## Where Does Math Come From?

#### The Nature of Mathematical Knowledge

Philip Kitcher thinks that mathematics is surprisingly like empirical science. Few mathematicians would agree; philosophers too, from Socrates on, have held the opposite opinion. In mathematics, they have said, we are able to solve problems and construct proofs by pure thought, without any need to check out how the land lies. Yet we can use geometry for surveying; hence, somewhat mysteriously, the products of reasoning apply to the world. Philosophers have also said that anything that you prove in pure mathematics *must* be true. To put it metaphorically, not even God could create a world in which a theorem that we have demonstrated is false. Men such as Plato, Aquinas, Leibniz, Kant, Russell, and Wittgenstein have said such things. Almost the only famous dissenters from this established tradition were (until recently) Descartes and John Stuart Mill. Kitcher is on their side.

These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling by using a proof in geometry to argue for the transmigration of souls. As reported by Plato in *Meno*, the boy who invents a proof of a theorem did not experiment on the physical world, but used only his mind in response to Socratic questions. Hence he must have had inborn knowledge of the proof and he must have got this knowledge in a previous incarnation.

Mathematics has never since been a subject for such philosophical levity. Russell taught that no philosophy is worth much unless it could first answer Kant’s question, “How is pure mathematics possible?” The liberal Mill suspected that the official view of mathematics was a Tory plot. As he wrote in his *Autobiography*, the notion that we can have knowledge

independently of observation and experience, is, I am persuaded, in these times, the great intellectual support of false doctrines and bad institutions. By the aid of this theory, every inveterate belief and every intense feeling, of which the origin is not remembered, is enabled to dispense with the obligation of justifying itself by reason, and is erected into its own all-sufficient voucher and justification. There never was such an instrument devised for consecrating all deep-seated prejudices. And the chief strength of this false philosophy in morals, politics and religion, lies in the appeal which it is accustomed to make to the evidence of mathematics….

Mill urged that the truths of mathematics are very general empirical facts about the world. We think that they are necessarily true, less because they are persuasive than because they are so pervasive.

Kitcher lacks Mill’s political passion but he knows more mathematics. Like Mill, he tries to deflate rationalist pretensions. How is pure mathematics possible? asks Kant. In just the same way as any other form of knowledge, Kitcher replies. Yet even he can suffer from the ancient philosophical fetishism that makes mathematics extraordinary. On the first page of his introduction he says that “for the ordinary person, as for the philosopher, mathematics is a shining example of human knowledge, a subject which can be used as a standard against which claims to knowledge in other areas can be measured.” What ordinary people has he in mind? For most of us mathematics is a shining example of esoteric expertise, and the idea of measuring claims to knowledge against it is nonsensical.

Once he gets into his book, however, Kitcher does some useful debunking. From Socrates on, a certain kind of mind (my own included) has been obsessed by the experience of inventing or grasping a proof, of suddenly seeing that a particular conclusion is inevitable. We then get the feeling that mathematical knowledge is built up by compelling insights, and that it concerns a reality more stable and profound than the ephemeral round of daily life. In the jargon of philosophers, mathematical knowledge is “a priori”—prior to any particular experience—and what we prove is “necessarily true.” Kitcher begins his skeptical counterattack by observing that we are taught much of our mathematical knowledge by rote, whether it be simple sums or differential equations. We learn it at school, or from books. We are sure of it because teacher said so, or because the arguments in the book seem right. In this respect it differs little from other branches of learning.

Secondly, the experience of having proved something may be convincing, but it is often deceptive. The school-room and the frontier of research are alike in this: a child or a creative genius may equally think that a chain of argument is a conclusive proof and later find out that it has a flaw. No final flash of intuition settles things. In the case of an important discovery one person may hit upon a brilliant new proof-idea, but commonly a collective effort is needed to fill in the gaps and to check that there is not some hidden and ineradicable defect along the way. Referees for mathematics journals may bear a heavier burden than those who work in other disciplines, because imperceptible errors often elude an honest author and his colleagues who did the previous checking. The philosopher propounding a theory of knowledge on the basis of mathematical intuition and insight has been forgetting the dreary facts of mathematical life. Concentrating on one phenomenon associated with the experience of understanding a proof distorts our picture of the whole. Kitcher might have added that the strange, unnerving feeling of the key turning in the lock is not peculiar to suddenly solving a problem in mathematics. It is familiar to those who do difficult crossword puzzles or read thrillers, not to mention all sorts of painstaking research in the natural sciences.

Kitcher also introduces the historical dimension. Standards of rigor for proofs have not been constant. What one generation regarded as proven conclusively is revised or at least re-proved by sounder arguments a century later. Kitcher illustrates this throughout his book by examples drawn from the history of analysis, that central branch of mathematical thinking built upon the differential and integral calculus.

In turning to history Kitcher follows the lead set two decades ago by T.S. Kuhn’s famous studies of the physical sciences. Kuhn urged that we cannot understand a science if we abstract from its history. We should attend less to the formulated knowledge than to its production. Kitcher says that “to understand the epistemological order of mathematics one must understand the historical order.” I do not well understand “epistemological order” and I suspect it is a highly loaded question-begging concept. But at least Kitcher is telling us to look at the past. He is hardly daring when he writes, “I suggest that the knowledge of one generation of mathematicians is obtained by extending the knowledge of the previous generation,” but I think he holds in addition that the most perceptive organization of the chain of inferences that constitute mathematical knowledge will follow the sequence of mathematical discoveries in time.

This proposition is extremely improbable. Many central branches of mathematics have been transformed beyond all recognition. For example, some contributions to set theory were made by people who ignored the problems of nineteenth-century analysis that brought it into being. It is true that some students learn better by being told the historical background, but it is equally true that others prosper best in an abstract and utterly nonhistorical milieu. Recalling historical problems can redirect important lines of research, but such truisms fall far short of Kitcher’s sweeping assertion about the “epistemological order.”

Only when we turn to Chapter 6, on “mathematical reality,” do we see Kitcher’s central point. A longstanding idea holds that mathematics is about a domain of eternal objects: numbers, shapes, operations, transformations, and the like. According to this idea, mathematical truths are about relations between these abstract objects. We are able to know these truths because the objects and relations are grasped by the human mind. Kitcher’s retort to this romantic picture is a sophisticated and slightly historicized version of J.S. Mill’s down-to-earth empiricism.

Kitcher first asserts that as a matter of brute fact some structures do exist in the world. “Structure” is by now a pretty useless vogue-word, but he is saying for example that there really are innumerable pairs and triplets of distinct things (such as oranges) that can be put together to form collections of five objects. Such facts are at the empirical core of arithmetical beliefs. Yet not all twos and threes can be assembled, in any literal sense, to form fivesomes. Oranges in an orchard, yes, but stars, no, not even, except at great expense, two oranges in a Moroccan souk with three in a Californian supermarket. Droplets of mercury tend to fuse while rabbits multiply.

So we begin to refine the concept of arithmetical addition in order to make the number concepts hang together in a more rational way. We invent the timeless “if … then” statement that *if* one set has two distinct members and another has three distinct members, and the sets have no members in common, *then* the set consisting of exactly the members of both those sets has five members. Although this statement is modeled on the practice of physically putting oranges together, it no longer has any direct connection with physical assemblings, and is as true of stars, rabbits, and droplets as it is of nearby oranges. Obviously Kitcher can provide no history of the shift from literal putting together to “idealized” abstract addition, for that happened too long ago. His history of the calculus is intended as a real-life example of a similar process. So he takes us on a tour of developments from seventeenth-century Leibniz and Newton to the end of the nineteenth century.

He recalls that many of the problems that arose for the original analysts could not be well answered in the terms in which they were posed. Only analogies and half-reasons were to hand. Concepts had to be made precise, manageable, and abstract by various kinds of stipulation, never arbitrary, but always motivated by a hunch about what the answers “ought” to be. Kitcher holds that such developments were not a matter of merely clarifying the conceptual geography of a pre-existing mathematical continent. They actually created the geography. Yet that creation was no free-floating carving up of icebergs. The development of analysis is intimately tied to “structures” in the real world. Newton was solving the problem of planetary motion and of the free fall of bodies. Solutions that are now taught as elegant facts about the summation or transformation of series were first invented for the analysis of heat. Facts about electromagnetism forced us to invent some mathematics, and the structures that result from the mathematics are, Kitcher argues, idealizations of the structures that are to be found in our experienced world.

Many a philosopher has said that mathematical truths are the result of human conventions. Mill insisted instead that the same truths are just widespread empirical facts. Kitcher balances both claims. On the one hand empirical “structures” are liberally splashed around the world: these are organized facts about combinations, shapes, or waves. Our mathematical structures are abstractions of and idealizations upon these. In the course of historical discovery, we gradually tighten up the conceptual connections, to the extent that those connections can be reasoned about and can produce the feeling of a firm “necessity.” The necessity is of our own doing (and there the conventionalist is correct) but the structures, or something like them, are real enough (and there the empiricist is right). The process by which the community of research mathematicians transforms empirical to abstract structures is one of “question-answering, question-generation, … rigorization, and systematization.” These somewhat flabby phrases are best explained by example, an obligation Kitcher tries to honor in his résumé of mathematical analysis from the seventeenth century through the nineteenth.

Kitcher finds it easy to explain the fact that pure mathematics is time and again so readily applicable to physical objects. We have invented abstract mathematical structures precisely because we have discerned features of the world upon which we model them. Mathematics has unexpected applications because of the brute fact that structures that occur in one field of experience crop up in others. Max Born could tell his student Werner Heisenberg, “This clumsy stuff you’re groping for is really just matrix algebra: come borrow this textbook.” Presto, quantum mechanics was transformed by the insight that an abstract structure formed by idealization for one part of physics also models a structure found elsewhere in the world. There may be ground for awe at this fact, or admiration for the Author of Nature, but the widespread applicability of pure mathematics should not be a source of philosophical perplexity.

There is much good sense in Kitcher’s homespun attitudes. Nineteenth-century crystallography, for example, began as a study of real structures, but contains many ideas that provide rich idealizations for today’s topologist, the student of the most abstract and general spatial transformations. A few years ago, at the height of the “new math” madness in American elementary schools, the American Mathematical Society arranged translations of Russian textbooks that started their pupils with a firmer sense of reality—crystals, for example. Indeed Kitcher has written something similar to standard Soviet (materialist) philosophy of mathematics, supplemented by more enthusiasm for set theory than Marxists would endorse.

We learn from the preface that Kitcher had a solid English sixth-form and undergraduate training in calculus and classical analysis, and that he was educated at Princeton, the soundest of American graduate schools of philosophical logic. It is good that he brings the knowledge learned at the former to illustrate the philosophy he acquired at the latter. He enriches a literature that has recently been rather scholastic. But the reader will seek in vain for an understanding of the nagging worries that have vexed philosophers from the time of Socrates on.

There is, for example, what Wittgenstein called “the hardness of the logical must.” When a new theorem is proven by traditional means from well-understood axioms, we have the feeling that the conclusion must inexorably be true if the premises are true. I am not now thinking of imaginative proof techniques which may be said to extend a discipline, and to introduce new standards or possibilities of proof. I am thinking of the case in which a longstanding problem is solved by the ingenious use of traditional techniques. Is the source of this necessity mysteriously hidden in the axioms of the discipline? Kitcher’s account can make us understand how a loose body of mathematical intuitions about certain structures can be gradually tightened into a rigorous structure of necessary conceptual connections; but how does this apply to the yet uncharted parts of the field?

The axioms in question may model some empirical structure, in the way that Euclidean postulates roughly model properties of human-sized spatial relations. It may be that the unproved theorem is also *empirically* true of the same structure (pretend it is Pythagoras’s theorem, which is pretty well true of the space immediately around us). We can understand why the theorem is true as a matter of fact. The question is, why “must” it be true when the premises are true? It is no use invoking a metamathematics whose conventions explain the necessity with which a theorem follows from the axiom, for then we reconstruct the same question at a level of metamathematics. Wittgenstein managed to raise very similar questions about immediate intuitive examples, without any of the paraphernalia of axioms and so forth. To say that there are these residual worries is not to deny the interest of Kitcher’s approach, but to say that there are regions of the philosophy of mathematics that he leaves quite untouched.

His book is also sometimes insensitive to questions that have exercised some of the more powerful and fascinating mathematico-philosophical minds of our century. I shall give two seemingly very distinct examples. Gödel was certain that there are definite answers to some fundamental questions that cannot be answered on the basis of any premises or axioms now available to us. There is, he urged, a truth of the matter that we shall discover only as we evolve deeper concepts and methods. (Gödel put this view in connection with Cantor’s continuum problem, “How many real numbers are there?”) It is fine to disagree with Gödel as Kitcher does, but the reader will wish that he revealed more sense of the power and source of Gödel’s convictions.

Indeed one can attempt a history different in kind from Kitcher’s, which may encourage those who share Gödel’s attitude. One may extract from the history of analysis—or of ancient geometry—the successive discovery of techniques of proof, and conclude that this history shows the gradual human mastery of these techniques. This will suggest again that we have gradually mastered a continent of mathematics, constituted not by the mathematical “objects” that Kitcher distrusts, but by modes of proof that have an identity largely independent of empirical structures.

Such a notion leads to what is so often presented as an attitude to mathematics opposed to Gödel’s: the “constructivist” approach for which Kitcher also has no sympathy. At the beginning of the century the Dutch mathematician L.E.J. Brouwer revamped Kant’s idea that the theory of numbers has its roots in the experience of the succession of instants of time. Brouwer believed that all mathematics had to proceed by the intellectual construction of objects in a sequence of steps. We should never assume even that a proposition is true or false unless we know how in principle to construct either a proof or a refutation of the proposition. Recent philosophers have made this question into a matter of semantics, and Kitcher assumes that that arid direction is the correct one. But Brouwer was making a claim about the very substance of mathematics that is radically different from Kitcher’s empiricist suppositions. Kitcher’s “structures” are essentially spatial structures. His book suggests that without those structures (from combining oranges on up) we would and perhaps could have no abstract mathematics. The Brouwer-Kant line about mental constructions in time makes exactly the opposite assumption.

Lacunas of this sort mean that readers who have intense philosophical convictions, ancient or modern or both, will find Kitcher’s book somewhat superficial. Kitcher’s overall hope, perhaps, is that he has quietly undermined those very convictions. Certainly the merits of the book lie in the undogmatic good sense with which it discusses a lot of moderately controversial questions. “Whatever may be the practical value of a true philosophy of these matters,” wrote Mill, “it is hardly possible to exaggerate the mischiefs of a false one.” Kitcher’s book will wreak no mischief.

Nor do I think it likely that there is such a thing as “the” true philosophy of mathematics—of the whole of the vague-boundaried domain that we call mathematics. Wittgenstein said that he wanted “to give an account of the motley of mathematics.” That could be a good guiding aphorism. Quite different kinds of things can be said about different branches, aspects, or techniques of mathematical reasoning and its applications. (Commentators on Wittgenstein’s inchoate *Remarks* commonly try to find one theory about mathematics to attribute to him, which seems exactly contrary to his own ambitions.) Anyone who shares the sense that there is no unity of mathematical activity or sensibility will welcome Kitcher’s rather somber description of one part of the many-colored cloak that we call mathematics.