---

VIII.2 “Why Mathematics?” You Might Ask

Michael Harris

---

It seems to me that they have a poor opinion of our religion if they think it needs the protection of philosophy.

Lorenzo Valla, Dialogue on Free Will

1 A Metaphysical Burden

ANDRÉ WEIL [VI.93], speaking at the 1978 International Congress of Mathematicians at Helsinki, concluded his address entitled “History of Mathematics: Why and How?” with these words:

Thus my original question “Why mathematical history?” finally reduces itself to the question “Why mathematics?,” which fortunately I do not feel called upon to answer.

Proceedings of the ICM, Helsinki, 1978 (pp. 227–36, quotation on p. 236)

I heard Weil’s address, and the applause that followed, and remember imagining circumstances in which that final question could not be so easily evaded. For instance, in 1991 the House Committee on Science, Space, and Technology called upon the American Mathematical Society (AMS) to answer a very similar question: “What are the main goals in the mathematical sciences?” Weil knew his audience, and the committee of twelve mathematicians responding to the government body responsible for research budgets knew theirs:

The most important long-term goals for the mathematical sciences are: provision of fundamental tools for science and technology, improvement of mathematics education, discovery of new mathematics, facilitation of technology transfer, and support of efficient computation.¹

“Meaning is what makes things sell,” wrote Roland Barthes (1967), and the AMS adopted the posture of FOURIER [VI.25], who, according to a celebrated comment of JACOBI [VI.35], included in a letter to LEGENDRE [VI.24] of July 2, 1830,

…had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind.

It might seem that the AMS has left a place for “honor” in its third goal, but a later elaboration of that goal directs the reader toward “unexpected” applications of pure mathematics.

Few pure mathematicians are as indifferent to practical applications as HARDY [VI.73], who in A Mathematician’s Apology famously claimed that: “Judged by all practical standards, the value of my mathematical life is nil.” But it is fair to assume that, when they are addressing one another rather than government committees, most pure mathematicians (including those who represented the AMS in 1991) would choose a quite different list of “most important long-term goals.”

In this they have long been able to count on the protection of philosophy. It has been a commonplace since Plato to grant mathematics intrinsic value on metaphysical grounds.² The topos of mathematics as a source of certain knowledge was already well established by the second century, when Ptolemy wrote

Only mathematics, if one attacks it critically, provides for those who practice it sure and unswerving knowledge, since the demonstration comes about through incontrovertible means, by arithmetic and geometry.³

THE CRISIS IN THE FOUNDATIONS OF MATHEMATICS [II.7] of the early twentieth century, which culminated in GÖDEL’S INCOMPLETENESS THEOREMS [V.15], was largely motivated by a desire to make mathematical certainty safe from dependence on human frailty. As RUSSELL [VI.71] wrote in Reflections on my Eightieth Birthday:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. . . . Mathematics is, I believe, the chief source of the belief in eternal and exact truth.

Quoted in Hersh (1997)

Russell’s hope to ground certainty in logic is largely a thing of the past—as Marvin Minsky wrote in another context, “without an intimate connection between our knowledge and our intentions, logic leads to madness, not intelligence” (Minsky 1985/1986) ⁴—but his words continue to echo. After Jean-Pierre Serre was named first recipient of the Abel Prize, he was quoted in Libération (May 23, 2003) to the effect that mathematics is the only producer of “totally reliable and verifiable” truths. And Landon T. Clay III, announcing the creation of the $ 7 000 000 Millennium Prize Fund, linked his decision to devote much of his personal fortune to the support of pure mathematics to “the decline in religious certitude … the pursuit of proof continues to be a strong motivating force in human actions.” ⁵

The mind saves its honor, as Jacobi would have it, but only through indenture to a higher power. I would like to express my opinion that the bargain implicit in comments like those just quoted, placing pure mathematicians on the front lines in defense of metaphysical certainty or some other normative concern of philosophers, is an unnecessary burden that fails to do justice to what is uniquely valuable about mathematics. It also fails to protect pure mathematics from the real existential dangers it faces, of which budget cuts are only the most obvious expression. Mathematics is not likely to collapse for lack of a coherent account of its certainty, but it may well collapse for lack of an account of its value.

2 Postmodernism versus Mathematics?

One danger that should not worry mathematicians is that of postmodernism. Many thousands of pages have been written on this topic, although it is not clear that the word designates anything specific. I will nevertheless add a few pages of my own, because the term has come to be used as shorthand for a radical relativism that is thought to call into question not only certainty but rationality in all its forms.⁶ One thus finds mathematicians who are skeptical of certainty in Russell’s sense but who nonetheless express hostility to something they call “postmodernism” as they try to defend reason and the value of mathematics as a rational activity.

Applied to architecture, postmodernism designates a reasonably precise tendency. As a trend defining the spirit of the times, it has been called “the cultural logic of late capitalism,” differing from modernism by emphasizing space rather than time, multiple perspectives and fragmentation rather than unity of meaning and totality, pastiche (sampling)⁷ rather than progress, and much more along the same lines. As a movement in philosophy it is most typically (if abusively) associated with Michel Foucault, Jacques Derrida, Gilles Deleuze, Roland Barthes, Jean-François Lyotard, and more generally the “French theory” of the 1960s and 1970s. Postmodern prose is eclectic, ironic, self-referential, and hostile to linear narrative. The variant known as posthumanism celebrates the fading of conceptual and material boundaries between human beings and machines.

We are all postmodernists insofar as we have experienced the degradation of public discourse under the influence of advertising slogans, and are therefore likely, in spite of ourselves, to read Jacobi’s invocation of “the honor of the human mind” as a precursor of that genre. Mathematicians can even claim to be the first postmodernists: compare an art critic’s definition of postmodernism—“meaning is suspended in favor of a game involving free-floating signs”—with the definition of mathematics, attributed to HILBERT [VI.63], as “a game played according to certain simple rules with meaningless marks on paper.” ⁸ Mathematics could nevertheless (or for that very reason) safely ignore postmodernism, were it not that the latter is supposed to have no room for certainty, metaphysical or otherwise.⁹ So it is not surprising that authors who are considered postmodernists have had some perplexing run-ins with science and mathematics.

The typically controversial postmodernist account of science sounds like this:

Science and philosophy must jettison their grandiose metaphysical claims and view themselves more modestly as just another set of narratives.

Terry Eagleton’s caricature of postmodernism, quoted in Harvey (1989)

As far as mathematics is concerned, relativism of this kind has more to do with English-language postmodernism than with the French original. One might have thought that mathematical progress from axioms to theorems and from lesser to greater abstraction or generality constituted a prime example of the sort of “master narrative” that French postmodernists regarded with suspicion, and a particularly tempting target given the special role Enlightenment thinking reserved for mathematical explanation; but that seems not to have been the case. Although the most prominent French philosophers associated with postmodernism were metaphysical skeptics in other regards, they had no quarrel with mathematics’ metaphysical pretensions per se; but they did question their relevance to the human sciences. For Derrida, thinking of LEIBNIZ [VI.15] in particular, “[mathematics] was always the exemplary model of scientificity” (in Of Grammatology, p. 27), and Foucault claimed that:

Mathematics has certainly served as a model for most scientific discourse[s] in their efforts to attain formal rigor and demonstrativity; but for the historian who questions the actual development of the sciences, it is … an example … from which one cannot generalize.¹⁰

The Archeology of Knowledge (pp. 188-89)

At least one of postmodernism’s canonical French texts does take on the issue of certainty in science and mathematics directly. Alluding to the trilogy of Gödel’s theorems, uncertainty in quantum mechanics, and fractals,¹¹ Lyotard saw in contemporary mathematics

a current that calls into question precise measurement and prediction of the behavior of objects at the human scale … postmodern science … produces not the known, but the unknown.

Lyotard (1979)

Various authors have reminded readers that Gödel’s theorems and the uncertainty principle (and chaos) are statements about formal systems in mathematics and particle physics (and nonlinear differential equations), respectively, and as such have no bearing on metaphysics.¹² The arguments are often eloquent but altogether beside the point, and of little comfort to seekers of certainty like Russell. Metaphysical certainty, whatever it may be, cannot be any less binding than a mathematical proof. Gödel’s theorem, that it is impossible to prove, within a formal system, that that formal system is consistent, can reasonably be taken to mean that metaphysical certainty cannot be guaranteed by mathematical means alone.¹³ But Serre, in his comments to Libération, surely meant something more than the tautology that mathematical truth is totally reliable and verifiable by the standards of mathematics. The struggle to pin down this “something more” to find what one might call the “essence” of mathematics, is why the philosophy of mathematics keeps visiting the scenes of its many past defeats.

Even if Lyotard does not make the case very well, one can detect a “postmodern” sensibility in much of recent science, from Stephen Jay Gould’s insistence that evolution is highly contingent, to complexity theory, to the study of consciousness as an “emergent” phenomenon. What these developments have in common is a rejection of reductionism and related top-down “master narratives,” not because they are wrong but because they are irrelevant and useless. It would be going too far to describe this kind of science as a new Kuhnian paradigm (the notion is, in any case, widely criticized as oversimplified), but it is noticeably different from the disciplines that inspired the analytic philosophy of science. As for mathematics, there have been suggestions that it too has postmodern aspects—for example, Jürgen Jost has written a book entitled Postmodern Analysis and some specialists now claim to be working in “postmodern algebra”—but I do not see any genuine signs of this sensibility. Indeed, I am not even sure that it makes sense to draw the line between modern and postmodern. Hilbert’s definition of mathematics as a game does sound like something from Derrida, but if Hilbert’s foundational program (“wir müssen wissen, wir werden wissen”) is not a prime example of high modernism, then what is? On the other hand, the abandonment of all forms of foundationalism in an anthology of Tymoczko (1998) is a rejection of “master narratives” within philosophy of mathematics, and indeed the blurb calls the anthology “postmodern.” ¹⁴

3 Sociology Aims for the High Ground

While Weil is supposed to have discounted Gödel’s metaphysical menace by making it into a joke—“God exists since mathematics is consistent, and the Devil exists since we cannot prove it”—his fellow Bourbakist Dieudonné attempted a counterattack:

Just as physicists and biologists believe in the permanence of the laws of nature, solely because they have observed this up to now, … the mathematicians called—wrongly—“formalists” (… at present the near totality of mathematical researchers) are convinced that no contradiction will appear in set theory, none having manifested themselves for 80 years.¹⁵

This is either an inductive (empirical), sociological, or pragmatist argument. All these trends are indeed present in postmodernism, more typically in English sociology of science than in French philosophy:

The compelling force of mathematical procedures does not derive from their being transcendent, but from their being accepted and used by a group of people. The procedures are not used because they are correct, or correspond to an ideal; they are deemed correct because they are accepted.

David Bloor, in Wittgenstein: A Social Theory of Knowledge (Macmillan, London, 1983)

The Sociology of Scientific Knowledge (SSK) movement, of which David Bloor was a founder, is firmly rooted in postwar philosophy of science in the analytic tradition. The later Wittgenstein’s discussion of mathematics, and knowledge more generally, in terms of “language-games,” “forms of life,” and learning to follow rules emphasizes social factors, and SSK is enthusiastically Wittgensteinian. Of course, Wittgenstein’s work is notoriously unsystematic and lends itself to a variety of interpretations. I find it wrong to see the Wittgenstein who wrote “Grounds for doubt are lacking!” as a skeptic. Beyond the social factors to which Wittgenstein drew explicit attention, he clearly perceived “something more” specifically in mathematics (“the hardness of the logical must”), to which our language and philosophy are not able to do justice.¹⁶

Can sociology succeed where philosophy has failed? Bloor’s militant “naturalist” response to the question of “whether sociology can touch the very heart of mathematical knowledge” (Bloor 1976) is less an exercise in debunking metaphysics than an attempt to seize the metaphysical high ground for sociology. An otherwise subtle ethnographic study by Claude Rosental of the resolution of a conflict among logicians betrays a similar sensibility, as does his suggestion that training in mathematics and logic might have constituted a “serious handicap” to carrying out his project (Rosental 2003). The classic declaration of the latter kind is due to Bruno Latour and Stephen Woolgar:

[W]e do not regard prior cognition … as a necessary prerequisite for understanding scientists’ work. This is similar to an anthropologist’s refusal to bow before the knowledge of a primitive sorcerer. There are, as far as we know, no a priori reasons for supposing that scientists’ practice is more rational than that of outsiders.

Latour and Woolgar, Laboratory Life, pp. 29-30 (Princeton University Press, Princeton, NJ, 1986)

But one can also imagine sociologists paying serious attention to mathematicians’ accounts of their experience, addressing in the process the question that Weil did not. For example, Bettina Heintz, in fieldwork at the Max-Planck-Institut in Bonn, which was billed as the first study of mathematics from the perspective of constructivist sociology of science, worries about “going native” and “overidentifying with the dominant culture.” But her subject is the eminently sociological one of determining how mathematicians reach consensus, and her methodology, far from treating practicing mathematicians as “primitive sorcerers,” records their epistemic perspectives sympathetically and at length. One has the impression that, in spite of the limitations of her methodology, Heintz is more interested in accounting for “real mathematics,” to which we shall return below, whereas Bloor and Rosental are preoccupied with marshaling evidence to counter the metaphysical preoccupations of philosophers.

Under siege from Gödel’s theorem, Popper’s attack on verificationism, Kuhn’s theory of scientific revolutions, Lakatos’s dialectical approach to the contents of knowledge in Proofs and Refutations, as well as Wittgenstein, certainty in Russell’s sense has largely been scrapped.¹⁷ As for the social, philosophical, and spiritual needs that the notion of metaphysical certainty was designed to address, they remain. Thus, on the one hand, those with tendencies that I have described as postmodernist continue to express skepticism regarding certainty, seemingly unaware that their target is now little more than an advertising slogan that has little to do with the real concerns of mathematicians; while, on the other hand, analytic philosophy has sought to substitute more flexible notions. The term “warrant,” for example, is used in an attempt by Philip Kitcher to develop a consistent account of mathematics on empirical rather than aprioristic grounds. Kitcher recalls FREGE’S [VI.56] frustration with the mathematicians of his time, observing that, “When Frege emphasizes the possibility of complete clarity and certainty in mathematical knowledge, he is advancing a picture of mathematics that is almost irrelevant to the working mathematician” (Kitcher 1984). However, Kitcher and the SSK remain obsessed by the problem of “how our mathematical knowledge [is] acquired” (Kitcher 1984), where knowledge is taken to be true and justified belief.

Reading Heintz (2000), one learns that now, as in Frege’s day, mathematicians themselves widely consider these problems to be outdated or beside the point. The most controversial aspect of the SSK’s “strong programme,” formulated by Bloor and Barry Barnes, is the “thesis of symmetry”: the insistence that truth or falsity not be taken into account when investigating how a scientific claim comes to be accepted as knowledge. Heintz’s fieldwork suggests that this is compatible with the view prevailing among mathematicians regarding acceptance of a mathematical proof, a “kind of consensus theory of truth” (Heintz 2000).¹⁸

A striking instance of “how a mathematical proof comes to be accepted as knowledge” is playing out even as I am writing these lines. Grigori Perelman’s announced proof of THE POINCARÉ CONJECTURE [VI.25] is undergoing unprecedented scrutiny in a small number of specialized centers, with the hope of determining the truth or falsity of Perelman’s claim. This is going on completely beyond the spotlight of sociology, as far as I know, and with no guidance from philosophy, even though the $ 1 000 000 prize offered by the Clay Mathematical Institute is in no sense Platonic, ¹⁹ and the rules for awarding the prize presuppose the fallibility of the mathematical community, in terms very similar to those that Heintz’s informants expressed spontaneously (see the third and subsequent paragraphs at www.claymath.org/millennium/rules_etc). The case is exceptional, however; “certifying knowledge,” in Rosental’s sense, is as such relatively unimportant to mathematicians, and Perelman’s close readers would probably describe what they are doing as attempting to understand his proof rather than “certifying” it as knowledge (for the sake of the community, or a generous benefactor, or philosophers or sociologists).²⁰

4 Truth and Knowledge

“By far the larger part of activity in what goes by the name philosophy of mathematics is dead to what mathematicians think and have thought, aside from an unbalanced interest in the ‘foundational’ ideas of the 1880–1930 period, yielding too often a distorted picture of that time,” announced David Corfield, presenting his efforts to develop a “philosophy of real mathematics.” ²¹ Corfield contrasted the traditional apriorist’s concerns, “How should we talk about mathematical truth? Do mathematical terms or statements refer? If so, what are the referents and how do we have access to them?” (Corfield 2003), with the list of questions Aspray and Kitcher consider typical of the “maverick tradition” in philosophy of mathematics: “How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation?” (quoted by Corfield 2003).

The mavericks, well represented in Tymoczko’s anthology, have moved a welcome step away from certainty. Nevertheless, the philosophers and philosophically minded sociologists I have mentioned—with the partial exception of Corfield, to be explored below—still often write as though mathematicians were creating Truth or Knowledge,²² almost as a favor to philosophy or sociology, to show how such a feat is possible. Or just to show that it is possible.²³ We mathematicians, on the other hand, are quite convinced that we are creating mathematics, and it is the “why” of that activity, without the ennobling assimilation to the generic objects of interest to epistemology, that, as Weil understood, required no explanation in Helsinki.

“Whoever undertakes to set himself up as a judge in the field of Truth and Knowledge is shipwrecked by the laughter of the gods,” wrote Einstein. Mathematicians tend to respond with dismay rather than laughter, and then only to blunders so egregious as to be universally recognized as such.²⁴ Although those who find fault with philosophical speculation regarding the nature of mathematics seem to be under an implicit obligation to propose a speculative alternative, experience suggests that the practice of mathematics renders one unfit to do so. This, more than the fear of ridicule, is the main reason I would not venture my own speculative philosophy of mathematics. If it is hard “for those who are used to thought processes stemming from geometry and algebra” to “develop the sort of intuition common among physicists” (R. MacPherson, quoted in Quantum Fields and Strings: A Course for Mathematicians by Deligne (volume 1, p. 2)), bridging the gap between mathematicians and metaphysicians is probably hopeless. There are superficial parallels, to be sure: a metaphysical abstraction such as “essence,” like a mathematical abstraction such as “set,” designates nothing in itself, but rather refers to a canonical body of specialized texts in which the term plays a central role. 1 would like to argue that the nothing designated by “set” is somehow different, and more fruitful, than the nothing designated by “essence.” But the means at my disposal for making such an argument take the form of mathematical reasoning, which leads me, at best, to a vicious circle.²⁵ More bluntly, and for reasons akin to those Serre invoked in his Libération interview, I cannot be satisfied with an answer that is less certain than the sort of answer mathematics provides; for a mathematician, a pragmatist answer to Weil’s question is an admission of defeat. And yet I am aware that (metaphysically certain) grounds for distinguishing mathematical certainty from pragmatic certainty are lacking!

Another, possibly more profound, reason to steer clear of speculation is that, whereas philosophy presents itself as a dialogue extending over millennia, so that to understand each new contribution one would ideally be familiar with all previous contributions, mathematics is in principle supposed to be derivable by pure reason from a small number of axioms. A philosophical proposition, in other words, remains attached to its origins and context; a mathematical proposition floats free. This principle, an important constituent of the aura of metaphysical certainty surrounding mathematics, does not in fact bear much resemblance to mathematics as it is actually practiced—“one of humankind’s longest conversations,” as Barry Mazur puts it. I am nonetheless painfully aware that my personal “conversation” with the philosophical tradition is thoroughly unreliable, and my choice of footnotes is primarily the fruit of a random walk (or random surf, or remix) among scraps of the literature I have happened to encounter.

If I am nevertheless writing about philosophy, it is in large part because of a question that was put to me in 1995, during a presentation of Wiles’s proof of FERMAT’S LAST THEOREM [VI.10] to an audience of scientists. An October 1993 article in Scientific American entitled “The death of proof” had called Wiles’s proof a “splendid anachronism,” citing Laszlo Babai and his collaborators, among others, in support of the thesis that, in the future, deductive proof in mathematics will be largely supplanted by computer-assisted proofs and probabilistic arguments. That same month the Notices of the American Mathematical Society (40: 978–81) published Doron Zeilberger’s manifesto “Theorems for a price,” predicting a rapid transition from rigorous proofs to an “age of semi-rigorous mathematics, in which identities (and perhaps other kinds of theorems) will carry price tags” measured in computer time and proportional to the degree of certainty desired, to be followed in turn by “abandoning the task of keeping track of price altogether, and … the metamorphosis to non-rigorous mathematics” (John Horgan, Scientific American October 1993:92–102).²⁶

Feeling called upon to answer the question Weil avoided, I argued that the basic unit of mathematics is the concept rather than the theorem, that the purpose of a proof is to illuminate a concept rather than merely confirm a theorem, and that the replacement of deductive proofs by probabilistic or mechanical proofs should be compared not to the introduction of a new technology for producing shoes, say, but rather to the attempt to replace shoes by the sales receipts, or perhaps the cash profits, of the shoe factory. The audience had its own question: Was I talking about certainty? Of course not. That option has been philosophically discredited, as I have tried to explain. And other normative prescriptions fall victim no less easily to the laughter of philosophers. On the other hand, I see no pragmatic reason why probabilistic or mechanical proofs would not suit the five goals on the AMS committee’s list just as well as deductive proofs, nor any sociological reason why they should not be as effective in commanding consensus in the event of a paradigm shift. So what was I talking about?

Such a question, at this point in the essay, practically begs to be answered by an advertising slogan. For example:

The practice of making what one writes “reliable and verifiable” fosters critical thinking in general.

This is a popular argument for teaching proofs, and is probably even true, but how would one go about verifying such a claim? I am very much tempted to say that the concepts that serve as material for “one of humankind’s longest conversations,” deserve to be appreciated on their own terms. Note that nothing is more “emergent” than a conversation. But that would be unfaithful to the spirit of Mazur’s book, one of whose strengths is its refusal to conform to a linear narrative. In any case, on its own the argument does not seem to be sufficient: a similar argument could be made in favor of religious faith.

5 “Ideas, Even Dreams”

Rather than hazard an answer to Weil’s (non)question here, I will take a cue from Corfield and suggest that one can best account for the value of pure mathematics by attending to what mathematicians write and say. A handful of commonplace words appear consistently, invested with unexpected power, when mathematicians attempt to account, formally as well as informally, for their value judgments, and these collectively constitute an answer to the question Weil left hanging.

WEYL [VI.80] wrote a book with the provocative title The Idea of a Riemann Surface²⁷ and referred in his preface to Plato. The word “concept” that was central in my reply to the audience is closer to this use of the term “Idea” as used by any number of philosophers, including most of those mentioned in this essay. A square, or a RIEMANNIAN MANIFOLD [I.3 §6.10], would be a concept or “Idea” in this sense, and this is how the word “concept” tends to be used by mathematicians, who generally reserve the word “idea” to designate something else. In Plato’s Meno, the proof of the doubling of the square—draw diagonals and fit the resulting triangles together—which the slave “remembers” under Socrates’ coaching, is taken by Plato to be contained in the “Idea” of the square. For a mathematician, drawing the diagonals and moving the triangles are the ideas.

That a contrast can be drawn, as I did in 1995, between “illuminating concepts” and “confirming theorems” is something of a truism among mathematicians and even some philosophers. Even by 1950 Popper had argued that “a calculator … will not distinguish ingenious proofs and interesting theorems from dull and uninteresting ones” (quoted in Heintz 2000). Corfield correctly states that “what mathematicians are largely looking for from each other’s proofs are new concepts, techniques, and interpretations”; they are not merely “establishing the truth or correctness of propositions” (p. 56). However, although he devotes a chapter to the “extremely complex subject” of “mathematical conceptualization,” he does not dwell on concepts (or “Ideas”) as such; and neither will I. It is almost impossible to talk in general terms about mathematical concepts without getting caught up in the debate over their reality (and provoking the laughter of the philosophers). Those who write about mathematics (mathematicians included: see Hersh (1997)) have an irritating tendency to claim that most mathematicians are Platonists, whether or not they have committed themselves explicitly to a philosophical position. Maybe it can be argued that Platonism is implicit in the syntax of mathematical statements; maybe this is what Weil meant by his claim, quoted by Bourguignon (2001), that most mathematicians “spend a good portion of their professional time behaving as if they were [Platonists].” ²⁸ In practice I would guess that most mathematicians are pragmatists, in the spirit of the remarks of Dieudonné quoted above.

On the other hand, there is no doubt whatsoever that the “ideas” that matter to mathematicians are real. A mathematician, according to a joke attributed to Weil,²⁹ can be defined as someone who has had two ideas (mathematical, of course). But then, Weil worried, so-and-so would be a mathematician. In a celebrated account by POINCARÉ [VI.61] of the role of the unconscious in a mathematical discovery, the climactic moment was the arrival of an idea (“the idea came to me”) as he placed his foot on the steps of the omnibus (“L’idée me vint,” Poincaré (1999)).

More to the point, consider Hacking’s justification of his own commitment to a realist ontology of electrons: “So far as I’m concerned, if you can spray them then they are real” (Hacking 1983). By the same token, if you can steal ideas, then they are real. Every mathematician knows that ideas can be and often are stolen. Polemics then ensue, considerably juicier than the epistemic controversy studied by Rosental.

Nothing in the life of mathematics has more of the attributes of materiality than (lowercase) ideas. They have “features” (Gowers 2002), they can be “tried out” (Singer³⁰), they can be “passed from hand to hand” (Corfield 2003), they sometimes “originate in the real world” (Atiyah in the preface to Arnold et al. (2000)) or are promoted from the status of calculations by becoming “an integral part of the theory” (Godement 2001). At some point they come into being: it is generally understood, for example, that “new ideas” will be needed to solve the Clay Millennium Problems. They can also be counted. I once heard Serre introduce the proof of a famous conjecture by saying that it contained two or three real ideas, where “real” was intended as high praise. The ambiguity did not concern the number of ideas—there were three, which Serre enumerated—but whether all three were original to the author. Ideas are public: necessarily so, in order to be stolen, or to be presentable as Serre did in his lecture. Poincaré’s idea was a sentence (“the transformations of which I had made use to define Fuchsian functions were identical to those of non-Euclidean geometry”); the slave’s idea in Meno was a line in the sand.

Early in his unpublished memoirs Récoltes et Semailles, Grothendieck wrote that “ideas, even dreams” were, in Allyn Jackson’s terminology, the “essence and power” of his mathematical work (Jackson 2004). An idea is typically symptomatic of “insight,” and the capacity for insight is generally called “intuition.” Mathematicians have borrowed all of these terms from philosophy but use them to completely different ends. Philosophers tend to follow Kant in attributing intuitions—the ones that without concepts are blind—to transcendental subjects or their more down-to-earth offspring. Intuition in this sense is a poor substitute for certainty, as even the mavericks recognize. “Intuition … is frequently a prelude to mathematical knowledge,” wrote Kitcher. “By itself it does not warrant belief.” Poincaré called intuition “the tool of invention,” a “je ne sais quoi” that holds a proof together, but he contrasted it with logic, “the tool of demonstration,” which “alone can provide certainty.” Saunders Mac Lane expressed himself in much the same terms nearly a century later. David Ruelle considered reliance on (visual) intuition a characteristic feature of human (as opposed to extraterrestrial) mathematics.³¹

In each case intuition belongs to the private sphere, and is relegated to the “context of discovery,” as opposed to the “context of justification” deemed worthy of philosophy’s full attention. When mathematicians refer to “intuition” in the sense I have in mind, it is crucially public.³² As in the quotation from MacPherson a few paragraphs back, it can be transmitted from teacher to student, or through a successful lecture, or developed collectively by running a seminar and writing a book on the proceedings. It has something in common with a “style of reasoning,” but on a smaller scale. Grothendieck resorted to perceptual metaphor when describing Serre’s ability to communicate something akin to intuition:

The essential thing was that Serre each time strongly sensed the rich meaning behind a statement that, on the page, would doubtless have left me neither hot nor cold—and that he could “transmit” this perception of a rich, tangible, and mysterious substance—this perception that is at the same time the desire to understand this substance, to penetrate it.

Récoltes et Semailles, p. 556

“Even those who try to articulate, to classify, the fruits of the imagination, and who are committed to the existence of an inner experience concomitant with it, admit to dark difficulty in describing it,” wrote Mazur, elaborating an unusual array of literary and rhetorical strategies to chip away at the difficulty (Mazur 2003). This much is certain: this inner experience of imagination, or of understanding, is what drives people to become mathematicians, and it is why Weil could count on his audience’s silent assent. Heintz recorded some of her informants’ attempts to describe this inner experience: “[In mathematics] you have concrete objects before you and you interact with them, talk with them. And sometimes they answer you.” She even talks about the “idea” that helps put the pieces together. “And suddenly you see the picture,” she was told. Yet all this raw ethnographic data is presented in a chapter whose title, “Beauty and experiment: discovery of truth in mathematics,” betrays her relentlessly epistemological preoccupations (Heintz 2000).

“The specific ways that mathematical truths move from person to person, and how they are transformed in the process, are as difficult to capture as the truths themselves,” wrote Mazur (2003), in what could have been a comment on Grothendieck’s remarks on Serre. The central notion in Mazur’s book is that of “imagination.” I have chosen the terms “idea” and “intuition” not for their intrinsic importance, though I believe each of the terms points to ways of talking about the famous “flash in the middle of a long night” that ends Poincaré’s The Value of Science: “But this flash is everything.” What strikes me about these terms is how their pervasiveness in mathematicians’ conversations—the sense that they, more than the definitive theorems, are “everything”— contrasts so starkly with their near exclusion from philosophical consideration, even though the words themselves can be seen on practically every page of philosophy of mathematics. Maybe their very banality makes them appear philosophically trivial. Or maybe the problem is that the same words serve so many distinct purposes. Corfield uses the same word to designate what I am calling “ideas” (“the ideas in Hopf’s 1942 paper”) as well as “Ideas” (“the idea of groups”) and something halfway between the two (the “idea” of decomposing representations into their irreducible components for a variety of purposes, p. 206). Elsewhere, the word crops up in connection with what mathematicians often refer to as “philosophy,” as in the “Langlands philosophy” (“Kronecker’s ideas” about divisibility, p. 202), and in many completely unrelated places as well. Corfield proposes to resolve what he sees as an anomaly in Lakatos’s “methodology of scientific research programmes” as applied to mathematics by “a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas” (p. 181). He sees “a kind of creative vagueness to the central idea” in each of the four examples he offers to represent this shift of perspective; but on my count the ideas he chooses include two “philosophies,” one “Idea,” and one which is neither of these.

Other value-laden terms are no less important. In the wake of BOURBAKI [VI.96], quite a few philosophers (Cavaillès, Lautman, Piaget, and more recently Tiles) have made serious attempts to make sense of “structure” in mathematics. I have read a number of philosophical attempts to account for mathematical aesthetics, though none has left much of an impression. The practically universal use of dynamical or spatiotemporal metaphors (“the space X is fibered over Y,” etc.), and the pronounced tendency to present proofs as series of actions playing out in time (“now choose an orbit passing arbitrarily close to the point x”) have attracted little attention from philosophers.³³ These phenomena may be linked to the curious preference of many mathematicians for blackboards over contemporary audiovisual technology, which in turn draws attention to the neglected (and emergent) aspect of mathematical communication as performance, a word that manages to be typically postmodern and premodern at the same time.

For his part, Corfield does not talk much about “intuition” and is ambiguous about what he means by “ideas,” but his discussions of “natural” and “importance,” in the context of an analysis of the debate on the relative merits of groups and groupoids, are philosophically insightful while remaining faithful to the use of the terms by “real” mathematicians. His treatment of “postmodern algebra,” where “diagrams are not just there to illustrate, they are used to calculate and to prove results rigorously” (p. 254), also has street credibility. It is true that much of his book remains concerned with “maverick” questions, such as accounting for plausible reasoning. But there is no question that Corfield likes mathematics, and for the right reasons; his book, unlike most treatises in philosophy of mathematics, is definitely part of the “conversation.”

Morris Kline called the “loss of certainty” entailed by Gödel’s theorems an “intellectual tragedy” and actually counseled “prudence” in designing bridges “using theory involving infinite sets or the axiom of choice” (Kline 1980). The word “tragedy” seems misplaced but the pathos is real, as it was for Russell. Pathos and its twin, resolute optimism, have found an unlikely home in the philosophy of mathematics:

If this conception of mathematics [as “human knowledge of structures gained by employing reason beyond the bounds of logic”] can be sustained, mathematics could once again serve as a source of an image of reason liberated from formal imprisonment, freed to confront apocalyptic post-modern visions.

Mary Tiles, Mathematics and the Image of Reason, p. 4 (Routledge, London, 1991)

Whether or not it carries weight with congressional committees, I find this goal appealing, but it is a goal for philosophers, not for mathematicians. I’m willing to apply the “principle of charity” to philosophers if they will do the same for me. Corfield wrote (p. 39):

Human mathematicians pride themselves on producing beautiful, clear, explanatory proofs, and devote much of their effort to reworking results in conceptually illuminating ways. Philosophers must not evade their duty to treat these value judgments in mathematics.

They also have a duty, it seems to me, to account for terms like “idea” and “intuition”—and “conceptual” for that matter. An answer to the question “Why philosophy?” might well begin there.

Postscript

In December 2004 my university joined a number of other institutions in France and elsewhere in hosting a traveling UNESCO-sponsored exhibition entitled “Pourquoi les mathématiques?” Hoping to learn the answer before my submission deadline, I spent a few hours at the exhibition. It was clever and engaging, presenting a variety of (pure) mathematical ideas with a sprinkling of practical applications, but in no way did it address the “Pourquoi?” of the title. An organizer was on hand, and when I turned to her for guidance she explained that the French title was a solution to a problem of translation. The English title, which came first, was “Experiencing mathematics.” This, she assured me, has no adequate French translation, so “Pourquoi les mathématiques?” was chosen as the best substitute.

Maybe the solution to the problem of my title is simply to accept the translation in the opposite direction. Even the most ruthless funding agency is not yet so post-human as to require an answer to “Why experience?” ³⁴

Acknowledgments. I thank Cathérine Goldstein and Norbert Schappacher for pointing me in the directions of the Rosental and Heintz books, among other source material, and for vigorously criticizing my project as well as its execution. I also thank Mireille Chaleyat-Maurel for explaining the title of the UNESCO exhibition and Ian Hacking for critically reading an earlier version of the manuscript with tolerance and rigor. David Corfield receives thanks for several helpful clarifications. Barry Mazur is thanked especially warmly for many suggestions, much encouragement, for help with the title, and most of all for showing, in his Imagining Numbers, that there is at least one way out of the fly-bottle.

Further Reading

Arnold, V., et al. 2000. Mathematics: Frontiers and Perspectives. Providence, RI: American Mathematical Society.

Barthes, R. 1967. Système de la Mode. Paris: Éditions du Seuil.

Bloor, D. 1976. Knowledge and Social Imagery. Chicago, IL: University of Chicago Press.

Bourguignon, J.-P. 2001. A basis for a new relationship between mathematics and society. In Mathematics Unlimited—2001 and Beyond, edited by B. Engquist and W. Schmid. New York: Springer.

Corfield, D. 2003. Towards a Philosophy of Real Mathematics. Oxford: Oxford University Press.

Godement, R. 2001. Analyse Mathématique I. New York: Springer.

Gowers, W. T. 2002. Mathematics: A Very Short Introduction. Oxford: Oxford University Press.

Hacking, I. 1983. Representing and Intervening. Cambridge: Cambridge University Press.

———. 2000. What mathematics has done to some and only some philosophers. Proceedings of the British Academy 103:83–138.

———. 2002. Historical Ontology. Cambridge, MA: Harvard University Press.

Harvey, D. 1989. The Condition of Postmodernity. Oxford: Basil Blackwell.

Heintz, B. 2000. Die Innenwelt der Mathematik. New York: Springer.

Hersh, R. 1997. What Is Mathematics, Really? Oxford: Oxford University Press.

———, ed. 2006. 18 Unconventional Essays on the Nature of Mathematics. New York: Springer.

Jackson, A. 2004. Comme appelé du néant—as if summoned from the void: the life of Alexandre Grothendieck. Notices of the American Mathematical Society 51:1038.

Kitcher, P. 1984. The Nature of Mathematical Knowledge. Oxford: Oxford University Press.

Kline, M. 1980. Mathematics: The Loss of Certainty. Oxford: Oxford University Press.

Lakoff, G., and R. E. Núñez. 2000. Where Mathematics Comes From. New York: Basic Books.

Lloyd, G. E. R. 2002. The Ambitions of Curiosity, p. 137, note 13. Cambridge: Cambridge University Press.

Lyotard, J.-F. 1979. La Condition Postmoderne. Paris: Minuit.

Maggesi, M., and C. Simpson. Undated. Information technology implications for mathematics, a view from the French Riviera. (This paper is available at http://math1.unice.fr/˜maggesi/itmath/;apparently not posted before 2004.)

Mancosu, P., ed. 1998. The current epistemological situation in mathematics. In From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford: Oxford University Press.

Mazur, B. 2003. Imagining Numbers (Particularly the Square Root of Minus Fifteen). New York: Farrar Straus Giroux.

Minsky, M. 1985/1986. The Society of Mind. New York: Simon and Schuster.

Poincaré, H. 1970. La Valeur de la Science. Paris: Flammarion.

———. 1999. Science et méthode. Paris: Éditions Kimé.

Rosental, C. 2003. La Trame de l’Évidence. Paris: Presses Universitaires de France.

Tymoczko, T., ed. 1998. New Directions in the Philosophy of Mathematics. Princeton, NJ: Princeton University Press. (First published in 1986.)

Wittgenstein, L. 1958. Philosophical Investigations, volume I. Oxford: Basil Blackwell.

———. 1969. On Certainty. Oxford: Basil Blackwell.

 

1. From “Pilot assessment of the mathematical sciences (prepared for the House Committee on Science, Space, and Technology),” Notices of the American Mathematical Society 39 (1992):101–10.

2. The present essay is mainly concerned with metaphysical certainty. Descartes wrote in Principles of Philosophy (chapter CCVI) of “certainty … founded on the metaphysical ground that, as God is supremely good and the source of all truth, the faculty of distinguishing truth from error which he gave us, cannot be fallacious so long as we use it aright, and distinctly perceive anything by it,” and cites “the demonstrations of mathematics” as his first example. Plato (in Republic, VII, 522–31) saw mathematics rather as a source of “knowledge of that which exists forever.” Certainty and its cognates are some, but only some, of the apparent blessings of mathematics that so impressed certain philosophers as to “infect” the whole of their work, as Ian Hacking (2000) argues.

3. See Lloyd (2002), in which is cited Ptolemy’s Syntaxis, I, chapter 1 16.17–21.

4. Compare René Thom’s comment in connection with his criticism of attempts to reduce mathematics to set theory: “In attempting to attach meaning to all the phrases constructed in ordinary languages, according to Boolean rules, the logician proceeds to a phantasmic, delirious reconstruction of the universe” (reprinted in Tymoczko 1998).

5. Transcript of interview by Francois Tisseyre conducted on the occasion of the Paris Millennium Meeting, May 24, 2000, graciously provided by the Clay Mathematics Institute.

6. For example, Lakoff and Núñez (2000) write of a “radical form of postmodernism which claims that mathematics is purely historically and culturally contingent and fundamentally subjective.” No examples are given of texts espousing this point of view.

7. “Because his … artistry comes from combining other people’s art, … the DJ is the epitome of a postmodern artist” (www.jahsonic.com/postmodernism.html).

8. Otto Karnik, in “Attraction and repulsion,” article in Kai KeinRespekt, p. 48, Exhibition Catalogue of the Institute of Contemporary Art (Bridge House Publishing, Boston, MA, 2004). The Hilbert quotation is easy to find but is probably apocryphal, which does not make it any less significant. Mathematics and the Roots of Postmodern Thought, by Vladimir Tasi, is an extended speculation on postmodernism’s mathematical antecedents; see my review in Notices of the American Mathematical Society 50 (2003):790–99.

9. For example, “[Derrida’s] thought is based on his disapproval of the search for an ultimate metaphysical certainty or source of meaning that has characterized most of Western philosophy.” From the Encyclopedia Britannica Online (www.britannica.com).

10. “Why don’t you ask a physicist or a mathematician about difficulty?” was Derrida’s response to a 1998 New York Times question about deconstruction: see Jacques Derrida, Abstruse Theorist, dies at 74, New York Times, October 10, 2004. Appeals to the presumed value of even the most abstruse mathematics, in order to legitimate obscurity elsewhere, are common. I first encountered such an argument in an article by composer (and former mathematician) Milton Babbitt entitled “Who cares if you listen?” (High Fidelity, February 1958): “Why should the layman be other than bored and puzzled by what he is unable to understand, music or anything else?” With this sort of talk, the justification of pure mathematics on aesthetic grounds is turned upside down. That is why I address aesthetic answers to the question of my title—which are by far the most popular among my colleagues—only in a footnote.

11. A cliché for the succeeding generation of literary critics: for a sample emphasizing chaos rather than Gödel, see N. Katherine Hayles (ed.), Chaos and Order (University of Chicago Press, 1991).

12. Much of Prodiges et vertiges de l’analogie by Jacques Bouveresse (Raisons d’Agir, 1999) is devoted to just this sort of reminder.

13. Predictably, religion steps in to fill the gap: see www.asa3.org/ASA/topics/Astronomy-Cosmology/PSCF9-89Hedman.html#16. John D. Barrow takes the implications of Gödel’s theorems for physics seriously, while denying that they necessarily limit scientific objectivity (see, for example, “Domande senza risposta,” in Matematica e Cultura 2002, edited by M. Emmer, pp. 13–24 (Springer, 2002)).

14. The anti-foundationalism of Tymoczko’s anthology is largely inspired by Gödel’s theorems.

15. Weil’s joke is quoted in at least eighty-five sites found via Google; no primary source is given. Dieudonné’s comment is naturally from Pour l’Honneur de l’Esprit Humain, pp. 244–45 (Hachette, 1987). Borel’s remarks on the “self-correcting power of mathematics,” in his contribution to the discussion of the article “Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics” by A. Jaffe and F. Quinn, express a more modest form of pragmatism (Bulletin of the American Mathematical Society 29 (1993):1–13).

16. Quotations from Wittgenstein (1969, paragraph 4; 1958, paragraph 437).

17. Lakatos’s posthumous “A renaissance of empiricism in the recent philosophy of mathematics,” presents a long series of quotations by mathematicians and a few philosophers, including Russell in 1924, acknowledging that mathematics is uncertain, after all. Naturally, most of those cited refer directly or indirectly to Gödel’s theorem. The article was reprinted in Tymoczko (1998). “Only dogma or theory has made people say that mathematics as a whole has a peculiar certainty,” writes Hacking (2000). Certainty persists, however, in the titles of philosophy books, e.g., Marcus Giaquinto’s optimistic The Search for Certainty: A Philosophical Account of Foundations of Mathematics (Oxford University Press, Oxford, 2004).

18. Heintz quotes yu. I. Manin—“A proof only becomes a proof after the social act of ‘accepting it as a proof’” —as well as René Thom’s “community” theory of truth. One can of course always ask whether Heintz selectively quoted mathematicians whose positions support her thesis. This question can be asked of any sociological study, and it is best to let the sociologists work out their methodological issues. An important remark, however: though Heintz’s original goal was to account for the formation of consensus among mathematicians within a science studies framework—with questionable success, but that is another matter—she does not defend a particular school within philosophy of mathematics. In this she differs from Bloor, for instance, who identifies himself explicitly as an empiricist.

19. This article was written in late 2004. The proof is now accepted as correct, and in 2006 Perelman was offered a Fields medal, which he declined. He has also refused the Clay Mathematical Institute prize.

20. “Having shown how the production of certified knowledge in logic could constitute an object of a sociological investigation and analysis, a vast field of research takes shape” (Rosental 2003). I suspect that identifying and accounting for the priorities expressed by mathematicians themselves would constitute a much richer field of research.

21. The quotation is from Corfield’s Towards a Philosophy of Real Mathematics (Corfield 2003). Compare it with Ian Hacking’s comment that “the most striking single feature of [twentieth century philosophy of mathematics] is that it is very largely banal” (Hacking 2002). For Hacking’s philosophy of mathematics, see his What Mathematics Has Done.

“Real mathematics,” for Corfield, who is remarkably well-informed about trends in the most diverse branches of mathematics, is “real” in the same way as “real ale.” I readily agree that skepticism toward this sort of realism is self-defeating.

22. See, however, Hacking: “The truth of a sentence (of a kind introduced by a style of reasoning) is what we find out by reasoning using that style” (Hacking 2002).

23. Many of the authors in Tymoczko (1998) also look to the (real) practice of mathematics for philosophical insight, but Truth and Knowledge keep creeping in. Arriving in France in 1994, I was astonished to discover that the concerns of twentieth-century French philosophers of mathematics are entirely different. Following Husserl, the French concentrate largely on the phenomenological experience of the individual mathematical subject. It is only a slight exaggeration to say that the French-language and English-language traditions in philosophy of mathematics have become mutually incomprehensible. Fortunately, mathematicians writing in French and in English have no trouble citing each others’ works.

24. As Serre put it in his comments to Libération, “Si vous ne voulez pas que les choses soient parfaites, ne faites pas de maths.” Heintz’s book is an inquiry into the roots of this apparent universal tendency to consensus, and finds it in the institution of the proof; Rosental treats a (highly unusual) case in which universal consensus apparently failed. The Einstein quotation is in Kline (1980).

25. “Truth is always the possibility of its proper destruction,” according to the (nonpostmodern) French philosopher, Alain Badiou, taking Gödel’s theorem as an example (www.egs.edu/faculty/badiou/badiou-truth-process-2002.html).

26. In the pop posthumanist scenarios promoted by Hans Moravec, Ray Kurzweil, and the like, computers acquire all human capabilities, including the ability to generate and prove theorems—for some reason this is always considered a landmark—by the middle of the twenty- first century. The distinction between humans and computers subsequently fades away rather rapidly, making Zeilberger’s prediction moot.

A more recent, and much more nuanced, discussion of prospects for automatic theorem proving has been posted on the Internet by Maggesi and Simpson (undated).

27. Weyl used the word Idee in his title but applied the term Begriff (concept) elsewhere in the text. Both terms arrived in English as “concept.”

28. Plato saw things quite the other way around: “Their language [speaking of mathematicians] is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed toward action” (Republic VII.527a, my emphasis).

29. I heard this joke reported by several people who claimed to have heard it from Shimura, and I believe but am not certain that I too first heard it from Shimura.

30. Quoted at www.abelprisen.no/en/prisvinnere/2004/interview_2004_7.htm1.

31. Kitcher (1984, p. 61); Poincaré (1970, pp. 36–37); Mac Lane, in his contribution to the discussion of the Jaffe–Quinn article cited in note 15, Bulletin of the American Mathematical Society 30 (1994): 178–207; Ruelle’s quote is from an article entitled “Conversations on mathematics with a visitor from outer space” from Arnold et al. (2000).

32. This is also true of the normative program of intuitionism associated with BROUWER [VI.75], but that is definitely not what I have in mind.

33. Nuñez’s article “Do real numbers really move?” (in Hersh 2006) makes interesting points regarding mathematicians’ use of metaphors of motion, though he limits his analysis to examples specifically related to the mathematics of motion. Plato specifically disapproved of mathematicians’ use of action metaphors.

34. Or, as Weyl put it, “with [mathematics] we stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.” Note the word “essence” (see Mancosu 1998). I thank David Corfield for this quotation.