Introductory Note-2

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Introductory Note Max Black The Philosophical Review, Vol. 59, No. 1. (Jan., 1950), pp. 77-78. Stable URL: http://links.jstor.org/sici?sici=0031-8108%28195001%2959%3A1%3C77%3AIN%3E2.0.CO%3B2-Q The Philosophical Review is currently published by Cornell University.

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FREGE AGAINST T H E FORMALISTS. I A translation of part of Grundgesetze der Arithmetik (Volume 11, Sections 86-103) Introductory Note Frege's Grundgesetze der Arithmetik (Jena, Vol. I, 1893 ; Vol. 11, 1903) was the culmination of his studies in the foundations of mathematics. Practically ignored at the time of publication,' it is today widely recognized as one of the few classical works of decisive importance in the development of mathematical logic. Russell, who was almost alone in recognizing the importance of Frege's work, said,a "Frege's work abounds in subtle distinctions, and avoids all the usual fallacies which beset writers on Logic. His symbolism, though unfortunately so cumbrous as to be very difficult to employ in practice, is based upon an analysis of logical notions much more profound than Peano's, and is philosophically very superior to its more convenient rival." But it is easy to exaggerate the clumsiness of Frege's symbolism; his diagrams may intimidate at first, yet one quickly becomes accustomed to their use. And cumbrous in formulation or not, Frege's work satisfies standards of rigor which even Principia Mathematica does not always achieve. Frege's program, to derive pure mathematics from logic by rigorously formal inference, does not lend itself to summary reproduction or appraisal. The technical sections of his treatise are, however, interspersed by discussions of more general interest, which can be read without detailed reference to the remainder. These include a polemic against the confusion of psychology with logic (pages xiv to xxv of volume one), an analysis of the conditions to be met by valid definition (sections 28 to 33 of volume one),' and a detailed criticism of other writers' theories of irrational numbers (sections 55 to 164 of volume two). W e are concerned here only with the last. The main terms, with their translations, are the following : Bedeutung Denotation4 Begriff Concept Actual number (to which reference is made in the Eigentliche Zahl customary conception of arithmetic)

' Compare p. xi of the preface to Grundgesetze, where Frege comments upon the absence of interest in his earlier writings. The Principles of Mathematics (Cambridge, 1903)~p. 501. 'This is summarized in J. JZrgensen, A Treatise of Formal Logic (Copenhagen and London, 1931), I, 153-156. Frege uses "Bedeutung" and "Sinn" in senses peculiar to him. In my previous translation of his Ueber Sin@ und Bedeutung (Philosophical Review, LVII, [1g48] 207-230) I used "reference." Subsequent criticism has persuaded me that "denotation" is, on the whole, a better choice.

T H E PHILOSOPHICAL R E V I E W Formale Arithmetik

Formal arithmetic (arithmetic in Heine and Thomae's interpretation) Figure5 (the physical vehicle of a sign, considered Figur as having no meaning) Gebild Structure6 Proposition Gedanke Quantitative ratio Grossenverhiltniss Inhaltliche Arithmetik Meaningful arithmetice (Arithmetic as customarily interpreted) Calculating game (what arithmetic becomes if its Rechenspiel signs are treated as mere figures) Arithmetical operation (addition, subtraction, etc.) Rechnungsart Sinn Sense Vorstellung Conception Zahlenfolge Numerical sequence Numerical sign Zahlzeichen Sign Zeichen

I must thank Charles W. Marshall and Steven Orey for their help in preparing a first draft and Norman Malcolm for useful criticism. MAX BLACK Cornell University

Frege uses the same term for the pieces of a game such as chess. B A n alternative might be "intuitive arithmetic," but this mistakenly suggests some affinity with the position of intuitionists like Brouwer and his followers. 78