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Why Empiricism Won't Work1 James Robert Brown University of Toronto Thought experiments present an enormous challenge to empiricism. So much so that some of us have (cheerfully) thrown in the towel and embraced good old fashioned platonism. I'll try to explain why, or at least, why one brand of empiricism won't work. Thought experiments provide us with scientific understanding and theoretical advances which are sometimes quite significant, yet they do this without new empirical input, and possibly without any empirical input at all. How is this possible? The challenge to empiricism is to give an account which is compatible with the traditional empiricist principle that all knowledge is based on sensory experience. Ernst Mach (1960, 1976) thought we have "instinctive knowledge" derived from e x t e n s i v e b u t not yet articulated-experience, and perhaps even innate. And this instinctive knowledge is conjured up when we imagine ourselves in some thought experimental situation. For example, in Stevin's wonderful inclined plane thought experiment, we are asked to consider what would happen to a chain draped over a prism-like pair of inclined frictionless planes. (Fig. l(a)) Would it slide to the left? to the right? or remain static? When the links are joined (Fig. l(b)) we see immediately what the answer must be: it will remain static. It is our instinctive knowledge that there couldn't be a perpetual motion machine that provides the crucial empiricist ingredient in coming to the right conclusion.
PSA 1992, Volume 2, pp. 271-279 Copyright O 1993 by the Philosophy of Science Association
Mach provides one empiricist approach to thought experiments, Roy Sorensen (1992) provides another. His view (which he calls naturalist and which is in some ways Machian) is that there is a difference of degree, but not of kind, between thought experiments and real ones. Knowledge derived from either is based on experience. "...thought experiment is experiment (albiet a limiting case of it), so that the lessons learned about experimentation carry over to thought experiment, and vice versa." (Sorensen 1992, p. 3) Mach and Sorensen are not the only empiricists looking at the issue. John Norton (1991) provides a different account, one which seems quite popular. The idea is simple. A thought experiment, says Norton, is an argument (deductive or inductive) with empirical premisses (which are sometimes suppressed), and with a lot of strictly irrelevant but picturesque detail (which gives it the experimental flavour). In Norton's own words, Thought experiments are arguments which (i) posit hypothetical or counterfactual states of affairs and (ii) invoke particulars irrelevant to the generality of the conclusion. (1992, p. 129) He provides an empiricist gloss: Thought experiments in physics provide or purport to provide us information about the physical world. Since they are thought experiments rather than physical experiments, this information does not come from the reporting of new empirical data. Thus there is only one non-controversial source from which this information can come: it is elicited from information we already have by an identifiable argument, although that argument might not be laid out in detail in the statement of the thought experiment. The alternative to this view is to suppose that thought experiments provide some new and even mysterious route to knowledge of the physical world. (ibid.) An illustration seems called for. It will not only help to us to understand Norton's account, but will show its elegance, its naturalness and its obvious appeal. Here's a very simple example, Einstein's elevator, which shows that light bends in a gravitational field.
1. An observer in an elevator cannot distinguish between being accelerated and being in a uniform gravitational field. (Principle of Equivalence) 2. If a light beam were to enter one side of the elevator and the elevator were accelerating, then the light beam would appear to bend as the elevator moved up. :. In a gravitational field the light beam would also bend.
This argument gives us the conclusion that light bends in a gravitational field. And it seems to fit the Norton pattern, a deduction from empirical premisses. There are different versions of the elevator thought experiment. Norton reconstructs the version that establishes the Principle of Equivalence, itself. It's worth looking at since the argument in his example is more complex than in the very brief illustration that I just gave; it even includes a philosophical premiss. Norton sees the argument as follows (1992, pp. 136-138): 1. An observer in an elevator cannot empirically distinguish between being
accelerated and being in a uniform gravitational field.
2. This situation is typical; the details of the observer in the elevator are not relevant. 3. Verification Principle: States of affairs which are not observationally distinct should not be distinguished by the theory. :. Being uniformly accelerated and being at rest in a uniform gravitational field should not be theoretically distinguished. :. Principle of Equivalence: Being uniformly accelerated is identical to being at rest in a gravitational field. Norton calls (2) an "inductive step". I'm not sure that the kind of pattern recognition that is involved here can reasonably be so characterized, but I'll let this point pass for now. We should also note that premiss (3) is certainly not an empirical premiss, even though many empiricists are happy to embrace it. However, we do anive at the conclusion (the Principle of Equivalence) via an argument from already accepted premisses, and that's Norton's main point. Maxwell's demon is a classic thought experiment that has a quite different structure than the others discussed here. It's an example of a thought experiment playing a mediating or illustrative role. In the 19th century James Clarke Maxwell was urging the molecular-kinetic theory of heat (Maxwell 1871). A gas is a collection of molecules in rapid random motion and the underlying laws which govern it, said Maxwell, are Newton's. Temperature is just the average kinetic energy of the molecules; pressure is due to the molecules hitting the walls of the container; etc. Since the number of particles in any gas is enormously large, the treatment must be statistical, and here lay Maxwell's difficulty. One of the requirements for a successful statistical theory of heat is the derivation of the second law of thermodynamics which says: in any change of state entropy must remain the same or increase; it cannot decrease. Equivalently, heat cannot pass from a cold to a hot body. But the best any statistical law of entropy can do is make the decrease of entropy very improbable. Thus, on Maxwell's theory there is some chance (though very small) that heat would flow from a cold body to a hot body when brought into contact, something which has never been experienced and which is absolutely forbidden by classical thermodynamics. The demon thought experiment was Maxwell's attempt to make the possible decrease of entropy in his theory not seem so obviously absurd. We are to imagine two gases, one hot and the other cold, in separate chambers, brought together; there is a little door between the two containers and a little intelligent being who controls the door. Even though the average molecule in the hot gas is faster that the average in the cold, there is a distribution of molecules at various speeds in each chamber. The demon lets fast molecules from the cold gas into the hot chamber and slow molecules from the hot gas into the cold chamber. The consequence of this is to increase the average speed of the molecules in the hot chamber and to decrease the average speed in the cold one. Of course, this just
means making the hot gas hotter and the cold gas colder, violating the second law of classical thermodynamics.
The point of the whole exercise is to show that what was unthinkable is not so unthinkable after all. It is, we see on reflection, not an objection to Maxwell's version of the second law that it is statistical and allows the possibility of a decrease in entropy. Maxwell's demon helps to make some of the conclusions of the theory more plausible; it removes a barrier to its acceptance. In one sense, Norton's account fits this example perfectly. We start with the statistical theory and we derive the probabilistic version of the second law; so we have a deductive argument. And the demon, as Norton says, is a "particular [which is] irrelevant to the generality of the conclusion." In fact, the demon is utterly unnecessary; we can derive the conclusion without invoking it at all. However, such an analysis misses the aim of the thought experiment. The point of Maxwell's demon is not to prove a conclusion hitherto unestablished, but instead to provide us with that elusive thing, insight and understanding. After the demon thought experiment we see how something is possible. The mechanism of sorting fast and slow molecules is not a physical possibility, since obviously there are no demons, and given the makeup of the world, there couldn't be. But we now have a grasp of the physical situation which we lacked before. Norton is right in one sense in this example when he claims that the picturesque details play no role in the argument: the demon is irrelevant to the derivation of the conclusion. But the demon is not irrelevant to the understanding of that conclusion, and that's the thing Norton's account misses. I realize that to talk of "understanding" is to wade into murky waters and to court mystery mongering. There is a form of explanation in science which tries to answer the "how possible" question. Maxwell's demon is an example. In general, narrative explanations in history and in biology are paradigmatic of this form. The explanations tell a story in which the events to be explained make sense. We see how things could come about-not how they actually do or did come about. For instance, on the one hand we have the theory of evolution and on the other we have the phenomenon of giraffes with very long necks. How did this come about. The Darwinian tries to answer
the "how is it possible?" question with a story about droughts, leaves on tree tops, early species of giraffe with relatively shorter necks, and those individuals among them with longer necks surviving at a higher rate to reproduce while the shorternecked ones perished. The evolutionist isn't committed to this at all. The point of the story is only to show that an evolutionary route does exist that produces long-necked giraffes. Similarly, Maxwell's demon thought experiment is a narrative explanation which answers the "how is the decrease of entropy possible?" question by telling us a story in which it makes clear sense. The truth of the how possible story-either think Darwin's or Maxwell's-is quite irrelevant for the purposes of understanding. (I truth explains the "success of science" in a similar way. See my 1993.) Narrative explanations are common in the social sciences and biology. They are rare is physics; thought experiments like Maxwell's may be their only instances. Still, they do play an important role, one which cannot be analyzed as an argument in the Norton mould. In his approach to thought experiments, Norton has a two-fold advantage. Empiricism has a long and successful history of explaining scientific activity. So it has the upper hand in the plausibility ratings when we turn our attention to a new field like thought experiments. And second, Norton says that thought experiments are often disguised, not explicit arguments. So the real claim is that they can be reconstructed along his empiricist lines. Existence claims like this are devilishly difficult to defeat. I doibt that an actual refutation could ever be delivered. The most I can hope to do is make the possibility of a reconstruction look implausible. That's what I'll now try to do starting with a look at an apparently tangential matter. Though we are concerned with thought experiments in the natural sciences, much can be learned from some remarkable examples in mathematics where pictures or diagrams play a role. The common view of diagrams in mathematics is this: they provide a heuristic aid, a help to the imagination when following a proof. And they are thought of as no more than this. In particular, diagrams cannot justify; they are not to be confused with real proofs, which are formulated in words and symbols. At most illustrations play a psychological role, and should never be used for making inferences. The standard account seems right for examples such as the Pythagorean theorem. In Euclid's Elements the diagram which accompanies this theorem and its proof is merely an aid, psychologically helpful, but not necessary for the justification of the theorem. However, there are a few rare and remarkable examples where something quite different is going on. The following theorem is from number theory; it has a standard proof (by mathematical induction) which uses no diagrams at all. But it can actually be proven with a diagram. (Take a moment to study the proof, to see how it works.) Theorem: Proof:
Of course, there is lots of interpreting going on to make this a proof. For example, we must consider the individual unit squares as numbers and we must bring some background information from geometry to the effect that a square with sides of length n has area n2. But these sorts of interpretive assumptions are no less innocuous than those made in a typical verbd/symbolic proof. Here's another example, this time a result about infinite series. 1
2 22 23
The usual proof of this theorem relies on standard E-6 techniques. But nothing like that is used here. Instead, we can simply see a pattern; we can see that the inner boxes are getting smaller, that they will eventually exhaust the unit square (without remainder), and so we see that their sum is equal to the whole square, hence equal to 1. The moral I think we should draw from examples like these is simple but profound: we can sometimesprove things with pictures. In spite of the fact that the number theory diagram seems to be a special case (n = 5), still we can see all generality in it. And the proof does not work by merely suggesting the "real" proof, since in the diagram there is nothing which corresponds to the passage from n to n+l which is the key step in any proof by mathematical induction. Those who hesitate to accept these pictures as genuine proofs might think that the diagrams merely indicate the existence of a "real" proof, a proof by mathematical induction or by using &-6 techniques, respectively. Perhaps they even wish to appeal to the well known distinction between discovery and justification: the picture is part of the discovery process while true justification comes only with the verbal/symbolic proof. But consider: would a picture of an equilateral triangle make us think there is a proof that all triangles are equilateral? No. Yet the above picture makes us believe-rationally believe-that the theorem is true and even that there is a verbal/symbolic proof of the theorem to be found, if we were to hunt for it. The picture is evidence for the existence of a "real" proof (if we like to talk that way), and the "real" proof is evidence for the theorem. But we have transitivity here; so the picture is evidence for the theorem, after all. This tangent into the mathematical realm has a point. Let's see how well Norton's empiricism does with such examples. Pretty clearly, these two examples fly in the face of Norton's account. A standard, traditional proof in mathematics is an argument; it's a derivation (though often quite sketchy) of the theorem from first principles. Proofs by induction or by E-6 techniques fit the bill. However, the two proofs by diagram that I've given here are not like this at all; there is not the hint of an argument about them. Instead, I suggest, we accept the theorem because we grasp an abstract pattern. This is a kind of intellectual "perception." I favour a platonistic ac-
count of what's going on similar to Godel's view (1944, 1947) (see my 1991), but I won't push this now; it is enough to know that the Norton proposal doesn't work in these mathematical cases. I would even go a step further and say that Norton's view of thought experiments is the analogue of the standard view of diagrams in mathematics. In both cases the picture or the visualized situation is allowed to be psychologically helpful, but plays not role in the real justification of the mathematical theorem or the scientific theory; justification, according to him, can only come via an argument from premisses which have already been established. Of course, Norton is interested in what goes on in science, not mathematics, so there are no morals to be drawn yet. But an interesting question does obviously arise: Do any thought experiments resemble the mathematical cases? As you might imagine, I am now going to argue: Yes. Consider, first, my favourite example. This is Galileo's wonderful thought experiment in the Discorsi to show that all bodies, regardless of their weight, fall at the same speed (Galileo, 1974, pp. 660. It begins by noting Aristotle's view that heavier bodies fall faster than light ones (H > L). But what would happen if a heavy cannon ball is attached to a light musket ball?
Reasoning in the Aristotelian manner leads to an absurd conclusion. On the one hand, the light ball will slow up the heavy one (acting as a kind of drag), so the speed of the combined system would be slower than the speed of the heavy ball falling alone (H > H+L). One the other hand, the combined system is heavier than the heavy ball alone, so it should fall faster (H+L > H). We now have the absurd consequence that the heavy ball is both faster and slower than the even heavier combined system. Thus, the Aristotelian theory of falling bodies is destroyed. But we are far from finished. We still want to know: which falls faster? The right answer is now obvious. The paradox is resolved by making them equal; they all fall at the same speed (:H = L = H+L). If we try to analyze this example from Norton's perspective we will run into trouble. The fust part of the thought experiments fits his account well. We have an argument, starting from Aristotelian prernisses, and ending in a contradiction; so it's a reductio ad absudum of those premisses. This much is certainly acceptable to any empiricist. It's
the second part of the though experiment that's problematic for Norton. When we see that H = L = H+L, we grasp it immediately. It is not based upon empirical experience; in fact all our actual observations are to the contrary: heavy objects generally do fall faster than light ones. And it does not follow from other premisses. We are intellectually primed by the discovered absurdity in Aristotle's theory. But Galileo's conclusion does not follow from that absurdity (except in the trivial sense that anything follows from a contradiction-but, clearly, that's not what is going on here). In sum, I've tried to make a case against Norton's empiricism in two ways. First, there are thought experiments such as Maxwell's that try to answer "how possible?" questions and lead to scientific understanding; as explanations they are not arguments, but narrations. Second, there are thought experiments such as Galileo's which result in something like an immediate perception; they, too, are not arguments, but instead are vehicles to directing our attention so that we can simply see for ourselves-and I do mean see.
Note l1 wish to thank my co-symposiasts, David Gooding and Nancy Nersessian, and also several of my Toronto colleagues, especially Ian Hacking, my commentator, for their remarks on an earlier draft. I am also glad to acknowledge the financial help of S.S.H.R.C.
References Brown, J.R. (1991), The Laboratory of the Mind: Thought Experiments in the Natural Sciences, New York and London: Routledge. - - - - - - . (1993), Smoke and Mirrors: How Science Reflects Reality, New York and
Godel, K. (1944), "Russell's Mathematical Logic", reprinted in Benacerraf and Putnam (eds.). Philosophy of Mathematics, Cambridge: Cambridge University Press, 1983. Godel, K. (1947), "What is Cantor's Continuum Problem?", in Benacerraf and Putnarn (eds.). Philosophy of Mathematics, Cambridge: Cambridge University Press, 1983. Galileo, (1974), Two New Sciences, (Trans. from the Discorsi by S. Drake). Madison: University of Wisconsin Press. Kuhn, T.S. (1964), "A Function for Thought Experiments", reprinted in The Essential Tension, Chicago: University of Chicago Press, 1977. Leff, H. and A. Rex (eds.) (1990), Maxwell's Demon, Princeton: Princeton University Press.
Mach, E. (1960), The Science of Mechanics, (Trans by J. McCormack), sixth edition. LaSalle Illinois: Open Court.
- - - - - . (1976), "On Thought Experiments", in Knowledge and Error, Dordrecht: Reidel.
Maxwell, J.C. (1871), Theory of Heat, London: Longmans. Norton, J. (1991), "Thought Experiments in Einstein's Work", T. Horowitz and G. Massey (eds.). Thought Experiments in Science and Philosophy, Savage, MD: Rowman and Littlefield. Sorensen, R. (1992), Thought Experiments, Oxford: Oxford University Press.