a role in its execution. It turns out that this nonveridical, actor-centered paradigm enables us to understand the function of the frontal lobes in a way that eluded us for as long as our approaches were limited to veridical tasks. We also used this paradigm in a study conducted with patients with frontal lobe lesions, and the effects of left frontal lesions and right frontal lesions turned out to be quite different. Until recently, the frontal lobes were thought to be “symmetrical” in their function. In order to elicit the very robust functional asymmetries between the frontal lobes, we had to resort to these non-veridical actor-centered techniques and then the functional differences between the left and right frontal lobes became quite striking, at least in males. By contrast, in females no evidence of such differences exists [19]. Up to now, it was assumed that functional organization of the frontal lobes in males and in females was the same, but with these non-veridical, actorcentered approaches, it was possible to show that they are actually quite different. To conclude, it appears that, indeed, the frontal lobes are engaged in nonveridical, actor-centered decision making, and it is necessary to take it into account in order to fully understand its function. Once we employ appropriate experimental procedures examining non-veridical, actor-centered cognition, both gender differences and hemispheric differences in the functional organization of the frontal lobes become quite apparent. To summarize, we have introduced two critical distinctions. One distinction, between veridical and actor-centered cognition, characterizes the way posterior cortex

160 Fig. 9.11 CBT card (adapted from [19])

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and frontal cortex work. The other distinction, between cognitive novelty and cognitive familiarity, characterizes the way in which the right hemisphere and the left hemisphere work. It is our belief that these two distinctions are salient in subtyping many faces of intelligence.

Acknowledgments Our sincere gratitude goes to Mr. Jeffrey Donaldson for extensive assistance in preparation of this manuscript.

References 1. K. Brodmann: Vergleichende Lokalisationslehre der Grosshinrinde in ihren Prinzipien dargestellt auf Grund des Zellenbaues (Barth, Leipzig 1909) 2. D. Wechsler: Wechsler adult intelligence scale, 3rd edition (The Psychological Corporation, San Antonio 1997) 3. E. Goldberg: The wisdom paradox: How your mind can grow stronger as your brain grows older (Gotham Books, New York 2005) 4. E. Goldberg: Gradiental approach to neocortical functional organization. Journal of Clinical Experimental Neuropsychology 11(4):489–517 (1989) 5. E. Goldberg, L.D. Costa: Hemisphere differences in the acquisition and use of descriptive systems. Brain Language 14(1):144–73 (1981) 6. K. Vogeley, K. Podell, J. Kukolja et al.: Recruitment of the left prefrontal cortex in preferencebased decisions in males (fMRI study). Ninth Annual Meeting of the Organization for Human Brain Mapping (New York 2003) 7. Y. Kamiya, M. Aihara, M. Osada et al.: Electrophysiological study of lateralization in the frontal lobes. Japanese Journal of Cognitive Neuroscience 3(1):88–191 (2002) 8. J.M. Gold, K.F. Berman, C. Randolph, T.E. Goldberg, D. Weinberger: PET validation of a novel prefrontal task: Delayed response alteration. Neuropsychology 10:3–10 (1996) 9. A. Martin, C.L. Wiggs, J. Weisberg: Modulation of human medial temporal lobe activity by form, meaning, and experience. Hippocampus 7(6):587–93 (1997) 10. R. Shadmehr, H.H. Holcomb: Neural correlates of motor memory consolidation. Science 277 (5327):821–5 (1997) 11. R. Henson, T. Shallice, R. Dolan: Neuroimaging evidence for dissociable forms of repetition priming. Science 287(5456):1269–1272 (2000) 12. T.G. Bever, R.J. Chiarello: Cerebral dominance in musicians and nonmusicians. Science, 185 (150):537–9 (1974) 13. SD. Glick, R.C. Meibach, R.D. Cox, S. Maayani: Multiple and interrelated functional asymmetries in rat brain. Life Science 25(4):395–400 (1979) 14. S.D. Glick, D.A. Ross, L.B. Hough: Lateral asymmetry of neurotransmitters in human brain. Brain Research 234(1):53–63 (1982) 15. D.M. Tucker, P.A. Williamson: Asymmetric neural control systems in human self-regulation. Psychological Review 91(2):185–215 (1984) 16. S. Grossberg: Competitive Learning: From interactive activation to adaptive resonance. Cognitive Science 11:23–63 (1987) 17. S. Grossberg: Neural networks and natural intelligence (MIT Press, Cambridge 1988) 18. E. Goldberg: The executive brain: frontal lobes and the civilized mind (Oxford University Press, Oxford 2001)

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19. E. Goldberg, R. Harner, M. Lovell, K. Podell, S. Riggio: Cognitive bias, functional cortical geometry, and the frontal lobes: laterality, sex, and handedness. Journal of Cognitive Neuroscience 6(3):276–296 (1994)

Discussion J.J.A. Mooij: Is it necessary to have self-consciousness to have autobiographical knowledge and memory; and, if self consciousness is necessary, is episodic memory a feature unique to humans? E. Goldberg: The “party-line” holds that episodic memory is a uniquely human territory; as a dog owner, I posit that it is not. This is my conclusion based less on my own work and more on work of others in the field, because my research exclusively involves humans, I do not do animal studies. I have not done animal work in forty years. But as a dog owner, I make all kinds of very interesting observations, including that of the existence of episodic memory in the canines. R. Durrer: If we assume that the left hemisphere is for storing already known things and the right hemisphere is for acquiring new knowledge, and, as you mentioned, in children the right hemisphere is used for language, while in adults, it is the left one, which hemisphere would be used for acquisition of a new language in an adult? Furthermore, if someone has a substantial lesion in the right hemisphere, would that person be unable to learn anything to a significant degree, or could the left hemisphere take over part of this function? E. Goldberg: First of all, this distinction is not absolute binary distinction, it is a continuum, so any complex cognitive process obviously includes both hemispheres, but to a different degree. There is a continuous involvement, and continuous transition. There are no precipitous transitions, so that up to a certain point a certain task is mediated exclusively by one hemisphere, and this mediation precipitously switches to the other hemisphere. It is gradiental in nature. In this spirit the second language, indeed, is shown to be more bilaterally represented; functional imaging studies have shown that, and in fact I talk about it in one of my books, The Wisdom Paradox (2005). In an adult individual the second language does show somewhat more bilateral representation than the first language, and it involves the right hemisphere to a greater extent. The right hemisphere is involved in language in children to some degree, but not exclusively so; it is not as if the left hemisphere did not play a role in language in children at all, that would be a fallacy, but again, it is a gradiental transition. In reference to the effects of lesions, for many years this entrenched notion existed, which is reflected in our terminology today. We refer to the left hemisphere as the dominant hemisphere, and the right hemisphere as the subdominant hemisphere. The notion was that language mediated processing the rules, and everything else is of secondary importance.

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Traditionally, clinicians were much more concerned about the ability of the right hemisphere to make up for the lost function of the left hemisphere when the left hemisphere is damaged than the other way around. But in reality, we know that the loss of the right hemisphere function can be quite catastrophic at early stages of life, in children for instance. These are precisely the kind of observations that gave rise to the noveltyroutinization distinction that I began to explore, the observations that in children, damage to the right hemisphere is more catastrophic than it is in adults, and is much more catastrophic than is damage to the left hemisphere, which sort of seeded this idea that at very early stages of cognitive development, the right hemisphere plays a critical role, and that role gradually recedes. E. Carafoli: You have mentioned the predominance of norepinephrine and norepinephrine-signaling in the right hemisphere as opposed to dopamine predominance in the left hemisphere. Are we already at the stage where you can be a little more specific about particular areas in each hemisphere? E. Goldberg: It is easier to answer this question for dopamine than for norepinephrine because mesocortical dopamine pathways are quite circumscribed. Both dorsolateral and orbitofrontal neocortex receives projections from the mesocortical dopamine pathway. The mesolimbic pathway projects into the amygdala, the nucleus accumbens, and the hippocampi. The nigrostriatal pathway projects into the striatum. Most of the studies where this asymmetry is implicated talk about the mesolimbic and mesocortical pathways, the ones which come out of the ventral tegmental region. It is not known to me if and to what extent these asymmetries are demonstrable for the pathways that originate in substantia nigra. However, let me say this with respect to the mesolimbic and mesocortical pathways. Eric Kandel demonstrated the role of dopamine in long term memory formation. This may have very interesting ramifications for hemispheric specialization, and play a significant role in the story why the left hemisphere and not the right hemisphere is the repository of long term knowledge. It is possible that the formation of long term storage, to the extent that it is dopamine mediated, is more efficient in the left hemisphere, because there is more of it in the left hippocampus than in the right hippocampus. On the other hand, the norepinephrine pathways are much more diffuse. They tend to favor the frontal lobes broadly over the posterior cortex, but they are less circumscribed. Which is why it is easier to answer this question for dopamine than for norepinephrine. G.P. Feltrin: Which is a more critical predictor of recovery of brain function: the amount of brain matter lost, or the site of the lesion? E. Goldberg: Both! This is not a flippant response – both factors matter. Clearly, the brain is a highly heterogeneous organ, so the site of the lesion matters, at the same time, based on the studies of micro-vascular diseases in multi-infarct dementia, the total volume of damage also matters.

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G. Setti: My first question is in reference to the study you mentioned earlier – the study that showed a difference between individuals studying music in a conservatory, and musically naïve individuals. Is it possible that in that particular case individuals who went to the conservatory did already have musical knowledge or predisposition, and therefore, in those individuals the transfer of cognitive control from the right hemisphere to the left hemispheres had already begun; for genetic or educational reasons, for instance. The second question concerns aging. Is it possible to maintain an equal rate of aging for both hemispheres through exercise? E. Goldberg: To answer the second question, it is theoretically possible through exposing oneself to cognitively novel challenges. Think about the great intellectuals of our time, and the past century. Think about Einstein playing the violin and sailing his sailboat, and Winston Churchill painting his acrylics, both as avocations. They embraced activities as far removed from their “daytime jobs” as possible. This notion of challenging one’s brain through novel challenges is very important. Let me now answer the first question, the question about the direction of causation, in effect. Music is a kind of privileged, exalted vocation. Being a musician requires certain talent, but this effect has been demonstrated also for much more mundane activities, such as the Morse code, which is no longer much in use, but which was in use fifty, sixty years ago. If we design tasks involving the Morse code, for instance, in most individuals they will activate the right hemisphere, but in individuals using the Morse code vocationally, they will activate the left hemisphere. If you design a task differentiating between the dot and line sequences and put Morse code naive individuals through a task like that, you’ll demonstrate a right hemispheric preference, but for Morse code operators, people who used to deal with these codes professionally, it will show the left hemispheric preference. This familiarity versus novelty effect has been demonstrated for variety of tasks. I brought up the study of musicians only as a way of assailing the “neuro-myth” number three which links artistic, creative activity exclusively to the right hemisphere. But, generally speaking, this relationship between cognitive novelty and cognitive familiarity in the two hemispheres has been demonstrated across a wide range of tasks.

Cycles of Re-Creation – A Psychoanalytical Approach to Music

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Joerg Rasche

Concert Program J.S. Bach: Prelude and Fugue e flat minor, Well Tempered Clavier I, 8 Frederick Chopin: Fantasy-Impromptu Robert Schumann: Kreisleriana op. 16

10.1 Introduction Creation and Re-Creation Creation – this is a common issue for both cultures. Natural science is interested in the origins of life, its emergence and differentiation and ability for self reproduction and variation. I want to mention here my friend Pier Luigi Luisi (now in Rome) who dedicated his scientific life to this fascinating question.1 The humanities on the other side of the spectrum are interested in the origin of mind and psyche, of emotions and our ability for relationship. For a psychoanalytic approach both interests belong together, they appear as the two sides of the same fascinosum and of the same greater reality. I would like to call our different approaches to this one reality, by natural science or by humanities as “complementary” – following an idea of Wolfgang Pauli [1]. Re-creation – this is what we are looking for, if we in a laboratory try to reproduce something, be it only the reproduction of results identical to an experiment we did before. Our mind is something creative, it needs creation and re-creation. We like to go to concerts or exhibitions, we like good cooking, having sex, somehow nourishing our body-mind-complex. We like to sleep, we like being loved and to love, and to be securely contained, we also like to feel free, we like having children, 1 Prof. Luisi is not only a scientist, but also the founder of the interdisciplinary Cortona Conferences of the ETH Zurich. He also writes and publishes novels for children and adults.

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and we like to integrate all these kinds of experiences in our conscious mind. Our mind, our psyche is more than the sum of its particles. We seem to have an inborn and essential longing for understanding and bringing together our personal experiences, to build some greater meaningful something, related to our ego or our self. Our mind does it by means of causality or analogy or just synchronicity – some coincidence which makes sense for us without being explainable by Aristotelian logical principles. We will see that this has to do with emotions and will be a circular process of our embodied mind. Re-creation does not mean repairing or creating something new. It is a kind of second act. In re-creating we follow a model or pattern we obtained before. We are not the creators, but maybe re-creators. We are re-creators because we love and admire what is before us, or because we lost it or believe that we have lost it. Then recreation is refreshing, making us feel alive again. Re-creation is an act of healing. It has to do with re-vitalization or re-constellation, re-formation, of giving life again to something by ourselves and in ourselves. It has to do with something lost, the past, a la Recherche du Temps Perdue,2 with memory, mourning and inner reconstruction of the lost. It is a transformation and follows the patterns of a cycle. In this sense, as we will realize, re-creation is a basic pattern of our thinking, of our mind, of the perception of different realities. Our mind, our brain is working by re-shaping and integrating. We are re-creators. Good music tells us a story like this. Good music is about psyche, about recreation in both meanings – it is refreshing and comforting, and it is a re-creative process. Music is an arrangement of sounds and movements that speak directly to our body, our mind and emotions. We do not need words. But we can hear more if we try to listen with all our senses. Then we even may see something non-visible with our inner eyes. This is what Bernhard from Clairvaux said: If you want to see – listen! Musical compositions often tell a story – not through words but by shaping sounds and structuring rhythms, melody, and harmony. Music cannot be translated because its meaning cannot be described thoroughly using language. Language is always talking through secondary semantics. Language knows a lot: Language provides us with such wonderful ambiguous words like “re-creation.” Music is different, it is what it is – and by that it tells stories about a primary reality, the reality of relations. It is a result of a cultural evolution. Maybe music and language became complementary approaches in our culture in the individual and collective re-creation of our psychic world. Complementary means: At the end the approaches exclude each other. As W. Pauli said in 1951 in the sense of quantum physics: As well as particle and wave exclude complementary descriptions in physics, also the two sides of reality: the quantitative and the qualitative, the physical and the psychological are complementary aspects. Behind this there is the common one world, but you must decide which side you want to experience. In the psychological world again you may see a similar relation: Language is about meaning, music is about relationship. 2

The work of Marcel Proust is about memorizing and inner re-construction. The psychoanalyst Hanna Segal wrote about him and the roots of arts in the mourning process. See [2].

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One may see this also in the complementary functions of the two hemispheres of our brain – language and music are primarily processed by different sides. But this would be, again, the natural scientist’s view. To understand the psychological difference and complementarities between music and language one must also analyze music itself. And there the problem will arise that you cannot really speak about your experiences. Today we know a lot about how our psyche emerges and is formed by our relations. Musical experience here plays an important part, long before the language. It begins a long time before birth with the acoustic perceptions in the mother’s belly. After birth comes the interaction of the very early “affect attunement” between mother and child. It is about the integration of affects/emotions with sounds, rhythms, primordial melodies, and meaningful gestures, as a basic process for the so called “mentalization” of perceptions. European music became a cultural vessel and expression for these fundamental psychological processes. This music is obviously still very important in the processes of re-creation of our modern psyche. European music has developed a wide spectrum of musical forms, such as rhythms, consonance dissonance, combinations and interactions among different structural patterns, polyphonic interactions, relations between inner voices, and so on [3]. An example is the music of J.S. Bach (Prelude and Fugue in E flat minor, Well Tempered Clavier I, 8). Listening to or performing this introverted music provokes an unconscious remembering and inner re-creation of early patterns of interaction and integration of emotions and experiences in our psyche. It is a peaceful dialog of inner voices. One very old, archetypal pattern and symbol of the psychic cycle is the so called Uroborus – the snake or dragon holding its own tail in its mouth. It is a powerful symbol of wholeness and being closed up and contained – along with its uncomfortable aspect. It is a dragon, a very powerful, non-individuated, not really kind beast. It shows us that we have to free ourselves out of the timeless and dangerous happiness of the early symbiosis. It fits that on cemeteries the Uroborus is often shown as symbol for eternity (the entire symbolism is shown in [4]). One important step in our psychological development is the hatching out of the early mother–infant common orbit. This step is not easy and often painful for both: the mother and the child. But the patterns of this early step towards individuation will be repeated in later life. It is a psychological cycle, beginning with being contained and totally protected, in a happy unconscious state, but necessarily leading to the first separation, to early loss, alienation, maybe even paranoid confusion, then mourning, grief, and finally re-creation of inner objects. There will always be a desire for the primordial feeling of wholeness [5]. This cycle also establishes the patterns of cognition, of our thinking. Also in later life our experiences, our fights and complexes will often follow the same rhythms. Psychoanalysis is about following these tracks. It is astonishing how detailed some music of the romantic period is in expressing this kind of re-creating processes. For me this music is an ideal non-verbal vessel for this kind of non-verbal emotional and structuring processes. This is be demonstrated in music of the Romantic era: Chopin’s Fantasy-Impromptu and the piano

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cycle “Kreisleriana” op. 16 by Robert Schumann. They are examples of the basic archetypal pattern of the creation and the inner fights of an individuated psyche. “If you want to see – listen!” (Bernard of Clairvaux). This paper has four parts. Now I will give you an example of my Jungianpsychoanalytical approach to music, demonstrating shortly something about the Fantasy-Impromptu by Frederick Chopin. The third part will be a short theoretical one, and the fourth part will be dedicated to Schumann’s Kreisleriana. I will finish with playing one movement of the Kreisleriana.

10.2 Chopin, Fantaisie-Impromptu op. 66 Chopin writes a dramatic process in four parts: A-B-A-C. A: “Allegro agitato”, c-sharp-minor – the piece starts in an obviously problematic psychological condition. The “musical subject” – this is the imaginative person who is suffering, acting and speaking through the music- is very active, driven (“agitated”) and not at ease with himself. There are two different rhythms at the same time, showing a split in the mind. One hand has to play even faster than the other one. The upper voice goes nervously up and down. It seems that the subject wants to escape and to do something, but it cannot (Fig. 10.1). Later on comes a transient consolidation (with an inner melody), but the first part ends in some desperate scales turning upwards and falling or crashing downwards three or four times. So the end of the first part shows disillusion, desperation and anger (for more details see Supplement 1). B: “Largo – moderato cantabile” – the second part (Fig. 10.2) opens an entire different world. Like lifting a veil a mystery of warm colors embraces the musician and the listeners. A melody emerges that has its routes in A but now goes up and on to a beautiful Chopin melody. The melody seems to have no end, it goes on like

Fig. 10.1 Chopin op. 66, Beginning

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Fig. 10.2 Chopin op. 66, Beginning of the middle section

a singing perpetuum mobile. For me it is like an uroboric circle. After some never ending repetitions the music seems seductively to ask: Do you believe me? Do you trust me? In the first part I am hearing-seeing an agitated (maybe male) ego that cannot find resolution and trust in himself. In the second I see the hallucination or vision of a female entity, the image (imago) of the mother, the promising and seductive anima. She seems to have and to offer all the happiness and satisfaction that the poor agitated first subject is so painfully missing. (See Supplement 2.) We will see what happens. Both characters together build an exposé of the drama. Will they come together, and if they do – how? The musical composition unfolds the stage for the inner process like a dream or a fairy tale. After the vision of the second part, the first part A comes again. We have to realize that nothing has changed. Again there is the restless flight, the vain and fugitive inner melody, and the breakdown at the end. We must conclude that the content of the second part B could not be integrated as it was. In its timeless beauty it had to remain a wonderful fantasy, a vision, maybe a memory. You will never find the real anima in your real life. Also the timeless happiness of childhood will never come back. But now Chopin shows that he is a great artist and man. The third part (the repetition of the first) is followed by a Coda C: a fourth part bringing the conclusion. Maybe we expect to hear again something of the comforting second part B. But we hear protest and resignation. In a unique condensation of emotions there is sadness and energy, greed, grief, and mourning. The melody shows both the half-tone and the whole-ton-steps. The rhythm now is clear and so to say cleaned. This means acceptance of “what is.” Acceptance of the loss. The movement then goes slowly down, poco a poco diminuendo, showing a reduction of the emotional content to a pure almost abstract pattern of energy. It is a moment of “timeless time.” (Fig. 10.3) As in Zen meditation we are waiting for something but we cannot say what it could be. Like a miracle some melody from

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Fig. 10.3 The last measures. The melody of the middle section appears again

deep down in the lower voice then emerges. It appears to be the endless melody of the second part B, re-created from the deep inside of the musical subject. So the beautiful female something is still there, it is not lost forever. There is no winner. Both characters are present at the end of the piece. They are balanced. Indeed the moment of resignation was essential, the mourning for the lost, to recreate it as an everlasting inner reality (an inner object).

10.3 About the Inner Re-Creation of Our World Music and language are connected in interesting ways. We want to speak about music, but we must realize that there is something in music we cannot say with words. On the other side language shows some attributes which are “like music.” There are a lot of para-semantic aspects like accompanying gestures, variations in sounds, and so on. In the Romantic period (the age of Chopin and Schumann) one said that music and language feel a desire, they are longing for each other. In our

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times many linguists, philosophers, and psychologists have studied the implications of this awareness. I only mention here Theodor W. Adorno, Hildemarie Streich, Victor Zuckerkandl, Peter Gülke, Roland Barthes, Julia Kristeva, and – last but not least – C.G. Jung and Wolfgang Pauli [1, 6, 7, 8, 9, 10]. The great physicist Pauli even wrote that “you can’t understand the world without playing the piano” [11, 12]. He argued that there are the same basic patterns underlying the world of music and of the material world. (See Supplement 3.) Following Pauli, music is about psyche, but it follows the same pattern as nature. Music is an arrangement of sounds and movements that speak directly to our body, our mind, and emotions. We do not need words. As musical compositions, the Well Tempered Clavier, the Chopin Impromptu or the “Kreisleriana” design pure musical processes in the form of cycles. There are no words. I am personally a friend of understanding what happens. I try to compare the patterns of the music (also of their emotional content) with patterns I know from psychoanalysis. This is a psychoanalytical approach to music, not so far from the scientific approach to natural phenomena. I want to emphasize that I am not talking about any pathology. For me music is so important because it shows to me something general and ubiquitous, something archetypal in human experiences, human emotions and relationships. The great artists, the great composers were able to find striking formulations for these experiences and transformations, which are experiences of all of us. Therefore we like their music. We cannot really remember the very early experiences of our first months and years because our memory and speech does not reach so far back. But they are still embodied and treasured in our unconscious. They follow archetypal patterns. In later years our emerging psyche will follow the same patterns if there are new experiences and emotions to be integrated. The cycles of our psychic processes will be very similar.

10.4 Schumann, Kreisleriana op. 16 I will now discuss something about Schumann’s Kreisleriana. The title “Kreisleriana” refers to some writings of E.T.A. Hoffmann, a German writer of the Romantic period. In his later years he lived in Berlin and became a kind of local Berlin hero. “Kreisler” is a figure in Hoffmann’s novels, a musician with strange ideas and attitudes – just the ideal masque for the young Robert Schumann himself3 [13]. His early piano cycles were like laboratories for his self re-creation. “Kreisleriana” was chosen as a title by Schumann to underline the very fantastic attitude of his compositions.4 Maybe the name “Kreisleriana” means more. For English speakers the German word “Kreis” is close to “crazy.” In German language “Kreisleriana” also maybe associated with “Kreis” – the circle, the cycle. Indeed 3

Kreisler appears also in other texts of Hoffmann, e.g., in Lebensansichten des Katers Murr (1819/1821) – there an “invisible girl” is the hidden center of the whole story. 4 Schumann composed the “Kreisleriana. Fantasies for the pianoforte” 1838. I am referring to the text of the first edition.

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Schumann’s Kreisleriana consist in musical circles. But there is another hidden meaning. One of Hoffmann’s Kreisler-texts is about a girl who is not visible – “das unsichtbare Mädchen.” This invisible girl is the hidden center of a very romantic network and spider web of secrets and adventures, and music plays a big part since Hoffmann himself was a composer too. 10.4.1 The “Invisible Girl” This is a wonderful synonym for what Jungian psychologists call the “anima” – the mysterious female being in our unconscious psyche. And it is an eternal image for the woman, the first wife, the first girl in our life, and her everlasting promises, her beauty, her smell, her seduction to trust her and to trust the world. Of course the model of the anima is the mother and all the mysteries we usually connect with her (or with “them” because in Jungian psychology the mother archetype is ubiquitous and contains more than our personal mother). The anima in our psyche is the guarantee for stability and trustworthiness of our existence. And by this she is the symbol (the psychic function) for relationship and for new beginnings. As “anima mundi” she inspires scientists in their research. A very common symbol in this context is – again – the circle, the “Kreis.” We all know that nothing in our life is as easy. Often the anima, as she appears in dreams or ideas, is not simply a good, willing and supporting fantasy. Sometimes she is more a kind of desire, we miss her, we suffer from emptiness or hunger for something or somebody we cannot name. Sometimes the anima in dreams and in life appears more like a bitch, playing with us and seducing us to adventures that will be, at least and if we have the fortune, a challenge and test for ourself. Then the uroboric circle shows its dangerous side. Also the memory of our childhood often is not such an early paradise. Often the desire for a kind of “invisible girl” is, to use a musical phrase, a basso continuo of our life. I want to mention again the physicist Wolfgang Pauli. A basso continuo in his life was the quest for “his inner girl.” She was a dancer. He met with her in his dreams; he spoke about her in his analysis with C.G. Jung. He wrote a fantasy about her called “The piano lesson” where she appears as a piano teacher. There this lady says: You can’t understand life without playing the piano. “The censors (this means the scientists) want to understand the world without playing the piano. This is absurd, isn’t it?”5 . Pauli was one of the pioneers of quantum physics. His scientific imagination was strongly connected with musical and archetypal fantasies. Some concepts of quantum physics seem to be analogous to musical phenomena. Very important is the concept or meaning of “time.” Music is the art of time – only in time the entanglement in the cloud of probability will be solved by a decision for one side of the complementary concepts. In music there is no causality but only synchronicity. The electrons rotate and behave like Pauli’s dancer or his “invisible girl.” Time and space are very connected in music as well as in physics. 5

In [12], p. 322 (§23), author’s translation.

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The ERP experiments work only “in time,” but the supposed interactions of the relevant particles are even faster than the speed of light. In some moments of music the time seems to stand still. There then is “only” the mystery of interaction or even intuition between the voices. Maybe music is a kind of “language” to describe what happens also on the basics of our material world. But: if you want to see – listen!

10.4.2 The Second Movement “Kreisleriana” itself is a cycle of cycles of re-creation. I want to present something of the second movement of Schumann’s Kreisleriana.

10.4.3 The Girl’s Melody It begins in B flat-major (B-Dur) with a very famous melody (Fig. 10.4). It is like a bow, an arc of harmony and consonance, going up and down and returning in itself like a circle. It is a meaningful gesture: Like in a dream a wonderful peaceful gesture saying something without a word. It is something like being held embraced, a hug, and like stretching when waking up in the morning, passive and active at the same time, a never ending circle – it is the Uroborus, the primordial feeling of unity and wholeness. Everything is good and will be good.

Fig. 10.4 Kreisleriana 2, Beginning

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10.4.4 Perturbations and Re-Consolidations But life means to fall out of the early paradise into existence. Now comes a series of perturbations and re-consolidations. In the next measures (9 ff.) lower (male) voices appear, at first softly but with increasing prominence, conveying some deeper sensations (or emotions). The deep voice climbs up and joins the upper voices to sing together. The next measures bring movements like stretching and turning inside, as in a dream, and finally bring a kind of apotheosis of the harmonious sound of the beginning. This sounds like a promise. In fact, Schumann wrote this music as a present for his absent girl friend and later wife Clara. It was not clear that Robert would be allowed to marry her. Maybe the theme of this movement was originally created by Clara. We may hear it as a portrait of this “invisible girl.” I call it the “girl’s melody.” The movement now brings some complications and irritations. Step by step the harmonious image will be complicated, dark, and problematic, even with colors of stress and panic. A first “intermezzo” is a kind of disturbing “scene of childhood,” or it might seem as a fugitive huntsman riding through the forest and disappearing. The girl’s melody nevertheless soon comes again in its comforting beauty. A longer and more intensive second intermezzo (Fig. 10.5) leads into chromatic scales and even panic fears. After this the girl’s melody seems to vanish – it is no longer to be heard, it is so to say now “invisible.” We may ask what happened. Unfortunately we do not have words, only notes and sounds. But we can follow the pattern of the composition itself. This is the way music works in us – we do not need words, but unconsciously we follow the pattern of the music, the ups and downs of the melodies, the interaction of the voices, the tensions and solutions of the harmonies. (See Supplement 4.)

Fig. 10.5 Kreisleriana 2, 92–96

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10.4.5 Introversion The second intermezzo leads to a panic. How can this shocking experience be integrated? The “girl’s melody” has disappeared. Now a remarkable sequence of introversion occurs (Fig. 10.6, 119 ff.). The primary melody seems to be entangled in a contrapuntal network with crossings of chromatic voices, and even crossings of the hands of the player. The music shows very clearly and directly that here some voices (this means some identities) are changing their positions. By the way, in language it would not be possible to express this so clearly. It is a kind of “physical” process that says “what it is,” not what it “means.” The different voices may symbolize different persons in an inner drama, different inner representations in one and the same psyche. The changing of the positions of voices is a changing of their special meaning: If the upper voice goes down under another voice it loses its supremacy. The former lower voice now comes to the top. Obviously both have the same value. This is what composers call the “double counterpoint.” The two voices are complementary. One effect of this composing method is that the musical form seems to become independent from other (outer) gravitation patterns. The interaction of the shifting voices has its tensions and solutions, its beauty and meaning in itself. The “music” and its value are now founded in the interaction of the voices alone. The music seems a bit to rotate around itself. It is a very “abstract” way of composition in which some old masters and J.S. Bach were experts. It is a kind of “objective music.” It seems to be somehow sad, but also abstract like mathematical procedures or equations. The drive, the inner flow of time is slowed down; time seems to be condensed into a pattern of mutual meaningfulness or synchronicity.6 “Consequently” in the following measures (126 ff.) even the sense of harmonic tension seems to become lost. It is as if Schumann (or the “musical subject” who lives in this music) would loose the orientation on where to go [14, 15, 16]. Music here shows a moment of questioning and alienation. The “girl” is not only invisible, she is lost. In this moment Schumann goes very far – it is a kind of psychotic experience of not knowing where to go and where to belong. 10.4.6 Reappearance of the Girl’s Melody But now a miracle happens (130 ff.). The original melody appears from somewhere – in a very foreign tune, unexpected, as a memory from outside with a very different “celestial” sound. It is an experience not so far from that in Chopin’s piece, at the end, when the inner melody reappears. It is like an inner decision. We recognize the melody immediately, even if it/she now appears in the foreign tune and on a very higher level of sound. We recognize the “Gestalt” of the melody even in the very foreign context of f-sharp-major (Fis-Dur). By the way, this shows how our brain works by combining memories and in certain cases images, “Gestalten,” to fill gaps and to help us out of alienation and disturbances. The girl’s melody 6

This is similar to some moments in compositions by Mozart and Bach, as described by Peter Gülke [7].

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Fig. 10.6 The crucial moments

here appears like a dissipative structure in chemical situations far away from equilibrium. It helps, and soon the melody will be found again in the original tune (134 ff.). We feel at home again. The flow of time starts again. The emotional impact is again there, maybe stronger than before.

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10.4.7 The Heartbreaking Moment But now something strange happens to the melody itself. Its end phrase (134/135) sticks at the starting note (B flat). It seems arrested, and cannot really move on. This change happens twice. Finally, in measures 138/139 the line, the Gestalt, of the melody itself changes. It cannot go up as before but falls back to its first tone. It is a heartbreaking moment. This now is the deepest point of the inner process of the entire second movement of Kreisleriana. It will be the turning point for a new appearance of the original theme, but now somehow transformed, cleaned and re-born out of the introversion. It will be a kind of resurrection of the original but transformed melody of the invisible girl. It is like the Uroborus re-born as an inner reality, as anima or Beziehungsfunktion – the ability of keeping relationships. Finally, the melody will even be transposed in the upper sphere like an apotheosis. I am showing this in detail for a special reason: Music of this density and quality may demonstrate something about how our mind works. The “language” of music is not an “indicating” semantic speech but an intrinsic one. It is closer, if you allow, to physics and mathematics – it is about pattern and at the same time close to the hermeneutic sphere of our emotions. It is what it is. Seeing by listening, and by following the tracks of the notes on the paper like the tracks of an electron in the fog chamber we may experience and re-construct how our mind is building itself by combining, memorizing, and emotionally working through. The crucial, heartbreaking moment in measure 138/139 may show what psychologically really happened. The early object of love has gone or had to be left behind. The musical piece is one about grief, mourning and inner re-creation. Maturation means leaving the mother’s sphere. The heartbreaking moment is the one when the musical subject realizes that he is alone. But by acknowledging this fact he becomes able to re-build and re-create her in his inner mind. The memory is still there. Mourning is essential for individuation. And it is essential for the transformation of lost objects into everlasting inner objects, and for the transformation of relations into the inner ability for relationships (see Supplement 5). This music is moving also because it designs something of our situation as modern humans in the alienated world. It is about the common feeling of not being sheltered or protected by a benevolent greater entity, by Mother Earth or by God. This question is bigger than developmental psychology or the aims of natural science. For the Romantic composers Chopin and Schumann the inner security and trustfulness of J.S. Bach were no longer the self-evident basis of their life. Today we are the followers of the Romantic crisis. Therefore, as I believe, their musical compositions may speak directly to our heart.

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Supplements (1) Chopin A: When the initial passages appear for the second time they show a slight difference. In measure 10 the narrow primal half-tone step (Halbtonschritt) becomes a little wider, it is now a whole-tone-step with another harmony and meaning. The harmonic drive (a kind of “gravitation”) now goes towards the parallel major scale E-major. This leads to the emergence of a hidden melody (m.13) where the musical subjects seems to find some stability. But the stability is lost very soon leading to passages of poor and miserable lamentation (chromatic scales m.21 ff.). The consolidation was only a transient one. In m.25 the agitated motives start again, but now instead of becoming consolidated leading to some desperate scales turning upwards (m.29 ff.) and falling or crashing downwards three or four times. So the end of the first part shows disillusion, desperation, and anger. (2) Chopin B: It is obvious that in the beautiful largo another subject is speaking than in the first part. Whereas the first was an agitated, unhappy one, the second one is somebody rich, promising, seductive. The first was very active in time, and with its complicated rhythm even one hand had to play faster than the other. The second is singing like in the present, in the fulfilled time of here and now. As both are parts of the same psyche of the composer I would say that both represent different complexes – this means different elements of the same psyche connected with different emotions and attitudes. In the first I hear-see an agitated (maybe male) ego that cannot find resolution and trust in himself. In the second I see the hallucination or vision of a female entity, the image (imago) of the mother, the promising and seductive anima. She seems to have and to offer all the happiness and satisfaction that the poor agitated first subject so painfully lacking. (3) Reality is more than the sum of its particles – in natural science we may think of an animal, which is certainly more and something different than its cells, its fur, ears, legs, and teeth. There is also its parental generation and its offspring, its ecological environment essential for its existence, and the scientific approach. The idea of an essential network and mutual interconnectedness is very common today in biology, but also (as a logical pattern) in mathematics or physics. What about our psyche? I want to make a short digression to introduce to you some idea linking psyche and music. For the old Greek psyche was part of nature: a butterfly, farfalla, Schmetterling dancing in the sunny air. This for the old was the “psyche.” Psyche means butterfly. They experienced “psyche” as a kind of reality in their outer world and in themselves, as human beings and part of the greater nature. Paradoxically, “psyche” was everywhere in the landscape, dancing from flower to flower, and at the same time a very personal inner experience and challenge. Butterfly-psyche became a symbol, something linking the unconscious and the conscious, and “explaining” and answering many questions. For example, that this beautiful butterfly is only the latest

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metamorphosis of a very poor and ugly animal that spends most time of its life being a blind worm under the earth. By this butterfly became a metaphor for resurrection and the immortality of the soul. The butterfly is also a very free animal. If caught it soon dies. In the moment when it sits down on a brunch or a leave and opens its wings you may immediately see a face. Often the wings of a butterfly show colorful eye-like patterns. Something is staring at you and immediately disappears when the butterfly closes its wings again. Something is watching you. This maybe a moment of immediate shocking and fugitive awareness. Who am I? Even in our days the symbol of the butterfly is still alive – following the chaos theory one flip of the wing of a butterfly anywhere in the world may cause a hurricane. Obviously, in this saying (facon de parler) we are unconsciously following an archetypal image. “Psychology” in the sense of C.G. Jung is the “science of the butterfly archetype.” There is some “psyche in the nature” and some nature in the psyche. This for Jung and Pauli was the common root of psychoanalysis and physics. There is only one world. Their elaborated scientific texts can be found in Naturerklärung und Psyche (1952) (see footnote 2). There Pauli speaks also about the complementary relation between physis and psyche. He asks for a kind of language that would fit both: to express the unity of the world, the unus mundus. Maybe music can provide some hints for this language. (4) Certainly this piece is about some of Schumann’s personal problems in a very impressive formulation. Biographically, he was a boy who had lost his mother in his third year, when she gave him to another family for some years. In his 15th year he lost his sister who committed suicide. And he was not at all sure that he would get Clara as his wife. For the Jungian approach indeed this biographical detail is not so important because music works on an archetypal level and formulates experiences that we all have. We all had the problem of losing and having lost somebody, and we have all lost at a certain moment the first object of our love. If we follow this music we unconsciously follow the emotions that are hidden in the patterns of this music. (5) I am not a physicist, but maybe there is something analogous or homologous: The collapse of the cloud of probability – der Zusammenbruch of the Wahrscheinlichkeitswolke – is similar to what happened to Schumann. There are similarities or analogies between physical and psychological processes. A real event or observation – a kind of decision – leads to something or somebody you can trust. It may be an inner image of a real person, or something with wave (or particle) attributes that allow experiments with reliable, repeatable results. Music and psyche then are not only about similar problems – they are about the same problems. Our psyche is part of the nature. Our patterns of perception are the same as the patterns of nature itself .We have to learn and to study them in our childhood and later on, in cycles of re-creation, in our labs, as well as in our life. . . Unus mundus – Keine Zeit. There is only one world. We are part of it by and with our ability for relationships. We are living through cycles of re-creation.

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References 1. C.G. Jung and W. Pauli: Naturerklärung und Psyche (Rascher, Zürich 1952) 2. H. Segal: A Kleinian approach to clinical practice (Free Association Books, London 1981) 3. J. Rasche: Das Lied des grünen Löwen. Musik als Spiegel der Seele. Book with CD (Walter-Patmos, Düsseldorf 2004) 4. E. Neumann: The Great Mother – Die Große Mutter (Rhein Verlag, Zürich 1956) 5. J. Rasche: Mutter–Kind–Musik. Psychoanalytische Deutung von Schumanns Kinderszenen op. 15. Analytische Kinder- und Jugendlichenpsychotherapie. Emotionale Entwicklung und Neurobiologische Grundlagen, Vol. 133, 2007 6. S. Bayerl: Von der Sprache der Musik zur Musik der Sprache. Konzepte zur Spracherweiterung bei Adorno, Kristeva und Barthes (Köningshausen und Neumann, Würzburg 2003) 7. P. Gülke: Musikalische und historische Zeitschichten. In: Klein, Kiem, Ette (Eds.) Musik in der Zeit, Zeit in der Musik (Velbrück, Weilerswist 2000) 8. V. Zuckerkandl: Kreis und Pfeil im Werk Beethovens, Eranos Jahrbuch 1964 (Rhein Verlag, Zürich) 9. V. Zuckerkandl: Die Tongestalt. In: Eranos-Jahrbuch 1960 (Rhein Verlag, Zürich) 10. H. Streich: Musikalische und psychologische Entsprechungen in der Atalanta fugiens von Michael Maier. In: Eranos Jahrbuch 1973 (Rhein Verlag, Zürich) 11. W. Pauli: Die Klavierstunde. Eine aktive Phantasie über das Unbewusste. Atmanspacher, Primas (Wertenschlag-Birkhäuser 1995) 12. J. Rasche: Kinderszenen. Irrationales in der Musik, in: Atmanspacher, Primas (Ed.), Der PauliJung-Dialog und seine Bedeutung für die moderne Wissenschaft (Springer, Berlin 1995) 13. E.T.A. Hoffmann: Kreisleriana (1822) (Reclam, Stuttgart 2005) 14. R. Nagler: Der konfliktuöse Kompromiß zwischen Gefühl und Vernunft im Frühwerk Schumanns. In Metzger and Riehn (Eds.): Musik-Konzepte Sonderband Robert Schumann I/II, Edition Text und Kritik (München 1992) 15. R. Barthes, in “Rasch”, in: Metzger and Riehn (Eds.) Musik-Konzepte Sonderband Robert Schumann I/II, Edition Text und Kritik (München 1982) 16. C. Rosen: Musik der Romantik (The Romantic Generation, 1995) (Residenz Verlag, Salzburg 2000) p. 746

Discussion G. Berlucchi: Could you comment on the relationship between music and the prosody of speech and of aesthetic versus logic? J. Rasche: Our spoken language (speech) is usually accompanied by meaningful gestures, variations of the sound of the voice, of facing and by other kinds of parasemantic, emotional, or “musical” communication. This is essential for the beginning of language in our childhood. Maybe from the beginning of modern times, from the Renaissance, or especially with Newton, language and music have developed in different directions. To say it very concisely: our modern language is about meaning, music is about relationships. Both ways of communication have become complementary cultural tools and expressions and we need both. But there is a problem: If I am writing or speaking about music I can only use what my language provides me with. But with my speaking I will never capture the meaning of music. Music has no grammar, no semantic triangle. It follows a different kind of logic, it is based on proportions, interactions, resonance, tension, and solution.

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As Pascal said, the heart has its own reasons, which are unknown to our rational mind. The “logic of affections” and emotions may be regarded as a link between language (speech) and music. This would also be the background for the “music” in the prosody and in our understanding of the meaning of words. It starts very early with the “affect-attunement” between mother and infant. G. Setti: I have two questions that are probably linked. The first is on the recreational/emotional part. I do not have technical knowledge of music, I only have emotional reactions to it. I need to listen to it a second and a third time, and so on. That means that in my mind I need to “play” it just before the instrument does. Why is it so? The second question is on the strong cultural/social side of music. As a personal experience, when I had the chance to listen to musical performances in different types of cultural environments (such as, for instance, in India) I could not become emotionally involved in spite of repeated efforts. J. Rasche: Repetition helps in integrating forms, movements, images, melodies, and interactions. Maybe one can say that “emotions” are like a glue that keeps experience and patterns together in our mind. If you hear a musical phrase for the first time the emotions will be different than if you listen to it for the 20th time; the emotional impact becomes modified. Even the experience of listening to the same music for the 20th time will have an impact. The knowledge of what will come next will add a feeling of continuity and reliability. In psychoanalysis we speak of countertransference. The analyst himself unconsciously lays something into the uttering of the patient and creates his own feelings and fantasies about its meaning and its possible solution. Maybe he will be bored by all the repetitions, but he may also realize that there is a kind of ritual happening, which may be helpful for the patient to survive a trauma. Repetition stabilizes the mind. About India: European music is about individuality and community, therefore, the “musical subject” (the “subject” in the musical composition) is a representative of the history of European individuality. This is different in traditional cultures like those of India (the Ragas) or Indonesia (Gamelan music), where tradition is much more important. The ragas are endless variations about a limited number of musical archetypes, and there is no “evolution” in the Western sense. Therefore, for us, traditional Indian music may provoke an atmosphere of timelessness and eternity, which is the aim of this traditional music. G. Giacometti: I was very interested in your mention of Pauli’s writings of his dreams and about Kepler. I have a question for you as a physician: Do you believe that Pauli was affected by the syndrome that today is called the “bipolar syndrome”? J. Rasche: Pauli was not affected by the bipolar syndrome; he was neither manic nor depressive. But he might have had a problem with emotions similar to those of C.G. Jung. Both needed an intense close personal relationship to feel good and secure. And both were fascinated by images, numbers, and proportions. In his book “Psychology and Alchemy” Jung published a lot of dreams of Pauli but the quality of emotions was mostly left out. It is a pity that natural science today often seems to be more fascinated by patterns and numbers than by emotions; maybe emotions as in-

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ner experiences cannot be objects for natural science. Pauli wrote in his 1954 paper about Kepler that the physical world and the psyche are complementary concepts. F. Brunetti: The word “re-creation” reminded me of the childhood experience of the “free time” between school hours. “Re-creation” means also the pleasure of creativity and connects the research activity in the artistic as well as in the scientific intuition. Two characteristics of this creative dimension of human mind are the intensive pleasure and the richness of meanings for the person who lives the experience, during the re-combination of memories and their deep sublimation. This leads me to ask you whether music, as you have discussed it in your presentation, is to be considered as a form of art or of science? J. Rasche: Re-creation is a wonderful word, and our language is a treasure of such cultural experiences, contained in wording like this one. The double meaning of recreation is a fine example. Today in brain research we learn how our mind works. The psychoanalytical approach can add some insights about memory, the structure of “complexes” as “working tools” of our psyche, or about the role of emotions and archetypal patterns in learning and in the exploration of our world. But there is still something missing. To use Pauli’s words: “You cannot understand the world without playing the piano.” If we are not open to the living beauty of our world we will only get only a very reduced picture of it. And the very basic patterns underlying musical structures are the same as those of the physical world. Somehow, we are going back to a Pythagorean view where art and science belonged together, except that today we look at them with a different concept of individuation.

Section Beauty and Creativity

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Symmetry: A Bridge Between the Two Cultures

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Marcus du Sautoy

When I was a kid I didn’t want to be a mathematician at all. My dream was to become a spy. This dream was fueled by my mother who had been in the foreign office before she’d had children. But becoming a mother was apparently incompatible with being a diplomat and so she had to leave her job. But she told me and my sister that they had at least allowed her to keep the black gun every member of the foreign office was required to carry as a member of the diplomatic service. I became convinced that my mother must in fact have been a spy and that she might be recalled to active duty at any moment. Somewhere in the house she was hiding this gun. I used to spend a good deal of my childhood hunting the house for the gun. But I could never find it. They’d obviously successfully taught my mother the art of concealment during her training. So I resolved that if I were to have my own gun the only way was to join the foreign office and become a spy myself. I decided the best qualification for joining the diplomatic corps was to learn as many languages as possible. So I started taking every language I could at school: French, German, Latin. The BBC started a series of programs teaching Russian on television. I thought that was ideal for a spy. So I got my French teacher to help me follow the course. But I started getting deeply frustrated with trying to learn these languages. There were so many irregular verbs and spellings that didn’t seem to have any logic to them. The Russian course was a disaster. I couldn’t even get past the word for hello, which was a cocktail of consonants with no vowels to help you negotiate this mouthful. Unfortunately the word was also the name of the series. Then when I was about 12 or 13 my aspirations for the future took a different turn. In the middle of a lesson my mathematics teacher suddenly shouted “du Sautoy, I want to see you after the class”. I immediately thought “oh god, I’m in trouble”. After the class he took me round to the back of the maths block. “I’m really in trouble now” I said to myself. But then he got out his cigar (which he used to smoke around the maths block because the other teachers couldn’t stand the smoke in the common room) and he said “I think you should find out what mathematics is really about. Mathematics is not about all the multiplication tables and long division we do in the

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classroom. It is actually much more exciting than that and I think you might enjoy seeing the bigger picture”. He gave me some names of some books that he thought I might enjoy, which would open up what this world of mathematics was all about. One of the books was “A Mathematician’s Apology” by G.H. Hardy [1]. At the time I was very interested in music; I was learning to play the trumpet, I enjoyed the theatre and reading. Science hadn’t really captured my imagination. Education at the time in Britain had been deeply affected by C.P. Snow’s concept of the Two Cultures. You either got your hands messy with chemicals blowing things up in the lab or else you hung out with the arty crowd playing music and putting on plays. But never the twain shall meet. I’d felt drawn to telling stories not building bridges. But I also had a desire for things that made logical sense, for solving puzzles, for a rational perspective on the world. “A Mathematician’s Apology” suddenly opened up a bridge between these two competing desires, these two cultures. As I read Hardy’s book, there were sentences which revealed to me that mathematics had a lot in common with the creative arts. It seemed to be compatible with things I loved doing: languages, music, reading. Here, for example, is Hardy, writing about being a mathematician: “A mathematician like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” [1] Later he writes: “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” It was also Hardy’s book that first introduced me to C.P. Snow. “A Mathematician’s Apology” opens with a beautiful biography of Hardy written by C.P. Snow, who was a friend of Hardy’s in Cambridge. It documents the beautiful story of Hardy’s relationship with the Indian mathematical genius Ramanujan. It brought to life the mathematicians behind the mathematics in such a passionate way that it further piqued my interest in this world of mathematics in which Hardy seemed to be like an artist or a poet. For Hardy, mathematics seemed to be a subject with a sense of aesthetics. I was very curious to find out more about this strange subject. Hardy’s book had revealed to me the creative side of the subject, but it was another book that my teacher recommended to me that revealed a connection between mathematics and my other passion: languages. It was simply called “The Language of Mathematics” [2]. I’d been trying to learn all these languages, finding it very frustrating because of all the irregular verbs and illogical spellings. But this book seemed to be saying that mathematics was also a language and a very powerful language to be able to describe the world around us. And although it contained lots of surprises and strange twists and turns, everything made perfect logical sense. One language that particularly struck a chord in me was the language of symmetry. It seemed to be a language that transcended the two cultures’ boundaries. Symmetry pervades not just mathematics but so many different parts of our culture and the natural world around us. It is Nature’s language. In school we were told we were either scientists or humanists. I wanted to be in both camps. But I realized that the language of symmetry had the potential to bridge the divide of the two cultures.

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11.1 Symmetry: The Language of Nature Nature has been using symmetry as a means of communication ever since organisms have been interacting with each other. Symmetry is an important part of the evolutionary language. Symmetry marks out the intentional, something with design, something with meaning, or a message against the background noise. For example, the eyesight of a bee is extremely limited. As a bee flies through the air in search of food, it has to find some way to make sense of the onslaught of images it is bombarded with. Evolution has tuned the bee to recognize shapes full of symmetry because this is where it will find the sustenance that will keep it alive. The flower is equally dependent on the bee for its survival. It has evolved to form a symmetrical shape in the hope of attracting the bee. Nature enjoys hiding mysterious symmetries at the heart of many parts of the natural world – fundamental physics, biology, and chemistry all depend on a complex variety of symmetrical objects. The six-sided symmetry of the snow-flake, the eight-sided symmetry of the medusa, and the simple reflectional symmetry in the human face are some of the obvious manifestations of how much Nature loves symmetry. The symmetry in the natural world serves as a function for each object and is not simply a thing of beauty. The honeycomb built by bees is made up of hexagons only because this six-sided figure is perfectly adapted to packing things efficiently. The spherical symmetry of a bubble is related to the fact that this shape produces the lowest energy state. Nature is very lazy. The sphere, the most symmetrical threedimensional object, is the shape with the smallest surface area containing a fixed volume of air and hence has the least amount of energy. The sphere is also a very strong shape, which is why we use it for diver’s helmets and submersibles under the sea. Symmetry has been at the heart of our theory of the natural world ever since Plato associated symmetrical objects with the building blocks of Nature. He believed that the fundamental elements of earth, wind, fire, and water were intimately connected to the cube, octahedron, tetrahedron, and icosahedron, respectively. The dodecahedron, the fifth Platonic solid, Plato associated with the shape of the universe. This looks ridiculous for our modern age, but intriguingly symmetry is very much at the heart of the modern view of the molecular and cosmic world. Recent theories in cosmology even indicate that the universe has a dodecahedral shape. It is quite incredible that Plato struck on the idea several thousand years ago. But it is not only on the cosmic scale that one finds symmetry. Just as Plato believed, symmetry is also the key to the molecular world. The crystalline structure of many molecules reveal a whole host of different symmetrical possibilities. The structure of the diamond, for example, is built around the symmetry of the tetrahedron and the symmetry of the object is the key to the crystal’s remarkable strength. Molecular biology also exploits symmetry. The protein case that surrounds many viruses exploits symmetrical shapes like that of the icosahedron. The DNA at the heart of the virus essentially acts like a program to recreate copies of the virus. But often the length of the DNA is very short, so the program has to be very efficient.

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The symmetry of an object like the icosahedron provides a very simple program for constructing the whole of the object from a simple building block. If we dig even deeper we find symmetry also at the heart of the subatomic world. The menagerie of fundamental particles revealed by the physicist’s supercolliders only make sense when you start to see them as facets of some strange higherdimensional symmetrical shape. In contrast to the inorganic part of the natural world, animals find it quite difficult to achieve symmetry. This is part of the reason why we are attracted to people with a lot of symmetry, because if they can waste energy making themselves symmetrical, it is a reflection of good genes. Symmetry is an indication that someone will make a good mate. It is Nature’s language providing an animal with a way to declare: “my genes are so strong that I can waste time making myself symmetrical”. Symmetry is a reflection of a comfortable life. Tests have revealed that chickens in a battery farm which have to work very hard to survive create very unsymmetrical eggs. In contrast, free range chickens which enjoy the freedom of the farm lay much more symmetrical eggs than their battery farm cousins. Symmetry is a fundamental concept across the sciences. But it is also important to many parts of human culture. From architecture to music, from poetry to painting, symmetry underpins many of the structures used in the creative world.

11.2 Symmetry: A Blueprint for the Arts Ever since humans have been shaping the environment around them they’ve been attracted, like the bee, to things with symmetry. Some of the first symmetrical objects carved out by humans relate to their addiction to playing games. One of the oldest games discovered by archaeologists was found in the ancient Babylonian city of Ur and dates back to 2500 b.C. It is an early version of Backgammon but rather than cube shaped dice, players tossed tetrahedrons, shapes made from four equilateral triangles. Two of the four corners were painted black and the player counted the number of black dots pointing upwards. Symmetry is, of course, fundamental to creating a fair set of dice. No configuration should be favored over any other. At the same time as Babylonians were tossing tetrahedral dice in Ur, stone balls were being carved with symmetrical patterns by Neolithic tribes in northern Scotland. These stone balls have circular patches carved into the sides. The sculptors experimented with the different ways these patches can be arranged around the sphere. For example, mirroring the discovery of the Babylonians, the tetrahedron provides a way to place four circles in a perfectly symmetrical arrangement. But the Neolithic artists discovered a range of other sophisticated configurations related to the other Platonic solids. We had to await to the Ancient Greeks for a systematic analysis of how many different dice can be built. It was Plato’s friend Theatetus who first provided a proof that the five solids – the tetrahedron, the cube, the octahedron, the dodecahedron,

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and the icosahedron – are the only ways of piecing symmetrical faces together where each face has the same shape and all the vertices of the shape look the same. The great Greek mathematician Archimedes developed the problem of symmetrical shapes by considering how many different shapes can be built when more than one symmetrical shape can be used as a face. For example, the classic football is built from piecing together pentagons and hexagons. Again no vertex looks different from any another so the shape has a lot of in-built symmetry. Archimedes discovered that there are 13 different ways to build such solids, now called the Archimedean solids. But the work in which he discussed his constructions was lost in antiquity. What is striking is that it was artists in the Renaissance who rediscovered these 13 shapes. The challenge of painting three-dimensional solids was a perfect testing ground for artists exploring the new techniques of perspective. Thus we find symmetrical shapes appearing in the paintings of Piero della Francesca, Leonardo da Vinci, Luca Pacioli, and Dürer. For the first time, drawings were made of the different ways different shaped faces can be pieced together to make up the 13 Archimedean solids. It is the connections between art and mathematics, a union of the two cultures, which pushed both on to a new level. One can see this dialog between art and mathematics at work again at the beginning of the 20th century. Then mathematicians began to explore the symmetry of shapes in higher dimensional spaces, beyond the three spatial dimensions in which we live. At the same time as a new mathematical language is being developed to explore the shape of hyperspace, the cubist movement is developing techniques to express visually on a two-dimensional canvas notions of multi-dimensional views of the world. Some artists very specifically sought out scientific and mathematical influences. Dali embeds many modern mathematical ideas in his work. For example, the painting of the crucifixion of Christ painted in 1954 depicts Christ being killed on a four-dimensional cube which has been unwrapped into three dimensions. Just as a three-dimensional cube can be unwrapped into six square faces sitting in a cross configuration in two dimensions, the hypercube unwraps into eight cubes arranged in as an intersection of two crosses. The symmetry of the four-dimensional cube is also key to the Grande Arche at La Défense in Paris. Dutch architect Johann Otto von Spreckelsen chose a different way to represent the shape in the three-dimensional world. Picking up on the way the Renaissance artists would represent a three-dimensional cube on a two-dimensional canvas by drawing a square inside a square, von Spreckelsen’s arch depicts a small cube embedded inside a larger cube (Fig. 11.1). But symmetry is not a concept that is universally loved by those practicing the creative arts. This is Thomas Mann in The Magic Mountain describing a snow flake: “he shuddered at its perfect precision, found it deathly, the very marrow of death.” For some artists the rigidity of perfect symmetry boxes one in, forces things to be too predictable. Many artists, for example, dismiss Escher as a second rate painter because his obsession with things symmetrical cramped creativity, leaving no room for artistic expression. But this reaction to symmetry in the arts is certainly not universally held. For other writers like the poet Pushkin, there is no escaping the importance of the con-

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Fig. 11.1 A three-dimensional cube is represented in twodimensions by drawing a square within a square

cept: “Symmetry is a characteristic of the human mind” he wrote in a letter to Prince Vyazemsky. The French poet Paul Valéry also recognized that symmetry is at the heart of the way we perceive our surroundings: “The universe is built on a plan, the profound symmetry of which is somehow present in the inner structure of our intellect.” Moreover, it is perhaps not surprising that Pushkin and Valéry are drawn to express their art through poetry rather than prose. The rigid logic of its rhyming structure and its rhythmic patterns make classical poetry one of the literary forms that most resonates with creating mathematics. Indeed, the Persian poet Omar Khayyam was also a masterful mathematician, exploring the intricacies of solving cubic equations. His famous Rubiyat uses a poetic form full of mathematical structure and patterns. The two cultures are embodied in one person.

11.3 Music: The Process of Sounding Mathematics The art form that perhaps resonates most closely with mathematics is music. Symmetry is a major theme across musical composition. The master at weaving mathematics through his music was, of course, Bach. Indeed Bach’s student Mizler used to describe Bach’s music as “the process of sounding mathematics.” A classic example of Bach’s mathematical mind at work can be found in the Goldberg Variations. It is a veritable song to symmetry. Published in 1741, the piece consists of 32 movements. It begins and ends with a simple aria, which sets down the motif that Bach will twist and turn over the 30 variations that are sandwiched between the arias. By coming back to the aria at the end of the variations Bach has already evoked one of the most symmetrical shapes in the mathematicians portfolio: the circle. Arranging the 32 movements around a circle one finds that the movement which is diametrically opposite the first aria is called The Overture by Bach, a name usually reserved for the beginning of a piece.

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It immediately highlights the fact that a circle does not really have a beginning or an end. Where the mathematics comes to the fore is in Bach’s use of symmetry to establish variations on a theme. The idea of a variation almost cries out for a mathematical interpretation. The variation should be a transformation of the original theme. So there should be some structural relationship between the two phrases almost like the logical connection between two lines of a proof. The 30 variations on the aria are divided into ten groups of 3 and every third variation is a canon. It is in the sequence of canons that you really hear Bach’s use of symmetry at work. The idea of a canon became a popular technique in Baroque music to create two lines of music where the second voice was a copy of the first voice, only that it starts playing some period after the first voice starts. Anyone who has listened to a broadcast via an analogue radio and a digital radio would have inadvertently set up a canon because the digital signal very often suffers a delay as the computer takes time to reassemble the signal. The classic canons are, of course, children’s songs like Frère Jacques. But this simple idea of simply repeating the second voice was limited. So composers started looking for other ways to play with the second voice in addition to a simple translation in time. And it is here that Bach’s mathematical skills are let loose on the basic theme. One of the first ideas he has is to shift the second voice in pitch as well as time. So with each new canon the second voice climbs up a note. As we move through the sequence of canons, the second voice gradually spirals up until something rather magical happens on the eighth canon. Suddenly we hear the octave and it is almost as if the canons have joined up in a second circle. In mathematics this evokes the image of a torus or bagel shape, which is a circle spun through a second circle. Or perhaps the spiral is a better image where the canon lines up with the first canon but with an octave between. Not content with these shifts in pitch and time, Bach also exploits mirror symmetries. In the fourth canon, as the first voice ascends, the second voice is reflected in the horizontal line of the stave and descends. There is even symmetry at work in Bach’s choices of rhythmic structure. In each canon, Bach makes a choice of two beats, three beats, or four beats in the bar. Alongside that, he also decides how this beat is divided: into quavers, triplets or semiquavers. There are nine different combinations one can make with these possibilities and Bach makes sure that the first nine canons cover all the bases. It is as if Bach is spinning through all the possibilities on a combination lock where one wheel controls the beats in the bar and the other how the beat is divided. For example, variation 24 or the eighth canon combines three beats in the bar with the beat divided into triplets. With so much symmetry running through the rhythmic structure, pitch, and musical phrases, Bach then surprises his audience by completely throwing the sense of symmetry when it comes to the tenth canon. It is called a quodlibet or musical joke and is a contrapuntal piece based on two common folk tunes of the day. The breaking of the symmetry actually goes to highlight how much symmetry has run throughout the rest of the piece.

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Many artists enjoy this device of breaking symmetry. On a trip to the Buddhist temples in Nikko in Japan, I was quite struck how the architects had created eight beautifully decorated symmetrical columns at the entrance to the temple, except that one of the columns had been set upside down. This was no building error but part of the design. The 14th century Japanese Essays in Idleness by Yoshida Kenko articulates the ethos at the heart of the gate in Nikko: “In everything. . . uniformity is undesirable. Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth.” Bach’s quodlibet is like the upside down pillar. We had to wait until the 20th century to produce composers who would pick up Bach’s mantle of music as the process of sounding mathematics. Perhaps my favorite composer that enjoyed using mathematics in his music is Olivier Messiaen. Messiaen was highly influenced by Schoenberg’s 12 tone system where scales were thrown away and substituted with permutations of the 12 notes of the chromatic scale. One of the set of permutations Messiaen explores in the 12 tone system gives rise to an incredibly sophisticated mathematical setting, which was only discovered towards the end of the 19th century. In his Ile de Feu 2 for piano, he chooses two arrangements of the 12 notes, which are obtained by performing a rather special shuffle of cards called the Mongean shuffle. Effectively you take the pack of cards in one hand then reorder them by alternatively placing each card under or over the stack you build in the other hand. These permutations are the generators to create a special symmetrical object called the Mathieu group M12. This symmetrical object is one of the building blocks of symmetry and has 95,040 different symmetries. Its discovery was the first inkling of how complex the world of symmetry would turn out to be. But these shuffles of the 12 notes are also at the heart of Ile de Feu 2. Quite independently Messiaen realized in sound one of the most intriguing mathematical objects discovered in the 19th century. Although Messiaen was second to the discovery of this mathematical object it still illustrates that composers and mathematicians are on remarkably similar wavelengths with sensitivities to the same sort of structures. At the heart of Schoenberg’s 12 tone system are the symmetries of a rectangle, which are used to produce variations on the 12 note theme chosen by the composer. The Greek composer Iannis Xenakis, a student of Messiaen, took this idea one dimension higher, exploiting the symmetries of the three-dimensional solids to generate interesting variations. In several compositions Xenakis used the symmetries of the Platonic solids as a tool for composition. Nomos Alpha is a piece for solo cello that seeks to capture the shape of the cube in sound. By using the 8 vertices of the cube as markers for different sound elements for the cello and then performing symmetries on the cube, Xenakis produces a sequence of musical parameters for each section of the piece. In fact, Xenakis uses two cubes that spin in different ways. One cube keeps track of things like dynamics, whilst the other takes care of sound textures, for example, pizzicato or glissando sounds.

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Each section consists of eight subsections obtained from cycling round the eight points on the cubes in a set order. The relationship between each section is provided by the symmetries that permute the corners of the cube around in different ways. So the cube actually restricts the possible permutations because the rigidity of the cube means that I can’t rearrange the eight vertices in any order I want. And this is why the structure of the piece will reflect in some mysterious way the rigidity of the cube. Xenakis extended his idea of symmetries in music in his composition Nomos Gamma for 98 instruments in which the pyramid and other shapes join the cube in designing the structure of the piece. Perhaps its not surprising that Xenakis was drawn to three-dimensional geometry for inspiration. Xenakis originally trained as an architect, another profession that exploits the power of mathematics to generate interesting structures. He worked closely with Le Corbusier in Paris, and the Philips Pavilion they built for the 1958 International Fair in Brussels was inspired by hyperbolic geometries created by mathematicians during the 19th century. But Xenakis also believed that such structures could be realized and translated into many different artistic media. In his first major composition Metastasis, Xenakis translates this hyperbolic geometry into music. The score for the piece looks remarkably like the blue prints he made for the Brussels pavilion. The piece is scored for 12 wind instruments, percussion, and 46 strings. Each instrument has its own part. It is the strings that are ultimately responsible for exploring the geometry at the heart of the piece by using glissandi to chart the shape of this hyperbolic space. Architects like Xenakis have been plundering the mathematicians’ box of symmetrical shapes for inspiration for centuries: from the ancient pyramids in Egypt to the spherical Imax cinema in La Villette in Paris, from dodecahedral flats built in Ramot in Israel (a nightmare to buy furniture for) to the shadow of a fourdimensional cube at La Défense in Paris. Every architecture firm now has a cohort of mathematicians who can exploit the twists and turns of non-Euclidean geometry or cut up toroidal surfaces to make up the new landscape of shapes that adorn our cities. There is probably one building more than any other that pushes the mathematical possibilities of symmetry to the limits: the Alhambra in Granada. Sitting like a jewel atop the hills that make up the Andalusian town of Granada, it is one of the most beautiful palaces built by the Moors in Spain. What draws most visitors to the Alhambra are the elaborate stucco decorations that adorn the walls of the palace. Although each room provides the artist with yet another canvas to express their art, the writings of the Muslim hadith, which interpret the Koran, have put some limitations on the artist. It is forbidden under Muslim law to depict the image of any living creature. So instead the artist has been forced to express the majesty of creation through more geometric games. And that is what makes the Alhambra such a feast for a mathematician making the journey to southern Spain. But at the heart of the Alhambra are some of the big questions of symmetry. What is symmetry? What are the limitations of symmetry? How can we say that two walls that look completely different actually have the same symmetries?

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To answer these questions we had to wait for a new mathematical language to be developed, which would enable us to get to the heart of this fundamental concept that runs through the two cultures.

11.4 The Mathematics of Symmetry It was a French revolutionary called Evariste Galois who would create this new language to articulate the subtleties of the world of symmetry. But his mathematical ideas were so original that no one could understand them at first. Frustrated by the disinterested reception of his work, he turned instead to revolutionary politics – with tragic consequences. On the 30th of May 1832 in the early morning mist a peasant, walking by a pond on his way to market on the Rive Gauche in Paris, discovered Galois writhing in agony. The young man had been shot with a single bullet to the stomach. It was clearly a dueling wound. Duels were a common way of resolving disputes over women, politics, insults, even geese, in 19th century Europe. Local newspapers would often carry notices of up-coming duels and their terms. It is still not clear what the duel was about. Was it politics? Was it a shared lover? Was it even a sham duel staged to spark a new revolution? Was it the authorities getting rid of a troublesome revolutionary? We will never know. He was taken to the Cochin hospital where he died a day later, refusing to take the last rites offered to him by the hospital’s priest. “Don’t cry” he said to his brother who was with him during the last hours “I need all my courage to die at 20.” But it is not for his contributions to revolutionary politics that Galois is now remembered. For several years Galois had realized that he had made a stunning mathematical breakthrough. But despite him sending his work to the leading mathematicians of the day no one could penetrate his ideas. The night before the duel he recognized that this was probably his last chance to articulate his ideas. He stayed up the whole night trying to explain his breakthrough in a letter to a friend. As dawn broke, he packed up his manuscripts and left to meet his destiny. Maybe the lack of sleep contributed to his bad aim. But contained in that letter, which some have described as one of the most important pieces of writing in the history of mathematics, was the beginnings of a new language called group theory that would finally help mathematicians to articulate this universal concept of nature: symmetry. At the heart of Galois’s vision is the belief that symmetry is about motion, in contrast to Thomas Mann’s view of it as deathly and without life. For mathematicians, post Galois, the symmetries of an object should be thought of as the actions you can do to an object so that the object looks the same as before you touched it. I like to call a symmetry “a magic trick move”. For example if I take an equilateral triangle and get you to close your eyes then a symmetry is essentially a way to pick up the triangle and place it back down again so that when you open your eyes again you cannot tell whether the triangle has moved.

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For example, consider the symmetries of the six pointed star-fish and the equilateral triangle depicted in Fig. 11.2. To try to identify the symmetries of an object, we can draw an outline around the shape and then ask: how many different ways can I pick up the shape and place it down inside its outline? The six-pointed starfish has lots of rotational symmetry because there are lots of ways I can spin the object which fit it back inside its outline. There are five different rotations that I can make of the starfish. Using the letters attached to the tentacles I can articulate these symmetries via a language for symmetry. For example, the rotation that moves tentacle a to tentacle b we will call B. The rotation that takes tentacle a to tentacle c we will call C. The five rotations therefore have the names B, C, D, E and F. There is one additional symmetry that mathematicians include. If symmetry consists of the things that I can do to an object which make it fit back inside its outline, then actually leaving the object where it is also deserves to be included. So we shall call this rotation A because it sends tentacle a to a.

Fig. 11.2 Six-pointed starfish and equilateral triangle

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These six symmetries A, B, C, D, E, and F are all the symmetries of the starfish. By putting a twist on the tentacles I have destroyed what most people commonly identify as symmetry, namely reflectional symmetry. The equilateral triangle on the other hand does have this more obvious reflectional symmetry. There are three lines of reflectional symmetry. For example, I can pick up the triangle and flip it over so that points y and z exchange and x stays where it is. Similarly, the other two reflectional symmetries fix y and z, respectively, whilst exchanging the other two vertices. Again we can give these reflectional moves names. The reflection fixing x we call X. The reflection fixing y we call Y and the reflection fixing z we call Z. In addition to these three reflectional symmetries we have two rotational symmetries. One spins the triangle by a third of a turn anti-clockwise. Call this symmetry move U. The other spins the triangle clockwise by a third of a turn. Call this symmetry move V. As with the starfish, there is an additional symmetry which consists of just leaving the triangle where it is. Call this symmetry I. Both objects have six symmetries. The symmetries of the starfish consist of the six rotations called A, B, C, D, E, and F. The symmetries of the triangle we have named I, U, V, X, Y, and Z. So given that these two objects have the same number of symmetries, should we say their symmetries are identical? What mathematicians realized is that it is the interactions between these symmetrical moves that is as important as the moves themselves. The essence of the symmetry of an object is captured by seeing what happens when you combine one symmetrical move with a second move. The combination produces a third independent symmetry. So, for example, in the starfish if I perform rotation B and then combine it with rotation C. I have a rotation that actually is the same as rotation D. Mathematicians realized that one could define a sort of multiplication or calculus expressing the way these symmetries interact. So, for example, one could write the observation about how these symmetries combine as an equation: B∗C = D . Given that the starfish has six symmetries, one could record in a 6 × 6 table the different ways that all the symmetries interact. Here is the table describing the interactions between the rotations of the starfish: ∗

A B

C

D E

F

A

A B

C

D E

F

B

B

C

D E

F

A

C

C

D

E

F

A

B

D

D E

F

A B

C

E

E

F

A B

F

F

A B

C

C

D

D

E

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So the entry in the ith row and jth column of the main table shows the name of the symmetry obtained by doing the ith symmetry followed by the jth symmetry. It is this table describing the grammar of the interactions of the symmetries that captures the identity of the symmetry of the starfish. By comparing this table with the table for the six symmetries of the equilateral triangle we can finally reveal the different symmetrical identities of the two objects. The table expressing the symmetries of the triangle is given below: ∗

I

U V X Y

Z

I

I

U V X Y

Z

U

U

V I

Z

X

Y

V

V

I

Y Z

X

X

X

Y Z

I

V

Y

Y

Z

V I

Z

Z

X Y U V

U X

U

U I

Mathematicians now have a way to say that two objects have the same symmetries. If there was a dictionary that translated the names A, B, C, D, E, F for the symmetries of the starfish into the names of the symmetries I, U, V, X, Y, Z of the triangle such that the two tables were identical, then we would say the symmetries of each object are identical. In this case, there is a way to reveal that such a dictionary is impossible. In the table for the starfish the grammar has a symmetry. It does not matter in which order you do the symmetries, the answer will be the same. For example, C∗D = F = D∗C . We say in this case that the group of symmetries of the starfish is commutative. But this is not true for the symmetries of the triangle. Now it is very important in what order the symmetrical moves are performed. For example, U∗X = Z whilst X∗U = Y . Or in other words, if we rotate the triangle anticlockwise then reflect in the vertical line through the triangle this combination leaves the triangle in a different position than if first I’d reflected the triangle and then spun it. It is the language of group theory first developed by Galois in the depths of revolutionary France that enables modern scientists to articulate the concept of symmetrical identity. It is with this language that mathematicians at the end of the 19th

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Fig. 11.3 These two mosaics from the Alhambra look very different but they have the same symmetry

century were able to express the fact that the two designs found in the Alhambra (Fig. 11.3), although physically very different, have identical symmetries. Once again, symmetry consists of the ways I can lift these tiles, move them and place them back down again so that they fit perfectly in the original outline. In each picture there are points around which I can spin the image by a sixth of a turn and see all the tiles magically align up. There are other points around which I can make a third of a turn and a half turn and see the images lining up. Neither wall contains reflectional symmetry. So although the images used by the artist on each wall are very different, mathematicians can identify the group of symmetries as identical. This new language enabled mathematicians to articulate the abstract concept of symmetry underlying a physical object. It is almost like discovering the concept of number: I can show you three bottles and three glasses and although they look very different, the threeness is something abstract that is common to both of them. Both the glasses and the bottles are expressions of the concept of the number 3. The same thing happens with the idea of symmetry thanks to Galois’ work. Two objects can look very different, but can still be expressions of the same underlying group of symmetries.

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The staggering thing is that using this language we can now actually say how many groups of symmetries are possible on a wall. There are in fact only 17 different underlying groups of symmetries that can be depicted on the two-dimensional wall, a fact that was proven at the end of the 19th century. There can be a lot of different expressions of these symmetries. But underlining each one of them is one of these 17 different groups of symmetries. Any attempt by the medieval artists to conjure up a new group of symmetries was doomed to failure. Whatever games they played, the wall would have symmetry identical to one of the 17 on the mathematicians list. Although many artistic and scientific disciplines have played with the ideas of symmetry, it is mathematics that provides a language to navigate this world, realize its limitations, but also provided access to new symmetrical objects that no one had previously dreamed imaginable. Indeed Galois’ work led to the realization that just as molecules can be broken down into atoms like sodium and chlorine, or numbers can be built out of the indivisible primes, symmetrical objects can also be decomposed into indivisible symmetrical objects. Christened simple groups, these symmetrical objects are the atoms of the world of symmetry. Galois’ breakthrough meant that it might be possible to create a periodic table of symmetry containing a list of all these simple groups of symmetry. Such a classification had the prospect to be as influential as the periodic table of elements has been to chemistry. Prime numbers, the building blocks of all numbers, are behind some of the first simple groups to feature in the classification. For example, take a flat 15-sided shape or polygon. Galois understood that the symmetries of this 15-sided polygon can be built out of the symmetries of two smaller shapes sitting inside the large shape, namely a pentagon and a triangle. How can the 15-sided polygon be rotated through a 15th of a turn using the rotations of the pentagon and triangle? First rotate the pentagon by two fifths of a turn; then pull back in the opposite direction by rotating the triangle by a third of a turn. The combined effect is a rotation of a 15th of turn. The reason this works is because: 1 2 1 = − . 15 5 3 So the group of symmetries of the 15-sided polygon are built out of the symmetries of a pentagon and a triangle. But the rotations of these prime-sided shapes cannot be broken down. So just as prime numbers are the building blocks of all numbers, it turns out that prime-sided shapes are some of the first building blocks of the world of symmetry. But it turned out that prime-sided shapes were not the only indivisible symmetrical shapes. For example, take the classic football made up of pentagons and hexagons. This shape has 60 rotational symmetries or 60 magic trick moves, which rearrange the shape so that all the pentagons and hexagons line up again. 60 is a very divisible number. It is one of the reasons that the Babylonians used it as the base for their number system and why we have 60 minutes in the hour. But despite the high divisibility of the number, Galois proved that the group of 60 rotations of the football are as indivisible as if it were a prime sided shape. Sure, there are rotations of a pentagon sitting as subset of the symmetries of the football.

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But try to divide by the symmetries of one of the faces and the result makes no sense. There are no shapes whose symmetries can be combined with those of the pentagon to realize the symmetries of the football. The symmetries of this shape turned out to be the tip of the iceberg. There are many other shapes whose groups of symmetries are indivisible but to create these shapes one has to move away from the physical world of three dimensions and enter the abstract world of hyperspace. For example, the symmetries of hypercubes give rise to a new family of indivisible shapes.

11.5 Symmetry in Hyperspace What do mathematicians mean by a cube in four dimensions? To play with such shapes, mathematicians needed to create a new language. In the 17th century, Descartes produced a dictionary that changed geometry into numbers. This dictionary is used by everyone who employs a map or negotiates a route with their SAT NAV. Every location on the surface of the earth can be translated into a pair of numbers which denote the distance east–west and north–south from the origin of this map located at Greenwich. So, for example, the GPS location of the Istituto Veneto di Scienze Lettere ed Arti is (45.43, 12.33). A geometric position translated into numbers. If I wanted to locate my place in space rather than on a two-dimensional surface I would need to use three numbers. Using these coordinates we can translate shapes into numbers. A square, for example, can be described by the coordinates of its corners: (0, 0), (1, 0), (0, 1), and (1, 1). Mark these locations on a piece of graph paper and you have the corners of a square. The corners of a cube are obtained by adding an extra dimension. So the eight corners of the cube can be described by the eight coordinates of numbers starting at (0, 0, 0), (1, 0, 0), (0, 1, 0) . . . continuing to the extremal point at (1, 1, 1).

So what about a four-dimensional cube? Although the pictures run out, the numbers do not. One side of this dictionary carries on. So a mathematician will describe a four-dimensional cube as the object whose corners are given by the coordinates with four numbers starting at (0, 0, 0, 0), (1, 0, 0, 0) and stretching out to the furthest

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point at (1, 1, 1, 1). Using the numbers I can explore the geometry and symmetry of this shape. So, for example, a four-dimensional cube or what is known as a tesseract has 16 corners, 32 edges, 24 square faces and is constructed out of 8 cubes. The symmetries of this shape turn out to be related to a new infinite family of indivisible symmetries to add to the prime sided shapes. With this new language to navigate hyperspace, mathematicians began compiling a stunning list of new symmetrical objects that form the building blocks of symmetry. It culminated in the 1980s with the discovery of a strange object called the Monster which is first seen when one enters the 196,883-dimensional space and has more symmetries that there are atoms in the sun. But with its discovery came the realization that mathematicians had finally discovered a complete list of all the indivisible symmetrical objects possible. The periodic table of symmetry called the Atlas of Finite Groups was complete. It represents one of the most incredible intellectual achievements of the 20th century. But the story is not over by any means. My own research is dedicated to understanding what symmetrical objects you build from these indivisible building blocks. We have a list of the atoms of symmetry, but what are the molecules we can synthesize from this list? Although I never realized my dream to learn Russian and become 007, by learning this mathematical language of symmetry I have found myself like a secret agent navigating a different land. I grew up in an educational system in Britain that seemed to encapsulate the division of the two cultures articulated by C.P. Snow. And yet for me this language and the concept of symmetry has provided a powerful bridge to unite the two cultures. But unless we try to communicate the power of the mathematical language to those outside our academic disciplines we are in danger of creating a beautiful bridge that stands in the middle of the river disconnected from both river banks. The problem of the isolation of mathematics in our culture is perfectly encapsulated by one of the stories C.P. Snow tells very early on in his essay The Two Cultures. He describes Smith, an Oxford don who is going to visit Cambridge to dine with a friend in one of the colleges. C.P. Snow writes: “He addressed some cheerful Oxonian chit-chat at the one opposite to him and got a grunt. He then tried the man on his right hand and got another grunt. Then, rather to his surprise, one looked at the other and said ‘Do you know what he is talking about?’ ‘I haven’t the least idea.’ At this, even Smith was getting out of his depth. But the President, acting as social emollient, put him at his ease by saying ‘Oh those are mathematicians! We never talk to them’.” And this is one of the problems with our subject. The languages we create are extremely powerful to take one into worlds of 196,883-dimensional space, to spin monstrous snowflakes with more symmetries than there are atoms in the sun. But Goethe expressed his frustrations with mathematicians long before C.P. Snow’s anecdote: “Mathematicians are like a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.” But can we draw those who are not equipped with this language into this world?

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This is what Hardy was trying to do in his apology, and very effectively too. His book certainly sucked me into the mathematical world. But although Hardy’s book is a beautiful example of a mathematician crossing the boundaries and communicating across the Cambridge high table, he did not make it easy for other professional mathematicians to follow in his path. Although Hardy’s book and his other writings did so much to popularize the subject of mathematics in the early 20th century, to break down the sense of two cultures, the opening sentence of his book actually did more harm in cementing the reluctance of those doing science to move out of their ivory tower. “It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics and not to talk about what he or other mathematicians have done.” This opening sentence is partly responsible for why there are two cultures. Hardy made it very difficult for a practicing mathematician to transcend his or her subject and explore connections with other disciplines, to do anything other than maths. But the climate is slowly changing. Mathematics is as much about communication as it is about discovery. An idea can hardly be said to exist unless it is brought alive in the minds of others. I have spent my academic life creating mathematics. But in tandem with doing mathematics I have also dedicated my energies to paying my mathematics teacher back for opening up the magical world of mathematics to me. Through articles in newspapers, series on the radio, programs on TV, and writing popular science books, I have tried to tell some of our best mathematics stories, to prove Hardy wrong that you can create maths and talk about it at the same time. One of the most exciting recent projects has been working with the theatre company Complicite on a new play that tries to unite the two cultures of maths and drama. Called A Disappearing Number it brings alive the mathematical story of Hardy’s collaboration with the Indian mathematician Ramanujan, the very story that C.P. Snow wrote about in the introduction of “A Mathematician’s Apology”. It is through such projects that the two cultures can find common ground to share their different perspectives on the world. Finding Moonshine: A Mathematicians Journey Through Symmetry [3] contains more details of the mathematical story of symmetry. My website www.maths.ox.ac.uk/∼dusautoy includes information about my work to bring maths to the masses.

References 1. G.H. Hardy: A Mathematician’s Apology (Cambridge University Press, Cambridge 1940) 2. F. Land: The Language of Mathematics (John Murray, London 1960) 3. M. du Sautoy: Finding Moonshine: a mathematician’s journey through symmetry (Fourth Estate, London 2008)

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Discussion G. Veneziano: Perhaps I could add something from the point of view of physics: you mention the world of elementary particles and their symmetries and what we use very much is not the concept of symmetry breaking in the way the biologist speaker was using it, but that of spontaneous symmetry breaking. Namely you have a system the description of which is completely symmetrical. For instance, you have the symmetry of the sphere, take a ferromagnet, but the ground state is not symmetric because it picks up one direction of space. Of course which direction it takes is irrelevant, because all directions by the symmetry are equivalent, but yet there is a difference that a direction is taken. So we make a lot of use of that concept, of symmetries of course, which can be either broken explicitly, namely they are not perfect and then we can do an expansion of the solution in terms of the small breakings or they are broken in that more subtle way which is called spontaneous. I do not know if you have anything like that in mathematics. M. du Sautoy: Each symmetrical object, the tiles in the Alhambra, for instance, are examples of symmetry breaking: you may start with the boring tiles you may have in a bathroom, maybe you have a very beautiful bathroom, mine is very boring, it only has squares in it, and that has a lot of symmetry. But then the interesting thing for me as a mathematician is to break that symmetry, to put colors on the thing, or to make twists in it, so I break a reflectional symmetry and that gives rise to new groups of symmetries, and that is what is exciting. For example, you do it with Lie groups. They are a sort of group of symmetry that you use a lot in physics. How were Lie groups discovered? They were discovered by Galois who said: Let’s look at how you can permute seven things around, seven playing cards. You can permute the playing cards any way you want and you get a group with seven factorial symmetries. But he said: Let us put more structure in this thing. Suppose these seven cards are actually lying in some geometry and now let’s act on them by something, which turns out to be a matrix. What he discovered by putting some structure in there are these beautiful new examples of the first Lie groups. As for me, I like breaking symmetry because it produces new exotic things, which are still symmetrical but in different ways. That is why the artist Xenakis used the symmetries of the cube to do his variations in the piece I played you. He had eight different themes and what he does throughout the piece is to permute those themes around. But he does not just want to permute them arbitrarily, he wants to put restrictions on it, so he says I want to put those variations, those musical ideas, on the eight points of the cube and then I’m going to permute the cube around. Actually, there are only 48 different arrangements of these themes rather than the 8 factorial that there could have been if he had not looked for that structure. So you may think that I do not like breaking symmetry because I love symmetry, but it actually produces its own beautiful symmetry, which is how we got things like the Lie groups. . . S. Wise: What is the relationship between the mirror and symmetry?

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M. du Sautoy: Most people’s initial response to the question of what is symmetry is to think of mirror symmetry. The left–right reflectional symmetry that we see in many animals or the reflection thrown up in the water. But the concept of symmetry is much richer than just that of the mirror. Symmetry is about the internal relationships in an object or structure. So, for example, rotational symmetry goes beyond the mirror and shows how spinning an object can reveal connections across the structure. Many flowers have no mirror symmetry but the petals have rotational symmetry. Once mathematics had produced a language to articulate more clearly an answer to “what is symmetry” we started to see that it is a concept that underlies a whole plethora of structures both physical and abstract. G.O. Longo: Is symmetry in need of an observer who recognizes it and extracts potentialities from it? Has symmetry to do with redundancy and meaning? Like guessing a whole circle from half of it? And saying: oh yes, I understand, it is a circle. M. du Sautoy: I am a Platonist at heart so I do not believe that mathematical concepts like symmetry need an observer or human mind for them to exist. Of course it is humans who singled out objects like the dodecahedron or the Monster, a symmetrical object in 196,883-dimensional space, as important objects to look at. But there is a strong sense that they were there all along waiting for someone to discover them. However, where an observer is important in the story of symmetry is perhaps in explaining why there is so much symmetry across the natural world. In evolutionary biology, symmetry has certainly been a language that animals and plants have used to communicate important messages about their genetic heritage. The question of redundancy in symmetry is an important one because that is the key to how symmetry is used in many error-correcting codes exploited by the telecommunications industry. The use of symmetry has allowed codes to be written which are highly efficient at correcting errors without having to transmit too much extra information. The key is exactly the point you make about the incomplete circle. Symmetry allows you to fill in what is missing. K. Moebius: In biology complex protein systems often exhibit remarkable degrees of symmetry – at first glance. Their biological function, however, relies on subtle breaking of this symmetry. Can you quantify such small deviations from perfect symmetry by “simple” mathematical tools? G. Veneziano: In physics we use a lot of symmetries, whether exact or broken. But there is also the concept of a spontaneously broken symmetry in which the ground state does not respect the symmetry of the theory. Is there something analogous in mathematics? M. du Sautoy: I will answer these two questions together because they are related to the same ideas. The mathematics of symmetry is full of interesting objects that in some sense arise from symmetry breaking. The sphere might be thought of as a perfect symmetrical object. If one then traces the outlines of a dodecahedron on the surface and ask for the symmetries to preserve the dodecahedral structure, then we have lost symmetry. But the resulting symmetry group is equally interesting and im-

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portant. The tiles in the Alhambra can be thought of as symmetry breaking because the inclusion of additional structure or design limits the possible symmetries. For example, many of the walls have no reflectional symmetry. All the permutations of a pack of cards can be thought of as a group of symmetries, but often it is the permutations generated by special shuffles that can produce interesting new symmetries. The mathematics of group theory is about classifying all the possible symmetrical structures and so should be a powerful tool in any system, whether it be in biology, physics, or chemistry, where elements of symmetry are important. Whether it is a “simple” tool depends on your perspective. H. Fogedby: Do you have any comments on self-similar objects like fractals? M. du Sautoy: Fractals are objects that have infinite complexity at all scales. If you magnify them then they never seem to get simpler. This is a good example of an unexpected sort of symmetry, namely symmetry of scale. When you magnify a fractal, sometimes you can see the same shapes appearing on a smaller scale. The internal relationship in the object between the large and small scale is a sort of symmetry. Instead of reflecting or rotating, we are looking at an invariance by dilation. So symmetry can have an important role to play in understanding the internal structure of a fractal. G. Setti: I wonder how far our minds have been programmed to recognize symmetry in the course of evolution and by abstraction then it went into more advanced concepts of symmetry. M. du Sautoy: I think this question gets to the heart of the development of the whole of mathematics. Survival goes to those who can understand and predict the behaviour of their environment. The concept of number grew out of a need to decide whether to fight or fly: if there are more of them than us then run away. Also in hunting prey, a sense of geometry is essential in judging whether you are in range to pounce. Symmetry is a basic evolutionary language too, communicating important messages of genetic superiority. Also in the chaos of the jungle something with symmetry is likely to be an animal that you can eat or that might eat you. So you had better take notice. But the wonderful thing about mathematics is that as soon as we had become sensitive to these important concepts of number, geometry and symmetry we were exploring beyond what was entirely necessary for our evolutionary survival. A more playful side of human nature kicks in. But of course the remarkable thing is that the modern world is built on the implications and abstract explorations that we have made throughout the history of mathematics. E. Carafoli: In ancient Greek the meaning of “symmetron” was best rendered by expressions that signified harmony, good proportions, stability, rhythm, all of them, in a way, connected with beauty. On the other hand, others have claimed that symmetry has something constrained, something immobile to it, whereas asymmetry has more freedom, more play and is thus more dynamic and more attractive. Any comments? M. du Sautoy: Symmetry has a very important role to play even if you are creating something with asymmetry because symmetry is important in setting up the expec-

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tations that as an artist you then will play with. The Goldberg Variations end with Bach’s musical joke or quodlibet where the symmetry is destroyed. But it only has such strength because of the constraints that have been introduced throughout the rest of the piece. I think part of the problem of artists’ perceptions of symmetry is that they are not aware of the wealth of rich, complex and unexpected structures that we have discovered in our exploration of symmetry. Symmetry is so much more than simple left–right reflectional symmetry or the rather tedious repeated tilings that one sees in Escher’s work. The language of group theory developed by Galois opened up the world of symmetry to reveal it as a rich playground full of unexpected and surprising objects that are dynamic and exciting and full of life.

The Dynamics of Beauty

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Giuseppe O. Longo

La beauté est là, dès l’aube. Levée bien avant nous. Fidèle, elle attend. Son haleine se répand dans le moindre silence, dans l’air autour des amandiers. Elle attend que s’ouvre en nous le chemin où elle pourra venir sans se blesser. Elle attend des heures entières, et le mouvement de son attente est celui du jour qui pointe, fleurit puis décline, mourant à nos pieds, méconnu, délaissé. Chaque jour ainsi, quelqu’un vient, quelqu’un qui tient entre ses mains un fin couteau de pluie ou bien un seul pétale de rose, de ceux que l’on glisse entre les pages d’un livre épais, plus léger que l’air sur le ventre des moineaux. (Beauty is there, since dawn. She got up long before us. She is waiting, faithfully. Her breath spreads in the faintest silence, in the air around the almond trees. She is waiting for a path to open in us that she can follow without being hurt. She waits for hours and the movement of her wait is that of the day that dawns, blossoms, then declines, dying at our feet, relinquished, unappreciated. Every day like this, someone comes, someone holding a sharp knife of rain in his hands, or one petal of a rose, such as those that one inserts between the pages of a thick book, lighter than the air on the chest of a sparrow.) Christian Bobin, L’homme du désastre

12.1 Introduction The debate on the “two cultures” is regaining strength as we approach the 50th anniversary of Charles P. Snow’s famous lecture entitled precisely The Two Cultures. The opposition between scientific culture and humanistic culture has sometimes assumed overheated and almost grotesque tones. Actually, the relationships between the so-called two cultures are very tight and their scopes and aims are closer than one can suppose. First, they meet in, and are produced by, human beings; and, second, they share the goal of seeking adequate and meaningful representations and models of the world and of man. Although they speak different languages and one is driven by tendentially objective and communicative goals and the other by subjective and expressive goals, finally both turn out to be intersubjective. I would like to put forward some considerations on beauty in the hope that it turns out to be a bridge between the two cultures. 207

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Fig. 12.1 Almond trees in bloom

Some beautiful almond trees are depicted in Fig. 12.1. The perception of beauty is produced by an emotional catastrophe René Thom [2]

12.2 The Spatial Dynamics: Near–Far I would like to open these notes by showing you a picture of my long-haired dachshund Alcibiade, which I consider a prime example of beauty (Fig. 12.2). I am confident that many will agree that Alcibiade is a beautiful “thing.” Here is another picture of Alcibiade, taken from a great distance. It is obvious that in this case it is more difficult to appreciate the beauty of my dog (Fig. 12.3). Finally, I offer you a picture of Alcibiade taken from a few millimeters: in this case, too, it is difficult to recognize a beautiful “thing” in the object photographed (Fig. 12.4). What I am trying to suggest with this example is that the perception and the recognition of beautiful objects are conditioned by, and depend on, spatial constraints: object and subject of the perception must lie within a proper distance range. Outside that range – too close or too distant – it becomes impossible to recognize

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Fig. 12.2 My dachshund Alcibiade

Fig. 12.3 Alcibiade from a great distance

Fig. 12.4 Alcibiade close up

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beauty. Of course the “proper” range depends on the objects being observed and on the observers, on their interests, instruments, culture, and so on. In this connection, I like to quote a passage from an essay by the English writer William Hazlitt (1778-1830), Why Distant Objects Please (1822) [3]. According to Hazlitt: Distant objects please, because, in the first place, they imply an idea of space and magnitude, and because not being obtruded too close upon the eye, we clothe them with the indistinct and airy colors of fancy. In looking at the misty mountain-tops that bound the horizon, the mind is as it were conscious of all the conceivable objects and interests that lie between; we imagine all sort of adventures in the interim; strain our hopes and wishes to reach the air-drawn circle, or to “descry new lands, rivers, and mountains,” stretching far beyond it: our feelings, carried out of themselves, lose their grossness and their husk, are rarefied, expanded, melt into softness and brighten into beauty [. . . ] Whatever is placed beyond the reach of sense and knowledge, whatever is imperfectly discerned, the fancy pieces out at its leisure; and all but the present moment, but the present spot, passion claims for its own, and brooding over it with wings outspread, stamps it with an image of itself. Passion is lord of infinite space, and distant objects please because they border on its confines and are molded by its touch. When I was a boy, I lived within sight of a range of lofty hills, whose blue tops blending with the setting sun and often tempted my longing eyes and wandering feet. At last I put my project in execution, and on a nearer approach, instead of glimmering air woven into fantastic shapes, found them huge lumpish heaps of discolored earth. I learnt from this (in part) to leave “Yarrow unvisited,” and not idly to disturb a dream of good! [3]

Also the great Italian poet Giacomo Leopardi (1798-1837) deems that in gazing at beautiful things distance is important, as confirmed by the following lines from La sera del dì di festa (The Evening Of The Holiday) [4] Dolce e chiara è la notte e senza vento E queta sovra i tetti e in mezzo agli orti Posa la luna, e di lontan rivela Serena ogni montagna (The night is sweet and clear, without a breeze, and the moon rests in the gardens, calm on the roofs, and reveals, clear, far off, every mountain)

If we imagine approaching those mountains until we discern every detail (rocks, pebbles, blades of grass, leaves, heaps of earth swarming with insects), we realize that the greatest part of the spell has vanished, perhaps to re-form again at a different level, creating a new beauty (or a specific interest) that might enthrall the field biologist, but not the ordinary wanderer. Talking of distance, the starry sky has always fascinated human beings. Again, Leopardi in Le ricordanze (Memories) [4] tells us about his gazing at the “lovely stars of the Plough:” . . . tacito, seduto in verde zolla, Delle sere io solea passar gran parte Mirando il cielo, ed ascoltando il canto Della rana rimota alla campagna. (. . . I used to sit silent on green grass, spending the greater part of the evening, watching the sky, hearing the croaking of frogs far off in the countryside!)

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And in the Canto notturno di un pastore errante dell’Asia (Night-Song Of A Wandering Shepherd of Asia) [4] the moon and the stars are not only gazed at intensely, but also provoke deep questions, a point to which we shall come back later: Spesso quand’io ti miro Star così muta in sul deserto piano, Che, in suo giro lontano al ciel confina; Ovver con la mia greggia Seguirmi viaggiando a mano a mano; E quando miro in ciel arder le stelle; Dico fra me pensando: A che tante facelle? Che fa l’aria infinita, e quel profondo Infinito seren? che vuol dir questa Solitudine immensa? ed io che sono? (Often as I gaze at you hanging so silently, above the empty plain that the sky confines with its far circuit: or see you steadily follow me and my flock: or when I look at the stars blazing in the sky, musing I say to myself: ‘What are these sparks, this infinite air, this deep infinite clarity? What does this vast solitude mean? And what am I?’)

Also in the famous Infinito (The Infinity) [4] the sense of mystery and the yearning for beauty stem from vagueness and distance, even from inaccessibility, compensated for by the poet’s ability to create imaginary worlds: Ma sedendo e mirando, interminati Spazi di là da quella, e sovrumani Silenzi, e profondissima quiete Io nel pensier mi fingo; ove per poco Il cor non si spaura. (But sitting here, and watching here, in thought, I create interminable spaces, greater than human silences, and deepest quiet, where the heart barely fails to terrify.)

It is interesting to notice the presence of another element associated with beauty, i.e., terror, an emotion that the Czech poet Rainer Maria Rilke (1875-1926) expressed in the first of the Duino Elegies [5]: Wer, wenn ich schriee, hörte mich denn aus der Engel Ordnungen? und gesetzt selbst, es nähme einer mich plötzlich ans Herz: ich verginge von seinem stärkeren Dasein. Denn das Schöne ist nichts als der Schrecklichen Anfang, den wir noch grade ertragen, und wir bewundern es so, weil es gelassen verschmäht, uns zu zerstören. Ein jeder Engel ist schrecklich. (Who, if I cried out, would hear me among the angels’ hierarchies? and even if one of them suddenly

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Giuseppe O. Longo pressed me against his heart, I would perish in the embrace of his stronger existence. For beauty is nothing but the beginning of terror which we are barely able to endure and are in awe of because it serenely disdains to annihilate us. Each single angel is terrifying.)

In this connection I like to mention in passing the so-called Stendhal syndrome, a sort of weakness or faint from an excess of beauty that seems to strike the most sensitive visitors in front of art masterpieces, particularly those of the Italian Renaissance: beauty is so overwhelming as to cause fright and swoon. The lines of Rilke are interesting because they concern the beauty of those immaterial beings called angels, whereas Hazlitt and Leopardi refer to concrete objects. It seems that beauty is a property of bodies, or embodied forms, especially visible, but also audible, perhaps tangible, things. Hence I suspect that Rilke imagined his angels as endowed with a perceptible and concrete body, splendid and blazing, in fact almost terrifying. It is interesting that in spite of his serious hearing problems, which led him to utter deafness, Ludwig van Beethoven (1770-1827) went on composing music, and what music, probably thanks to an inner representation of sounds lacking the concreteness that we usually attribute to music. We generally distinguish between natural and artificial objects, and consequently between “natural beauty” and “artificial beauty.” Later I shall come back to this point; for the moment I only notice that the two concepts are intertwined, often producing confusion. For example, referring to a beautiful natural object we sometimes say: “it is so beautiful that it looks artificial,” whereas of a beautiful artificial object we might say: “it is so beautiful that it looks natural.”

12.3 The Relational Dynamics: Object and Subject In all the above-cited passages, the authors adopted the point of view of the subject; in fact a more or less explicit allusion was made to the contribution given by the subject to the “construction” (or “recognition”?) of beautiful objects. It seems, therefore, that beauty has a subjective character (beauty is in the eye of the beholder, as the saying goes). Actually, the common nature of men and the experiences that they all share make the experience of beauty intersubjective. We shall come back to this point. For example, in the poem The Infinity [4] the contribution of the subject or beholder consists not only in gazing at the landscape, but also in helping actively to create (spaces, silences, quiet) “in thought.” What the eye cannot see is created by imagination before the mind’s eye. Hence the importance of the implicit, of the notseen, or not-exhibited: not only the full display of the object is stimulating, but also the hint, the partial or veiled vision, that allows a great freedom to the (re)creating imagination.

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Art, in particular literature, often adopts the strategy of saying and not saying, of showing and not showing. In literature the rigorous questions about how and why and the consequential logical chains are weakened or suppressed. Of course, literature has its own coherence and rigor, but these do not rule out, in fact they foster, contradiction and ambiguity. Actually in literature, and even more so in poetry, truth asserts itself by disclaiming itself, it appears and disappears, exhibits itself through problems rather than through solutions, displays itself not only by flaunting and parading but also by surrounding itself with silence and mystery. The sense of literature lies in what is not written or said as much as in what is said, and it is the sacred quality of the unwritten or unsaid (i.e., silence) that enlightens and gives sense to what is written or said. Ludwig Wittgenstein (1889-1951) maintained that the most important part of his Tractatus [6] was the unwritten one. And the unsaid can also be unsayable. All that is now said once was not said or will not be said in the future, perhaps will become unsayable again. And what is unsayable today, perhaps one day will be sayable. If and when, finally, the unsayable will be said, it will lose much of its charm: because, after all, the unsayable is the only thing for which we long to talk about. And the unsayable includes the answers to the fundamental questions, always looked for and never found out. The world has a sense and at the same time has no sense. This double and contradictory truth, that science could not stand, not even grasp, is fostered and strengthened by the enigmatic, obscure and allusive nature of literature and art in general. In this connection, a remark by the Italian poet Eugenio Montale (1896-1981) seems relevant to me: It was not my purpose to seek obscurity, but nobody would write poems if the problem were to be understood. The problem is to clarify that quid that words alone cannot reach. This does not happen only to those poets that are considered obscure. I believe that Leopardi would burst out laughing if he could read what his commentators write about him. [7]

The quid Montale refers to is sense. We do not know, and maybe we shall never know, whether the world has a sense in itself: it is us that ascribe or deny sense to the world. But it is interesting to note that Lucretius’ hopeless position, according to which the world has no sense at all, a position that some nowadays adopt, is at odds with our deepest needs, expressed by such fundamental questions as: who are we, why do we exist, where are we going? Like many others, I believe that beauty helps us give a sense to the world. I believe that Howard Phillips Lovecraft (1890-1937) was right in maintaining that beauty is more important than truth. In particular the object perceived as beautiful by the subject can be a human body (I do not say a human being or a person, since beauty as referred to the latter has a much wider scope, including metaphorical, spiritual, intellectual, and moral aspects). In this connection, I quote the following verses of Titus Lucretius Carus (99-55 BC), from the fourth book of his great poem De rerum natura (On the Nature of Things) [8]: Sic igitur Veneris qui telis accipit ictus, sive puer membris muliebribus hunc iaculatur seu mulier toto iactans e corpore amorem, unde feritur, eo tendit gestitque coire

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Giuseppe O. Longo t iacere umorem in corpus de corpore ductum; namque voluptatem praesagit muta cupido. (Thus, one who gets a stroke from Venus’ shaftsWhether a boy with limbs effeminate Assault him, or a woman darting love From all her body- that one strains to get Even to the thing whereby he’s hit, and longs To join with it and cast into its frame The fluid drawn even from within its own. For the mute craving doth presage delight.)

Here a further important element is to be found, i.e., the link between the beauty of bodies and the physical appeal, which often leads to carnal intercourse, to procreation, and species propagation. Thus there exists a link that connects beauty, love, sex, heat, reproduction, and evolution: useful and delightful at the same time. The beauty of bodies seems a fundamental evolutionary factor having a remarkable survival value. Charles Darwin (1809-1882), too, proposed a theory about “sexual selection” linking attractiveness to reproduction success. As an example, he believed that the peacock’s tail (Fig. 12.5), that queer, exaggerated appendix, could represent a danger or a drawback in the struggle for life, conspicuous and heavy as it is. Why, then, did the peacock burden itself with such a cumbersome, garish and striking tail? Darwin’s reply was that such a showy appendix can only serve the purpose of enticing the peahen: it is a sexual trait, and an aesthetic one, representing a link between beauty and evolutionary usefulness. A beautiful tail, although it represents a drawback in the struggle for life, offers an evident reproductive advantage. As Darwin put it: The sexual struggle is of two kinds: in the one it is between the individuals of the same sex, generally the males, in order to drive away or kill their rivals, the females remaining passive;

Fig. 12.5 The peacock’s tail

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while in the other, the struggle is likewise between the individuals of the same sex, in order to excite or charm those of the opposite sex, generally the females, who no longer remain passive, but select the more agreeable partners. [9]

Many examples of this kind are to be found in the animal kingdom. Not only humans, but also animals are susceptible to the beauty of bodies. On the other hand we, too, are susceptible to the beauty of animals and plants: we are attracted not only by lap dancers or by Leonardo’s paintings or by handsome actors and good-looking actresses, but also by the tiger’s muzzle (Fig. 12.6), the galloping horse, the foliage and branches of the oak. Beauty appears to be widespread in the living realm, and this beauty points at our kinship to the biological world in which we are embedded. This world displays a luxuriant wealth so different from the anorexic rigor of logicians, who fear the multiplication of entities more than the devil. It was the Franciscan friar and scholastic philosopher William of Ockham (1285-1349) who introduced a principle of ontological parsimony, called Ockham’s razor, meaning that one should always opt for an explanation involving the fewest possible number of causes, factors or

Fig. 12.6 The tiger’s muzzle

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variables: Entia non sunt multiplicanda sine necessitate (one should not multiply entities beyond necessity). His razor shaves away all that logic considers superfluous. But nature ignores Ockham’s razor, that hideous barber’s instrument that logicians wave to drearily eradicate every bit of redundancy from thought: in fact species and individuals multiply merrily in spite of any attempt at reductio ad unum (reduction to one single element). If we adopted Ockham’s razor in front of the living beings, we could ask absurd questions such as: what is the use of the elephant since we already have the hippopotamus? Or even: what is the use of Peter since we already have Paul? Abundance, which we observe at its highest degree in the living realm, is also present in all human activities and products: art, fashion, gastronomy, architecture, literature. . . An unjustifiable yet charming multiplication of entities, of species and individuals is to be found everywhere in our society, in its laborious, muddled, and entropic institutions: from school to the administration of justice, from public transportation to trade unions. Everywhere except in science. As I have said, the reciprocity relationship between object and subject implies that no beautiful object can exist without a subject able to create and admire its beauty. This is true in particular when the object is a human being. To quote Leopardi again, consider these verses from Il passero solitario (The Solitary Bird) [4] Tutta vestita a festa La gioventù del loco Lascia le case, e per le vie si spande; E mira ed è mirata, e in cor s’allegra. (Dressed for the festival young people here leave the houses, fill the streets, to see and be seen, with happy hearts.)

Here a very common symmetric mechanism of spectatorship-exhibitionism is described. Dante (1265-1321) expressed this relationship in his Divine Comedy ([10], fifth Canto of Inferno) with a famous verse: Amor, che a nullo amato amar perdona. (Love that exempts no one beloved from loving.)

Such a symmetric relationship tends to intensify and in some cases reaches the highest of love and even leads to tragedy. Love, passion, jealousy, obsession: a fatal progression that has been the theme of paintings, novels, operas, and theater pieces. Along with symmetric relationships (whose physiological correlate might be the socalled mirror neurons), also complementary relationships exist, which mingle and oppose the symmetric ones, to the effect that in the spectator-exhibitionist pair bipolar relation one of the two components can gradually prevail over the other: a stifling courtship can dampen the beloved one and this in turn provokes an even more intense courtship. “In love’s war, he who flieth is conqueror” as the saying goes, and the Italian poet Francesco Petrarca (1304-1374) celebrated this complementary mechanism rooted in beauty at length. As an example, consider the following verses

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from the canzone Ne la stagion che ‘l ciel rapido inchina (At the Moment When the Swift Sky Turns) [11]: Ahi crudo Amor! ma tu allor più m’informe A seguir d’una fera che mi strugge La voce e i passi e l’orme E lei non stringi, che s’appiatta e fugge. (Ah cruel Love, instead you drive me on to follow the sound, the path and the traces, of a wild creature that consumes me, one I cannot catch, that hides and flees.)

The interaction between subject and object in the field of aesthetics exhibits a strict analogy with the relationship between subject and object in he field of epistemology. In the latter a constructivistic or interactionist view is emerging that goes beyond both realism (reality exists independently of the subject, who knows it by gradually adapting to it or reflecting it) and idealism (all reality is contained a priori in our mind). In science, from Galileo (1564-1642) onward, realism has taken the lion’s share. Galileo’s position could be defined as naive realism; for him science is a long-standing effort to translate the “Book of Nature” into a written book (that should take the place of the Scriptures). But such a translation cannon succeed, as no translation effort really succeeds, partly because, in spite of Galileo’s belief that the Great Book of Nature be written in mathematical characters (namely triangles, circles, and similar geometrical figures), we do not really know whether Nature is a book and what its language might be. And even assuming that the Book of Nature be actually written in mathematical characters, one might ask what these characters would be like: circles and triangles, or the figures of some non-Euclidean geometry, or fractals, or the characters of some not yet discovered alphabet. . . Here we meet the unsolved problem of the apparent ability of mathematics to describe the physical world, “the unreasonable effectiveness of mathematics in the natural sciences,” as Eugene Wigner (1902-1995) put it [12, 13]. But we cannot even touch upon this problem. In spite of such “unreasonable effectiveness,” nobody knows the language of Nature; in fact nobody has access to “the thing in itself:” between me and things there is always a creative filter (as Gregory Bateson called it [14]). This filter consists of our body and mind, i.e. sense and cognition, which are inseparably entangled. What this filter gives us is a more or less elaborated map of something that is out there, but the map is by no means a mirror image of that something, as many philosophers and scientists have argued. I wish to mention what Ludwig Wittgenstein wrote: The fact that the motions of the planets can be described by means of Newton’s mechanics does not tell us anything about reality. [6]

The world that we consider as “given” is largely our own construction, or better a co-construction, a mutual and reverberant aesthetic stimulation (having a multimedial nature!) in the etymological sense of the word (aisthánomai = to feel) that emerges when the interaction between ourselves and reality takes place. According to Francisco Varela (1946-2001), such constructive interaction between subject

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and object is always at work. The sensory–motor processes, perception and action, are inseparable from cognition and are at the basis of what Varela calls “enactive view:” For the dominant computational tradition the starting point for understanding perception is utterly abstract: i.e. the problem of information processing in recovering pre-existing properties of the world. On the contrary, the starting point of the enactive approach is the study of how the observer steers his own actions within local situations. Since these situations change continuously as a result of the perceiver’s activity, the system of reference to understand perception is no longer a pre-existing world, independent of the observer, but rather the sensory-motor structure of the cognitive agent. [. . . ] It is this structure, the way the observer is embodied, rather than some preexisting world, that determines how the observer can act and be modulated by the environmental events. [. . . ] Reality is not deduced as data, it depends on the observer, not because the observer “constructs” it according to his own imagination, but because what is considered as the relevant world is inseparable from the structure of the observer. [. . . ] Hence perception is not simply placed in the environment and subject to its constraints, but also contributes to the enaction of that environment. [. . . ] Organism and environment are tied together in a mutual description and selection. [15]

If in the above passage “reality” were substituted by “beauty,” Varela’s words would preserve their explanatory effectiveness. I have said that an object is not beautiful in itself, rather with reference to a subject (but we shall see that this subjectivity extends to an intersubjectivity). In this sense beauty belongs to that vast territory explored by Gregory Bateson in which we also find information, order, complexity, and in general all communication phenomena, in which the presence and activities of the subject are essential. Hence beauty has a relational character.

12.4 Co-Evolutionary Dynamics I now come to the core of my contribution. I propose the following systemic and evolutionary definition of aesthetics, considered as a combination of sensitive abilities and activities, in particular in front of beautiful objects:

Aesthetics is the subjective (but shared) perception of our bond with the environment in which we are immersed and of which we are part; such a bond is characterized by a deep and balanced dynamical harmony and displays itself in the recognition of beautiful objects through senses and emotions.

I now make a step forward by asserting that, in my naturalistic perspective, aesthetics is closely linked with ethics, for which I propose the following definition:

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Ethics is the subjective and intersubjective capability of conceiving and carrying out actions that keep our bond with the environment healthy and balanced. Such actions maintain and increase beauty.

Ethics and aesthetics are the two sides of the same coin since they arise from the strong systemic and evolutionary co-implication between species and environment and both are “mirror images” of that co-evolution within ourselves. Thus aesthetics is the (inter)subjective perception of the balanced (vital) immersion in the environment, and ethics is the (inter)subjective perception of the necessary respect for, and suitable action within, the environment. As a consequence, ethics permits us to keep aesthetics and aesthetics functions as a guide in acting ethically. Both ethics and aesthetics are rooted in our evolutionary history, in fact in the co-evolution between the environment and ourselves. Such a primordial evolutionary connection concerns beautiful objects, whereas beautiful concepts, e.g., those that some find in mathematics, are much more recent philogenetically, hence recognizing their possible beauty is more difficult, involves rationality, is less spontaneous, and requires a specific individual training. In the case of natural objects, a great part of such a training was carried out in our place by the past generations. As can be noticed, in the proposed definitions there is a shift from the subjective position to the intersubjective one. Such a shift is justified by the common nature and history of human beings, from the fundamental physical and chemical components, through the shared genetic and biological roots, up to the common existential experiences. The electron charge, the gravitational force, the proton mass and so on; all of these characteristics have conditioned the rise and development of life on the earth and have been impressed upon us. This justifies what Vitruvius (80-23 b.C.) and later the alchemists asserted, that there is a strict correspondence between the macrocosm (the world) and the microcosm (man) (Fig. 12.7). Moreover, it seems that life had one and the same origin on the earth, in fact the DNA of all living beings is the same. Given these premises and given that all individual experiences are very similar, one can argue that all men are more or less equal and, therefore, they feel and think more or less in the same way. (Of course, this is not in contrast with the uniqueness of every single man.) If it were not so, our continuous attempt to communicate, to converse, and to narrate would never succeed. But it does succeed. We cannot access the secret closet of other minds, but on the basis of the analogy with our own minds we can conjecture with a certain degree of success what other people think. This conspecific intersubjectivity can be extended to aesthetics and ethics as well: beautiful things and good deeds are recognized as such by (almost) everybody. We love sunsets, snowy mountains, the gaudy coats of tropical fish, the graceful bodies of our mates, because in the course of ages our ancestors have contemplated thousands and thousands of such images, which became impressed upon them as ineradicable stamps full of meaning and value. In this sense, the beauty of a thing is not perceived, rather it is recognized. We cannot define beauty, but we can rec-

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Fig. 12.7 Vitruvian man

ognize it when we see it. Hence Plato (427-347 b.C.) could be right in maintaining that knowledge is a matter of recollection and not of learning, observation or study. To recognize beauty would then be equivalent to refer the particular object contemplated now and here to the endless sequence of similar objects that, interacting and overlapping, have gradually built the abstract idea of beauty, of which that particular object is a representative specimen. The bond between aesthetics and ethics is confirmed by many clues. The Greeks, our trusted mentors in almost every domain, coined the expression kalòs kagathòs, meaning “beautiful and good.” In mathematics this association is reinforced by the persuasion, nurtured by many specialists, as quoted by Carafoli in his paper, that beauty is a sign of truth (or perhaps of consistency). We shall come back to this point. The tie between beautiful and good is also confirmed by the opposite association, between ugly and wicked, not only in the popular lore and traditions, but also in mythology, fairy tales, and legends. One of the most striking examples is offered by Frankenstein’s monster (Fig. 12.8), created in 1818 by Mary Shelley (1797-

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Fig. 12.8 Boris Karloff as Frankenstein’s monster

1851) [16]. Of course both associations have plenty of counter-examples: Lucifer (beautiful and wicked), and the Beast (ugly and benevolent) of the tale “Beauty and the Beast.” Going back to my naturalistic definitions, it is important to remark that the equilibrium of the total system of which we are a part is dynamic and not static. As a consequence, ethics and aesthetics evolve in time and depend also on the artificial objects constructed by men. As they are invented and built, such objects become part of the environment where we live and have our experiences. Hence ethics and aesthetics have a historical character, both for individuals and for mankind, and evolve due to every new experience. It seems that such experiences activate specific brain circuits. Many would agree that today aesthetics suffers from a deep crisis. Some people even wonder whether homo technologicus – i.e., the hybrid creature made of man and technology that is taking the place of homo sapiens sapiens – still needs beauty and in particular art. I have no answer to that question, but I believe that a possible cause of such a crisis is the arbitrariness of codes. From information theory we have learned that in general signs and codes are arbitrary, in the sense that the association between an object (significatum) and the sign (significans) that represents it is utterly free. For example, the object “tree” has nothing to do with the word “tree” or with the words “albero,” “arbre,” “fa,” “Baum,” etc. Once we became aware of that arbitrariness, we came to wonder whether the representational codes used in art and considered as

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“natural” were not perchance arbitrary too. And we realized that their alleged “naturalness” was simply due to tradition, usage, antiquity. As a consequence, the historical dimension was ignored and all codes were declared equally arbitrary. A code is as good as any other, and the most capricious representative codes can be introduced. In music, in architecture, in figurative arts, and partly also in literature, the traditional aesthetics (no more to be called natural), was disrupted, and gave way to cubism (see Fig. 12.9), futurism (see Fig. 12.10), abstractism, conceptualism, combinatoriality, dodecaphony. . . Subsequently, of course, this intoxication of arbitrariness underwent a more careful consideration. The whimsical and quaint codes introduced just for a change often proved less effective, less pleasant, less valid and less suggestive than the traditional

Fig. 12.9 Factory at Horta de Ebro (Pablo Picasso, 1909) © by SIAE 2009

Fig. 12.10 Dynamism of a dog on leash (Giacomo Balla, 1912) © by SIAE 2009

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ones. (I am told that performers of dodecaphonic music often suffer from serious psychosomatic diseases.) Thus the old “natural” codes have been partly reappraised, as they are better suited to our physiology and psychology, that evolved in strict interaction with the making of those codes. Of course we cannot go back to the past, so the present situation is a mixture of old codes rejected and new ones not yet fully established.

12.5 The Dynamics of Questions Let us consider again the Canto notturno di un pastore errante dell’Asia (NightSong Of A Wandering Shepherd of Asia) by Leopardi [4]: E quando miro in cielo arder le stelle; Dico fra me pensando: A che tante facelle? Che fa l’aria infinita, e quel profondo Infinito seren? che vuol dir questa Solitudine immensa? ed io che sono? (Or when I look at the stars blazing in the sky, musing I say to myself: ‘What are these sparks, this infinite air, this deep infinite clarity? What does this vast solitude mean? And what am I?’)

When we contemplate the beauty of the starry sky, we are led to ask fundamental questions such as: who are we, where are we going, where do we come from? These questions are likely to remain unanswered, and therefore are considered as “meaningless” by science. Still we cannot help asking them. Over such fundamental questions great minds have labored through the centuries and large portions of the intellectual and artistic activity of mankind have been conditioned by questions about existence, morality, and the sacred. According to George Steiner, to avoid these questions, to censor, or to repress them would be equivalent to abrogating the specific condition and dignitas of our human essence. It is the dizzines of querying that vitalizes a thoughtful, considerate life: in a word, a human life. Here are a few important questions and considerations deriving from the approach I have adopted to beauty (and to epistemology as well): • Are there neurophysiological or evolutionary (but also physical or chemical) limits to our observation, analysis, and perception ability that might somehow hinder or negate our pretensions of knowing and perceiving (epistemology and aesthetics)? • Are there, as Blaise Pascal (1623-1662) claimed, limits to our reason? For Pascal, the acknowledgement of such limits is the most important achievement of reason, and this foreshadowed a sort of Gödel-type theorem ante litteram: “The perfection of reason lies in acknowledging its own imperfection.” [17]

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• Also: not only there are things that we ignore, but there are also things that we do not know that we ignore; as a consequence we cannot measure the extension of thought with respect to reality. • Mystery is everywhere and the deepest and most beautiful emotion is the sense of mystery. According to Albert Einstein (1879-1955), mystery is the source of true art and true science. The one who does not possess this ineffable sense of mystery cannot even be a genuine scientist. • Can we get a broader and deeper understanding of the world and of ourselves thanks to the hybridation with the “machines of mind” (such as the computer and the Internet)? • Shall we become “superintelligent” thanks to our machines? Or will the machines themselves become superintelligent, leaving us behind? • And last (but we might go on putting questions): how firm are the foundations of our mental constructions, e.g., mathematical theorems, physical theories, philosophical systems, etc., of which we are so proud? Who can guarantee that they are not splendid illusions? Like many others, this fundamental question lacks an answer that is not founded on ideology or faith. Moreover, along with the well-formed and well-articulated thoughts built by our rationality, there are impure, turbulent, half-bred thoughts into which creative and visionary people, be they scientists or artists, dive and re-emerge having in their hands and minds magnificent creations: paintings and poems, theorems, or machines that can shape the history of the world. We are traversed by these underground chaotic flows and whirls of thought that seem to promote our creativity. In A Midsummer Night’s Dream [18], William Shakespeare (1564-1616) tells us that The lunatic, the lover and the poet Are of imagination all compact

He could have added the most creative scientists to the lot. In other words, perhaps it exists a tangle of “obscure” reasons that help to control our relationships with the world and with ourselves. These reasons have been considered as blind, debased forces like Instinct, or as blind and virtuous ones, like Life and Desire; in both cases they lack any sign of intelligence. Being blind, actually, they do not know what they are doing: they do not follow any rule, any reason, any principle. Many thinkers, including Gregory Bateson and John Searle, have argued, from different points of view, against such a disparagement, condemning the rationalistic reductionism that tries to suppress “the reasons of the heart” (Pascal wrote that the heart has its reasons of which reason knows nothing [17]), i.e., the complex fundamental cognitive activity that allowed living beings to evolve and to develop an intelligence that on top coincides with rationality and computation, but at the root is pure biology and existence. Referring to Pascal’s reasons of the heart, Martin Heidegger (1889-1976) wrote: The inner and invisible domain of the heart is not only deeper, hence more invisible, than the “inside” of the computational representation, but also encompasses a region that is wider than

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that of the simply reproducible objects. In the invisible ultra-interior region of the heart, man is primarily urged towards what must be loved: the ancestors, the dead, the infants, the not yet born. [19]

It can be added, I believe, that such inner domain is the region of the obscure, primeval, dawning intuition, the region where art, poetry, and the auroral vision of mathematics arise. Also the constructions that are to be completed subsequently by rationality are permeated by emotions and sentiments. Actually there exists a deep emotion associated with thinking and with the primitive, still not refined acts of rational knowledge. Perhaps Heidegger’s ultra-interior region is a space whose uncertain boundaries correspond to the limits of our own biology: perhaps it is by virtue of that space, not completely enlightened by rationality, that we can, if not really understand, at least compute quantum mechanics.

12.6 Remarks on Symmetry and the Golden Ratio Some of the participants to this symposium, e.g., Carafoli and Du Sautoy, have mentioned symmetry, and I would like to add a few comments on that topic. First of all, the presence of symmetry in nature probably stems from an economy principle, or minimum principle: to describe a symmetric figure (in particular a symmetric organism) less information or instructions are needed than to describe a non-symmetric one. Moreover, a symmetric object exhibits a structural redundancy that could correspond to a functional redundancy: the two brain hemispheres, ad in general all organs occurring in pairs, have partly vicarious functions. Perhaps our liking for, and inclination towards, symmetry arises from the frequent observation of natural symmetric objects. We often introduce symmetric features into our artifacts, but in its constructions nature does not always follow the path of symmetry, and not always the beauty that we recognize in natural objects has to do with symmetry. An alternative is offered, e.g., by the so-called “golden section” or “golden ratio.” Let me refer to one of the most interesting and enigmatic paintings by Piero della Francesca (1412-1492), The Flagellation (Fig. 12.11). In this masterpiece, Piero reached the climax of Renaissance perspective. The big (almost) central column divides the painting into two parts, whose surfaces are in the “golden ratio” 1,618. . . that Piero knew very well. Actually he wrote at length about the golden ratio in his treatise on regular solids and it is only natural that he wanted to insert it also in his paintings. The hieratic aspect, the symbolic character and the oneiric timelessness of the figures convey the impression that The Flagellation sends us several encoded messages. Actually riddles, ciphers, and puns were common in the Renaissance, and according to some recent conjectures it seems that this painting contains many.

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Fig. 12.11 The Flagellation (Piero della Francesca)

Just a few words about the golden section, or golden ratio, defined as the (irrational) number √ 1+ 5 = 1.618033988. . . Φ= 2 The algebraic relationship that defines this constant is a+b a = . a b Thus a and b are in the golden ratio if the ratio between the sum of the two quantities and the larger one is the same as the ratio between the larger and the smaller one. Besides symmetry, in nature we also find this harmonious dissymmetry, e.g., in the structures of some shells, in the dimensions of leaves, in the distribution of branches in trees, in the distribution of seeds in sunflowers or of scales in pine cones. . . It is as if the entangled tendrils of life were somehow reflected by these mathematical structures, by these geometrical rhythms, by these arithmetical periodicities. The Greeks (and before them the Egyptians) knew this number and used it in architecture, e.g., in the Parthenon (Fig. 12.12) and in the Great Cheope’s Pyramid (Fig. 12.13), in painting, and in sculpture. Since it is so frequent in nature, the golden ratio was deemed worthy of being introduced in man’s works. Also in more recent times some architects, like Le Corbusier and Giuseppe Terragni, have used it

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Fig. 12.12 The Parthenon

Fig. 12.13 Cheope’s Pyramid

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in designing rational buildings, and some musicians, like Béla Bartók and Claude Debussy, have resorted to it in some of their compositions. Galileo asserted with assuredness that the Great Book of Nature is written with mathematical characters, but today we know that those characters are not only the polygons and regular solids that he referred to, but also richer and more suggestive structures, such as non-Euclidean geometries and fractals, that can express a creative dynamism that is to the elementary stillness of Euclidean geometry as the living world is to the stiffness of the mineral kingdom. Perhaps it is only our inclination to simplification, our continuous search for reductivist formulas that make us see still, calm Platonic solids even where nature exhibits complex and dynamical forms. The discovery of fractals has led us closer to a different and perhaps more suitable geometry of nature, reflecting the dynamism of the Fibonacci sequence: F(n) = F(n − 1) + F(n − 2) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55(= F(10)), 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765(= F(20)), 10,946, . . . that is strictly connected with the golden ratio. The Fibonacci numbers and the golden ratio are to be found also in the living world: the growth mechanisms are such that in many flowers the number of petals is a Fibonacci number. For example, the lily has 3 petals, the ranunculus 5, the chicory 21, the daisy often 34 or 55. . . The seeds of the sunflower are arranged along two series of spirals going in opposite directions, and the number of spirals of opposite direction differs for 21 and 34, 34 and 55, 55 and 89, or 89 and 144 seeds. . . The same holds true of pineapple and of some kinds of cabbages, especially the romanesco cabbage, of strawberries, and of pine cones. The presence of the Fibonacci sequence and the golden ratio in these living structures is another example of that mysterious correspondence between mathematics and reality that so many physicists and philosophers have been wondering about and that remains an unsolved enigma. I believe that in asserting that symmetry is the key, or one of the keys, to beauty, especially to that beauty that science tries to pull out of nature, we adopt a sort of simplification, a reductionism that eliminates a great part of the phenomenal richness to reduce it to our explicit mental schemes. Moreover, we should not forget that in many cases what matters is symmetry breaking: symmetry breaking is the source of information, of evolutionary structures and of history. Benoît Mandelbrot (1924), who is considered as the founder of fractal geometry, was interested in very concrete problems: how can we describe common objects such as a cloud or a mountain? A cloud is by no means approximated by a sphere and a mountain is quite different from a cone or a pyramid. The traditional geometric forms, regular polygons and polyhedrons, do not describe in a suitable way the natural objects that surround us. So Mandelbrot began to seek a new geometric language, permitting to approximate the coast of an island or the form of a tree or of a cabbage in a suitable way. About the romanesco cabbage, Mandelbrot wrote:

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Fig. 12.14 The romanesco cabbage

If it did not exist in nature, the romanesco cabbage would have been invented by a fractal theorist: among all everyday objects, it is the best instance of a rough, but rich in invariances, surface. (See Fig. 12.14.) [20]

In this way fractal geometry was started. It was welcomed as the most important conceptual achievement since quantum mechanics. This is probably an exaggeration, but many consider fractal geometry as the “true geometry” (i.e., the true “description” or “representation”) of nature (see Fig. 12.15). It appears therefore that, thanks to these refined tools, we can describe nature better than with the classic regular polygons and solids. It is an idealization or reduction, of course, but it is certainly more adequate than the first Euclidean approximation. Between this new geometry and some natural forms there is a connection that has been investigated from a mathematical viewpoint by Mandelbrot and others. Such a connection sets and partly solves the problem of the aesthetic and artistic value of this geometry. Actually, if art is, at least to some extent, mimesis, then these mathematical structures, that often reproduce nature with unequalled effectiveness, have an artistic value, at least to some extent. There is nothing surprising about this, since figurative art can be considered as a vision of reality, although a filtered and distorted one, but certainly deep-reaching. The reality of which we are talking about is not only the “natural” reality as we perceive it with our senses: it could be a reality already filtered by other instruments, de-constructed and re-constructed, or simply invented (think, for example,

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Fig. 12.15 Examples of fractals

of the so-called virtual reality). Hence, the new visions and the devices that allow these visions, first of all the computer, have a great importance in art and their impact should be investigated carefully. These instruments offer us the possibility of entering artificial scenes and acting within them as if they were real. The notion of art as reproduction or imitation of reality is becoming more and more encompassing, as reality can also be simulated – or created – by us: this is the great conceptual revolution of (interactive) computer graphics. So we realize that the triangle

still hides many unexplored regions.

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Both science and art try to represent reality, including “artificial” reality, and this attempt is carried out on the basis of some a priori aesthetic and epistemologic requirements: harmoniousness and balance between the parts and the whole, global structure, self-similarity, and invariances of various kinds. Such requirements have been developed in time by the constructive co-evolutionary interaction between subject and reality. As a consequence, to some extent they represent an image of reality in the subject. Perhaps this image, albeit filtered, modified, and distorted, is the primeval root of the creative (or re-creative) emotion that characterizes the aesthetic (and then cognitive) nature of the relationship between subject and object.

12.7 The Beauty of Mathematics The attempt to represent reality concerns in particular mathematics. But art and mathematics exhibit a deep difference: actually we live in a world of forms and colors that we perceive with our senses, and only secondarily we dwell in a world of concepts that we understand with our minds. To understand mathematics we need to master difficult techniques and tools that we can only acquire through long and difficult training. Everybody can listen to a symphony or gaze at a painting, enjoying it at some level, whereas mathematics cannot be listened to or gazed at in the same way. The final test of an art work resides in the subjective (or better intersubjective) aesthetic pleasure that it gives, while, concerning mathematics, mathematician Morris Kline says: The question whether mathematics possesses or not a kind of beauty can only be answered by those who have a culture in this domain. Unfortunately, to master mathematical ideas one needs years of study and there is no via regia or king’s highway that can shorten the process. [21]

Although specialists know very well what mathematical research is like, it is almost impossible to explain it to those who have not taken the necessary steps of the initiation, in spite of the free-and-easy statements of some cheats that try to sell recipes and shortcuts to become mathematicians without effort and in a short time. Many use the word “mathematics” in a metaphoric and almost fanciful way. Art should not be confused with mathematics and in turn mathematics in art should not be confused with “true” mathematics. For instance, Le Corbusier uses the word “mathematics” in a metaphoric way: For the artist mathematics does not mean mathematical science. It is not necessarily computation, rather it is the sway of a rule; a law of infinite resonance, consonance, order. The rigor is such that the art work is a consequence of it, be it a drawing by Leonardo, the amazing precision of the Parthenon, the impeccably exact construction of a cathedral, the unity realized by Cézanne, the law that creates a tree, unitary splendor of roots, trunk, branches, leaves, and flowers. In nature nothing is due to chance. Once you understand what mathematics is in the philosophical sense, you will find it in every work. Rigor and precision are the means to find the solution, the reason of harmony. [21]

Le Corbusier insists on rationality and rules and does not mention the infringement of rules, that helps construct higher and more general rules. That infringement refers

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us back to Gregory Bateson’s “pattern that connects,” that is laden with a remarkable aesthetic value, and aesthetics is the deeply unifying theme of the co-evolutionary making of the biological and non biological components of the world. For Bateson that theme is the pattern that connects the crab with the lobster, the orchid with the primrose and those four with the horse and those five with man; actually all of them bear in themselves the primordial imprint of the environment and in turn are active agents in transforming the environment. Mathematics lends itself (also) to a rational understanding, hence it can be reproduced, proved and communicated. Art work, on the contrary, cannot be the object of a proof, it can only be the source of ineffable aesthetic shivers or of moving participation. In his talk, Ernesto Carafoli pointed to Schrödinger’s equation as the emblem of mathematical beauty. To that equation I wish to add one of Euler’s (see Fig. 12.16) celebrated formulas, perhaps his most famous one, that physicist Richard Feynman considered the most beautiful of mathematics: eiπ + 1 = 0 . This amazingly simple expression connects four of the most important entities of mathematics, i.e., the numbers 1, 0, e, π . But, again, to appreciate its beauty one needs to be a mathematician, because, as I said, we did not co-evolve with mathematics, but with the world, and our interaction with the world has been mediated more by the body than by the mind. For

Fig. 12.16 Leonhard Euler (1707–1783)

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sure, mind, too, has its evolutionary history, but the abstraction process that has led to mathematics is very recent. The co-evolution between us and the world is much more ancient, and it has impressed in us the coordinates (images) of sunsets and trees and animals and humans. Furthermore, it is important to observe that in doing mathematics we have some degrees of freedom in choosing the codes, the symbols or the coordinates. If we choose certain codes or coordinates instead of others, our formulas or equations obtain certain forms instead of others: as a consequence, the “beauty” of mathematical expressions depends on those choices, which are contingent and historical, sometimes utterly random. Natural forms, too, are contingent and historical, as evolution has taught us, but on the one hand, the physical and chemical determinants impose stronger constraints on them and on the other, that evolution is much longer. So these forms have a certain stability due to a long selection and confirmation process. The beauty of the natural world is much more robust and resists better to coordinate transformations. This is why we have been talking, and still talk, of “natural” codes.

12.8 Language and Thought I wish to close by hinting at the problematic, sometimes even hostile, relationship between thought and language. Let me refer again to Eugenio Montale, to whom Ernesto Carafoli also refers in his contribution. For him both poetry and nature show the surface and hide the essential. But there is an important difference: the poet knows the deep motives of his poetry and ignores those of nature. He wants to know them, however, and wants to overstep the threshold of mystery and arrive at the essence of the world (see the poem I limoni, The Lemons [7]). But this is impossible, and the poet must linger at the surface of things and pay attention to the musical aspects of poetry. It is as though a thin veil separated him from the ultimate essence: he seeks the absolute expression able to tear the veil, making the world itself appear in front of him. Not being able to reach this vision, Montale resigns himself to the musical expression of poetry. There is a wide gap between the urge of creation and its linguistic expression. We seek the words to say “it” but almost always in vain. The great American dancer Isadora Duncan (Fig. 12.17) used to say: “If I could say what I felt then I wouldn’t have to dance it” [13]. This gap shows itself both in everyday life, and when we try to give form to philosophical reflections or to artistic intuitions. The frontier represented by language is assaulted by emotions, sensations, insights, flashes, but, as George Steiner points out, it continues to be an almost impenetrable wall. Only the great writers, sometimes, are able to open a breach in that wall and bring back a shred of real beauty. The creative outburst does not come out well; the thought that we like to imagine genuine and pure does not spring out, does not smile on us: it weakens and fails, becomes exhausted. It cannot help being bridled by the rules of language, with all

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Fig. 12.17 The dancer Isadora Duncan

the load of grammar, with the dead metaphors, with the vocabulary established and violated by use. And that burden, which has nothing to do with thought, empties it of its native strength. I hope you will forgive me if I quote a passage from my novel La gerarchia di Ackermann, Ackermann’s Hierarchy [22], which seems relevant to me. The words that he uttered contained a heavy and terrestrial strength, independent of him and connected with syntax, so that, after the first start, his thought and would not count any longer and all those sounds would roll downhill with an unbearable thunder, following ancient gullies dug by prehistoric glaciers. The words kept withstanding him, kept taking him where they wanted. Moreover, he thought, we have only one mouth, so we have to utter one thing after the other, whereas inside there thoughts run together like little bluish flames along the neurons, synapses in billions, and they throng to be uttered all together. Their ranks emerge from a dark crater, swarming, angels or devils, and the native strength, perhaps the truth of things lies in their laborious teeming. But to be uttered, they have to slip into that narrow slot, and so they lose vigor and dimension, lose their traveling companions, they are left naked and speak of other things. Things should never be said, because something else comes out and dreadful misunderstandings are created. With our mouth we say ’infinity’ and that sort of inner sea-storm of little teeming waves scattered one against the other that seem to head towards the asymptotic brink of the abyss emerges as a surprising and somehow mean sound of four syllables, where nothing is left of the rippled underlying turmoil. So the fuzzy stammering of words moves us away definitely from the glimmering face of thought that we have seen for a moment, perhaps, and then has vanished. An immense plateau covered with snow, marked with the long scars left by ancient skiers [. . . ]

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Going back to Montale, let me draw your attention to the last verse of the poem L’estate, Summer [7]: Occorrono troppe vite per farne una. (Too many lives are needed to make one.)

It is a substantial declaration of helplessness vis-à-vis the mystery of life and of the world. Our limited ability and intelligence allow us to understand some things, but many other things remain out of our reach. After all, artists, and men in general, try to find an antidote for the finiteness and unicity of the lives that have been allotted to them; we can live only one life and we wish we had many, to devote them to different deeds and explorations. We cannot defeat the inexorable arrow of time, but we can create alternative worlds where to live, albeit for a short time. Hence, if we contemplate our life from the viewpoint of desire and eagerness, we see it as made up of countless other lives, that are, however, subordinate to the one that appears real to us. And still we do not give up, we continue seeking, pursuing, courting the deep sense, the intimate, unreachable essence of nature and life. Although we do not hope to succeed.

References 1. C. Bobin: L’homme du désastre (Fata Morgana 2003) 2. R. Thom: Stabilità strutturale e morfogenesi (Einaudi, Torino 1980) 3. W. Hazlitt: The Complete Works of William Hazlitt. Ed. P.P. Howe, 21 vols. (J.M. Dent & Sons, London 1930–1934) 4. G. Leopardi: Tutte le poesie e tutte le prose. L. Felici, E. Trevi (eds.) (Newton Compton, Roma 2007) translation by A.S. Kline 5. R.M. Rilke: Duineser Elegien (Reclam Universal-Bibliothek, Ditzingen 01/1997) p. 155, translation by A.E. Fleming 6. L.J. Wittgenstein: Tractatus Logico-Philosophicus (Routhledge & Keagan Paul, London 1922) 7. E. Montale: Opere complete (Mondadori Meridiani, Milano 1996) 8. Titus Lucretius Carus: De rerum natura. Ed. A. Fellin (UTET 2005), translation by W.E. Leonard 9. C. Darwin: The Descent of Man, and Selection in Relation to Sex (Princeton University Press, Princeton 1981) 10. D. Alighieri D: La Divina Commedia. Ed. Chiavacci Leopardi A.M. (Mondadori, Milano 2007) 11. F. Petrarca: Canzoniere (Donzelli, Roma 2004) p. CXIX-1015 12. E. Wigner: Symmetries and Reflections: Scientific Essays of Eugene P. Wigner (Indiana University Press, Bloomington & London 1967) 13. E. Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, vol. 13, No. I, February 1960 (John Wiley & Sons, Inc., New York) 14. G. Bateson: Steps to an Ecology of Mind: Collected Essays in Anthropology, Psychiatry, Evolution, and Epistemology (University Of Chicago Press, Chicago 1972) 15. F.J. Varela: Un know-how per l’etica: Lezioni Italiane n.3 (Fondazione Sigma-Tau, Laterza, RomaBari 1992) 16. M. Shelley: Frankenstein, or The modern Prometeus. The 1818 text / Mary Shelley, edited with introduction and notes by Marilyn Butler (Oxford University Press, London 1994) 17. B. Pascal: Pensées (Kessinger Publishing, Whitefield 2004) p. 268 18. W. Shakespeare: A midsummer night’s dream. Ed. L. Buckle (Cambridge University Press, Cambridge 2005)

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19. G. Zanarini: L’emozione di pensare (CLUP-CLUED, Milano 1990) 20. H.-O. Peitgen and P.H. Richter: The Beauty of Fractals. Images of Complex Dynamical Systems (Springer, Heidelberg 1986) 21. M. Emmer: La perfezione visibile (Teoria, Roma 1991) 22. G.O. Longo: La gerarchia di Ackermann (Mobydick, Faenza 1998) translation by the author

Discussion E. Carafoli: You have mentioned the concept of “enactive” perception of Francisco Varela, in which reality depends on the perceptor. As if, in other words, the act of perception would create reality. As you know, Varela frequently quoted Machado’s famous verses in the poem Caminante (The Traveler): Caminante, no hay camino, se hace camino al andar. Traveller, there is no path, you make the path as you go.

Doesn’t the concept carry with it the danger of sliding into the idealistic idea that nothing exists if we do not experience it, that there is no path unless we do not make it as we go? G.O. Longo: Well, yes, there might be such a risk, but in all positions there is some risk. Also in realism, where the risk is to become naive realists, i.e., to believe that there exists a world out there that is mirrored in us without any filter or distortion. It is a risk that I am willing to take. I call myself a moderate constructivist, i.e., I certainly believe that there exists a world out there, but I perceive that world by filtering it through my perceptive and cognitive systems, i.e., through all the evolutionary path and all the systemic relationships that have built those systems. We are the outcome of an evolutionary process, and this is interesting not only because it allows us to discuss and to polemize, but especially because we have undergone a co-evolution with the environment since we have grown up inside it. I think that this is the vision that Varela expresses in an articulated and suggestive form. C. Riedweg: I wish to thank you for your talk, so full of interesting ideas. The speaker before you took an essentialist, I would say an Aristotelian approach, whereas you are more of a constructivist. I was impressed by the link between aesthetics and ethics, which is perhaps less constructivistic and more platonic. How does ethics contribute to beauty? G.O. Longo: In my opinion, the link that I have indicated between ethics and aesthetics is of paramount importance, since it goes back to evolution, or rather co-evolution. Ethics and aesthetics are the two sides of the same coin since they arise from the strong systemic and evolutionary co-implication between species and environment and both are “mirror images” of that co-evolution within ourselves. Thus aesthetics is the perception of our balanced immersion in the environment and ethics is the perception of the necessary respect for, and suitable action within,

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the environment. As a consequence, ethics permits us to keep aesthetics and aesthetics functions as a guide in acting ethically. Both ethics and aesthetics are rooted in our evolutionary history, in fact in the co-evolution between the environment and ourselves. The balance I refer to is not static, but dynamical, as it undergoes an evolution that some people call progress; actually it is a sequence of equilibrium states, of adaptations to the changing conditions. The environmental crisis that we witness today is probably a consequence of the fact that we have lost the sense of aesthetics, hence the sense of ethical action. We are not able to preserve the systemic harmony and the global system is entering a pathological state of unbalance. The system possesses excellent homeostatic properties and is able to cure itself, i.e., it can absorb any perturbation and to reach a new state of equilibrium, provided the perturbations are not too strong. If, however, the perturbations are too severe, the system collapses, i.e., cannot reach a state of equilibrium. I think that the perturbations we give the system today are too strong.

Scientific and Artistic Creativity: In Search of Unifying Analogies

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13.1 Introduction This chapter will attempt to demonstrate that the structure of the creativity process is the same in the artistic and scientific cultures, even if the two cultures are intrinsically different. The differences are clear: the practitioners of the scientific culture aim at acquiring impersonal and objective knowledge that will withstand the scrutiny by others, while the practitioners of the artistic culture aim at generating personal, subjective knowledge that does not need verification. Science makes progress, art changes, but does not make progress: the scientific theories of the early Greeks are now little more than historical curiosities, whereas Praxiteles’ sculptures have the same value and importance today as they had more than 2000 years ago. Science is right or wrong, art cannot be right or wrong. On this accepted background, this chapter will work to show that both cultures, different as they may be, nevertheless have the same aim. They try to understand reality and to make sense of it. The statement is not obvious: by traditional consensus, the search for truth and the generation of beauty have been considered the distinctive goals of the two cultures: Samuel Taylor Coleridge [1] summed up the concept in his incisive prose nearly 200 years ago: The proper and immediate object of science is the acquirement, or communication of truth: the proper and immediate object of poetry is the communication of immediate pleasure.

Coleridge did not talk directly of beauty. He talked about pleasure, but the idea of beauty was implicit in his statement. In making it he is likely to have had in mind not just poetry, but the other forms of art as well. For a long time, Coleridge’s proclamation has reflected general feelings on the matter of art and science. However, in more recent times feelings have changed: this chapter will try to demonstrate that the absolute dichotomy of goals between the artistic and scientific cultures is a thing of the past. It will not be too difficult to demonstrate this for the case of art: a lot has happened since the times of Coleridge to convince us that art does not only aim at 239

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creating beauty (pleasure), but is also a means to approach knowledge (truth). Paul Klee said it most beautifully when he stated that Art does not reproduce what is visible. Rather, it makes (it) visible.

A statement that is germane to the frequently quoted declaration by Pablo Picasso that Art is the lie that makes us realize the truth.

Klee has extended the concept in his famous metaphor of the tree [2], according to which the artist is the trunk, the reality is the root from which the artist receives the lymph that is the necessary inspiration, and the foliage is the artistic product, which is not a copy of the reality, but its interpretation by the artist, who is the mediator.

The dual objective is less obvious in the case of the scientific culture, however, the chapter will show that the search for beauty is an important, even a necessary aim of science as well.

13.2 Beauty and Science Starting with science, one could begin by touching on a curious psychological trait many scientists appear to have, which is the secret wish to be somehow considered as artists as well. Examples of this psychological attitude are plentiful, beginning with the statement by Max Planck that scientists in their work should use essentially artistic imagination. The list of statements implying beauty as the object of science, of course coupled to the search for testable truth, is nearly endless. One could, for instance, quote the unusually bold statement by another stellar figure in the scientific panorama of last century, Jacques Monod [3]: A beautiful model or a theory may not be right, but an ugly one must be wrong.

So, apart from their obvious properties of being either correct or incorrect, scientific theories, which are the ultimate products of the scientific culture, may be beautiful or ugly as well. For non-scientists this may perhaps be a surprising proposition, but most practicing scientists would tend to agree with Monod. I have worked all my life in the area of biology-biochemistry, and I would thus like to offer a fitting example in support of the proposition of Monod that comes from this area of science. Obviously, the double helix of DNA, with which everybody is presumably familiar, could have instead been chosen. But the theory part of it, elegant as it was, was only the proposition that the demonstrated structure of the DNA was a plausible means for its replication. The example in Fig. 13.1 is instead in his entirety a theory that demanded experiments to prove it. This is the theory, proposed by Peter Mitchell, which has gone down in history as the chemiosmotic theory [4]. It explains in molecular terms how energy, for instance the energy of light, can be captured and transformed in biology. The theory postulated that energy capture occurs through the separation of electrical charges across biological membranes, as shown graphically in the figure. No explanations are offered here for the symbols and arrows in

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Fig. 13.1 The chemiosmotic theory of Peter Mitchell [4]

the figure, as this chapter is not the place for going into this type of detail: it will be enough to grasp the essence of the phenomenon, and to understand that the theory offered a very simple and elegant explanation for a key problem of biology, and thus of life. A problem that researchers had tried to solve for decades with cumbersome proposals that did not withstand the test of experimental demonstration. As is generally known, membranes separate the content of cells from the outside world and, inside cells, different compartments from one another. Some membranes, for instance those of an intracellular organelle called the mitochondrion, contain proteins that are oriented transversally in the membrane, and may thus allow the separation of charges as they either transport either hydrogen atoms or electrons. According to Mitchell’s theory in this way these proteins “charge up” the membrane with energy. This is the essence of the theory, which shook the foundations of bioenergetics like an earthquake: it is hard to think of other scientific theories that have had an equally revolutionary impact. Peter Mitchell was aware of the simplicity and elegance of his concept, and was always very proud of its beauty. He thus resisted fiercely the inevitable attempts to introduce changes in his elegantly simple construction: such adjustments had eventually to be made, but they were altogether minor, and did not alter the beautiful simplicity of the concept. However, if Mitchell’s theory is a striking example of beauty, mathematics, which as a science is perhaps the closest to philosophy and to art, is the best arena for proclamations on the connection of scientific truth and artistic beauty. It is indeed from mathematics that the concept of beauty as a necessary ingredient of scientific work has originated: even more, mathematics has originated the idea that the quest for beauty may even take primacy over the scientific accuracy. One could sum up all this with the tranchant proclamation of Godfrey Harold Hardy [5] that Beauty is the first test; there is no permanent place in the world for ugly mathematics.

These are strong and important words, especially as they open in a very direct way the problem of the hierarchical relationship between beauty and scientific correctness in mathematical research. On this, perhaps the most impressive declaration

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is that by another towering figure in the scientific panorama of last century, Paul Adrian Maurice Dirac [6]: I think that there is a moral to this story, namely that it is more important to have beauty in one’s equation than to have them fit experiment. . . It seems that if one is working from the point of view of getting beauty in one’s equation, and one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one’s work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.

Thus, beauty appears to take primacy over scientific soundness. The conclusion leads directly to a very general question: what is beauty? How does one decide on it? On this point, and specifically on beauty in mathematics, I would like to quote a statement by Ilya Prigogine at a Solvay Conference in Brussels more than 20 years ago. In an informal conversation during one of the coffee breaks he had said that equations are not only right or wrong, but beautiful or ugly as well. The obvious question one of us had asked at that point had been: how is one to decide whether an equation is beautiful or ugly? He had disarmingly replied that there was no way one could do it, unless he was a mathematician. A statement that opens an obvious problem. What does one need to know, to decide whether a given cultural product is beautiful or not? Let us look at a painting that is universally known: Botticelli’s Spring (Fig.13.2). Most people have seen it a number of times and each time they have presumably experienced feelings of pleasure.

Fig. 13.2 The Spring (Sandro Botticelli)

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Most educated people would presumably be ready to say that this painting is beautiful. Very beautiful indeed. But let us now look at the image of Fig. 13.3, which shows one of the versions of the famous equation of Schroedinger [7], generally considered by mathematicians as very beautiful. Briefly stated, it describes the space-and-time dependence of quantum mechanical systems.

Fig. 13.3 The equation of Schroedinger

Other equations could of course also have been chosen as examples, for instance, and foremost, Einstein’s equation, which appears on the cover of this volume. It is perhaps the most famous of all equations, and its impact has become widely known also to non-scientists. In its beautiful simplicity it explains a large number of things related to fundamental phenomena of the universe: it probably is the ultimate in terms of mathematical elegance, as it explains so much with so few basic postulates: in it one sees the principle of parsimony, which may also be a significant parameter of beauty, applied at its best. However, Einstein’s equation would have been too obvious, and this is why Fig. 13.3 shows instead an equation that mathematicians consider very beautiful, but which is presumably unknown to the general public. But this now becomes the question: how many non-mathematicians actually retrieve feelings of pleasure by looking at this equation? How many non-mathematicians could state that it is indeed beautiful? Maybe none. Yet both objects in Figs. 13.2 and 13.3 represent beautiful cultural products, and should thus appeal to all cultured individuals. But they do not: most people would be ready to say that Botticelli’s Spring is beautiful, but remain cold in front of Schroedinger’s equation. Clearly, this is a paradox. Can a way be found out of it? Perhaps only by concluding that we believe we appreciate the beauty of the products of artistic creativity, on which most people are ready to swear, but in reality we do so in a way that is overwhelmingly, if not exclusively, connected with superficial emotions; i.e., not in the way professionals would do. In other words, when most people hear a performer playing Mozart, they are on a different level of appreciation with respect to that of a Furtwaengler or of an Abbado, or, more modestly, of the normal music connoisseur. The main problem thus in reality remains that of defining beauty: could one go beyond subjective parameters linked to the individual set of values of the observer, in trying to define it? The problem goes back a long time: to Plato, who thought that beauty indeed was an intrinsic property of objects destined to generate pleasure in the perceiver. In some of his Dialogues he stated [8] that beauty had its own existence independent of the observer, although in one of them (Hippias Major) he appeared to have mitigated the stance by stating that a measure of participation was involved in the appreciation of beauty. Plato’s views have dominated aesthetics for 2000 years, but at the time of the enlightenment a view that was completely at odds with Plato’s emerged that shifted the principles of beauty from the object to the observer. Beauty became subjective, no longer intrinsic to the object, but linked to

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the cultural background of the observer. Perhaps the first unambiguous declaration along this line of thinking was the statement by David Hume [9] that Beauty is no quality in things themselves, it exists merely in the mind which contemplates them.

Just as Plato’s concept had dominated aesthetics before the enlightenment, the idea that judgement was the basic feature in beauty now took center stage: the crucial event was the appearance of Immanuel Kant’s Kritik der Urteilskraft in 1790 [10]. Importantly, in Kant’s conception the aesthetic judgement must be “disinterested,” i.e., not linked to desire (to the agreeable, in his definition). Paradoxically, the existence of the beautiful object could even become irrelevant. As is well known, the ideas of Kant have shaped the thinking on aesthetics for a long time and cannot be discussed in any detail in this chapter. However, given its focus, one point must certainly be discussed: whether, next to the cultural and educational experience of the perceiver, objective parameters pertaining to the object also play a role in the definition of beauty. Do such parameters exist? Simplicity was just mentioned. But could, for example, another objective parameter be symmetry, which is often claimed to be linked to beauty?

13.3 Symmetry and Beauty In ancient Greek the meaning of συμμετρια was best rendered by expressions that signified harmony, stability, rhythm, good proportions; all of them clearly connected with beauty. For instance, is it not commonly accepted that symmetry is an important factor of beauty in a human face? Perhaps the boldest statement on symmetry and beauty is one by Hermann Weyl [11], who wrote Beauty is bound up with symmetry.

But is it so, really? Or is one to think that symmetry has something rigid, something static and predictable to it that makes it less beautiful, whereas asymmetry is more dynamic, less predictable and could thus in the end be more attractive? Dagobert Frey [12] has incisively summed up these concepts: Symmetry signifies rest and binding, asymmetry motion and loosening, the one order and law, the other arbitrariness and accident, the one formal rigidity and constraint, the other life, play, and freedom.

Thus, opinions diverge on the matter of symmetry and beauty, and a number of other authorities, from Ernst Gombrich [13] to Rudolph Arnhein [14] have indeed voiced doubts on symmetry as an exclusive parameter of beauty. Even Immanuel Kant [5], long before, had expressed his dislike of symmetry. Given his standing in the panorama of human culture a statement he made could be quoted here: All stiff regularity (such as borders on mathematical regularity) is inherently repugnant to taste, in that the contemplation of it affords us no lasting entertainment [. . . ] and we get heartily tired of it.

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Thus, whether symmetry, as opposed to asymmetry, can be accepted as the only possible parameter of beauty is evidently questionable. Biology has also had something to say on it: it has indicated that objective parameters of beauty indeed exist, and that symmetry could be important among them. Numerous observations on animal behavior argue in favor of symmetry. For instance, convincing experiments on the common Barn Swallow, show that the female pays attention to the symmetry in the tail ornaments in selecting the male partner [15]. However, as will be briefly discussed below, biology has cogently indicated that subjective beauty exists as well; perhaps asymmetry could have a place in it. However, apart from symmetry, biology has also identified another parameters of objective beauty, i.e., the golden ratio: as it is perhaps the most important, it will be discussed in some detail in the next section.

13.4 The Quest for Truth in Artistic Creativity 13.4.1 Music, Mathematics, and the Golden Ratio The other side of the coin with respect to the proposition that the quest for beauty is an essential corollary of the scientific culture, is the search for truth in the operation of the artistic culture. That the artistic culture has aspects that are intimately related to science is perhaps best shown by music, as its relationships to mathematics had already been recognized in antiquity: the Pythagoreans had discovered that musical sounds are regulated by harmonic ratios that could be expressed by integer numbers. This important scientific concept gained fresh momentum in more modern times, particularly with Johann Sebastian Bach’s pupil Lorenz Mizler. In 1738 he founded a semi-secret Society of Musical Sciences (Bach was its 14th member), which had the declared aim of studying the interplay between music and mathematics; it was Mizler who stated that Music is the sound of mathematics.

Interestingly, the bylaws of the Society required members to operate to bring back music to its mathematical (Pythagorean) origins. Johann Sebastian Bach himself produced for the Society the Variations on the Theme Von Himmel hoch da komm ich her,which, together with the Goldberg Variations, are constructed on abstract rules of arithmetic symmetry. The explorations on the interplay between music and mathematics have expanded continuously since the times of Mizler’s Society up to our days, with special attention on the golden ratio (sectio aurea), on whose presence in music a lot has been written. Briefly stated, the golden ratio allows the division of a segment in two parts, in which the ratio between the entire segment and the largest part is equal to that between the largest and smallest parts (Fig. 13.4).

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Fig. 13.4 The golden ratio

The numerical value of the ratio is 1.618033, and its symbol is the Greek letter Phi, from the name of Phidias, who used it extensively in his sculptures. A rectangle in which the proportions of the sides are those of the golden ratio is a golden rectangle. Removal from it of a square generates another golden rectangle, and so on. By drawing an arch inside each of the squares a golden spiral is generated, as indicated in Fig. 13.5.

Fig. 13.5 The golden rectangle and the golden spiral

The golden rectangle and the golden spiral are forms of the golden ratio that appear in nature: indeed, the golden ratio is not a simple ratio of numbers. It is found in the proportions of the human body, in the arrangement of leaves, seeds, and petals in flowers, in the form of sea shells, in the spiral of the cochlea of our inner ear, even in the form of the galaxies. It is also related to the famous series of numbers of Fibonacci, i.e., the numerical series in which each term, apart from the first two, is the sum of the two preceding ones (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.). The series was discovered in the 12th century by Leonardo Fibonacci, as he was trying to determine how many rabbits could be raised in one year starting from a single pair. It has some astounding mathematical properties, but the point of interest for the present discussion of the golden ratio is that the ratio of two consecutive Fibonacci numbers tends to approach more and more the golden ratio of 1.618 as the numbers become bigger. That so many patterns in nature should follow the rules of the golden ratio and the Fibonacci series is naturally a challenging finding. Thus, it is not surprising that the golden ratio has been claimed to contribute to the harmony of nature. This

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Fig. 13.6 The Great Pyramid in the Giza valley plateau and the Parthenon

chapter is not the place for a discussion of the philosophical implications of the golden ratio and of the series of Fibonacci, but it is certainly pertinent to point out that they are a very serious argument in the hands of those who believe that the world of numbers existed before men discovered it. The golden ratio appears everywhere in art: from architecture to music. In music it appears, for instance, in the relationships of intervals and distances between notes. Many composers, from Beethoven to Chopin, to Mozart appear to have used it, although they may have done it intuitively. Numerous other composers, however, from Debussy, to Strawinski, Bartok, and to Stockhausen, have made conscious attempts to use it. In architecture, one could quote the proportions of many architectural masterpieces, like those of the Parthenon and, perhaps, those of the of the Great Pyramid in the Giza valley plateau1 (Fig. 13.6). In modern times Le Corbusier has been the architect with the highest interest in the golden ratio. He has elaborated its principles theoretically and has applied them extensively to his work. Mario Botta deals with this extensively in his chapter. The golden ratio appears in the paintings of numerous artists of the past, including Leonardo da Vinci, who dealt with it in very influential writings, Botticelli, and Piero della Francesca, who also wrote extensively on it. In his chapter, Giuseppe O. Longo deals with Piero in detail, analyzing The Flagellation, perhaps his most famous painting, and the one in which the golden ratio is most obvious. It also appears in the work of scores of modern painters, among them Georges Seurat, Piet Mondrian, and Salvador Dalí. The deliberate use of the golden ratio is particularly explicit in the painting by Dalí shown in Fig. 13.7: the ratio of the dimensions of the painting are exactly those of the golden ratio. Even more: Dalí included in the painting in a dodecahedron, a Platonic solid that is precisely related to the golden ratio. It is present in science as well, from geometry to physics, e.g., in that new form of matter which are the quasi crystals. But for the subject matter of this chapter it is very important to mention biology again, to emphasize that convincing neurobiolog1

The square which is the base of the pyramid has sides of 230 m, the height of the pyramid is 146 m. The ratio of these two dimensions is 1.573. . . , which is very close to the golden ratio, However, whether these dimensions reflect the awareness of the golden ratio by the Egyptians nearly 5000 years ago, or whether they are simply coincidental is still a disputed problem.

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Fig. 13.7 The Sacrament of the Last Supper (Salvador Dalí) © by SIAE 2009

ical evidence now supports the idea that human aesthetic perception subconsciously privileges proportions that obey to the rule of the golden ratio. Recent experiments on activity in selected brain regions imaged in subjects presented with sculptural masterpieces in which the proportions of the bodies obeyed the golden ratio rule, or with modified versions of the same sculptures in which the golden ratio rule was violated, are particularly telling; the registered activity in the brain clearly favored the sculptures in which the proportions were those of the golden ratio. Of great interest was also the observation that subjects asked only to observe the sculptures as if they were walking in a museum activated brain areas different from those which became activated when they were asked to express an aesthetic judgment and found the sculptures beautiful [16]. Evidently, aesthetic perception relies on objective parameters, one of which is the golden ratio (another parameter, as mentioned above, could be symmetry, as inferred from similar brain imaging experiments [17]. On them, subjective parameters are superimposed that are linked to the experiences and set of values of the observer. Paradoxically, one of them could even be asymmetry. 13.4.2 Visual Arts Apart from mathematics and music, an obvious case that supports the arguments made above is that of architecture, an activity in which art and science – beauty and truth if one adheres to the cliché – are intimately fused almost by definition. The chapter by Mario Botta later on in this book deals with this in detail, underlining the importance of the golden ratio in architectural creation. However, one could say that architecture is a special case, and thus it would be convenient to substantiate the discussion with arguments that have more general validity. One could start from

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a bold statement on painting made by John Constable [18]) more than 200 years ago: Painting is a science, and should be pursued as an inquiry into the laws of nature.

This statement is ideally complemented by the famous declaration, also related to painting, by Paul Cézanne [2]: Treat nature by means of the cylinder, the sphere, the cone, everything brought into proper perspective.

The emphasis of Cézanne on geometry immediately brings to mind cubism, of which Cézanne, especially in his later work, is considered to be the precursor. Cubism was initiated in 1907 by Pablo Picasso and Georges Braque, who decided to move away from the conventional representation of homogeneous forms, breaking instead the composition into geometric planes that interpenetrated each other, distorting and cutting up the figure into a series of flat facets devoid of any sense of depth. The three-dimensional representation of the traditional painting of the Renaissance and beyond, which was based on a single point of observation – that of the observer standing in front of the object – was now replaced by a multitude of points of view, as if the observer, instead of standing still, were moving, experiencing simultaneously different views, in which various aspects of the figure, eyes, ears, noses, would be simultaneously seen in profile and in front. Figure 13.8 shows

Fig. 13.8 Girl with mandolin (Pablo Picasso, © by SIAE 2009, left panel). Woman with a guitar (Georges Braque, © by SIAE 2009, right panel)

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Fig. 13.9 Nude descending a staircase (Marcel Duchamp) © by SIAE 2009

this very clearly. The two paintings by Picasso (Girl with Mandolin, left) and by Braque (Woman with a Guitar, right) shown in the figure are good examples also because they are more accessible than later cubist paintings. The two artists, even if becoming very abstract, were still somewhat conditioned by the physical appearance of the subject, which is the same in the two paintings. They could be compared with much more extreme later cubist paintings, for instance, the famous Nude Descending a Staircase of Marcel Duchamp (Fig. 13.9) that created a scandal of such proportions when it was first presented in Paris in 1912 that the artist was asked to either withdraw it or to change its title. Cubism was a revolution that spread quickly to other forms of art, for instance to literature, initiating a tendency to present characters and experiences from a multitude of perspectives, also using technical expedients like minimal punctuation. The names of Apollinaire, Cendrars, Duchamp, Jacob, Cocteau, and of a number of others immediately come to mind. In painting, cubism was the triumph of geometry, with all figures dissected into planes and shown geometrically. The art critic Louis Vauxelles described the work of Braque saying that “it scorned form, and reduced everything to geometric schemes and cubes.” Here, given the leitmotif of the chapter, biology should be mentioned again, as modern studies on perception have shown that the cubist way is indeed the way in which the visual images of objects are formed; not from one single glance, but from a large number of glances that are then united in the human mind.

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The statement of Cézanne on geometry, even if it was made in 1904, can also be taken back to the Renaissance and the tradition of painting based on the imitation of nature and on creating the illusion of a three-dimensional perspective. However, even in these paintings the idea of geometry was frequently present, except that it was somehow hidden, not obvious, so much so that one must be very perceptive to discover it. In most cases, as one looks at a painting of the Renaissance, one does not realize that there is something related to geometry behind it that the painter used as his basic material. The example shown in Fig. 13.10, which comes from a painter of

Fig. 13.10 Head of Leda (Leonardo da Vinci). Final painting, upper left panel. Studies for the head and the hairdo, upper right panel. Detail of the final painting, lower panel

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mythical stature, Leonardo da Vinci, is particularly telling, as its connection to the statement of Cézanne is very evident. The figure shows some studies of Leonardo for the head of Leda In the upper left portion of the Figure there is the head of Leda as we see it in the final painting today. Next to it, there are some studies of Leonardo for her braids and hairdo. Clearly, here we are in the world of geometry, which Leonardo had evidently found necessary to explore to arrive at his final representation of the head of Leda. The traces of Leonardo’s preliminary play with geometry can only be guessed in the final painting. However, the hairdos of the two putti in the lower portion of the figure are explicit remnants of Leonardo’s original journey into the world of geometry. Geniuses of the caliber of Leonardo had traveled similar paths millennia before him. The pre-historic paintings of the Altamira cave, which are about 14,000 years old, or those of the Lascaux and the Magdalenian caves, speak for themselves. Figure 13.11 shows in the left panel a bison from Altamira, which someone has defined as the Sistine Chapel of prehistory, and next to it, another bison from one of the late Magdalenian caves and a horse from the cave of Lescaux.

Fig. 13.11 A bison from the Altamira cave (left panel), a bison from one of the late Magdalenian caves (central panel), and a horse from the Lascaux cave (right panel)

These paintings do not talk the language of geometry as clearly as in the study by Leonardo, which is not too surprising, as geometry as a written and codified science was not even there in those times. But hints of it are unmistakenly there, and in any case it is obvious that the unknown geniuses who painted these animals wanted to convey something that would be more revealing than the simple representation of the bisons in the way their fellows in the community saw them. They evidently sought to transcend the orthodox representation of these creatures, to reveal their deeper nature. Clearly, these ancient artists wanted to increase knowledge and understanding, in addition to creating beauty.

13.5 Interplay Between Science and Art: An Example from the Architectural Work of Michelangelo Another example, taken in this case from the architectural work of Michelangelo, further emphasizes the interplay between the two cultures (Fig. 13.12). In a sense,

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Fig. 13.12 Drawings for the wall fortifications of one of the gates of the city of Florence

the example is inverted in scope with respect to that of Leonardo da Vinci, who had visited the world of geometry before going back to the human figure. Michelangelo does the opposite. In his sketches for the wall fortifications of one of the gates of the city of Florence, he starts from the rigid inanimate geometry of the walls (left panel), but then adds to them as defense devices curvilinear objects that are curiously but clearly reminiscent of the claws of a strange animal (right panel); a way to blend something animate, not rigorously scientific, in the rigid inanimate geometry of the walls.

13.6 More on Art as a Path to Knowledge: Expressionism and Music The discussion above on visual arts would not be complete without a quick reference to expressionism, in which the idea of art as a way to understanding is most obvious. Expressionism was not concerned with the representation of the objective reality of subjects, but privileged instead emotional aspects, feelings, exploring the hidden recesses of the human psyche to expose its dark sides. It thus most clearly transcended the goal of communicating immediate pleasure, as in the statement of Coleridge quoted above, to reach deeply into the world of knowledge. Gottfried Schatz [20] recently analyzed a portrait Oskar Kokoschka had made of the great architect Adolf Loos (Fig. 13.13). In the analysis of Schatz, the portrait bares the soul of Loos, revealing his inner conflicts. The outer form of Loos shown in the portrait no longer shows the architect people conventionally saw: the portrait now instead reveals his internal self. The feverish grasp with which the hands are held together betrays in a particularly telling way his internal conflicts. A true and impressive piece of psychological science.

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Fig. 13.13 Portrait of Adolf Loos (Oskar Kokoschka) © by SIAE 2009

Music has already been discussed in some detail, but can also be used to extend the idea that art is a way to reveal meanings, and is thus a path to understanding. Music, unlike all other arts, soars above the realities of this world, and can thus only be related to the quest for truth when it deals with human emotions. In Wagner, for example, the leitmotifs so common in his music are musical passages used to explore the soul of the characters and to reveal their innermost thoughts. They can either reinforce or contradict the essence of the character, or of a situation. When in the Goetterdaemmerung the sinister character Hagen greets the arrival on the scene of Siegfrid singing of loyalty and friendship, the accompanying orchestra warns the audience of his duplicity by playing fortissimo the leitmotif of the Ring, the sign of greed and betrayal. Musicologists have analyzed technically the Ring leitmotif, which is modified but insignificantly in its subsequent appearances, pointing out its ability to create tension, and setting up expectations in the audience.

13.7 The Search for Truth in Poetry The idea of art as a path to knowledge can be further and impressively substantiated by digressing into the world of poetry. Here, the work of Eugenio Montale is particularly revealing. The ever-present leitmotif in his work is the idea of poetry as a means to attain the truth, the absolute truth, which is hidden somewhere. Some of his most beautiful poems are centered around the concept of a sudden illumination,

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of a flash of light that would put us, to use his own words, in the midst of a truth: in his poems the quest for beauty and the quest for truth fuse in a truly masterful way. One could, for instance, read the third stanza of I limoni (The Lemons) [21], the composition that opens his famous first collection of poems, Cuttlefish Bones. Although other suggestive examples could have been chosen, as this is a recurring theme in Montale, this poem bears the notation In limine, as if it were a personal manifesto of his poetry: Vedi, in questi silenzi in cui le cose s’abbandonano e sembrano vicine a tradire il loro ultimo segreto talora ci si aspetta di scoprire uno sbaglio di Natura, il punto morto del mondo, l’anello che non tiene, il filo da disbrogliare che finalmente ci metta nel mezzo di una verità

The translated stanza, which loses part of its beauty, as Montale is notoriously difficult to translate, would read like this: You see, in these silences in which things let themselves go and seem close to reveal their ultimate secret sometimes one expects to discover an error of Nature, the dead point of the world, the ring that does not hold, the thread to be untangled that would eventually put us in the midst of a truth.

The text explicitly shows that Montale expects that poetry will lead him to discover a mistake of nature, the ring that does not hold, as he says. Through it he hopes to find a path to attain the truth, which is hidden out there. Poetry, he hopes, will give him a varco, as he calls it in another famous poem, La casa dei doganieri (The House of the Customs Officers). A “varco” is an opening, a passage: in the night, far away on the horizon, a passing oil tanker sends intermittent flashes of light. “il varco è qui?”, he asks, “is this the opening?” Flashes of light, sudden illuminations, which he hopes to somehow exploit to see beyond the screen. To attain the truth. Here one can mention truth without hesitations, as the poet himself identifies the search for it as the goal of his poetry.

13.8 The Structure of the Creativity Process. The Essential Role of Intuition The discussion above has offered argument on the similarity of goals in the two cultures and on their interplay. It is now convenient to complement it with a discussion of the structure of the process that eventually leads to the creation of the scientific and the artistic products. If one does not consider the final products of the two cultures, but only the mechanism by which the creativity process arrives at

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it, the differences between the two cultures tend to disappear. The concept of intuition, which is an essential moment in the process of creativity, must be briefly discussed in this context. In a public lecture delivered in 1928 at the Athenaeum Club in Granada with the title “Imagination, Inspiration, Escape in Poetry” Federico Garcia Lorca [22] discussed in some detail the differences between imagination, which for him was based on reality, and inspiration, which describes the purely poetic phenomenon. In the lecture he did not specifically use the word “intuition,” but the term “intuition” used here corresponds to what he defined as inspiration. The concept he proposed was that the creation of a poem is not the immediate result of intuition. In his lyric description he said that intuition is a leap, a flight that takes the poet to a magic land, to a land “where he hears the flowing of great rivers, where his forehead feels the cool of the reeds that tremble in the midst of nowhere. Where he wants to hear the dialogue of the insects beneath the boughs, where he wants to penetrate the current of the sap in the dark silence of great tree trunks. Where he wants to understand the Morse alphabet spoken by the heart of the sleeping girl.”

The poem, he said, is the description of the journey back, of the return with the treasure: it is the classification and selection of what has been brought back. The image is captivating and beautiful; not surprisingly, as it comes from a person with the artistic imagination of Lorca. It tells us that intuition is the primum movens, without which there would be no poem of any beauty. But then, the production of the poem itself is hard work. In the last chapter of this book Michelangelo Pistoletto describes exactly the same sequence of events between intuition, which in his case was a dream,and the actual generation of the artistic product – the Rosa Bruciata – as he wakes up. This is the way in which artistic creativity operates. But the important point is that it is also the way in which scientific creativity operates: intuition offers a scientist a lead, but then the scientific product is the result of the hard struggle to substantiate it with a testable demonstration. The parallel is clearly striking.2

13.9 Intuition in Scientific Creativity Examples showing the importance of intuition in the process of creativity are plentiful in science. They are more obvious than in arts, probably because scientists do not have the ability to cloud, to hide this sort of things as artists normally do. One particularly telling example is the famous dream of Friedrich August Kekulé, one of the great chemists of the 19th century. He had been studying how carbon atoms could link up into long chains when he stumbled on the hydrocarbon benzene, which is made up of six atoms of carbon and six atoms of hydrogen. The composition of 2 However, on this point a difference between the two cultures must be pointed out. The product of artistic creativity is always unique, and therefore always valid. In science, instead, the process of creativity may lead to a product that is not unique, as others may have arrived at it at the same time, maybe using different paths.

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benzene had been known for a while, but it remained a mystery how these 12 atoms were linked to one another. He simply could not see it, but one night he had a dream, which led him to propose the immensely important structure of a six-membered ring of carbon atoms with alternating single and double bonds. This is how he described the dream in an extemporaneous speech he gave at a Meeting of the German Chemical Society organized in 1890 to celebrate the 25th anniversary of his first benzene paper [23]. I was sitting writing on my textbook, but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by the repeated visions of the kind, could now distinguish larger structures of manifold conformation; long rows sometimes more closely fitted together all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.

The vision of the snake seizing hold of its own tail wash the flash of intuition that gave Kekulé the idea of the circular structure of benzene, as depicted artistically in the cartoon of Fig. 13.14. Interestingly, at the end of his speech, Kekulé said “Let us learn to dream, gentlemen, and perhaps then we shall learn the truth.” The story of Niels Bohr is also very instructive. He had been trying for a long time to figure out the structure of the atom, designing model after model in a long series of unsuccessful attempts, until one night he had a dream, of which he spoke frequently afterward: he had the vision of the nucleus of the atom with the electrons spinning around it as in the solar system the planets would do around the sun. He woke up, and he realized that the dream had given him the model of the atom, which subsequent experimental work proved true.

Fig. 13.14 The dream of Friedrich August Kekulé and the structure of benzene

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Kekulé and Bohr are towering figures in the panorama of science. A much humbler example could also be quoted, that of Elias Howe, the man who invented the sewing machine around the middle of the 19th century. He had the idea of a machine that could sew pieces of cloth, but the needle was a problem, and none of the models he designed and tried, for instance one with needles pointed at both ends and with a hole for the thread in the middle, worked. Then, one night he dreamt of being taken prisoner by a group of wild natives, who started dancing around him with spears, and in the dream he noticed that the spears had holes near the tip. He woke up to realize that he had the solution to his problem, as the hole at the tip of the needle would catch the thread after it had passed through the cloth: the sewing machine was born. The unique moment of pure joy, of passion, defined as intuition, is thus essential in both arts and science. As should be obvious, this only applies to products of value, products that are qualitatively different, that cannot be simply planned and thus predicted. No intuition is necessary for routine, repetitive work in either culture. And of course no intuition is necessary for the brute force work that has become so common in today’s science. Obvious as it may be, this is an important point: intuition, in both art and science, is the discriminating factor between masterpieces and routine products. Take the Monna Lisa: anybody could have painted a reasonable portrait of a rather plump, not particularly beautiful lady of the Florentine nobility in the 16th century. But the intuition of Leonardo da Vinci of giving her that incredible enigmatic smile has made the portrait an absolute masterpiece.

13.10 Intuition in the Work of Poets: The “Gift of God” A number of years ago I had a dinner conversation with Vittorio Sereni, one of the best Italian poets of the second half of last century. In that conversation he told me of the labor of the poets, of the fatigue involved in the composing of a poem. He sounded convincing, and in any case he knew what he was talking about. Yet, there was something odd, nearly disturbing in the notion of fatigue, of repeated variants of the verses, of painful doubts on how to say something in poetic language. Thus I told him: “but Doctor Sereni, you are not going to tell me that Leopardi, when he wrote that immortal verse” Dolce e chiara è la notte e senza vento [25]

“agonized much on it before putting it down on paper.” He smiled and replied: “Of course not, but that verse was a gift of God. God often gives poets a perfect verse as a gift, and it could be the first, or the last, or one in the middle of a poem. It could even be a couple of verses. But then the poem must be constructed around it, around that precious gift, and believe me, this is hard labor.”3 3

The “Gift of God” may not even be a finished verse, but only an idea that will lead to it, and be an essential part of the concept behind the poem.

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That was a very illuminating statement. It impressed me profoundly, so much so that I now always search – in poems that fascinate me particularly – for that magic, perfect verse that tells me that the poet had been blessed with that priceless moment he could not have possibly planned, which had taken him to the magic land of Lorca. And the examples quoted above should have made clear that these “gifts of God” are essential in scientific creativity as well. One could push the discussion of “gifts of God” to poets a little further, using still another argument to support the idea that verses that are the immediate product of intuition are somehow born perfect. One reads them, mumbles them, and such is their beauty that he does not realize that they are frequently semantically incorrect. In the verse by Leopardi I quoted a few minutes ago, the night was defined as “dolce”, which means “sweet.” Now, a night can be warm, dark, clear, cold, long, etc., but the adjective sweet applied to it which was simply perfect to create the melancholically nostalgic atmosphere of the poem, is semantically questionable. In Italian, it is, however frequently used metaphorically to define meanings other than the orthodox ones. Thus, more convincing examples of the inappropriate use of semantics in “born perfect” verses would be useful. As Sereni has originated the argument used here, it is appropriate to choose one of his most beautiful poems as an example. The Strada di Creva (The Road to Creva) begins with three splendid verses [25]: Presto la vela freschissima di Maggio ritornerà sulle acque dove infinita trema Luino. . .

In English they would read like this: Soon the cool sail of May will return to the waters where infinite trembles Luino. . .

Here, the semantic inconsistency is unambiguous, both in the choice of the adjectives and of the verb: sails cannot be “cool,” a town cannot “tremble” and cannot be “infinite.” Again, however, the verses convey masterfully the expectation of the awakening of the season, the joy for the new beauty of the lake, the love and perhaps the nostalgia for the home place. Naturally, scores of “perfect” verses could be quoted in any language by educated readers. Here, the argument has been only introduced to corroborate the concept that for the magic moment of intuition normal rules do not necessarily apply. One does not discuss a gift of God: one accepts it, happily and mercifully.

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13.11 Summary That the operation of human mind would generate products that could identify diverse cultural categories was of course known well before Snow’s Rede Lecture of 50 years ago [26]; the declaration by Coleridge quoted at the outset of this chapter is but one example of this. It was, however, more a feeling than a documented concept. Snow’s merit is to have stated the problem clearly and rationally. Unfortunately, for a long time after his Rede lecture most of the emphasis was on the analysis of the differences between the two cultures, which are indisputably there. Only gradually, and not without difficulties, has interest shifted to the search of analogies and of the points of convergence, steering away from the divergences. This chapter has sought to show that this search is not only important, it enriches the field, as it adds important new dimensions to it. And it opens new avenues of study. In the mind of the author, a particularly important avenue is that of brain biology. The excursions in the area made in this chapter have been necessarily brief, but have nevertheless underlined the great potential of brain biology as a tool to unravel the basic mechanisms of aesthetic perception. The field is still in its infancy, but has already offered indications that objective and subjective parameters in aesthetics are not mutually exclusive: both have a role in shaping our approach to beauty. To end this chapter with a conciliatory note, one could say that Plato and Kant, after all, were both right.

Acknowledgments The author wishes to thank Arrigo Marcolini (Modena, Italy) and Gottfried Schatz (Basel, Switzerland) for many helpful and stimulating discussions. Gottfried Schatz has graciously granted permission to use a figure (Fig. 13.6) he had used in a previous publication. The author also wants to thank Laura Fedrizzi for her help in the organization of the manuscript.

References 1. S.T. Coleridge: Definition of poetry (1811). In: The Literary Remains of Samuel Taylor Coleridge, collected and edited by H.N. Coleridge, vol. II (William Pickering, London 1856) 2. P. Klee: Das Bildnerische Denken. Schriften zur Form und Gestaltungslehre, Vol. 1 (Benno Schwabe Verlag, Basel und Stuttgart 1956) 3. J. Monod: Symmetry and Function of Biological Systems at the Molecular Level. Nobel Symposium, A. Engström and B. Sandberg (eds.) (Almqvist & Wiksell, Stockholm 1968) 15–27 4. P. Mitchell: Chemiosmotic Coupling in Oxidative and Photosynthetic Phosphorylation (Glynn Research Ltd., Bodmin 1966) 5. G.H. Hardy: A Mathematician’s Apology (Cambridge University Press, London 1940) 6. P.A.M. Dirac: The Evolution of the Physicist’s Picture of Nature. Sci. American 208, 45–53, 1963

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7. E. Schroedinger: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28, 1049-107 (1926) 8. Plato: Phaedrus and Hippias Major. In: Dialogues (Appleton and Co., New York 1898) 9. D. Hume: Of the Standard of Taste. In: Four Dissertations (A. Millar, London 1757) 10. I. Kant: Kritik der Urteilskraft (Lagarde und Friedrich, Berlin und Libau 1790) 11. H. Weyl: Symmetry (Princeton University Press, Princeton 1983) 12. D. Frey: Studium Generale 2, 268–278 (1949) 13. E.H. Gombrich: Symmetrie, Wahrnehmung und künstlerische Gestaltung. In R. Wille (ed): Symmetrie in Geistes- und Naturwissenschaft (Springer, Berlin 1988) 94–119 14. R. Arnheim: A Review of Proportion. In: G. Kepes (ed): Module, Proportion, Symmetry and Rhythm (Georg Brazilier, New York 1966) 218–230 15. A. Pape-Moller: Evolution, 49, 658–670 (1994) 16. C. Di Dio, E. Macaluso, and G. Rizolatti: PloS One, 11, 1201–1208 (2007) 17. T. Jacobsen, R.I. Schuboth, L. Hoepel and Y.V. Cramon: Neuroimage 29, 276–285 (2006) 18. R. Lambert: John Constable and the Theory of Landscape Painting (Cambridge University Press, Cambridge 2005) 19. P. Cézanne, Letter to Emile Bernard, 1904 20. G. Schatz: Jenseits der Gene (Verlag Neue Zürcher Zeitung, Zürich 2008) 21. E. Montale: Ossi di Seppia (Rivoluzione Liberale, Torino 1925) 22. F.G. Lorca: Imagination, Inspiration, Evasion. Lecture at the Ateneo of Granada 1928 23. F.A. Kekulé: Benzolfest Berichte der deutschen chemischen Gesellschaft 23, 1302–1311 (1890) 24. G. Leopardi: La Sera del Dì di Festa. In: Canti (Le Monnier, Firenze 1835) 25. V. Sereni: La Strada di Creva. In: Frontiera (Edizioni di Corrente, Milano 1941) 26. C.P. Snow: The Two Cultures and the Scientific Revolution (Cambridge University Press, London 1959)

Discussion A. Miller: Unlike Picasso, Braque did not play any role in discussing cubism. E. Carafoli: Maybe Picasso was more talkative than Braque. Braque spoke through his work, and did it very eloquently. Perhaps we should not forget that Louis Vauxelles, when he coined the term “cubism” (apparently inspired by Matisse) did so to describe the work of Braque. A. Miller: One can speak about verification in art as well. Architectural images can be right or wrong. Generally, if an art form is fruitful it will engender other art forms and will thus be verified. E. Carafoli: In principle I agree, but only if we refer to special cases like that of architecture. Maybe to that of philology as well; philology, like architecture, has a strong component of scientific culture. As to other forms of art, I would disagree; what would the expression “right” as opposed to “wrong” mean if applied, for instance, to a painting or to a symphony? S. Califano: The Pythagoreans thought that geometrical forms had beauty and harmony, which stemmed from proportions and symmetry. For them, numbers also had geometrical properties, i.e., they could be square, rectangular, triangular. The insistence of the Pythagoreans on symmetry in geometrical representations may have had mystical origins, and could have derived from oriental religions. Their empha-

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sis on symmetry in the beauty of numbers led them to “hate” some of them. They “hated” number 17 more than any other number, to the point of calling it an “obstacle”, because its place was between number 16, which is a square, and 18, which is a rectangle. Numbers 16 and 18 are indeed the only numbers that form geometrical figures in which the perimeter is equal to the area, and number 17 separates them, disrupting the proportion of one to one eighth into unequal intervals. Beautiful and ugly numbers, then exist, and symmetry is a crucial factor in aesthetics. These themes were debated extensively in ancient Greece, and one could even go one step further and mention the five platonic solids, which for Plato were perfect forms, forms with intrinsic beauty. E. Carafoli: Your points extend nicely what I said on beauty in mathematics: I only mentioned equations, and you go one step further, finding beauty (and ugliness) in numbers themselves. The mention of symmetry was very interesting. A. Peccanda: You have not discussed the point of productivity in creativity. Science creates technical progress, which is correctly used in architecture and, possibly, in music as well. By contrast, in contemporary visual art there is only imitation and incorrect use of the technique. Therefore, science is productive, contemporary visual art only imitative. E. Carafoli: I find your statement and conclusion a little too absolute. Even in contemporary visual art, some would even say particularly in contemporary visual art, innovation and creativity are present and in some case very evident. The chapter of Michelangelo Pistoletto is a particularly striking example of this. F. Brunetti: Prior to contemporary times it used to be common that scientists would practice art, e.g., they would paint, and that artists would, for instance, use mathematics as a tool of creativity. It seems not to be so with contemporary authors. When did this sort of separation start, so that artists would no longer like to be scientists, and scientists would no longer want to be artists? To what extent does the language used shape the ideative/creative process? E. Carafoli: Perhaps you are overly pessimistic. Examples of today’s scientists with a deep interest in arts and active in one of them are still plentiful. Music is perhaps the most common case: a number of important scientists were, and are, accomplished musicians. Music, however even if the most common, is by no means the only case, as numerous scientists were also accomplished painters and sculptors. The opposite case, i.e., that of artists who have expressed more than an amateurish interest in science is more rare. Apart, again, from the case of music, as a number of well known musicians were also excellent mathematicians. Language is certainly a point: it is easier for a scientist to master reasonably well the language of arts, than for an artist to master that of science. But the most important point, that goes back to your observation on language, is that art can be practiced part time, as a hobby. Science cannot. G. Setti: It seems to me that while beauty in science and in the equations, and in the symbols that describe them, is objective, i.e., is something on which one would

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agree worldwide. This does not apply to beauty in arts, which it is closely linked to the diversities of cultures and societies. E. Carafoli: Culture, upbringing, personal experiences are indeed important in the approach to beauty in art. This, as I have said, is now the predominant opinion on the matter of beauty. It goes back to the enlightenment and to Kant. These subjective aspects, by contrast, are not important in the approach to scientific beauty. However, the point I have tried to make is that of objective parameters in artistic beauty that would transcend cultural diversities. I am convinced that they exist, and I have mentioned the golden ratio and symmetry. I must say that I am impressed with the recent discoveries on brain imaging. M. Bresciani: I wonder if it would be possible to use this definition of beauty: beauty in science and in literature has to be understood as a means to realize an updated representation of the world in which we live and have to interact. So beauty can be seen as a means of expressing precisely in a symmetrical way our knowledge of both the natural world and the realm of human relations. In other words, beauty is an aspiration to a harmonious order of the parts; born from a desire for knowledge and expressed in an image that thought creates and feelings stimulate. Fantasy, intuition, rationality: together they shape the linguistic models through which is configured a “reading of the world.” Both a poetic text and a mathematical equation bear the imprint of this “reading.” Mathematical and geometric criteria govern the features of any meaningful discourse. And when this discourse is meaningful, it is necessarily beautiful and strikes the cords of both reason and sentiment. When successful, both a scientific and a poetic result represent a fragment of self-ordered reality that moves us and enthuses us because of the intrinsic beauty of its imagery. Both languages aim to decode the external world. Even literature, at least that which is worthy and timeless, is born out of a project, a logical construction model to interpret reality. Good literature, which survives over time is born from the construction of an interpretive model of reality which, once it goes beyond abstract thought, in contrast to science, dives again into the world of sensory perception and embodies itself in narrative. That literature acts in the context of a cultural system continually updated and in its own way wanders with fantasy inside possible worlds and recreates de dicto a new reality. Hence, the necessity of always constructing new “styles”, new linguistic models only if they adhere to the conceptual structure of a new way of looking at the world, and only if they are not arbitrary and contribute to expand the boundaries of our knowledge. We had in a previous lecture an example of a parallel between modern science and the artistic avant-guard. This is one of the aspects that creativity in art can assume. The early 20th century was very stimulating in this sense. But other cultural paradigms can influence and interact, governing literary and artistic creativity. This is one of possible aspects of creativity in art. The early 20th century was vibrantly stimulated in this direction. However, there are also other cultural paradigms that may influence, interact, and govern literary and artistic creativity. One exam-

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ple for all: that literary masterpiece, La coscienza di Zeno, would not have been conceived if Svevo had missed his fortuitous encounter with Freud, just in time to meditate on the “child” inside us and to acknowledge its existence. He would never have written those extraordinary final pages, concluding a story that, in the opinion of some critics, may even appear somewhat disconnected from them. Those pages represent instead the premise, the unexpressed thoughts that put in motion the tasty, light and merrily ironic narration of the main character’s vicissitudes. It is through these that he earns the role of a person finally aware and utterly serene against the difficulty of living, with which the biological nature has to constantly measure up with. E. Carafoli: Your points complement nicely what I have said in my presentation. I like your expanded definition of beauty, which is certainly very attractive. I was very concise when I stated that artistic creativity aims at making sense of the world around us, but you said it in a more articulate way. I thank you for mentioning Svevo and The Conscience of Zeno. The encounter of literature and psychoanalysis has been tremendously important, and would provide enough material for a full Symposium. Perhaps some of it ought to have been included in this Symposium as an extension of the discussion on the interplay between music and psychoanalysis.

The Artworld and the World of Works of Art1

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14.1 Introduction On 18 October 2006, I entered the Marlborough Gallery in New York for the official opening of Fernando Botero’s exhibition on Abu Grahib. I visited the exhibition with a friend of mine, the painter Valerio Adami, and with his wife Camilla, a painter as well. Valerio talked to me about the way Botero draws hands and animals, and later introduced me to him. In wandering around the Gallery, I then ran into Arthur C. Danto and his wife. Danto is Emeritus Professor of Philosophy of Art at Columbia University and reviewed Botero’s exhibition in a long article, titled “The Body in Pain”, which was published by The Nation on 27 November. I remember how all of Valerio’s and Camilla’s remarks, and Danto’s article, echoed within my mind when I visited again Botero’s exhibition, this time in Berkeley, where in the meantime it was moved. The philosophical question behind this apparently (but perhaps more than just apparently) frivolous story is the following: where is the artistic element in all that? What gathered at the vernissage at the Marlborough Gallery was the “Artworld.” The critic, the fine connoisseur (which, as in the case of Valerio and Camilla, is himself or herself a painter), the artist, the artwork, the gallery, the art market, the newspaper that publishes the review. . . Maybe also the philosopher of art Ferraris, no less, that should have found inspiration from such an occasion for a theory that explains why, for instance, we are interested in those works of art and why the humiliated bodies painted by Botero – with which we feel an immediate identification (as Danto correctly remarked in his review) – are art and the pictures that circulate on the Internet are not; and why (leaving aside the sphere of tortures, and turning to the pleasures of life) we do not generally consider the Rolls Royce Silver Shadow used by Botero to go to dinner after the show as a piece of art. 1

This is a slightly modified version of the first chapter of my book, La fidanzata automatica, Bompiani, Milan 2007.

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No doubt, the Rolls Royce in question is a beautiful object, and one may even ask why (and that would be a question that would pose a challenge to any philosophy of art) it is not a work of art. In other terms, why Duchamp’s pissoir, for instance, is a work of art and Botero’s car is not?

14.2 From Mimesis to Ready-Mades It is somehow mandatory to start our analysis from this question. In 1917, Duchamp exhibited a urinal and said that it was a work of art. With such a move, he just gave voice to the widespread opinion, after the explosion of modern art, according to which anything can be a work of art. I do not know if behind such an idea there was the pleasure of the critics in acting as omnipotent judges or the joy of the makers of bathroom suites, who saw an unforeseeable expansion of the market for their products; be that as it may, exactly ninety years after Duchamp’s trick, common sense and experts agree at least on one dogma, the one according to which anything can be a work of art. Just like any idea on which there is too much general consent, the idea according to which anything can be a work of art gives rise to some suspects. To begin with, it is quite clear that such a thesis cannot express an aesthetic judgment such as “anything can be beautiful,” given that it would be just like saying that “anyone can be beautiful” as the result of a critic’s fiat and not at the cost of great sacrifices such as diets, plastic surgery, high expenses and, sometimes, even mortal risks – sacrifices that most of the times do not produce the wanted result, leaving those who attempted to improve their appearance ugly as before and making them somewhat ridiculous. Let us suppose that the thesis in question amounts to the following idea: in certain conditions, and leaving aside any aesthetical consideration, anything can become a work of art, namely anything can be counted among the artistic objects, provided that everyone can express his or her indifference or disgust about the object in question. In such a formulation, the thesis sounds sufficiently reasonable and liberal. However, is it really so? If not, what is then the difference between works of art and ordinary objects? Such a question seems to have replaced the Platonic version of the problem of the difference between objects and works of art. No one thinks – at least to my knowledge – that a urinal imitates an idea, and no one thinks that Duchamp’s urinal is the imitation of an imitation, as Plato perhaps would suggest; that is, no one would suggest a hierarchy that goes from the idea of urinal, to the urinal that the demiurge places at people’s disposal, to the urinal that Duchamp exhibits in a gallery, without even worrying about painting it. We all agree that the Platonic hierarchy doesn’t hold; however, leaving Plato aside, why Duchamp’s urinal is a work of art, whereas it was just an ordinary thing before being exhibited? What this question suggests is that, after many centuries, the philosophy of art has disposed of the problem of the mimesis, but only to fall prey of an even more puzzling question, the one concerning ready-mades: the problem is not that of knowing whether a work of art is a copy of

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a copy, but that of explaining why an object whatsoever can become, as if by magic, a work of art. If this was the case, it would be nice after all, for the price of the works would be paltry and we would be surrounded by masterpieces. As a matter of fact, in a lot of cases this is far from being the case2 and actually something strange happens. Let us consider, for instance, the 1964 exhibition in which Andy Warhol showed, in another New York gallery, a box of Brillo detersive, only bigger than the original ones and made out of plywood. Now, how is it possible for an ordinary box to become a work of art? In spite of such a question, people liked the Brillo box. And the question, after all, is not so puzzling, for in the end it is just a silly question: why on earth a box (that is designed precisely to catch the attention of buyers) should not be appreciated? However, once the pseudo-problem has been dissolved, a slightly subtler and really interesting question for the ontology of the works of art comes into sight. At the exhibition, there were also other works made by Warhol that were accurate reproductions of commercial boxes, including the Kellog’s corn flakes, the Delmonte’s peach halves, the Campbell’s tomato juice, and the ketchup Heinz. However, as a matter of fact, the common reaction of the largest part of the public was that such works were less beautiful than the Brillo box, and even nowadays – after the disappearance from the market of that particular Brillo detersive, whereas the other products can still be found in grocery shops – the value of the works still reflects that reaction. How is this possible? The answer is quite simple. The Brillo Box was objectively more beautiful than the boxes made for the tomato juice or the peach halves. The designer who painted it, James Harvey, who was himself an artist by the way, just designed a better – i.e., an intrinsically better – work than the other designers did with the other boxes. However, when we look for information on the Brillo Box on Google, the designer’s name is rarely mentioned, and, of course, he hasn’t made history as an artist. Most of information is about Andy Warhol, and this is not unfair, as it may seem, for it isn’t different from saying that it was Columbus who discovered America and not the sailor that first sight its land.

14.3 The Hermeneutics of Art and the Ontology of the Works of Art Nevertheless, even without refusing to acknowledge the merits of Columbus and Warhol, the core of an ontology of works of art does not lie in the question whether how a box can become a work of art. Such a question, as I have already said, is somewhat futile and – unless one wants to argue that certain boxes are indeed extremely beautiful – it can easily be answered by taking into account the role of the Artworld. Just like a widespread consent can convince everybody that the king has his clothes on, so a well-orchestrated press campaign can convince everybody that 2 As it is well known this is the starting question of Danto’s philosophy of art, as he himself acknowledges at the outset of The Transfiguration of the Commonplace [1] and as he reaffirms in The Philosophical Disenfranchisement of Art [2].

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a dirty trick is a workable electoral agenda. But the real question for an ontology of the work of art is precisely the following: how is it possible that a box is found more attractive than another? What features does that box possess that other boxes do not? Here we get closer to a more intrinsic aspect of the Brillo Box, for the question now concerns not only James Harvey, the first author, so to speak, but also Andy Warhol, the second author. As a matter of fact, the world is full of artists that have made use of everyday objects, however the number of artists that have succeeded in conveying feelings by means of detersive boxes – or, even more so (more on this later), by means of Marilyn’s face, electric chairs, skulls – is much smaller. In addressing the key question of the ontology of art – the one concerning the intrinsic beauty of the Brillo Box – one is reminded of that passage in which Aristotle points out that birds have different dialektoi, namely that they express themselves in different ways, but also that some of them can express themselves better than the others [3]. And this observation amounts to saying that there is a limit to the power of conventions. Why certain birds sing better than other birds? And why the Brillo Box is better than the Ketchup Heinz? Why indeed Warhol is a better artist than other artists? And, going back to the very first question, is it really the case that anything can be a work of art, including the number 9, Pythagoras’ Theorem, and a tropical storm? We may even wonder whether it is in the end true that the existence of readymades actually show that anything can be a work of art or if their existence shows exactly the opposite, namely that only certain objects, with certain features, can become works of art and others simply cannot, even if they are creations of Warhol, even if they are creations of God himself. Given that my aim in this article is that of providing the necessary, but not sufficient, conditions for an object to be defined a work of art, what I’m proposing is an ontology of the work of art that accounts for the intrinsic features of the works of art, namely of those features that do not depend for their existence from the consensus gentium or from the more definite and limited Artworld. This amounts to saying that the hermeneutics of art – the most practiced activity in the philosophy of art of the last century – must be supplemented by a theory that accounts for the intrinsic features of works of art, i.e., those features that allow some objects, but not others, to become works of art. After all, the starting point is quite simple: we have to limit the extent of the thesis according to which anything can be an artistic object provided that a community of experts decide that this must be the case (in such thesis is echoed one of the most anti-ontological tenets advocated by Nietzsche, the one according to which there are no facts, only interpretations [4]). If such a tenet is devastating from a moral point of view (just imagine it written down in a law court), and meaningless from an ontological point of view (i.e., it is just as meaningless as the slogan: “there are no cats, only interpretations”3), it is anything but innocuous in the aesthetic realm, for it means that an omnipotent judge (that in aesthetics would be a self-appointed judge) can decide that anything can be anything else. Now, it 3

Here I am drawing on considerations I have made in my 1998 book, “L’ermeneutica” [5].

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is clear that it cannot be so. “There has to be a judge in Berlin,” said the miller from Potsdam in the face of the arrogance of the King of Prussia. And I would say: “There has to be some merit we can ascribe to the works of art in themselves.” If we are interested in works of art – and of course this happens less often than one is willing to admit – is because of features they possess and not as a result of the lot of talk about them within the art world. If talk of art were enough, if we really could do without works of art, then we would listen only to the radio (which in effect may be a good means to enjoy at least another form of art, i.e., literature). Now, it is precisely this simple and quite traditional4 task of saying what a work of art comes down to that hermeneuticists5 have completely neglected, probably because they were under the combined action of the lazy reason exemplified in the slogan “there are no facts, only interpretations” (endorsed as a philosophical dogma) and of the thesis according to which anything can become a work of art (a quite profitable artistic and commercial dogma). Given these assumptions, it is clear that asking what kind of object is a work of art does not make any sense; however, if this is true, then there are a lot of other questions that do not make sense.

14.4 The Limits of Interpretation Maybe, it is too early to throw in the sponge; if we can say what kind of thing is a driver’s license or a monsignor (and no one is willing to say that “anything can be a driver’s license” or that “anything can be a monsignor”), then there is no reason whatsoever to deny the possibility to say, at least in general terms, what kind of object a work of art is. To begin with, are we really sure that vernissages and museums count as the ordinary type of artistic experience? Nothing is less obvious than this. Let us begin by considering the typical situation of the Artworld, in its most illustrative and significant function: visiting museums. In the Easter holidays, many Italians typically visit cities of art, or at least this what television and newspapers say. Cities of art are so defined precisely because they are full of monuments and museums that allow people to spend some of their time in a gratifying way. However, the real purpose of such journeys does not lie in some kind of cultural experience but rather in eating, drinking, fornicating, and finally sleeping. It is worth noting, after all, that some of the monuments that are visited by tourists were designed for purposes other than artistic contemplation (as in the case of churches, whose function may be still the religious one). The attitude of the public of art (in the case 4

I’m referring here, of course, to Aristotle [3], but also to contemporary theorists such as Ingarden [6] and Heidegger, at least in his “Origin of the Work of Art” [7]. 5 Gadamer [8] makes a comparison between art and activities such as games, the symbol, the party, but he never explains what kind of things works of art are supposed to be. Other philosophers that have talked a lot about art just dismissed the question as irrelevant (and this is not surprising as long as they did not have a plausible answer).

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of paintings and monuments) can be described as halfway between the vacationer (the majority), the devout, the well educated, and the bored. However, such an attitude has to do with a particular form of fruition, and this seems to count against the alliance between the Artworld and hermeneutics in creating works as much as one likes. The same attitude, in fact, cannot be had towards different kinds of works. For instance, it cannot be had towards concerts (with important differences between classical and pop music), and even less can it be had toward movies (with differences between mainstream and avant-garde movies) or novels. The thesis of the Artworld, in all these case, just does not apply. Indeed, what should not be neglected is the difference between avant-garde and commercial enterprises that require a lot of money. In the first case (as in the case of poetry), we can easily imagine the somehow parochial situation in which a small group of friends decide that a urinal is a work of art; after all, such a situation is not so different from the situation in which the members of a family decide that uncle John is a great impersonator, a good singer, or an irresistible joke teller. Things, however, are quite different when it comes to the making of a movie that requires a lot of investments and that only to cover its cost must count on a global market. In such a case the small group of friends and the vernissage are out of question and what are needed are objective criteria, whose existence just proves the thesis of the Artworld false. The Artworld, in other terms, is somehow overpowered by the real world. The theory according to which a detersive box can become art holds in effect only for a very small and selected group, just like a small group of students can convince themselves that their professor is a genius while he is not such, not only if one considers the international standards, but also if one considers the standards of the closest neighborhood. It suffices to take a walk and one may discover that in the nearby block nobody likes the urinal, that nobody cares about the Brillo Box, that the professor is a bore, or that one can easily do without the poetry of the Great Minor Poet of the 20th Century; what is more, such judgments are made by people who know little about literature and muses, but know much about movies and concerts and books that sell millions of copies (differently from the essays of the professor who only looks like a genius or the poetry of a great minor poet of the 20th Century), etc. Thus, the theory of the omnipotence of the Artworld fails miserably if one makes the simple move of leaving the small world of galleries and critics that philosophers typically identify with the Artworld. On the one hand, the relativity of the work of art seems to apply to a certain kind of things (such as contemporary art exhibitions) and not to others: can a telephone book really become a novel as a result of an appropriate hermeneutics? Of course, it cannot. However, it can certainly become a ready-made, and this is, in some way, worthy of note: if it were true that anything can become a work of art, why then can a urinal become a ready-made but not a symbolist poetry or an Italian comedy? On the other hand, the reason why one can consider certain works more beautiful than others is unclear if the judgment about the very nature of the works of art depends on a hermeneutic consent: would it not be better to establish an hermeneutic consent also in the case of aesthetic judgment,

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so that everybody is happy? And, again, if one holds that a urinal or a detersive box can become works of art as the result of a hermeneutic consent, why not also say what the agreement among critics is that makes a movie what it is? Such a claim is hardly believable (just as it is hard to believe that the agreement among critics can turn a Tintoretto painting or “Treasure Island” into a urinal) and in the end just shows that, even if in the history of art there have been works of art that became such as the result of a hermeneutic consent, there are a lot of other works that were created as such from the start and that could not be made otherwise (what other function have a piece of film have? Can one use it as a tie?). Then, a small amount of reflection suffices to discover that the thesis to the extent that anything can become a work of art is as absurd as the idea according to which anything can become beautiful. In other terms, and this is what I would like to emphasize, the things that have became works of art have always possessed specific features. This idea somehow suggests how to proceed in the analysis: look for the necessary (even if not sufficient) conditions in things that have become works of art and you will get an ontology of the works of art.

14.5 Art as the Class of the Works of Art Notice that we are talking about works and we will keep on referring to works. This is already a fundamental ontological desideratum. Without things such as works of art, it is not possible to identify something as art. We can certainly imagine works without art (such as, for instance, votive objects that are shown in museums but that originally were placed in graves), but we cannot imagine art without works of art, because there will be only pompous concepts without correspondent objects. In art, more than anywhere else, concepts are empty without intuition, so that an unacknowledged masterpiece is all but a masterpiece. What do people who strive to stay awake during classical concerts and the crazy and screaming audience at Beatles’ concerts have in common? What unites, beyond the appearances, the museum visitor who looks thoughtfully at a painting that he or she does not understand and the same visitor when he or she can finally watch his or her favorite TV show? What all these people and situations have in common is their inevitable reference to works (namely to a class of objects) and the fact that at least some of their feelings are directly caused by such a reference. The rule seems very simple: no works of art without art, as long as they are social objects – as I will show subsequently – namely they are what they are because there are persons who can use the category “art” and they are willing to apply it to certain objects. This is somehow the only hermeneutic ingredient of this account. But hermeneutics is possible only because there cannot be art without works, since works are what the category “art” applies to (and they are typically physical objects, as I will claim in the following) and, to use an old philosophical terminology, works are objects that are intentioned by an audience in search of aesthetic pleasures: art is

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always something that refers to a work, just like thoughts are always thoughts of something. And this is the ontological ingredient. As we have seen, once one maintains that there are no facts, only interpretations, and that anything can be a work of art, what is thus compromised is the pact between the public and the works or art, for if these two theses are true, the public is given the power to decide what can become a work of art, and even the power to decide that a work of art is not an object. It is obvious that in such a framework the ontology of art cannot make sense, and of course if such an account is true, and given my definition of art as the class of artistic objects, the word “art” itself would simply devoid of any reference. Nevertheless, as we shall see, not only works of art are objects, but such objects do satisfy the minimal conditions for being works of art. Such conditions are necessary (i.e., they limit the hybris of the interpreter) but not sufficient (and in this case what is limited is the hybris of the work, so to speak).

14.6 The Paradigm of the Object Art, then, before being a manifestation of the spirit (this is one among the many available definitions), is a class of objects, namely the class that comprises works of art. In other terms, art should be identified with the set (an open set, as long as it changes through time and space) composed by the medium-sized objects that we typically buy, exhibit, see, or listen to. Art is the class of the works and not what is said to be such by the “Artworld,” by means of hermeneutic procedures; the advocates of the latter view went as far as to hold that owners of art galleries and critics are part of the art, a thesis that sounds bizarre, for no one seems willing to buy a critic and put it in his or her living room. To sum up, art emerges from a set of works, and not the other way round. There is nothing like a descent of the spirit on the world that produces statues and crucifixes, poems and comedies, concerts for violins and orchestra and sepulchral bas-reliefs. Rather, it is from this class of objects, characterized by the fact of being middle-sized physical objects that become expressions of social acts, that art can arise, at least in some circumstances and in some civilizations. This is a crucial point in the ontology of art, and it is not indeed a circular explanation. From time to time, a number of objects is included in the sphere of art, and such a process has rarely to do with hermeneutics, for, again, one can always decide that a hurricane, or a movement of the head, or an art critic are works of art, but unsurprisingly nobody would be willing to believe such a person. By the same token, an artist who refuses to produce a work of art is not an artist. A violinist who teaches how to play violin but cannot play violin, like Hoffmann’s violinist, is just a romantic fiction, otherwise he would be only pathetic, just like those critics that claim to be artists. The key role of the paradigm of the object in the definition of art is corroborated by the fact that, in recent years, the market of visual arts has become one of the most

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profitable market, precisely because it is strictly connected to the sphere of works and their uniqueness; and what is more, such a market has its own junk bonds or Enrons, and they are represented by precisely those works of art that are defined as such in accordance with the thesis that anything can be a work of art. If there is a market for horrible daubs – which is no news at all; however nowadays this is explicitly theorized – it is certainly because apparently anything can become a work of art (it suffices to find the fool buyer); but, as a matter of fact, only a physical object can become a work of art, and even the buyer of a daub would have some difficulty in buying a person, a theorem, or a hurricane. The key role of the paradigm of the object gets even clearer in a comparison between visual arts and music. While visual arts are flourishing, with the production of objects that are under the control of their authors, merchants, and buyers, the old music industry is in decline and the reason for this decline can be already found in the standard definition of this kind of cultural industry: the record industry. The reason for its decline lies precisely in the fact that it has lost its control over the prevailing forms of recording (vinyl, tape, CD) of the last century; as a matter of fact, music is brought back to its pre-industrial status, when the profits for musicians came from their public performances (events that are located in space and time, that are subject to copyright law, for which the public pays tickets, etc.).

14.7 Diffuse Aesthetics A possible counterexample to the primacy of the works seems provided by the notion of diffuse aesthetics. The reasoning might go roughly like this: it is not necessary to put art in such a strict relation with works, for this would be just an obsolete way of accounting for art. Rather, art is realized in atmospheres and climates more than in objects; in other terms, it is realized in a diffused aesthetics. The idea of diffused aesthetics would then be in plain contradiction with the thesis according to which there is no art without works of art. I think that this reasoning is flawed. The label “diffuse aesthetics” is either without meaning or else it just amounts to saying that fashion, design, and cuisine can be considered as arts (and this, of course, makes sense as long as one is willing to include utilitarian arts in the framework of fine arts). It certainly cannot mean (because it would not make any sense) that a diffuse fog named “aesthetization” spreads over the world; but rather it simply means that the kinds of objects that may be considered as art can be extended. How this happens is quite simple: it suffices to say or write something like “Certain Rumanian painted eggs are real works of art,” “the carts of the Rio Carnival are real works of art,” etc. (whereas nobody could say, “Marilyn Monroe is a real work of art,” unless this judgment is referred to the portraits of Marilyn made by Warhol).

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However, I do not want to dispute the idea – which is indeed a modified version of what Goethe said about truth6 – that beauty is not located in a single point but rather is in the air, like an atmosphere. This is a highly plausible definition of beauty, but not of art, for beauty can indeed possess such an environmental feature (in fact, what in particular is beautiful in a landscape?). Such considerations count against the notion of “diffuse aesthetics,” at least if one is interested in the ontology of the work of art. Moreover, it seems that the very idea of “aesthetization” is based on the mistaken assumption according to which the world of our times is more aesthetic (in the sense of being more “interested in visible pleasantness”) than before. The evidence for this claim is supposed to be provided by the large number of everyday objects and packagings – i.e., various kinds of Brillo Boxes – made by famous designers. Now, it is quite clear that before the industrial era it did not make sense to package products as long as the term “product” had a quite different meaning from the current one, and as long as attractive packagings were probably useless before the introduction of supermarkets. However, to say that the pre-industrial world was not “aesthetized” is a case of myopia if not of blindness: armors, wigs, and pre-industrial clothes were absolutely more aesthetized than contemporary helmets and clothes; moreover, the salt cellars made by Cellini and Giambologna are much more aesthetized than those designed by Philip Starck, and this can be said of almost every product of the first period of the industrial era in comparison with contemporary products: whoever could feel the need, nowadays, to decorate his or her gun or a radiator, as it was normal to do in the 19th century?

14.8 Impossible Works To be sure, my account is bound to fail if anything could become a work of art, for this amounts to denying the existence of intrinsic features that are possessed by the class of the works of art. But of course this is not the case. There are indeed clear cases of rather ordinary objects that cannot become works of art, contrary to the conventionalist thesis. Let us call them “impossible works,” in analogy with the “impossible objects” of the psychology of perception. Let us take into account a first, modest evidence: Pythagoras’ theorem, the greatest prime number, the relations “at the left of” or “bigger than” cannot ever become works of art. And even if on the roof of the Mole Antonelliana, which is right behind me as I am writing these words, has been placed with a Fibonacci series made of neon lights, the work of art in question is not the ideal object (the Fibonacci series) but rather a physical object (the neon lights). Now, numbers and theorems are indeed objects. When I say “Pythagoras’ theorem” I am referring to a different object than the one designated by the name “Euclid theorem,” just like when I say “table” I am referring to a different object than the one designated by the word “chair.” However, 6

“It is not always necessary for the truth to be tangible; it is enough if it hovers over us spiritually and produces harmony, if it wafts gravely and kindly through the air like the pealing of bells.”

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it seems impossible to sell Pythagoras’ theorem or the greatest prime number. No doubt mathematicians may find a theorem “beautiful” or “elegant” but this does not suffice to make them works of art, for the latter kind of objects typically have an original, have a beginning in time and they change if their style is altered. In other terms, I am allowed to say that a theorem is beautiful in the same sense in which I am allowed to say that Sharon Stone is beautiful; in neither of these cases, however, am I allowed to say that they are works of art. Then, contrary to the theory according to which anything can be a work of art, we discover that there is a whole world of objects that cannot become works of art, neither if an appropriate decree were issued. In fact, suppose Duchamp had exhibited the Pythagoras theorem or the number 5. In what would the exhibition have consisted? Perhaps, a piece of paper on which he would have written the theorem or the number. But then in such a scenario we would have to identify the work of art in the piece of paper and not in the ideal object represented on it, given that ideal objects are not located in space and time. A work of art must be a physical object located in space and time and it must have an individual element, namely a style. When we talk about “works of art” we do not refer to the ideas but to their physical appearance, to their expression, and when we visit museums we do not look at artists’ ideas written on artists’ notes but we look at their works. Suppose that Goya and Picasso had painted two Pythagoras’ theorems. Such paintings would have been two different objects exactly like in the case of the Maya Nude and its remaking by Picasso, or just like the many different Depositions, Crucifixions, or Flights into Egypt of the history of art. What constitutes the work of art, then, is not the Pythagoras theorem but its transcription: in geometry the transcription has to do with the socialization of its ideal objects; in art, however, a specific inscription is what can become a work of art given determinate conditions. We have then ruled out the possibility that ideal objects can be artistic objects. But does this amount to saying that any physical object (i.e., objects located in space and time) can be a work of art? This is hardly believable. For instance, a statue whose height is 20 kilometers (or more) can hardly be defined a “work of art.” The criterion used in this case may appear arbitrary, however it is based on the simple consideration that such an extremely high statue could not be perceived in its entirety and would not be a suitable object of contemplation.7 When Christo wrapped up the Great Wall, he did so only with a portion of the wall and what really counts for his fame are the photographs of the wrapped up wall 7

It would even exceed the already demanding size represented by the Kantian mathematical sublime. “This explains Savary’s observations in his account of Egypt, that in order to get the full emotional effect of the size of the Pyramids we must avoid coming too near just as much as remaining too far away. For in the latter case the representation of the apprehended parts (the tiers of stones) is but obscure, and produces no effect upon the aesthetic judgement of the subject. In the former, however, it takes the eye some time to complete the apprehension from the base to the summit; but in this interval the first tiers always in part disappear before the imagination has taken in the last, and so the comprehension is never complete.” ([9], § 26). Hegel even denies the right to beauty of the Pyramids because of their size. As a matter of fact they are beautiful as it is clear from the amount of movies, photographs, novels, and travel packages that rely on their perceived beauty.

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more than the work itself. Works of art have in common with ordinary objects the feature of being of the appropriate size for humans. A one-millimeter statue, one that can be seen only by means of a microscope, is not a work of art in itself but it would be a performance that includes the use of the microscope; just like a copy of “Don Quixote” the size of a soup cube (that can be read only by means of a lens) is more a curiosum than a real book. The moral is that essentialism is a highly recommendable strategy, provided that one take into account the correct essences, and in particular provided that one is willing to find the essential in what is generally thought of as inessential. God is in the details and even in sizes: we are used to thinking about the question concerning size as irrelevant in art, however, when a work of art cannot be seen with the naked eye in its entirety and requires, let us say, travel on a satellite or the use of a microscope, it becomes something different from what we usually consider as a “work of art.” And such limits are not only spatial, but also temporal. When it gets late in the night, people want to go to sleep, and I think this is the authentic reason behind the Aristotelian proposal to the extent that tragedies should conform to the rules of the unity of time, place, and action. Furthermore, imagine a novel a million pages long. Readers in their 50s would ever dare to start reading such a book, and even younger people would probably avoid reading such a novel on the assumption that there is certainly more to life than books. And who might be the author of such a book? Even the Bible, which we are told was dictated by God himself, is not a book of such an incredible length. Works of art must be of the appropriate size for human beings: we cannot imagine a concert that lasts for one hundred years, and already the performance of Wagner’s Ring posed some problems (by the way: how many of you have listened to the entire Ring? I have not, but I have listened to shorter pieces of music). These limitations concern the fruition of the works of art. Nobody is entitled to ask me to spend a thousand years in reading a Finnish epic poem, but of course this does not mean that the Kalevala might not survive for centuries. However, this is a different story. In spite of the thesis to the extent that “anything can be a work of art,” I have claimed that a theorem and a number cannot be works of art and neither can physical objects with extraordinary sizes such as a statue whose height is 20 kilometers or 1 millimeter or a book of one million pages be works of art. One may object that some Martian may appreciate the incredibly high statue, or that the incredibly small statue may be found attractive by an ant, or that a baobab may like the idea of a very long novel. But it should be obvious that even in such cases these objects would not count as works of art, simply because art is a social fact that exists in virtue of, and only for, human beings. If this is true, then even a concert made with whistles for dogs can never be a work of art. Such a concert would certainly be a physical event, however it would be perceivable only by dogs and not by humans. But works of art are also social objects, namely objects that exist only in virtue of the fact that humans believe in their existence, and this is why the concerts of whistles for dogs, though a physical event, cannot be considered as a work of art. Of course, it would be different if there were a screen on which all the sounds of the concert

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were graphically represented. In such a case the symphony for dogs would turn into a work of visual art (a sort of conceptual work of art), but it would not be a musical work. No symphonies for dogs, then. We have to resign ourselves. Let us take stock. Pythagoras’ theorem, the number 5, the incredibly high and the incredibly small statues, the one million pages novel, and the symphony for dogs can never become works of art. My impression is that we effortlessly refuted the thesis according to which anything can be a work of art. If just one of the objects listed above cannot become a work of art, it is false that anything can be a work of art. A further thing that can never become a work of art is something chameleonic and ever changing. Let us imagine an ever-changing object. In the morning it is an insect, in the afternoon it becomes an admiral, in the evening it turns into a computer program, and in the course of the night it becomes Duchamp’s urinal. This would be the first day. The day after, it becomes a camel, a penny, a passport, rain, and the Monna Lisa. On the third day, the object would turn into other objects and so on. It would be the hermeneuticists’ favorite thing, given their conception of art as the result of infinite interpretation: indeed with such an object there would be a lot of hermeneutic work to do. Nonetheless, even if such a thing could exist, it could certainly not become a work of art, for works of art require definite identities, contrary to what the theorists of infinite interpretation might think.

14.9 Instruments There is still a limitation for the thesis according to which anything can be a work of art. Not only an ideal object cannot be a work of art, and not only an incredibly big or small object or an ever-changing object cannot become works of art, but also an instrument cannot be a work of art at least as long as it functions as an instrument. An object cannot be at the same time an instrument and a work of art. If one sets Carmen as his or her cell phone ringtone, after a few days, he or she would probably forget that music is a work of art and would think about it just as the ringtone of his or her cell phone. And indeed when we are confronted with two versions of the same instrument, one more beautiful than the other, it is not surprising that we may be more inclined to choose the less beautiful of the two if it has to be used as an instrument. The question in such a case does not concern the size but the function of the object. Instruments are typically medium size things (most of the time the dimension is determined by the extension of the hand, for in most cases the instrument can be handled). Differently from natural things, instruments are created for a purpose. In this sense, salt, sugar, and milk may be used in cooking and are typically sold as moderate-sized specimens of dry goods, they can be part of things, but they are not instruments. Screwdrivers, spoons, and wheels are instruments that can be used for different purposes from those they have been created for (screwdrivers may be used as weapons, sometimes prisoners sharpen spoon handles in order to use them as knives), however this is not always the case.

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Though works and instruments may have size in common, what seems to make the difference between them is that works are useless. The Kantian notion of “purposiveness without a purpose” refers to such a feature. Works of art seem to have a purpose but we immediately discover that they do not have one or (just like persons, at least from a Kantian perspective) that they have an internal purpose. In fact, if one used Duchamp’s urinal in its proper function, no one would consider it as a work of art (provided that this does not happen in a museum, for in this case one could not simply justify himself or herself by saying that, after all, it is just a urinal). And, in effect, Wharol’s Brillo Box can hardly be taken as one of the real Brillo boxes give that it is bigger and is made out of plywood. If we think about it, the distinction useful/useless is more appropriate for works of art than the distinction between true/false.8 From this point of view, artists like Mucha have always been problematic: are his posters art? It is not clear. On the one hand, they may not be considered as art because they include useful information such as the time of the concert and the price of the ticket. On the other hand, once some time has passed, the fact that they have on them some information ceases to be a limitation. Similarly, for the believer the Bible may contain information on God’s will, whereas for the non-believer it may be read and enjoyed just as a collection of beautiful metaphors. Moreover, let me refer to my personal experience. I quit reading novels 30 years ago, however I must say that I enjoy an aesthetic experience in reading encyclopedias, geographic maps, and history books. Such an experience, nonetheless, is not produced by fiction, but by the truth of history, geography, and astronomy. After all, natural beauty can hardly be considered as fiction. However, architecture and design may be counterexamples to what I have said so far. Is it really impossible to consider the unité d’habitation by Le Corbusier both as an instrument and a work of art? Can we really say that the writing machine Olivetti designed by Sottsass is not a work of art? Moreover, in many cases, the instrumental feature of an object seems to increase its aesthetic value and the predominance of the aesthetic element on the functionality seems indeed to decrease its attractiveness. However, the fact that we can find writing machines beautiful when no one uses them anymore, or that we find Brionvega black and white televisions interesting, or the we may like houses in which no one would like to live, is a clear sign of the independence of the works of art from instrumentality – an independence that in architecture and design is only less obvious.

14.10 Physical, Ideal, and Social Objects So far we have learned that not anything can become a work of art, and, what perhaps is more important, that there are different kinds of objects. We have provided, in other terms, a first sketch of an ontology. Let us then follow this promising path. 8

Indeed the distinction useful/useless seems to prevail over the other: a fake instrument, like a toy gun, is just a useless instrument, and when weapons are exhibited in museums they are typically made unusable.

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For common sense “objects” typically means physical objects. However, common sense is not always coherent for it seems to neglect the distinction between an ideal objects such as the Fibonacci series and its physical representation in a work of art. In addition to physical objects, Plato and other philosophers considered as objects ideal objects. My proposal, on the other hand, is to distinguish three kinds of objects (a distinction that may call to mind Popper’s distinction, however the one I am proposing here has a quite different form). First, there are physical objects, which exist in space and time, independently from the subjects; then, there are ideal objects, which exist outside space and time, independently from the subjects; finally, there are social objects, which exist in time and are located in small portions of space, dependent on the subjects. In the theory I suggest, works of art are both social and physical objects and ideal objects cannot be works of art. Works are social objects because they exist only in virtue of the fact that we think that they exist (this accounts for the hermeneutic element), just like aristocratic titles and documents. For a beaver there are no driver’s licenses and no works of art (and this is not due to the fact that they are devoid of any aesthetic sense). And when I say that works of art “exist only insofar we think that they exist” I mean that such a recognition must be intersubjective, namely it must involve at least two persons. Moreover, when I say that works of art “exist only insofar we think that they exist” I am talking about them as social and not physical objects: the existence of the pieces of paper that constitutes my driver’s license is independent from me and the policeman who checks it; however the driver’s license as a document does depend for its existence on our beliefs. In other terms, social objects are a quite different kind of objects if confronted with the visions and voices that one sees and hears if seized by the delirium tremens. However, what exactly do we mean when we say that works are physical objects? The hermeneutic conception just overlooks the importance of this question in saying that anything can be a work of art. As we saw, ideal objects, just like imaginations in our heads, cannot be works of art. For something to be a work of art, it must be an external object, located in space in time. My insisting on the idea that works of art must be physical objects is intended to emphasize the role of visual perception and of the other senses. Money, checks, or coins can be ugly but the only thing we care about in using them is their functionality; things, however, are quite different in the case of works of art. The role of visual perception, and of the senses in general, has to be taken into account in the definition of what is a work of art. The information conveyed by the numbers “15:57” can be learned by reading it on a clock or on paper, or by hearing it from someone, or from the speakers in the station, however the Monna Lisa can be perceived as such only by means of visual perception. It is true that there might be a sense in which a piece of music by Schubert may be perceived with eyes, but at most this means that one can read the correspondent score, and it is quite doubtful that this is the same as listening to it. Moreover, in the case of jazz (that typically has no score) and pop music (that typically is made for an audience that simply cannot read scores) this seems plainly false.

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14.11 Objects that Pretend to Be Subjects All these kinds of objects that are included in the inventory of what there is have a common trait: they do not possess any representation whatsoever, neither of themselves nor of other objects. There is, however, another kind of objects that do possess representations and such objects are what we call “subjects.” I have a representation of the table, but the table does not have a representation of me, and neither do mirrors or cameras: they have, at most, reflections and impressions, but not images in a mind. The ability to have representation – an ability that not every object has – has a lot of consequences, in particular the ability to think and the ability to have feelings. At this stage of our discourse, we can say that the world (the world in general and not the parochial world of the Artworld) is composed of the following ingredients (Table 14.1). Where is the place of works of art in such a framework? I have already said that works of art are undoubtedly physical objects, given that they are in space and time and are perceived by senses. On the other hand, I have also said that they are social objects, given that they exist only insofar human beings believe in their existence (and of course this is not to say that they exist for every human being; it is enough to say that they exist for many or for some human being). In this sense, works of art are just like taxes and holidays, aristocratic titles and years in jail, lottery games and public offices, money and restaurants. The question now concerns the way we can distinguish between different kinds of social objects. What is the difference between a painting and a restaurant, a poem and a law, a statue and a sentence? Does it lie in the fact that we enjoy works of art? But we certainly enjoy good restaurants. Does the difference lie in the fact that works of art have an inner meaning? But even laws are said to have an inner meaning, which may be different from their letter. Does the difference lie in the fact that works of art teach us something? To be sentenced some years of prison may teach us something as well, perhaps even more than contemplating a statue. The difference, I submit, lies in a quite peculiar feature of the works of art. A work of art gives the impression of a will to say something to us, the work itself, not its author, just as if it had representations, thoughts and feelings. In other terms, the work, which is undoubtedly an object, presents itself as a quasi-subject and causes

Table 14.1 Subjects and objects Subjects

Have representations

Objects Do not have representations

Physical: exist in space and time independently from subjects Ideal: exist outside space and time independently from subjects Social: exist in space and time dependently from subjects

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a form of spontaneous animism in its audience. If we find a dish uneatable, we typically lay blame on the cook; if a law does not work, we usually put the blame on the legislator, and if a sentence is unfair, we usually put the blame on the judge. However, if a painting or a novel is boring, we put the blame, at least initially, on the work itself, and only after we take into account the author. Thus, the answer to the above question – the one concerning the distinction between different kinds of social objects – that I would like to suggest is the following: works of art are objects that pretend to be subjects, or more simply, they are things that pretend to be persons. In what sense?

14.12 The Automatic Sweetheart In addressing very different questions (the role of God in the universe) from those addressed in the present essay, the American philosopher William James (1842– 1910) focused on the following problem: “I thought of what I called an ‘automatic sweetheart’, meaning a soulless body which should be absolutely indistinguishable from a spiritually animated maiden, laughing, talking, blushing, nursing us, and performing all feminine offices as tactfully and sweetly as if a soul were in her. Would any one regard her as a full equivalent? Certainly not, and why? Because, framed as we are, our egoism craves above all things inward sympathy and recognition, love and admiration. The outward treatment is valued mainly as an expression, as a manifestation of the accompanying consciousness believed in. Pragmatically, then, belief in the automatic sweetheart would not work, and is point of fact no one treats it as a serious hypothesis.” [10] I think it is a serious hypothesis. It looks like a thought experiment, but it is just the description of a real fact: from the point of view of the theory I’m suggesting, libraries, newsagents, concert rooms, galleries are full of automatic sweethearts that we ordinarily call “works of art.”9 And my contention is precisely that a work of art is nothing but James’ automatic sweetheart. Works, just like automatic sweethearts, are both physical and social objects that cause feelings (differently from other social objects such as train tickets), just like persons when are treated as such and not as mere functions; however, differently from real persons, works of art do not offer, and 9 After having presented these ideas at a conference, some years ago, I received a lot of suggestions about examples of automatic sweethearts and I would like to thank all – and in particular Anna Li Vigni – for their help. In literature and theatre: Landolfi, La moglie di Gogol [Gogol’s Wife], in Ombre [Shadows]; Rosso di San Secondo, Una cosa di carne [A Thing Made of Flesh], La bella addormentata [Sleeping Beauty]; Cormack McCarthy, Child of God; Wedekind, Hoffman (but one can even add Proust when he contemplates the sleeping Albertine, reduced to a vegetable). Cinema: Lubitsch, Die Puppe, Tarantino, Kill Bill, Almodovar, Talk with me. A sincere thanks; but I have to point out that such cases typically exemplify the passage from being alive to being an automaton (the coma, the idiot) and more rarely the passage from being an automaton to being alive. As a matter of fact, according to my theory, even a screwdriver (and, of course, even a bottle rack) can be an automatic sweetheart, namely can pretend to be a person even if it does not have any anthropomorphic trait.

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neither require, any sort of reciprocity. The predictable objection to the extent that landscapes, Mondrian’s paintings, or symphonies do not look like persons allows me to clarify at least one point. My contention is not that works of art should be anthropomorphic, but rather that persons are typically thought of as being intrinsically sources of feelings, that can be caused only accidentally by things such as landscapes, colors or documents (for instance, a fine). In other words, our expectations toward the works of art are very similar (though much weaker, of course) to our expectations toward persons. In this sense, works are objects that cause feelings, real feelings (I do not pretend to cry when I cry during a touching movie) though disinterested, for I am part of the audience and not an author. Feelings arise exactly because works of art are located in the external world, and in this respect they are different from drugs that cause internal states, or from a panic attack that differently from fear does not have an object. Works, however, differently from persons cannot return our feelings. They are ungrateful. And they better be so, because the feelings in question are indeed authentic, but also too much disinterested. According to my account, works of art and automatic sweethearts pass the same test, because automatic sweethearts are physical objects with a social value that cause disinterested feelings that cannot return. A screwdriver, a train ticket, a beer, a gun at one’s head, and a real sweetheart – if she is awake and in non-pathological conditions – do not pass this test: in fact, the screwdriver and the train ticket are not touching, the beer provokes an internal, chemical reaction and not an external one, the gun at the head makes passion interested (especially if it is our head!) and, finally, the real sweetheart do not pass the test because, hopefully, she recognizes us.

References 1. A.C. Danto: The Transfiguration of the Commonplace (Harvard University Press, Cambridge (MA) 1981) 2. A.C. Danto: The Philosophical Disenfranchisement of Art (Columbia University Press, New York, 1986) 3. Aristotle: De partibus animalium, B, 17, 66 a 35 4. F. Nietzsche: Nietzsche Werke: kritische Gesamtausgabe. Giorgio Colli und Mazzino Montinari (Eds.) (Walter de Gruyter, Berlin 1967) 5. M. Ferraris: L’ermeneutica (Laterza, Roma-Bari 1998) 6. R. Ingarden: Das literarische Kunstwerk (M. Niemeyer, Tübingen 1931) 7. M. Heidegger: Der Ursprung des Kunstwerkes, in Id.: Holzwege (V. Klostermann, Frankfurt am Main 1935–36) 8. H.G. Gadamer: Die Aktualität des Schönen (Reclam, Stuttgart 1977) 9. I. Kant: Kritik der Urteilskraft (Lagarde und Friedrich, Berlin und Libau 1790) 10. W. James: The Meaning of Truth: A Sequel to “Pragmatism” (Longmans, New York 1909) Green et al. (Eds.) (Ann Arbor, The University of Michigan 1970)

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Discussion E. Carafoli: I have a comment on the statement by Coleridge which I mentioned in my presentation, and which is relevant to the points you have discussed. According to Coleridge, the only aim of art is the creation of immediate pleasure, and I had disagreed with him. I had chosen Eugenio Montale as an example, quoting a very important poem, in which he explicitly stated that the raison d’être of his poetry, its most important aim, is the search for truth, rather than the creation of aesthetic pleasure – which is obviously also an aim of poetry. Maybe I could quote a statement by Thomas Henry Huxley, who proclaimed that “art and science are not two things, but to sides of one thing”. M. Ferraris: Yes, I agree, but note that by posing the question in this way, you have some further presupposition. First, you assume that modern art is paradigmatic, which is not sure at all, and I would say that it’s false. It’s clear that in modern art the struggle for knowledge is very important, but this was not the case in classical art, and this is a historical fact. Second, we usually don’t say things like “this concert was boring, but I learned so much.” Imagine that that happens though. Suppose that you go to a concert of Stockhausen, and even if you don’t like Stockhausen, you say “well, it was not so moving for me, I prefer Eros Ramazzotti, still I have learned something.” I think this is atypical for an aesthetical experience, because you are assuming that the paradigmatic model of art is in the range of avant-garde and of chef d’oeuvres or masterpieces. However, most of our aesthetic experiences are due to “lesser” artworks, such as objects of design, folk songs, and movies. An ordinary commercial movie can be very beautiful, but you wouldn’t say that you learned so much by it or that it showed to you a whole new reality. S. Califano: Does not your position imply that Mozart and a commercial movie are on the same level, that both are entitled to be art to the same extent? And this is absurd, isn’t it? M. Ferraris: Yes, my claim is that a commercial movie for us today is exactly what Mozart was for most of the people at the time. But this is not absurd at all. Mozart was nothing more than a commercial movie. A movie for the poor Emperor: would you give to the Emperor something more sophisticated? Clearly not. S. Califano: Well, the Kapellmeister of the Emperor was Salieri, and Salieri indeed can be compared to a commercial movie. But I do not think that the same holds for Stochausen or Berlioz. If the Emperor had one of them as Kapellmeister, maybe he would have developed a better understanding of art. M. Ferraris: No, because if the Emperor had had Stochausen or Berlioz as Kapellmeister, he would have killed Berlioz or Stochausen, and ask for Salieri to come back. He would have claimed “Give me back Salieri, I need something good for me.” S. Califano: In ancient times they did not have many sciences, but they had art. Do not you think that philosophy of art can be defined as a philosophy of history, in

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a way? These are concepts that are old now, and probably they have completely lost the meaning of the word “philosophy.” M. Ferraris: I don’t know, it’s clear that there was a distinction in philosophy that is no more valid. You are right in saying that it seems old, or maybe too recent, because in ancient philosophy the distinction did not exist. But to me it’s pretty clear that there is a difference between, say, the philosophy of medicine, which exists, and the philosophy of art. I mean, the fact that there is a difference between the philosophy of medicine and the philosophy of art – between art and physics for instance, the fact that when you are sad you read, say, a novel and when you are ill you go to the physician, and if you make the opposite it’s a mistake, it’s clearly a mistake – this doesn’t mean that there is a distinction between two cultures. And this is an important point. L. Boi: This is question and a comment at the same time. If I understood you right, you claimed that what distinguishes artworks from natural things is the fact that the latter have no cognitive aims, because there is no subject in them. M. Ferraris: Not really. I claim that natural things have no author. L. Boi (follow up): Right, there is no subject behind a natural thing, and thus it is not an artwork. But if we adopt such a criterion – which I think is too strict and too absolute at the same time – we are led to deny that there are natural “artworks.” However, think at a fractal mountain, a fractal cost, a star – you say that they by no means are artworks, because there is no subject, no aim behind them. But this seems wrong, because it implies that there is no dynamic activity of a creative sort in nature, that nature cannot display a productive activity yielding artworks. I do not mean that there are aims in nature, or there is a subject. Still, there is creation in nature. This is my fist point. Secondly, you said that a criterion for being an artwork is having a certain size, right? M. Ferraris: Roughly, this is the case. One of the criteria to be an artwork is having a “average” size. Therefore, a star or a galaxy cannot be an artwork. L. Boi (follow up): I think such a criterion is too limiting. I think we can consider much more things as artworks than this allows us to do. Consider Mahler, many of his symphonies are inspired by natural tunes – the music of stones. And most of African or Pre-Columbian art is inspired by cosmic, natural artworks. I am not convinced that such a criterion can be a discriminating principle for art. M. Ferraris: Thanks for asking this, and for the comment. It allows me to elaborate on certain points, which obviously I had touched all too briefly upon. Firstly, mine are necessary, but not sufficient conditions to define an artwork. That is, my claim is that if something is an artwork then it has an average size; that is, something possessing average dimensions can be an artwork, but this does not mean that every object of an average size is an artwork. Indeed, many average size items that are not artworks surround us. This microphone has all the physical characteristics of an artwork, it even has a social element, yet is not an artwork. This is the first point. The second point,

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which is the most important, is that I do not identify artworks with good artworks: the artistic with the beautiful. I certainly grant that there can be beautiful things in nature. Indeed, most of the nice things that exist in the world are natural rather than manmade – i.e. artistic – things. Let me note that the star system is essentially based on natural beauty – namely on the fact that natural bodies can be beautiful, and that we care about this. Even outside art the role of beauty is primary – think at evolutionary processes and the like – and it is not necessarily linked to art. Obviously, it might be claimed that God created everything, and inside the Creation there are a lot of artworks. Still, art can be detached from beauty anyway. It is desirable to have beautiful artworks. But one cannot fail to notice the huge variance of beauty canons in arts through history. There is no artwork that has not been “abandoned” or misunderstood during its history. Not even the Divine Comedy: nobody liked it during the eighteen century. Nonetheless, it had all the necessary conditions to be an artwork at the time too: it was something inscribed, something of an average size: one hundred cantos are not one hundred millions of cantos. The example of the cosmic beauty is a good example. Clearly, most of the productive and artistic activity of man is inspired by natural facts. Incidentally, Aristotle suggested that dance (from which nemesis stemmed) was meant to be an imitation of the celestial movements. There was a reference to the natural world, but it involved human actors, and it took place in a limited portion of space: that of a village or a temple. Certainly not in the huge galactic spaces where the movements of the stars take place – those were just evoked. I am not claiming that something like “the symphony of the universe” cannot be composed. It has been done already. What I claim is that a symphony of the universe must be a symphony anyway, and thus it must be something that we can listen to in a limited amount of time, and that takes place in a limited space – like a theater or a square. Something that goes at the speed of light, or that takes place in a huge portion of the universe cannot be a symphony.

Architecture Between Science and Art

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15.1 Introduction As I do not consider myself either a theorist or a critic, and not even a historian, I will simply talk about what I do. I will try to develop the matter going back in time to recover some premises that are essential for those who create. But I must also confess that, as a creative person, I find myself following the same line as the craftsman. I like the concept of craft, of something that is done in time, verified in history. Craft, the craft of building, comes from afar and has its own remarkable historic property, as many people have worked before us. So I find myself with a historical and generational development that continues, and into which I try to insert a positive testimony of my own time, of our time. I was not born an architect. I became one in the wake of other people’s work. I consider myself the son of post-Bauhaus, of rationalism, of 20th century culture. So I find myself faced with the current conception of spatial organization within an ongoing dialectic: certainly with the history of architecture, but, also, before that, with the way man places himself within the space. It is a concept that comes from very far and in some ways follows and motivates the development of humanity. But it is a concept that, as an architect working today, I must face with the tools of our culture and our education. This preamble leads to a first dialectic element, i.e., that the relationship between art and science, between creativity and objective conditions, is more and more characterized by a state of conflict. It is so in itself, as science follows the laws of reason and logic. It demands to be demonstrated, it requires proofs and objective, quantifiable results. Creativity then moves in its own very vast sphere in which many factors come into play, including the emotions which, as you well understand, are closer to the world of the artist than to that of the scientist. And yet science and creativity are strictly related to one another and must find a stable coexistence.

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The history of architecture, of the relationship with space, is so vast that various trends are woven into it. However, along a general line we can glimpse in it a tendency that relates to an initial rational kind of stimulus one works creatively on afterwards. It is a little like Le Corbusier’s way of working: a logical process that is interrupted to leave room for creativity, which, on its side, imbues rationality with life. It is a bit like a cobweb: it reveals its beauty at the moment it breaks, when the symmetry that prevents its appreciation is disrupted. In the same way the interruption of reason and the breakage of symmetry arouse a level of emotion that would otherwise be missing. This is to say that the relationship between science and art is in a state of constant development; it stems from a conflict that is absolutely necessary to the work of creative people, of artists.

15.2 Inadequacy of Technical Answers The architect obviously needs a confirmation to his own work. He must respond to the needs of rationality. But the research work along this line, which is in itself risky, which has no safety net, allows him to proceed on the edge of rationality, to interrupt and then rediscover it in another form, based on a different intuition. It is a progression in which the creative-artistic component constantly bounces back, forcing the rational component to come up with a feasible solution. I will show you some recent examples of my work in which, even paradoxically, the aim of building is not exactly set aside, but almost becomes a pretext for focusing on other elements within the functionality of the project, including the evocative value of the architecture and the feeling of moving through space. Elements that are the result of artistic factors, naturally supported by the technique. And this is unexpected, because the global culture produces technical-functional answers. That is, answers that are aimed at meeting various practical requirements (technical, sanitary, economic, alimentary, etc.) as if they were sufficient for meeting man’s needs. But man needs much more, and this additional need becomes evident in the organization of space. In this presentation I will try to show some examples where technical themes have served as a pretext for saying other things, for indicating that our current condition in the world is different, for saying that there are social, ethical, environmental, and energetic exigencies, which are the real problems we must face and try to answer. Thus, let us not think that the technical-functional solutions are the only, real answer. Quite to the contrary, architecture begins where the technical and functional problems have been solved and a newer perspective begins to open up, a Weltanschauung, a general view of the world that indicates to us other needs that are in the air and must be grasped in their ability to indicate the spirit of our time.

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15.3 The Need for Beauty and for the Infinite Architecture is the merciless mirror of our time. There is no architecture that is not a formal expression of history. Architecture shapes history itself and not the technical-functional demands of history. It is a striking and paradoxical thing that today, when we leave our cities and find ourselves living in the dimension of an extinct civilization, whose memory we do not even retain though it has almost come down to us, it is precisely in this dimension that we find ourselves at ease in spaces that were designed for other functions, other dynamics, other conflicts, and other loves. Another consideration of a historic nature must be made here: that the European culture has an added value with respect to the American and Asian models, which demand technical-functional answers. From a strictly distributive point of view, the Asian city is better organized than the European city. It works better. But then it becomes hell for other reasons, confirming the idea that that kind of solution is inadequate. This means that architecture communicates with man at precisely the moment when – the demands of a program, of an economy or of a certain kind of sociality having been met – he begins to confront the problems, the real problems, which have been identified and discussed here. And the real problems are the need for beauty, the need for the infinite, the need for emotions that must support us as we deal with everyday problems.

15.4 Le Corbusier’s Lesson The first picture is a tribute to that great master, Le Corbusier; to his ability to use science, technology and mathematics in man’s service. Le Corbusier is the greatest theorist I have ever known. He actually has even bent the rules of architecture’s golden ratio with his modulor [1] (Fig. 15.1). He has conceived a scale of proportions in which he applied the golden ratio to the proportions of the human body to establish that a man with his arm raised measures 226 cm, that the average height of a man is 183 cm, that his navel is at 113 cm. He has used these measurements to build residential spaces and also to

Fig. 15.1 The modulor

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design objects of everyday use, i.e., to establish that the height of a bench must be 113 cm, that of a table 72 cm, that of a chair 43 cm, and so on. His scholars criticized his modulor by comparing him to Albert Einstein, who instead found this mathematical distortion in favor of man so interesting that he defined it as “a scale of proportions that makes the bad difficult and the good easy.” Le Corbusier used this principle of proportions for space, for objects and for art. He applied this code to his artistic production, thus to the size and proportions of his sculptures, to the dimension of the plastic volumes he designed, even to the height of the pedestals (Fig. 15.2). By applying these rules to his first purist paintings he sought confirmation of the fact that the relationships between the parts had a mathematical proof. Continuing in his study, he searched for the diverse configurations that led him to formulate the scale for two different reading keys for the modulor: the blue and the red one, where these references arising out of the golden ratio are constantly compared with the dimensions of man. In the early 1950s when he made the town plan and designed many public buildings for Chandigarh, the city conceived by Pandit Nehru, which was only built later, he again stuck to the proportions of the modulor. In the 1920s he considered the sanitary problem in designing the maisons Citrohan, which were then built in Stuttgart in 1927. At the end of the Second World War he invented the Villes Nouvelles, which were the first series of his Unités d’Habitation in Marseilles. More than simple houses, they were genuine city buildings. Also in these cases, he confirmed that his great lesson consists in having been able to translate into architecture the events of life and of history of his time.

Fig. 15.2 Application studies of the modulor © FLC, by SIAE 2009

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Fig. 15.3 Application studies of the modulor © FLC, by SIAE 2009

It was precisely in Chandigarh (the city of silver) that he drew a stylized human figure with one arm raised above its head, flanked by two vertical measurements: the red series based on the height of the solar plexus (113 cm), and the blue series based on the entire height of the figure, double that of the height of the solar plexus (216 cm) (Fig. 15.3). A spiral, graphically drawn between the red and blue series, seems to mime the volume of the human figure. Here, beyond its functional applications, is the extraordinary idea that designs a hand “open for giving, open for receiving” (Fig. 15.4).

Fig. 15.4 Open Hand by Le Corbusier (photo by Enrico Cano) © FLC, by SIAE 2009

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This is a small sketch drawn by Le Corbusier on the flight from Karachi to Cairo when returning from India, where he constantly annotated these dimensions almost as if he wanted to confirm that such an unusual work, an open hand against the background of the new city, required infinitesimal checks to find a logical reason for a form that had nothing to do with rationality; it was a poetic gesture, a hymn to the city of Chandigarh and, especially, to the possibility that men would live together in a community.

15.5 A Comparison with History I now want to go back to some meetings I have had through my work on the problem of applying, or better, of reflecting, on the mathematical aspects. Figures 15.5 and 15.6 shows a 17th century chapel in Mogno, Val Lavizzara, in Canton Tessin, which was destroyed by an avalanche in 1986.

Fig. 15.5 Former 17th century little church destroyed by an avalanche in 1986

Fig. 15.6 The avalanche in 1986

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Fig. 15.7 Site plan showing the plan of old church (yellow) and the one of the new project (pink)

When the community asked me to rebuild the church, I immediately wanted to give a meaning to the place, to the topos. The new church actually stands on the axis of the old one, but it has a completely different configuration (Fig. 15.7). This project I presented set off a long, drawn out controversy based on the socalled “good common sense” that reconstruction must be a matter of rebuilding what was there exactly as it was. But repeating the self-same church in the same place would have been like denying history; in this case that there had been an avalanche. So I said no to the idea of rebuilding the church exactly as it was. The church was a 17nth century building whose only remaining trace was its mark on the ground. But the avalanche had occurred and had swept the church away, so the testimony of our time had to take into account what had happened. I therefore designed the new church with an elliptical plan, as Fig. 15.8 shows. It is a remarkable thing, because the elliptical shape on the axis of the old church indicates a desire to withstand a future, albeit improbable, avalanche. We begin by

Fig. 15.8 Geometry

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posing physical resistance, because one does not go into a valley to build a new church that will fall down, considering that the previous one has lasted for more than 300 years. It certainly is a difficult shape, because the ellipse has two centers. So it is an unquiet form, which would be suitable for a Baroque space. I therefore looked for a section in which the minor axis of the ellipse, when overturned, would become as long as the main axis. Thus, this logical construction emerged, which, based on the right inclination, has allowed the basic form to be calm, circular and with a single axis. This church (Figs. 15.9 and 15.10), which I finally built between 1994 and 1996, is nothing other than a slanted section where an elliptical form regains peace, becomes calmed. It is the reconciliation of the building, of the village, with the mountain; at the same time it is a historic reconciliation of that village, of that valley, of the inhabitants with the forces of nature they have always had to face. Something that goes well beyond mathematics occurs in that spatial position where the main axis of the ellipse becomes the same as the minor axis. But then a strange thing happened. This church was studied by the writer, art historian, and psychologist, Rudolf Arnheim (Berlin 1904-Ann Arbor 2007), with whom I entertained a nice correspondence because one of his students was writing a thesis on the subject of The Monumental in Architecture. Arnheim wrote to me posing a number of questions. “Dear Botta, is it true that you excavate into the thickness of the wall?” Actually, the walls of the church in Mogno are two and a half meters thick at the base. I wanted them like this to provide the ellipse with a depth, with a firm rooting in the ground. The idea of gravity interests me. Then Arnheim wrote again: “Is it true that the apse that opens up behind the altar is composed of 12 setbacks in the wall? 12 like the apostles, 12 like the months of the year?” So I counted them, and they were indeed 12. So I wrote: “Dear Arnheim, listen, I never thought of it.” And then he wrote again: “is it true that, starting from the upper circle, if you take the perpendicular axis to the circle you come exactly to the face of Christ on the wall?” I checked and it was indeed true.

Fig. 15.9 Church “San Giovanni Battista”, Mogno, Maggia Valley, Switzerland (1986/92–1998). a View of the front. Photo by Pino Musi. b View of the back. Photo by Pino Musi. c Skylight. Photo by Pino Musi

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Fig. 15.10 View of the apse and the altar. Photo by Pino Musi

What does this mean? It means that there are a posteriori verifications that go beyond those of the project. The position of the crucifix was chosen because it seemed right to me, and those ashlars that are each 20 cm deep highlighted the thickness of the wall at the right point. If they had been smaller, there probably would have had to be more of them. It means that the mathematical analysis made subsequently by Arnheim was able to bring out the creative, artistic dimension of my work. I could go on a long time because there are many stories that underline this concept, that is, the priority of creativity over rationality.

15.6 The Symbolic Value in the City Another example is the San Francisco Museum of Modern Art (SFMoMA) (Figs. 15.11-15.14). The building rises in downtown San Francisco, in a jungle of concrete where I could find no elements of reference. Furthermore, I had to design a very small building compared to the tower blocks behind. I literally did not know to which saint to turn to. Then I latched onto a house, which was nothing but a corner element that had temporarily survived the destruction of urbanization. I saw, however, that it had a strong angle, a dignified brick structure that I took up in the project; above all I noticed that it maintained a presence of reference on the street. So I started from the little that remained of an almost destroyed building for a building that was to become an image of the city. The big skyscrapers all around are certainly functional in their precision and technical cleanness in steel, metal, glass etc. But they are mute. It is not possible to say whether this or that building holds a prison, a home, a hotel, or offices.

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Fig. 15.11 MOMA, Museum of Modern Art, San Francisco, USA (1989–1995). The museum and the surrounding context. Photo by Pino Musi

Fig. 15.12 Skylight. Photo by Robert Canfield

15 Architecture Between Science and Art Fig. 15.13 Foyer. Photo by Pino Musi

Fig. 15.14 Night view. Photo by Pino Musi

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The city has completely lost its iconic capacity, its symbolic value. So, to the client’s request – “we want a building that expresses the idea of a museum” – I went back to the small church of Mogno and to the Romanesque churches as a point of historical reference in European cities. I took the idea of the rose window to San Francisco and built the museum around it. In this way a link is created with the city, through that small scrap of history, that surviving corner building, which, by the irony of chance, has now been retained. It was going to be demolished, but its symbolic value was recognized. The new has saved the old in the sign of a track, which is more familiar to us in Europe. As it happened, then a new building that is typical of our time like the Museum of Modern Art has helped the city recover a memory that would otherwise have been lost. This museum is a shape, I dare to say a face, a mask, a figure that confronts the city – this is the fog, this is the Bay Bridge, etc. – and becomes an icon despite its small size. The symbolic strength of this structure that turns to the city is evidently not a technical fact nor even a client’s request. It is an ideal factor taken from European thinking, from European architectural culture, to a big American city. It is a sign of historic recognition, an element of reference. When we move through our cities in Europe we immediately know that this is a market, a library, a church, the station, and so on. We instinctively recognize the symbolic value of the buildings and of the spaces beyond their function. They are elements of identity, of culture, of memory imprinted into the city itself. The exact opposite is the case with the Rovereto Museum of Modern and Contemporary Art (MART) (Fig. 15.15). The city council was disappointed when I produced my design, which is hidden in the back rather than being in the foreground on Corso Bettini. “How come? We want the most beautiful, the newest building in town and you have set it back in the second row?” The fact is that I had set the

Fig. 15.15 MART, Museum of Modern and Contemporary Art, Rovereto, Italy (1988–2002). Entrance from Corso Bettini between the historical palaces. Photo by Enrico Cano

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Fig. 15.16 Museum courtyard. Photo by Pino Musi

building back because it was an abnormal size and therefore needed space. I was not interested in putting the solid structure of the new museum alongside other solid structures. I was interested in creating a space. So I designed an avenue that entered; I opened the wall between two buildings and this became the door. In this way Corso Bettini became the foyer of the museum behind it, so that the existing old structures are forced to interact, to communicate with the new one (Fig. 15.16). It is not a conflict between volumes, but a progressive situation through which the visitor gains access to the heart of the museum. There is no longer a compact facade nor a door to the new collective spaces, to the urban spaces, that connect the structure of the town to the central part of the building.

15.7 Geometry that Organizes Space Another significant example is the Santo Volto church in Turin. I worked on this project developing the idea of some towers that convey light to the interior of a single, central space. Thus, I drew an octagon. The octagon gave me difficulties because, having an even number of sides, it never offers an entrance; even in baptisteries the entrances are always on a side. Then I began thinking about the number 7, which constantly returns in artistic literature; one needs only to think of the seven columns of the Temple of David. It is a magical number, never applied in architecture. I discovered that the heptagon has its own magic, because in the moment I enter into that space I always find a thorough situation. I was intrigued by the reading of Leonardo da Vinci’s Vitruvian Man (Fig. 15.17) by my friend Mario Helbing, a philosopher of science. He has identified a number of relationships on the basis of which the heptagon is really the template, the mathe-

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Fig. 15.17 Vitruvian Man

matical design stems from and whose focal point is philosophical, i.e., the centrality of man. This is interesting because it confirms that the interpretation of the Vitruvian Man is not so obvious. On looking closely, we actually find that the heptagon has the ability to enclose everything, to form a thorough situation (Fig. 15.18).

Fig. 15.18 Geometry

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So I designed a gear, a structure with seven sides, on the area that includes the service portion of the curia and of the parish. I placed seven towers around it, each one conveying light to the interior space, which is transformed into a double chapel (Figs. 15.19-15.21). It is a recent project, but it is a work that comes from afar in the sense that it arises from geometry and its ability to organize space in the best possible way. I did not start from a precise geometrical idea, I had actually initially worked on the octagon, but no matter how I tried, I could not find the serenity that was apparent on the drawing board. So I fell back on this new geometry, on the seven sides and their symbolic and historical value. And then something ought to be said about the face taken from the Holy Shroud. It is not a drawn image, but a wall in which the stones of Verona are turned into a combination of pixels that reproduce the likeness of the holy face through the relationship between light and shade. The apse behind the altar receives the zenith light, so the vertically laid stones are struck by the light from above and the others, slightly slanted down, receive the shade and thus create dark areas (Fig. 15.22).

Fig. 15.19 Plan of the ground floor

Fig. 15.20 Santo Volto Church, Turin, Italy (2001–2006). a View from the chimney. Photo by Enrico Cano. b Parvis, church and chimney. Photo by Enrico Cano

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Fig. 15.21 Assembly room and altar. Photo by Enrico Cano

Fig. 15.22 Detail of the small Verona bricks that form Christ’s face (Santo Volto). Photo by Enrico Cano

The design vanishes when looked at from nearby, letting the work on the masonry (the technical aspect) emerge, but from further away an image appears that recomposes the face on the shroud (the creative element) (Fig. 15.23).

Fig. 15.23 Santo Volto. Photo by Enrico Cano

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15.8 Presence in Landscape Another example is the Arosa wellness center in the Swiss mountains. A renowned tourist resort, a big hotel, a wellness center; these are the new frontiers of sociality. The competition called for a construction inside a precise space. So I decided not to build, or rather to build underground. I created a bridge that connects the hotel, then I arranged the services on four floors with the water high above and, going down, the area for the massages, the beauty center, and the fitness center (Figs. 15.24-15.26). But the thing that is really interesting is the outline of the surrounding mountains; so I designed these leaves of glass that, when raised, become structures that convey the light down (Fig. 15.27). The figure shows these pictures with the snow because in this way the ground disappears entirely, and what remains is a system of presences in the landscape that, quite evidently, are inspired by the idea of the trees. At night they are fascinating in their artificial light (Fig. 15.28).

Fig. 15.24 Wellness center Tschuggen Berg Oase, Arosa, Switzerland (2003-2006). Plan portraying the four levels of the spa

Fig. 15.25 Glass bridge between the Tschuggen Hotel and the spa. Photo by Enrico Cano

304 Fig. 15.26 Swimming pool in the “water world ”. Photo by Urs Homberger

Fig. 15.27 Snowy landscape. Photo by Urs Homberger

Fig. 15.28 Night view with the lit-up leaves. Photo by Urs Homberger

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On the architectural level the fact that a receptacle for a wellness center is placed as a completely new presence in the landscape, disrupting its traditional reading, becomes interesting. This naturally has nothing to do with wellness, but is a representation that enriches the mountain landscape, the relationship with the vegetation, the forest and the mountain peaks. . . It is nothing else than a creative invention borne out from technological support.

Suggested Reading 1. Le Corbusier (Charles-Edouard Jeanneret): Modulor (Editions de l’architecture, Paris 1948)

Discussion G. Giacometti: I liked the unique angle of the cut of the ellipse which becomes a circle. A curiosity: is the eye in the San Francisco museum obtained by the same principle of that of the church in Mogno? M. Botta: The two are similar, but not equal. In the church of Mogno the ellipse becomes a circle, while the opposite occurs in the Museum of San Francisco. E. Carafoli: Art and science are closely interwoven in architecture, in which science defines the technical aspects which make the architectural objects usable by the people. Art and science should thus be well balanced. Sometimes, however, one has the feeling that architects tend to privilege the artistic portion of their work, paying less attention to the technical-functional aspects that should make the product a pleasantly usable object as well. Could you comment on this? M. Botta: True, contemporary architecture tends to be more and more conditioned by slogans, with the result that the technical-functional aspects become less important. However, we should keep in mind that both the technical and the artistic aspects are components of a discipline which is a language, a means of expression that offers emotions to men. A balance is therefore necessary between the two components, which should complement each other.

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Michelangelo Pistoletto

The room will now obscured, but something bright will appear before us: the screen with the title of my presentation, which includes the word spirituality. This word was not included in the wonderful list of topics of the symposium that has originated this volume: the concept of time, infinity, nothingness, the function of numbers, the origin of thinking, the human brain, the relationships between artistic and scientific creativity. I think the concept of spirituality spans over all these themes. I will not specifically speak about spirituality, but its presence will perhaps be perceived from my work and it will thus be possible to understand what I mean. In ancient art, particularly Italian, spirituality was incorporated into the illustration of religious thinking. In the 20th century, however, the artist has stopped portraying religious and socio-political images. He has moved away from all what had until then been the contribution of to the social sphere; he has gradually retired into his own inner life and has become autonomous. In the 20th century art reached a level of intellectual autonomy that it had never had before. I started working as a painter by observing the art of the past from close up. My father was a painter and also restored ancient pictures. I began working with him on art restoration at the age of 14. Many figurative works have passed through my hands and under my eyes, including quite a few icons; that is, the golden backgrounds of Byzantine origin. It was precisely following that iconographic experience that I decided, in the mid 1950s, to direct my studies toward figurative art. And this happened at a time when the confrontation between figurative and abstract art had reached its highest level of tension. The meeting/clash between figurative and abstract art was actually the artistic-intellectual epilogue of an old history of conflicts between the religions of middle-eastern and western countries, divided precisely between iconography (adoration of the image) and iconoclasm (negation of the image). That ancient contrast reached its final stage with modern and contemporary art. Abstract art never existed in Italy because artistic expression depended fundamentally on the Christian/Catholic iconography. I realized that I had to resolve my

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artistic-existential and spiritual problem through figuration, as my education derived from the figurative culture of artistic representation. If the abstract artists had attained their expressive independence from any influence of an iconoclastic nature, I had to acquire mine through the image, but removing it from religious fideism. I therefore dedicated myself to the self portrait, with the initial goal of understanding my own identity. I will now show you a self-portrait from 1960 (Fig. 16.1). The reference to the Byzantine icons can clearly be seen. The background and the figure are the essential elements of the work. The background is monochrome, in gold color (but I have also done backgrounds in silver, copper, aluminum, and other materials). On this background I had painted the image of my person, life-size, expressionless, simplified to make its presence in the bright surrounding space symbolic, even if also alike. In short, the connection between the human being and the boundless space of the background is evident. In 1961 I replaced the silver background with a monochromatic very glossy black paint. In this way the background became a mirror, reflecting what was in front of the picture (Fig. 16.2). The painted figure detached from the background to appear almost three dimensional. For the first time I had painted, even if summarily, my image by mirroring myself directly in the picture rather than in the mirror, which painters always placed beside the canvas in order to paint their self-portraits. I then tried to make the reflection of the background as sharp as possible, and in 1962 I made my first paintings on stainless steel, polished to a mirror-like sheen. The figure on the now perfectly reflecting surface was now no longer painted, it was photographic. The use of photography, after many recon-

Fig. 16.1 Gold Self-portrait, 1960, oil, acrylic and gold on canvas, 200 × 150 cm Fig. 16.2 Self portrait, 1962-1998, serigraphy on stainless steel, mirror-like polished, 250 × 125 cm. Photo: J.E.S. 

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siderations and experiments, had become indispensable. I had found no other means that would offer the possibility of reproducing an image with the same objectivity of the images reflected in the mirror. In this way the photograph was incorporated into the mirror and the “mirroring pictures” were born. Other people and then others and others again were reflected in the work along with myself. So the artist was no longer alone in his self-portrait. In all the self-portraits of the past (apart from some experiments) the artists appear alone. Now, for the first time the world went into the picture along with the artist, so the place of loneliness became a place of plurality, of meetings, and of exchanges. The “mirroring painting” (Quadro specchiante) brings a new conception, of phenomenological nature, into the work of art. Nothing of what we see in the mirroring painting depends on my subjectivity; the work does not express any feeling, emotion ,or fantasy of mine. The picture is an object that combines two elements, the dynamism of the reflected figures and the stillness of the figures fixed in that dynamism. So we have the chance to capture the maximal stillness and the maximal mobility at the same time. These two extremes coexist in the painting. At every moment the mirrored movement is blocked by the immobile, fixed figure, which in turn enters into the motion, that is, in the succession of images, like a film running in the present. It is a frame that corresponds to a fragment of memory that is linked to the present, while the present on its part constantly attracts the future into the picture. A photograph just shot is already a memory of a past present, and the mirrored present constantly becomes past, but the photo on the mirror brings the past back to the present and sends it into the future. The black pictures of 1961 are entitled “The Present” (Il Presente) because the present is the phenomenon of time and space crossed by every event contained by the work. The people reflected are present in the same way as is the room in front of the mirror. In the mirror every image is born and dies at the same moment. There is no stopping in this dying and regeneration of the image, everything appears and disappears through the present. In a sense, the future is also already present in a certain sense. We know that people not yet born will one day appear in the mirroring picture, so they are potentially already in the mirror, just as I was potentially already in the mirror before I came into existence. So this work has the ability to bring complete time and complete space to us observers. A mirror here and now reflects the present just as a mirror placed at the same time in any part of the world or even on another planet does. So there is a present that eliminates space in that it happens everywhere simultaneously, and there is a present that creates space because it distributes itself in it. We are used to considering time as space, we think that our present compared to that in New York is separated by 6 hours, but my pictures in New York at this moment reflect the present even if we are not there. So the mirror is not just a piece of glass or material that reflects, it is total reflection. The mirror is the phenomenon of reflection that extends everywhere in front of the physical nature of the existent: the mirror thus has a universal dimension. We must, however, consider that the mirror does not exist in itself if there is no one in front of it. The mirror does not exist in the sense we give to the mirror itself; the act of mirroring assumes a value in that it produces a reflection of our thought.

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Umberto Eco writes that the mirror has no semantic value because “the mirror reflects things as they are,” therefore it does not give them a character. But I think instead that the mirror has essentially semantic properties. When I look at something, I see it directly. But when I look at the same thing in the mirror, I see it represented. It is the same thing but passed through the concept of the object itself. The mirror is representation par excellence, so it is the representative semantic reflective base on which languages are developed and articulated. The artist in the act of painting his self-portrait transfers the image from the mirror to the painting giving it the property of concept. The mirroring picture recognizes the conceptual property of the mirror itself. To me the mirror is the essence of language as it represents the language of representation in the most minute and at the same time most extended manner. From the mirror I draw a great number of indications, considerations, and verifications about the existence of things. In the mirroring work, culturally layered elements coexist: the picture, the image, the form, the proportions, etc., but that is not all. One can feel the sense of transeunte, the perception of the infinitely mobile, in the change that is grasped moment by moment. The “time” dimension is revealed directly in its phenomenological dynamics, and this happens in the great emptiness of the mirroring surface. But it is an emptiness in which everything that exists appears. We witness an extraordinary event before the mirroring picture: the coincidence of all opposing terms: static and dynamic, front and back, true and false, finite and infinite, near and far, present and absent, virtual and real, minimal and maximal, material and non-material, chaos and order, everything and nothingness, and so on. The mirroring picture is not a work that recounts or expresses emotions, however, it provokes emotion through reason. Emotions lie in the perception of a phenomenologically proven revelation. In fact, many feel a sense of wonder before the “mirroring painting,” even if they do not have any deep knowledge of it. So emotion and reason fuse together in the “mirroring painting” precisely because of its conception, which is as scientific as it is imaginative. The imagination enters actively into play in the observer who catches himself inside the picture and feels involved in the projective zone of art, exactly like the artist who has to create his own self-portrait. The visitor, like the artist, looking at himself in the painting realizes he is there by chance and understands that all other things are also there by chance. Everything happens by chance. Chance combines and upsets images, generates occasions, events and circumstances. Chance makes everything relative. In the “mirroring painting” relativity is the protagonist to the point of becoming absolute: absolute in the sense that has no comparisons because it contains total perfection. Relativity in the “mirroring painting” is total, so, it is absolute. The culturally layered elements in the work include spirituality and, in this sense, rather than retrieving any contrast between the ideas of transcendence and immanence, one perceives their common and reciprocal identities. Going on we come to another period of my work. We are now in front of a mirror without images fixed on it, simply reflecting a strong light. It is an interiorization of

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the mirror. In fact, the mirror has penetrated my mind to the point of making me feel as if I were a mirror myself. At a certain moment I asked myself: “if I were a mirror how would I create a self-portrait? How would I be able to identify myself as I did when looking at myself in the mirror?” In 1975 I made an installation entitled “The Rooms” (Le Stanze). The gallery space consisted of three identical rooms connected by three openings of the same size. I placed a mirror on the end wall of the third room as if it were another opening, and the space doubled. It became a succession of doors, real and reflected, and I counted the openings: 1, 2, 3, 4, 5; how much is twice 3? Twice 3 is 6, but there were 7 openings. “What’s happening?,” I asked myself, and immediately realized that one of the doors was the mirror, which has no double. The mirror produces the double of everything but not of itself. So I noticed that if I had been a mirror, I would not have been able to do a self-portrait because of the lack of another mirror. At that point I closed my eyes and said: “if I were a mirror and could see myself, what would I see?” I was frightened because I had seen “nothing.” A non-image that reflects a non-image is nothing. We are able to see the mirror solely because it reflects something or because it is bordered and has something around it, otherwise it would be nothing reflecting nothing. This is how the concept of nothing comes into being. So I then asked myself “what should I do?” and immediately I constructed an “action,” which I then called “division and multiplication of the mirror” (Figs. 16.3a–c). I took a glass mirror and cut it in two. In this way I finally gave the mirror its double. I then realized that the division of zero, that is, of nothing, does not make one but two: 1 plus 1. I started to move the two mirrors at an angle and these began to reflect each other and inside them a third mirror was born. The third mirror appeared inside the two mirrors like a filiation. Bringing the angle to 90 degrees gives us four mirrors; as the angle is gradually reduced and the number of mirrors increases, the progeny of mirrors deriving from the initial couple grows. As the mirrors are brought together, there is an exponential acceleration in this growth, ending with an infinite multiplication when they are in front of one another. At that point the mirrors touch and are no longer visible, thus, we can enter the work only with our imagination. The mental phenomenon emerges in this way in its entirety. Inside that work there is a dimension we can recognize only with our minds. The work is identical to us: it is externally physical and internally immaterial like the thought. In the mirroring painting the reflected image is intangible, it has no body. The mirror reproduces reality removing from it its physical weight. However, the reflected figures are caught by the eye, and seeing is one of the senses, like touch, taste, hearing, and smell. Seeing touches with the eye. But when we are faced with the two mirrors that touch each other frontally, the sensorial phenomenon is excluded and the experience of intangibility is generated. There is, however, a perfect correspondence between the virtual nature of the “division and multiplication of the mirror” and the reality of biological life. Indeed, in an organic system a cell divides into two parts, the two parts then divide and these parts divide again, and in doing so multiply. So multiplication, also of cells, occurs by division. The mathematician

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Fig. 16.3 a–c Mirror cage, 1973-1992, iron and mirror, five elements, two panels each, each panel 130 × 40 × 250 cm. Photo P. Pellion

will tell me that “division and multiplication occur together.” I object and say that division occurs before multiplication: if I had not divided the mirror I would not have multiplied it. I could not have started from multiplication, I had to start from division. This is to me a very important principle because, translated into terms of society, while division becomes sharing, the principle of multiplication gives rise to accumulation, invasion, appropriation, and exclusion.

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Fig. 16.4 “Oggetti in Meno” (Minus-Objects), 1965–1966. View of the personal exhibition, MAMAC, Nice, 2007. Photo A. Lacirasella

I now come to a group of works dated 1965–1966 entitled “Minus-Objects” (Gli Oggetti in Meno, see Figs. 16.4 and 16.5). Each work is different from the other. The exhibition of these works is reminiscent of a collective one: it is no longer unmistakably recognizable Pistoletto, like for the mirroring paintings. Pistoletto has divided himself into many different objects. Each work is different in form, material and meaning. All this is based on the concept of diversity. I achieved such diversity by adopting the criteria of immediacy, contingency, spontaneity. An idea is borne? Let’s do it! So in a few months I had filled my large studio with works that are all completely different. Why did I call this group of works “Minus-Objects”? The rationale lies in the peculiarity of the inventive and executive process. Each work bursts forth from the brain turning a potential into a reality. The process consists in extracting the tangible from the intangible. A potential that exists in my mind is turned into a physical object by means of the creative impulse. And as a potential is brought into reality, I now have one less unexpressed potential. So every work which is realized is removed from the sphere of the potential. Every work corresponds to one potential necessity less in that it is made real. Thus, every object is the total of the unexpressed less that object. These works are different from one another as every moment, in the following of time, is different from another moment and every existing thing becoming unique as if differentiates itself from other things. The potential of the mind is enormous and we actually grasp only a very limited amount of it. The “Minus-Objects” are an example of how we restrict our real possibilities. I dreamt the work “The Burnt Rose” (Rosa bruciata). I could have got up and not done it. I dreamt that I was constructing

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Fig. 16.5 “Oggetti in meno” (Minus-Objects), 1965–1966. View of the personal exhibition, MAMAC, Nice, 2007. Photo A. Lacirasella

this object, with cardboard that I was cutting and wetting to give it a form, sprayed red and finally set fire to the center of the rose. When I got up I could have done something else. Instead I went to find the carton, the red spray-paint and, as if the dream were a recipe, I made the work. This was a potential that, passing through the dream, had become an object. The idea of seizing the fleeting moment and of translating it into matter was transformed in the creation of all these works, which are different from one another, each with its own autonomous identity. One work consists of an ancient sculpture set in orange Plexiglas, another is a canvas on which I wrote “I love you” (Ti amo) in very large letters, in another my working clothes are displayed in a showcase, and then there is a fibreglass bath-sarcophagus next to a metal “Structure to Talk While Standing” (Struttura per parlare in piedi) near a “Painting for Lunch” (Quadro da pranzo) in wood, in which you can also sit. The “Cubic Meter of Infinite” (Metrocubo d’infinito), which is analyzed last, contains the meaning of the entire group of “Minus-Objects.” This work is directly related to the sequence of the “Division and Multiplication of the Mirror” I have just presented. Here there are six mirrors creating a cubic space among them. The mirrors are turned to the inside of the cube, which becomes a unit of measure that, by selfreflection, is reproduced infinitely; no longer only horizontally, but in a spherical sense. This physical cube produces inside itself an infinite number of immaterial cubes. The six mirrors are held together by a simple piece of string that has the same function as the nerves in a body. We do not see the interior, but we perceive it with our thought. I said that this work is actually the symbol of the “Minus-Objects.” It

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represents the infinity, less that cube, because without that cubic meter of mirrors, the infinite would not have been produced. But to be able to produce the infinity it has had to divest itself of its own infinite immateriality, subtracting from it a cubic meter of physical space. The “Cubic Meter of Infinite” is therefore the infinite less one, so it is also a Minus-Object. In 1998 I was asked to design an architectural work that was opened in 2000. It is entitled “Lieu de recueillement et de prière” and was created in Marseilles in the Paoli-Calmettes Hospital. The people of this city come from a large number of countries. It is not a mono-cultural, mono-religious society, but consists of Muslims, Buddhists, Christians, Jews, and non-believers. There was a Catholic chapel in the hospital, but it did not meet the needs of a multi-cultural, multi-confessional community. The hospital management understood the spirit of my work and asked me to conceive a multi-confessional, and also secular “place of meditation.” In Biella, in the Cittadellarte (city of the art), I have created the same “place” in a moveable tent, which I have then installed at different sites, including the Island of San Servolo for the 2005 Biennial Exhibition in Venice. The symbols of the various monotheistic religions – Buddhist, Muslim, Jewish, and Christian – are arranged in a circle in the flower-shaped space, with the “Cubic Meter of Infinite” in the middle. There is also an empty pedestal representing Laicity or other religions. In this “place of meditation and prayer” art takes the spiritual centrality (Fig. 16.6).

Fig. 16.6 Place for meditation and prayer, Paoli-Calmettes Institute, Marseille 2000. Photo M. Spiluttini

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I have placed the cube of infinite reflection at the center of the niches occupied by the different religions, considering art as the original factor in the entire development of the entire human race. Creativity, of which art is the essential expression, has produced human intelligence, which has separated itself from the intelligence of nature, and created the artificial world that has now assumed global proportions. The word “art” expresses the very concept of artifice, the root that gave birth to the sphere of “artificial life” on which the entire humankind now gravitates. So all the organisms that make up society on the basis of religions are born of art. These are the first ring of the social structure and in the small temple in Marseilles they surround the “Cubic Meter of Infinite”, which places art at its point of origin. In this position, art assumes a role of responsibility in front of human society. My current work is mainly directed toward the project called “Third Paradise” (Terzo Paradiso), which is centered precisely on the responsibility of art. The first paradise is the natural one, the second paradise is the artificial one that is tragically devastating nature. The Terzo Paradiso represents the move to another level of civilization in which artificial intelligence is reconciled and integrated with the natural world. Art may be thought of like original sin, or be seen as the original wonder of thought, depending on religions or different vision of the world. In any case it now has a basic responsibility. In the twentieth century the artist freed himself from all religious and political references, so he became capable of managing such liberty with his own expression and mark. Such freedom must now correspond to all equal responsibility. The freer one is, the greater responsibility one has. By assuming this responsibility myself, I have created a place called “Cittadellarte.” It is dedicated to the interaction between art and every other sector of human activity in order to set off a responsible transformation of society. In today’s society, based on hedonistic values, I think we must not give in to the avant-gardes that have conquered the autonomy of art. There is rather the need for a new avant-garde. The way I see it, art does not follow a straight road as the avantgarde of the past did: it indicates a turn. The mirroring painting shows precisely this turn. The mirror shows us the space behind us. In order to follow this path, i.e., to enter into the perspective opened up by the mirror, we have to turn around and carry on, looking in that direction. We must no longer proceed in the direction of the Renaissance perspective that would lead us directly to the conquering of the universe. At this point we find a mirror showing us all the responsibilities accumulated and neglected in the past. The perspective of the mirroring painting is bi-frontal and creates a balance between past and future. The mirror is not like Duchamp’s glass, which in making us look always in the direction of Renaissance perspective continues to drag us toward the old myth of progress. The mirror places the individual in front of himself and of all his responsibilities. I now want to present another passage of my work, the “Art-Sign” (Segno Arte, Fig. 16.7). It is a door whose shape corresponds to a person with arms raised, slightly open, and legs slightly apart. The person I photographed going through the door shows her navel: the navel is at the very center of the Segno Arte form. In 1490 Leonardo moved the legs of Vitruvius’s man, creating the space of an equilateral triangle between them, and

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Fig. 16.7 Through the art-sign. 1976-2007, sublimation on a stainless steel plate, mirrorlike polished, 250 × 125 cm. Photo J.E.S.

raised the hands slightly to complete the circle. I raised the arms of the person even further and, between her arms, I had her reflect the triangle formed at the base between her legs. Vitruvius and Leonardo have made the circle that contains the person by placing the compass on the navel. I then drew two circles that, suggesting the symbol of the infinite, meet at the navel. One of these touches the two feet and the other touches the two hands exactly where hands and feet touch the big circle taking in the entire figure. Joining the center of the upper circle to the tip of the two feet and connecting the center of the lower circle to the tip of the raised hands generates the two triangles that make up my Segno Arte, at the middle of which is the navel of the person going through the door. I have chosen a female image, because her belly is the place of reproduction. The navel is the symbol of life perpetuating itself. The umbilical cord actually connects us to all the human beings of the past, and will continue to connect us to the people of the future as long as humankind continues to exist. It is our responsibility to make it so that the symbol of the navel continues into the future. I just talked about the “third paradise.” The drawing that represents the “third paradise” is the “New Sign of Infinity” (Nuovo segno di infinito, Fig. 16.8), which also has its center in the procreative belly. I have added a central circle to the two

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Fig. 16.8 The new sign of infinity, 2005, a groove cut in the ground, San Servolo Island, Venice. Photo editing L. Ogyzko

circles of the traditional symbol of the infinite. There are therefore three circles. One represents nature, the other represents artifice, and between the two grows the belly, pregnant with the generations that will know how to make the artificial world coexist with the natural one. At this point I should talk more about Cittadellarte, but I suggest rather that those interested in it go to the internet website: www.cittadellarte.it

Discussion Pinton: All your work always translates into symbols. We can’t live without symbols. Do we have to translate these symbols into substance? M. Pistoletto: I really think that there is a symbiosis between symbol and substance. I arrived at the mirror trying to eliminate all symbols. I saw the mirror as a desymbolized symbol. I said earlier that art attained its complete autonomy in the twentieth century, something that had never happened before. Such autonomy was reached by concentrating all the symbols into only one: the subjective, personal one of the artist. That individual symbology was so compressed within the intimacy of the artist that it caused existential crises that even led to suicide. With the “mirroring painting” I have moved from subjectivity to objectivity, and I have also eliminated the last symbol that corresponded to subjectivity. This is how I was able to enter into the world of everybody, into the dimension of ordinary life. Then I began drawing new symbols from the mirror, destined to signify and produce a substantial process of social change. S. Califano: The beauty of Vitruvian Man is related to the fact that the distance from the navel is always related to the golden section, the same beauty as that of the Greek monuments. M. Pistoletto: I am trying to combine the idea of balance between the person and the world, as had happened in the classical periods. I think of the classical after the Romantic era like that of modernity, which could be referred to the middle ages in

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its progressive upward thrust: the rockets go to the moon as if they had left from the spires of Gothic cathedrals. I now feel the need for balance, proportion and, precisely because of this, I once more place the person at the center; a wiser person to whom I assign new responsibilities. Beauty today must be found in wisdom, this is the new aesthetic of art. F. Bertola: How does your work l’uomo di stelle (The Man of Stars, see below) fit into this presentation?

Self portrait of stars, 1973, photograph on plastic, 205 × 106 cm. Photo M. Scattaro

M. Pistoletto: The Autoritratto di stelle is a photographic silhouette of myself filled with a photograph of galaxies taken from the Mount Palomar observatory. I made this work thinking of a camera that on one hand captures my image and on the other that of the universe and blends them into one. This work, too, is reflecting, even if in a different way reflecting: I see the immense universe in my body, and the universe in its boundless dimension sees itself in my little body.

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S. Califano: These symbols, this conceptualization of your work are expressed in such an abstract way that I wonder how the average man can come to grisps with them? M. Pistoletto: I have described this symbology in depth in this talk which was directed to people working in the sphere of thought, but these elements enter into real activities at the Cittadellarte. I am also producing images and symbols that are more immediately understood. Anyone coming to Cittadellarte sees young people working on political, social, and economic problems. Regarding politics, for example, we created “Love Difference,” movimento artistico per una politica intermediterranea’, a movement with which we organize meetings in various Mediterranean countries. Cake shops named “Love Difference” are opening up in various places as meeting points for an exchange of artistic-cultural and political views. The symbol is the Mediterranean Sea, which is very intelligible. I also translated the emblem of the “third paradise” into a very simple image of immediate impact. This is currently being displayed at the Science Festival in Genoa. It is a big apple, the Mela Reintegrata (Restored Apple); the bite of the biblical apple is restored. Artifice returns to nature, original sin is redeemed. This symbolism is immediately understandable.

Name Index

Aihara, M., 152 Apollinaire, G., 250 Archimedes, 40, 48, 49, 189 Aristotle, 3, 16, 40, 48, 52, 90, 91, 268, 269, 285 Arnheim, R., 294, 295 Augustine of Hippo, vi, 101 Bach, J.S., 165, 167, 175, 177, 190–192, 206, 245 Balla, G., 222 Bartok, B., 247 Bateson, G., 46, 217, 218, 224, 232 Beethoven, L. van, 212, 247 Benussi, V., 133 Benveniste, E., 108 Bergson, H., 14, 24, 99 Berlucchi, G., 119, 180 Bertola, F., 319 Bever, T., 153 Bobin, C., 207 Boethius, S., 110 Bohr, N., 257, 258 Boi, L., 11, 37, 51–97, 113, 118, 121, 284 Borges, L., 39, 40, 42–48, 125 Botero, F., 265, 266 Botta, M., 247, 248, 287–305 Botticelli, S., 242, 243, 247 Bougakov, D., 147–164 Bradley, 42 Braque, G., 249, 250, 261 Bresciani, M., 47, 263 Broadman, K., 147, 148 Broglie, L. de, 64 Brunetti, F., 182, 262 Califano, S., 94, 261, 283, 318, 319 Carafoli, E., 48, 113, 163, 205, 220, 225, 232, 233, 236, 239–264, 283, 305

Carroll, L., 41, 42 Casimir, H.B.G., 55, 61, 63, 71, 75, 96, 121 Cendrars, B., 250 Cézanne, P., 231, 249, 251, 252 Chopin, F., 165, 167–171, 175, 177, 178, 247 Clairvaux, B., 166, 168 Claudel, P., 102 Cocteau, J., 250 Coleman, S., 78, 79, 92, 96 Coleridge, S.T., 45, 239, 253, 260, 283 Constable, J., 249 Copernicus, N., 30, 38, 119 Dali, S., 189, 247, 248 Dante Alighieri, 46, 216 Danto, A.C., 265, 267 Darwin, C., 214 Debussy, C., 228, 247 Descartes, R., 16, 53, 200 Diogenes, 39 Dirac, P.A.M., 53, 54, 64, 66, 67, 70, 71, 96, 242 du Sautoy, M., 120, 185–206 Duchamp, M., 133, 250, 266, 275, 277, 278, 316 Duncan, I., 233, 234 Durrer, R., 27–38, 115, 120, 121, 162 Dyson, F., 68 Eco, U., 310 Einstein, A., 4, 6, 14, 17, 31, 35, 37, 53, 55, 58–60, 66, 81–83, 92, 94, 96, 164, 224, 243, 290 Eliot, T., 20, 21 Feltrin, G.P., 163 Fermi, E., 7, 65, 71, 92 Ferraris, M., 265–285

321

322 Feynman, R., 4, 51, 65, 68, 74, 96, 232 Fibonacci, L., 228, 246, 247, 274, 279 Fock, V.A., 86, 94–96 Fogedby, H., 205 Frey, D., 244 Gadamer, H-G., 269 Galileo Galilei, 4, 30, 31, 50, 53, 119, 217, 228 Galois, E., 194, 197–199, 203, 206 Giacometti, G., 96, 181, 305 Glick, S.D., 154 Goldberg, E., 147–164, 190, 206, 245 Goldstone, J., 76, 77, 93, 94 Gombrich, E., 244 Grossberg, S., 155 Gunzig, E., 57 Guth, A.H., 77, 78, 80, 83 Hardy, G.H., 120, 186, 202, 241 Harvey, J., 267, 268 Hawking, S.H., 5, 8, 55, 56, 62, 77, 92, 96 Hazlitt, W., 210, 212 Hegel, G.W.F., 105, 275 Heidegger, M., vi, 14, 17, 18, 24, 99, 100, 102, 104, 113, 224, 225, 269 Heisenberg, W., 6, 51, 54, 55, 57, 64, 74, 75, 85, 86, 90 Helbing, M., 299 Higgs, P.W., 75, 77–80, 93, 94 Howe, E., 258 Hui Shi, 40 Hume, D., 244 Husserl, E., 14, 18 Jacob, M., 250 James, W., 267, 268, 281 Jordan, P., 64, 66 Jung, C.G., 171, 172, 179, 181 Jünger, E., 102 Kafka, F., 39, 43, 45, 47 Kamiya, Y., 159 Kant, I., vi, 40, 106, 244, 260, 263 Keats, J., 22 Kekulé, F.A., 256–258 Kittel, G., 107 Kline, M., 231 Le Corbusier, 193, 231, 247, 278, 288–292 Le Poidevin, R., 21 Lee, T.D., 96 Leibniz, G.W., v, vi, 102, 113, 114 Leonardo da Vinci, 189, 247, 251–253, 258, 299 Leopardi, G., 113, 119, 210, 212, 213, 216, 223, 258, 259 Linde, A., 78, 83, 96 Longo, G.O., 45, 204, 207–237, 247

Name Index Lorca, F-G., 256, 259 Lucretius, T.C., 213 Mandelbrot, B., 228, 229 Mann, T., 189, 194 Martin, A., vi, 224 Mazzarella, E., 99–114, 118 McTaggart, J., 13–15 Messiaen, O., 192 Michelangelo, 252, 253, 256, 262, 307 Miller, A., 12, 37, 90, 261 Mitchell, P., 240, 241 Mizler, L., 190, 245 Moebius, K., 204 Mondrian, P., 247, 282 Monod, J., 240 Montale, E., 213, 233, 235, 254, 255, 283 Montecucco, C., 144 Mooij, J.J.A., 13–25, 37, 145, 162 Mucha, A.M., 278 Nehru, P., 290 Newton, I., 4, 6, 14, 16, 30, 37, 53, 125, 180, 217 Nietzsche, F., 111, 268 Ockham, W., 215, 216 Odifreddi, P., 39–50 Oudai-Celso, Y., 47 Pascal, B., 52, 53, 181, 223, 224 Paul of Tarsus, 101 Pauli, W., 54, 59, 66, 70, 165, 166, 171, 172, 179, 181, 182 Peccanda, A., 262 Penrose, R., 5, 55, 56, 62, 93, 115, 118 Petrarca, F., 216 Phidias, 246 Picasso, P., 222, 240, 249, 250, 261, 275 Piero della Francesca, 189, 225, 226, 247 Pinton, 318 Pistoletto, M., 256, 262, 307–320 Plato, 11, 40, 47, 48, 51, 90–92, 100, 125, 140, 187, 188, 220, 243, 244, 260, 262, 266, 279 Prigogine, I., 242 Proust, M., 166, 281 Pythagoras, 28, 268, 274, 275, 277 Ramanujan, 186, 202 Rasche, J., 24, 144, 165–182 Ratzinger, J., 102 Reichenbach, H., 17, 24, 25 Riedweg, Ch., 11, 90, 236 Rilke, R.M., 211, 212 Royce, J., 42, 265, 266 Sartre, J-P., 14, 17, 99 Schelling, F.W.J., vi, 105

Name Index Schleiermacher, F., 108 Schoenberg, A., 192 Schroedinger, E., 243 Schwinger, J., 60, 68, 96 Sereni, V., 258, 259 Setti, G., 37, 114–122, 145, 163, 181, 205, 262 Seurat, G., 247 Shakespeare, W., 42, 125, 224 Shelley, M., 220 Sitter, W.de, 58, 77, 79, 80 Snow, C.P., v, 3, 4, 13, 125, 186, 201, 202, 207, 260 Socrates, 125, 140 Steiner, G., 223, 233 Sterne, L., 40 Stockhausen, K., 247, 283 Strawinski, I., 247 Sugita, Y., 126 Theognis, 101 Thom, R., 208 Torricelli, E., 51, 52 Tryon, E.P., 78

323 Vauxelles, L., 250, 261 Veneziano, G., 3–12, 24, 92, 115–118, 121, 203, 204 Vilenkin, A., 57, 58, 78, 80, 83, 96 Vogeley, K., 152 Volpi, F., 102 Von Spreckelsen, O., 189 Wagner, R., 254, 276 Warhol, A., 267, 268, 273 Wechsler, D., 147, 149 Weinberg, S., 78–80 Weyl, H., 16, 59, 116, 244 Wheeler, J.A., 84, 91, 96 Wigner, E., 64, 66, 77, 217 William the Silent, 106 Wise, S., 203 Witt, B.S. de, 96 Witten, E., 78 Wittgenstein, L., 41, 213, 217 Xenakis, I., 192, 193, 203 Yeats, W.B., 22

Vallortigara, G., 125–145 Varela, F., 217, 218, 236

Zeno of Elea, 39