Not Merely Good, True, and Beautiful
[I]f this is a “We-system,” why isn’t it at least thoughtful enough to interlock in a reasonable way, like They-systems do?
—Thomas Pynchon, Gravity’s Rainbow
If it has become urgent for mathematicians to take up the “why” question, to which we are now ready to turn, it is because the professional autonomy to which we have grown attached is challenged by at least two proposals for reconfiguration. The first proposal is neither new nor specific to mathematics. The economic crisis that began in 2008 placed universities under enormous stress. Cutbacks have been especially severe in humanities departments, where the elimination of entire programs has become routine, but the crisis has also intensified existing pressure to subordinate scientific research in Europe and North America to commercial applications. It is telling that universities in Britain are now the responsibility of the Department of Business, Innovation, and Skills.1 Here is how Gary Goodyear, Canada’s Minister of Science and Technology, put it in a recent television interview:
[W]e are also asking our scientists to appreciate the business side of science … as the Prime Minister has said many times, science powers commerce…. We have to move forward on applied science, the commercialization end … we are the first to say that there is far too much knowledge left in the laboratory, and knowledge that is not taken off the shelf and put into our factories is actually of no value” [my emphasis].2
Goodyear is unusually blunt, but he is hardly the first to talk this way. Already in 1907, Clarence Birdseye had written that “our colleges have become a part of the business and commercial machinery of our country, and must therefore be measured by somewhat the same standards.”3 Quoting Birdseye and other voices of what we might call “market instrumentalism,” Joan Wallach Scott pointed out in 2008 that “Business-men and politicians, then as now, have had little patience with the ideal of learning for its own sake…. Today the sums may be larger and their impact on university research operations greater, but the pressure to bring universities in line with corporate styles of accounting and management persists.”4
There is now a massive literature on the pressures facing university laboratories.5 These books mostly ignore mathematics, where stakes are not so high and opportunities for commercial applications are scarce, especially in the pure mathematics that is the subject of the present book. Nevertheless, even pure mathematicians, when presenting the case for research support to the “Powerful Beings”* who often go by the name of “decision-makers,” find it necessary to point to potential commercial or industrial spinoffs or at least to resort to what Steven Shapin calls the Golden Goose argument, to the effect that BECAUSE the potential benefits of scientific research are so often unpredictable, THEREFORE, Powerful Beings have every interest in funding research motivated by curiosity alone.6 Weimar Germany’s Kaiser-Wilhelm Institutes, supported by public and private funds, put the Golden Goose argument to practical effect by allowing scientists to “focus on basic research in response to problems posed by industry without concern for immediately applicable results.”7 Abraham Flexner, founder of Princeton’s Institute for Advanced Study (IAS), devoted a 1939 Harper’s article, entitled “The Usefulness of Useless Knowledge,” to the Golden Goose argument. After listing some of the unexpected historical benefits of purely theoretical research in the sciences, including pure mathematics—and announcing plans for the construction of Fuld Hall, the main building at the IAS—Flexner wrote that “we cherish the hope that the unobstructed pursuit of useless knowledge will prove to have consequences in the future as in the past.”8
Mathematical societies today see promotion of the Golden Goose argument as one of their primary responsibilities. Chapter 10 reviews some recent examples in France, Germany, Britain, and the United States. There is obviously no reason not to hope that the ways of thinking that arise in pure as well as applied mathematics might contribute to solving a few of the problems facing humanity.9 Leibniz’s thoughts as mathematician and philosopher tended to be abstract, but his motto was Theoria cum Praxi—“theory with practice.” That’s a very different ethical stance from equating our worth with our potential as Golden Geese, especially when the Powerful Beings to whom we promise our gold are primarily concerned with short-term economic goals to which we may not subscribe. Flexner promoted the IAS not as a Golden Goose refuge but “as a paradise for scholars who, like poets and musicians, have won the right to do as they please and who accomplish most when enabled to do so.” The physicist Peter Goddard, director of the IAS in 2011, when I was beginning this book, vigorously defended Flexner’s vision against unspecified but persistent challenges. Goddard argued that, with universities facing the “introduction of business principles and methods”—“systems of standardization, accountancy, and piecework,” he wrote, quoting Thorstein Veblen—the reasons for creating institutes like the IAS, of which there are now hundreds, “have greater force than they did a century ago.”10 It is not only dishonest but also self-defeating to pretend that research in pure mathematics is motivated by potential applications. Pure mathematics is a living community, and we’ll see that the motivations of pure mathematicians are rich and varied, but there is no evidence that market instrumentalism ranks prominently (or anywhere at all) among them.
The second proposed paradigm shift poses less of an immediate material challenge but is more interesting in that it arises from within the profession. The proposal stems from an observation with which no one would disagree. Proofs in some areas of mathematics are growing increasingly elaborate, involving in some cases hundreds of pages of calculations or expertise in too many areas to be fully understood by any one mathematician. Traditional methods exhaust the physical limits of human beings as presently configured and for that reason are deemed inadequate to confer legitimacy on some of the most remarkable theorems of the past few decades. The most notorious examples are the 1976 computer-assisted proof of the four color theorem by Kenneth Appel and Wolfgang Haken, which launched the ongoing debate on the epistemic status of computer-assisted proofs;11 the classification of finite simple groups, whose proof, completed in the 1980s but still undergoing adjustment, is estimated to fill five thousand pages; and the proof of the Kepler conjecture, completed in 1998 by Thomas Hales and Samuel Ferguson with the help of around 3 gigabytes of computer code, which no committee of human referees could validate in its entirety.12
Some mathematicians, disturbed by the uncertainty surrounding these proofs, envision a future in which the human theorem prover is yoked more or less securely to nontraditional devices that don’t suffer human shortcomings. The immediate objective of Hales’ Flyspeck project, “an undertaking … that has the potential to develop into one of historical proportions,” is the formalization of the proof of the Kepler conjecture. Formalization is an ideal of mathematical argument acceptable to logicians and to like-minded philosophers of Mathematics: instead of writing a proof in some approximation to ordinary language and appealing to intuition in the transition from one step to the next, a formal proof is a series of propositions written in a formal language from which all ambiguity has been eliminated, with each proposition obtained from its predecessors by strict application of one of a short list of permitted transformations. The idea is that individual propositions written in a formal language, and the transitions between them, can be certified licit by a proof-checker, a computer program written for this purpose. The long-term objective is naturally more ambitious:
We are looking for mathematicians … who are computer literate, and who are interested in transforming the way that mathematics is done.13
A substantial fraction of elementary mathematics has been successfully formalized in this way, and there’s no reason to doubt that the proof of the Kepler conjecture will be converted into one that can be checked by machine. What happens next—the “transforming” part—is not so clear. As far as I can tell, no one is suggesting that mathematicians actually spend time reading the formal proofs, only that we take comfort in the knowledge that a specialized mechanical entity has done so and has given its seal of approval.
What’s noteworthy about the comfort being offered is that it is collective and metaphysical, not merely personal and psychological. The search for such comfort has a long history, like the history of the attempts to impose a corporate model on research but here rooted in what historian Jeremy Gray calls “anxiety.” “[O]nce the safe havens of traditional mathematical assumptions were found to be inadequate,” Gray writes, “mathematicians began a journey that was not to end in security, but in exhaustion, and a new prudence about what mathematics is and can provide.” Gray is writing about the turn of the twentieth century, just before the Foundations Crisis, when the “new discipline of physics could compete very favorably with mathematics on utilitarian grounds … more critical mathematicians were aware that they therefore had to base their claims for the quality and value of mathematics on more intrinsic grounds.”14 “Useful uselessness” in Flexner’s sense no longer sufficed; a metaphysical anxiety now haunted our spare moments when we were not busy trying to convince the Birdseyes and Goodyears that we are not parasites.
I would argue that this particular anxiety is overstated, the ghost story mathematicians take down from the shelf when we want to share a morbid moment. But it remains a pressing concern for philosophers of both mathematics and Mathematics, as we’ll see later; and it’s one of the rare mathematical themes readily accessible to audiences outside our circle. Thus IAS professor Enrico Bombieri, in a public lecture in 2010, recalled the concerns about the consistency, reliability, and truthfulness of mathematics that surfaced during the Foundations Crisis and alluded to the ambiguous status of “computer proofs” (like the Appel-Haken proof) and “too-long proofs” (like the classification of finite simple groups)15—Bombieri’s IAS colleague Pierre Deligne once remarked that “what interests him personally are results,” unlike the two just mentioned “that he can, by himself and alone, understand in their entirety.”16
Another IAS professor, the Russian-born Vladimir Voevodsky, is more actively anxious about the future of mathematics. “If one really thinks deeply about” the possibility that the foundations of mathematics are inconsistent, he said on the occasion of the IAS eightieth-anniversary celebration, “this is extremely unsettling for any rational mind.”17 Voevodsky obtained his prestigious position at the IAS and his Fields Medal18 for his work in a field in which “too-long proofs” are common and in which the relatively small number of competent potential referees typically spend much of their time writing “too-long proofs” of their own, so he might understandably be concerned that proofs are not being read as carefully as they should. But Voevodsky is worried about something more central to mathematics. Consistency, in the technical sense, is what guarantees that the basic axioms don’t allow you to prove a statement as well as its negation. Without consistency—if you could prove both p and not p, as logicians put it—then the distinction between mathematical truth and falsehood collapses and with it all the philosophy that seeks its grounding in mathematical logic.
Philosophy’s reliance on the standards of mathematical reasoning is an old story. Descartes, for example, thought that “even in relation to nature there are some things that we regard as not merely morally but absolutely certain … Mathematical demonstrations have this kind of certainty….”19 “Mathematical knowledge,” wrote Heidegger, “is regarded by Descartes as the one manner of apprehending entities which can always give assurance that their Being has been securely grasped.”20 In the language of politicians, “mathematical certainty” is the certainty that brooks no contradiction. Thus, in the middle of the continuing euro crisis in 2012, one could read in Time that budget adjustments being demanded of the Greek government (with catastrophic consequences, by the way, for scientific research, mathematics included) were “not politically, socially, or even mathematically possible.” Under the influence of Frege and Russell, among others, twentieth-century philosophy focused on symbolic logic as the most secure branch of mathematics. The brief period of the Foundations Crisis, following Russell’s discovery of paradoxes in Frege’s interpretation of Cantor’s set theory, marked the culmination of this concern with certainty. The interaction of mathematicians with philosophical logicians was so intense at the time that both groups were convinced that mathematical certainty was built on logical foundations. Hilbert’s program went so far as to reinterpret mathematics as a kind of game, bound by a system of rules, inherited from the tradition (but metaphysically neutral), and rigid but flexible enough to provide a proof of their own consistency.
Unfortunately, the second incompleteness theorem of the Austrian logician Kurt Gödel asserts precisely that the consistency of a system of basic logical axioms cannot itself be guaranteed by a formal proof within the system.21 Voevodsky’s talk outlines his reasons for suspecting that inconsistency is more likely than consistency. Are you unsettled? Is your mind rational? To the anxious, Voevodsky offers a carrot and a stick. The carrot is that getting used to using (possibly) inconsistent systems of reasoning would be “liberating” in that “we could then use kinds of reasoning known to be inconsistent but closer to our intuitive thinking” to construct proofs that could then be verified to be reliable. His new project, to construct univalent foundations of mathematics, is a piece of the carrot.
The stick is hidden in this sentence: after finishing the project for which he won the Fields Medal, mathematics’ highest honor, in 2002, Voevodsky “became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era, which will be characterized by the widespread use of automated tools for proof construction and verification”22 [my emphasis]. In February 2011, at an IAS After Hours Conversation (a kind of before-dinner party), Voevodsky predicted it would soon be possible to design proof checkers based on univalent foundations that could effectively verify correctness of mathematical proofs written in the appropriate machine-readable language. In a few years, he added, journals will only accept articles accompanied by their machine-verifiable equivalents.
Bombieri ended his lecture with the surprisingly Kuhnian conclusion that “mathematics follows a kind of Darwinian evolution” and that “mathematical truth is not irrelevant, nor tautological; it is the glue that holds the fabric of mathematics together.” Bombieri, like most mathematicians, has his eye on the fabric. He writes that “the working mathematician”—a character we’ll meet repeatedly in the course of this book—“is guided by clear aesthetic considerations: Intuition, simplicity of arguments, linearity of patterns, and a mathematically undefinable aristotelian ‘fitting with reality.’ ”23
Voevodsky worries about the glue. Bombieri’s “intuition” or Deligne’s “understanding” do not offer the “absolute certainty” at which Descartes was aiming. In developing his “conceptual notation” [Begriffschrift] for mathematics, Frege worked hard “to prevent anything intuitive [Anschauliches] from penetrating … unnoticed.”24 For “the working mathematician” understanding and intuition are real. The project of mechanization, which can be traced through Frege back to Leibniz,25 proposes to substitute a model of real closer to Frege’s, one that lacks the defects of subjectivity. “I never would have guessed,” wrote Mike Shulman in 2011, “that the computerization of mathematics would be best carried out … by an enrichment of mathematics”26 [my emphasis] along the lines of Voevodsky’s univalent foundations project, to which Shulman is an active contributor.
I italicized the definite article in the Shulman quotation to emphasize how mechanization of mathematics—with or without human participation in the long term—is viewed in some circles as inevitable. To my mind, the fixation on mechanical proof checking is less interesting as a reminder that standards of proof evolve over time, which is how it’s usually treated by philosophers and sociologists,27 than as a chapter in the increasing qualification of machines as sources of validation, to the detriment of human rivals. In Yevgeny Zamyatin’s novel We, whose protagonist is a mathematician named D-503, the scourge of human subjectivity—what virtual reality pioneer Jaron Lanier calls “the unfathomable penumbra of meaning that distinguishes a word in natural language from a command in a computer program”28—is ultimately eliminated by an X-ray operation that turns people into machines.
Who finds machines more appealing than humans, and for what ends? Most mathematicians have long since recovered from the exhausting journey of the Foundations Crisis, but certification of mathematical knowledge is still a preoccupation among the “mainstream” philosophers of Mathematics, for whom participation by human mathematicians is optional. Convinced, like Hume, that no ought can follow from an is, philosophers of Mathematics skip straight ahead to the ought, disdaining the is as unfit for philosophical consideration.29 When I insist, like philosophers of mathematics (from my prephilosophical internal vantage point) on getting the is right—the “what” question—I am in fact addressing an ought that overlaps with the ought of the philosophers of Mathematics but is by no means identical, nor is the overlap necessarily very substantial.
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Some observers suggest the time might be ripe in mathematics for a paradigm shift, arguing either on epistemic or stylistic grounds.30 The proposed paradigm shifts discussed above, in contrast, are based on conceptions of mathematics as serving competing ethical ideals: productivity for the international market in the first case, a standard of certain truth in the second. The two ideals coexist in contemporary mathematics departments, but they have been in tension since the time of Plato. The classic account in is Plutarch’s Life of Marcellus:
To [his mechanical engines Archimedes] had by no means devoted himself as work worthy of his serious effort, but most of them were mere accessories of a geometry practised for amusement, since in bygone days Hiero the king had … persuaded him to turn his art somewhat from abstract notions to material things, and by applying his philosophy somehow to the needs which make themselves felt, to render it more evident to the common mind [hoi polloi].
For [this] mechanics*, now so celebrated and admired, was first originated by Eudoxus and Archytas, who embellished geometry with its subtleties, and gave to problems incapable of proof by word and diagram, a support derived from mechanical* illustrations that were patent to the senses…. But Plato was incensed at this, and inveighed against them as corrupters and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of abstract thought and descended to the things of sense, making use, moreover, of objects which required much mean and manual labour. For this reason mechanics* was made entirely distinct from geometry, and being for a long time ignored by philosophers, came to be regarded as one of the military arts.31
Plutarch adds that Archimedes himself “regard[ed] mechanics and every art that ministers to the needs of life as ignoble and vulgar.” Centuries later, Omar al-Khayyām, no less indignant than Plato, contrasted “the search for truth” with “vile material goals”:
[M]ost of those who in this epoch of ours compare themselves with philosophers drape truth with falsehood, and so they do not go beyond the borderline of deceit and the simulation of knowledge; and they do not use the amount of sciences they know except in vile material goals. And if they witness someone who is interested in the search for truth and has a predilection for veracity, endeavoring to reject falsehood and lie and to abandon hypocrisy and deceit, they consider him stupid and they laugh at him.32
“[A]s producers of cultural products the ultimate legitimacy of which is at the mercy of society at large, mathematical communities … at some point need to make a case for their own relevance, and take up a position within their social milieu.”33 But we have seen that “truth” is no longer as compellingly relevant as it was for Plato and Khayyām; and the priorities of Goodyear and Birdseye appeal as little to today’s pure mathematicians as they did to Archimedes. Other ideals are available.34 Mathematicians wishing to treat our discipline as an end rather than a means often fall back, as Bombieri did, on justification on aesthetic grounds. G. H. Hardy’s A Mathematician’s Apology, analyzed at length in chapter 10, provides the prototypical formulation: “real mathematics … must be justified as arts if it can be justified at all.”
A curious art, not recognized as such except by mathematicians, without listings in the Sunday supplements and without critics, as the philosopher Thomas Tymoczko noted.35 One obvious difficulty with this use of art is that the aestheticizing mathematician uses the word art as a synonym of beauty. Thus Cédric Villani, his freshly minted 2010 Fields Medal in hand, told a French parliamentary committee that “the artistic aspect of our discipline is [so] evident” that we don’t see how anyone could miss it … immediately adding that “what generally makes a mathematician progress is the desire to produce something beautiful.”36 It has been a long time since one could unproblematically substitute artist for mathematician in that last sentence. Catalan conceptual artist Antoni Muntadas interviewed more than one hundred major players in the art world between 1983 and 1991 and published excerpts from the transcriptions; the word beautiful is used only three times: twice incidentally, once to invite museum-goers to “let go of your preconceived notions that art has to be beautiful.”37 But I think the confusion is deeper. When mathematicians refer to beauty, there are reasons to suspect they really mean pleasure. But it’s a specific kind of pleasure they have in mind—mathematical pleasure—and if our goal is to explain how aesthetics can serve as an ideal for mathematics, we find we are practically back where we started.
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The utility of practical applications, the guarantee of absolute certainty, and the vision of mathematics as an art form—the good, the true, and the beautiful, for short,—have the advantage of being ready to hand with convenient associations, though we should keep in mind that what you are willing to see as good depends on your perspective, and on the other hand the true and the beautiful can themselves be understood as goods. Even when such justifications are given, it shouldn’t be assumed mathematicians and the Department of Business, Innovation, and Skills understand utility in the same way, or that truth means the same to mathematicians as to philosophers of Mathematics, and I’ve already argued that mathematicians who talk about beauty really mean something else.
The good and the beautiful will be examined at length in subsequent chapters, especially in chapter 10. I want to focus for a moment on the true, the source of some of the most persistent misconceptions about the point of mathematics. In the first place, we should get out of the habit of assuming that mathematics is about being rational, at least as this is understood by philosophers of Mathematics. David Corfield’s paper quoted in the previous chapter was originally entitled “How Mathematicians May Fail to be Fully Rational.” Corfield, we have seen, is a philosopher interested in mathematics rather than Mathematics, and his paper was a sympathetic exploration of what it meant to be faithful to the specific kind of rationality that implicitly guides mathematical practice. Whether or not one agrees with Corfield’s solution—namely, that mathematics is a “tradition-constituted enquiry” in the sense of Alasdair MacIntyre, to whom we return below38—there’s no reason to assume the rationality proper to pure mathematics is exhausted by the attempt to ground proof on unshakeable foundations. Quite the contrary. Mathematicians frequently write that they see a proof not as a goal in itself, but rather as a confirmation of intuition. Thus G. H. Hardy once claimed that “proofs are what [collaborator J. E.] Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils.” More recently, the eminent British mathematician Sir Michael Atiyah could write:
I may think that I understand, but the proof is the check that I have understood, that’s all. It is the last stage in the operation—an ultimate check—but it isn’t the primary thing at all.”39
The answers are less important than how they change the way we look at the questions. So the proof confirms that we have found the right way.40
This fits into a general debate among philosophers on the epistemic significance of heuristics. In his classic Homo Ludens, Johan Huizinga quotes a Dutch proverb: “It is not the marbles that matter, it’s the game.”41 To those of my colleagues who predict that computers will soon replace human mathematicians by virtue of their superior skill and reliability in proving theorems, I am inclined to respond that the goal of mathematics is to convert rigorous proofs to heuristics. The latter are, in turn, used to produce new rigorous proofs, a necessary input (but not the only one) for new heuristics.42
What passes for paradigm shifts in mathematics can plausibly be traced to heuristic rather than axiomatic innovation. Voevodsky’s univalent foundations project looks like,43 and in fact is, a sophisticated and highly technical research bridge between ways of thinking about topology, the kind of mathematics that has been a constant inspiration in Voevodsky’s work, and ways of thinking about what are usually called the foundations of mathematics, by which I mean how one goes about axiomatizing formal mathematical reasoning, a topic one would expect to be of interest to all mathematicians. The bridge is astonishing in many ways—I will try to do it justice in chapter 7—and it is already stimulating thinking in the two domains it joins. If and when univalent foundations is adopted as a replacement for today’s (largely informal) standard foundations, it will probably be on roughly Kuhnian grounds, but in a positive sense: rather than being provoked by an insoluble crisis, any change is more likely to be triggered by a demonstration of the new method’s superiority in addressing old problems or in accommodating neue Erscheinungen in Kronecker’s sense.44 This is why Grothendieck’s approach to algebraic geometry replaced André Weil’s “foundations” in the late 1950s and why Langlands’ unifying vision brought the somewhat arcane theory of automorphic forms into the center of mathematical life about ten years later. An account of paradigm shifts in contemporary mathematics might focus on such reworkings of “foundations,” understood not as the bedrock of certainty but as a common language. Yuri Ivanovich Manin, one of the most philosophically sensitive Russian mathematicians of his generation, points to the ambivalence of the word as used by mathematicians:
I will understand ‘foundations’ neither as the para-philosophical preoccupation with the nature, accessibility, and reliability of mathematical truth, nor as a set of normative prescriptions like those advocated by finitists or formalists. I will use this word in a loose sense as a general term for the historically variable conglomerate of rules and principles used to organize the already existing and always being created anew body of mathematical knowledge of the relevant epoch…. [F]oundations in this wide sense is something which is relevant to a working mathematician, which refers to some basic principles of his/her trade, but which does not constitute the essence of his/her work.45
Hermann Weyl hints that the quest for reliable foundations has spiritual roots: “Science is not engaged in erecting a sublime, truly objective world … above the Slough of Despond in which our daily life takes place.” Mathematicians will judge univalent foundations on pragmatic mathematical grounds—as we did when we adopted Grothendieck’s or Langlands’ new foundations for their respective subjects—rather than on the logical grounds of its conformity to norms of absolute certainty; as foundations rather than Foundations. But even if certainty were mathematicians’ primary target, there’s no reason to think this would satisfy philosophers of Mathematics. Paul Benacerraf’s 1973 article Mathematical Truth has landmark status in the mainstream tradition for its claim that philosophers cannot simultaneously rationally account for truth and knowledge in mathematics, whatever Foundations one chooses. Other philosophers of mathematics retort that logical Foundations are not the point:
When Paul Benacerraf limits the articulation of mathematical truth to logic and then complains that the ability of mathematicians to refer has been lost, it is no wonder; it is also no wonder that number theorists and geometers have not borrowed the language of logic to do their work.46
Faced with the task of identifying the point of mathematics, sympathetic authors fall back on terms like experience or practice or intellect or human thought, all of them ambiguous and resistant to logical formalization. Referring to what he called the “antiphilosophical doctrines” of philosophers of Mathematics, logician Georg Kreisel argued in 1965 that “Though they raise perfectly legitimate doubts or possibilities”—in other words, they help mathematicians keep their riotous impulses in check—“they just do not respect the facts, at least the facts of actual intellectual experience.” In his account of the Foundations Crisis, Jeremy Gray argues that “The logicist enterprise, even if it had succeeded, would only have been an account of part of mathematics—its deductive skeleton, one might say…. mathematics, as it is actually done, would remain to be discussed.” Philosophy of Mathematics remains encumbered by its “tendency to reduce [mathematics] to some essence that not only deprives it of purpose but is false to mathematical practice.” The Italian geometer Federigo Enriques put it this way in his lecture at the 1912 International Congress in Cambridge:
The act of free will that the mathematician embarks upon in the formulation of problems, in the definition of concepts, or in the assumption of a hypothesis, is not arbitrary. It is the opportunity to get closer—from more than one perspective, and by continuous approximations—to some ideal of human thought, i.e. an order and a harmony that reflect its intimate laws.47
Capital-F Foundations may be needed to protect mathematicians from the abyss of structureless reasoning, but they are not the source of mathematical legitimacy. Herbert Mehrtens views the Foundations Crisis as the culmination of a conflict between moderns and countermoderns [gegenmoderne]. Mechanization of mathematics is modern in Mehrtens’s sense; the countermoderns “wanted both to preserve in signs a residue of reference and to continue to find the certainty of mathematics “in the human intellect” and not “on paper” (today’s Papier is digital).48 Practically all mathematicians working today are countermodern in that sense. Sociologist David Bloor imagines that he is reinventing the countermoderns when he writes that “[e]ven in mathematics, that most cerebral of all subjects, it is people who govern ideas not ideas which control people.”49 Handing over mathematical communication to mechanical proof-checkers would eventually fix that, in an ultimate triumph for the moderns, but it would also withdraw mathematics from the purview of sociology altogether, while paradoxically vindicating Bloor’s questionable decision to turn his sociological eye to philosophy of Mathematics rather than to human mathematical practice.
So what is the point of human mathematical practice? In line with Hans Reichenbach’s classic separation of the context of justification, considered to be the proper domain of philosophy of science, from the (contingent and, therefore, philosophically irrelevant) context of discovery, Jeremy Avigad has argued that
[w]e have all shared such ‘Aha!’ moments and the deep sense of satisfaction that comes with them. But surely the philosophy of mathematics is not supposed to explain this sense of satisfaction, any more than economics is supposed to explain the feeling of elation that comes when we find a $20 bill lying on the sidewalk … we should expect a philosophical theory to provide a characterization of mathematical understanding that is consistent with, but independent of, our subjective experience.50
The obvious problem with this paragraph is that if the border between subjective experience and mathematical understanding were clearly delineated, there would be less need for philosophy of mathematics. But maybe that “deep sense of satisfaction” is a key to an understanding deeper than philosophers of Mathematics might encounter in their Foundational dreams. Embedded in Hilbert’s list of twenty-three problems is one answer to the “why” question. Explaining in 1900 how he compiled that list, Hilbert wrote that “[a] mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the convoluted paths to hidden truths, ultimately rewarding us with the pleasure [Freude] of the successful solution.”
The short answer to the “why” question is going to be that mathematicians engage in mathematics because it gives us pleasure, Hilbert’s Freude. “What,” ask David Pimm and Nathalie Sinclair, “is mathematics’s debt to pleasure? Specifically, what is the pleasure of the mathematical text? Might there even be an aesthetics of mathematical pleasure to be developed?”51 Already in the fourth century BC, the Greek mathematician Eudoxus was disapprovingly singled out in the first and last books of Aristotle’s Nicomachean Ethics for identifying “the good” with pleasure. The word plaisir [pleasure] occurs on 106 of the 900+ pages of Récoltes et sémailles, Grothendieck’s unpublished (and unpublishable) memoirs. Grothendieck is off the scale in many ways, but Bernard Zarca’s recent sociological survey of attitudes of mathematicians (see chapter 2, note 56) shows that in his pursuit of pleasure, Grothendieck fits squarely within the mathematical norm. Zarca’s subjects were given a list of sixteen “dimensions” of their work and asked to rate their importance. “The pleasure of doing mathematics” was rated “absolutely important” or “very important” by 91% of pure mathematicians, a significantly higher proportion than among applied mathematicians (88%) or scientists in related fields (76%) and much higher than the second choice—pleasure, again, this time in “sharing, by exchange and communication, the results one obtains”—given top rating by 66%.52 Some of the other dimensions referred to the good in the form of applications, while the true and the beautiful were curiously absent; but all three re-appeared, slightly disguised, on a list of four “virtues” of mathematics—“artisanal pleasure” was the fourth—that respondents were asked to place on a scale from 0 to 4. No surprise: for pure mathematicians, beauty comes first, followed closely by “artisanal pleasure,” then “understanding of the world,” with “utility” dead last. (For applied mathematicians, utility comes far ahead of the other three.) And since Zarca specifically inserted the word artisanal to forestall confusion with other kinds of pleasure (notably aesthetic, intellectual, or moral), we should assume that today’s pure researchers still see pleasure as mathematics’ primary virtue.53
I fervently hope this book makes a modest contribution to some future comprehensive and unexpurgated guide to mathematical pleasure. It deserves to be investigated, for example, why no one has ever put much energy into arguing, as Aristotle might have done, that moderation in mathematical pleasure is a virtue. In the meantime, we have not yet explained why we should be paid for the time we spend working on mathematics. If it’s so enjoyable, wouldn’t we do it for free? Well, maybe we wouldn’t be so efficient or creative if we had to work for a living—in our spare time, so to speak—at something we presumably don’t enjoy so much.54 It might help to put this question in perspective to remember that mathematicians are not the only scholars (or academics, to use a term better adapted to contemporary reality) under pressure to defend their pursuit of a pleasurable activity, at public expense, in the context of generalized austerity. In a letter to David Willetts, the minister responsible for universities, science, innovation, and space (within that Department for Business, Innovation and Skills), the British Philosophical Association (BPA) attempted to rise to the challenge in 2010, shortly after the government eliminated the Middlesex University philosophy department. While agreeing “that the money provided by the taxpayer for philosophical research should be put to good use” and that “it would be entirely appropriate for the government to expect us to be able to justify our view that that money is indeed being well spent,” the philosophers denied that “impact” can “meaningfully be measured over short periods of time or at the level of individual researchers or groups of researchers within a particular institution.” Instead, they offered an eclectic menu of some of the “enormous benefits that philosophy has been responsible for” (justice, democracy, the scientific method, computers, secular ethics, animal welfare …), in the process making painfully obvious the gulf between the disciplinary ethos of philosophers and today’s world of “business, innovation and skills.”55
The philosophers chose not to answer another question: why is it a matter of general interest, independently of the uncertain prospects of short-or long-term benefits to human welfare, to have a small group of people working at the limit of their creative powers on something they enjoy? If a government minister asked me that question, I could claim that mathematicians, like other academics, are needed in the universities to teach a specific population of students the skills needed for the development of a technological society and to keep a somewhat broader population of students occupied with courses that serve to crush the dreams of superfluous applicants to particularly desirable professions (as freshman calculus used to be a formal requirement to enter medical school in the United States). The prospect of taking pleasure in the research we freely determine would then be the lure that gets us into the classroom (the assumption that not all of us are eager to be there is a gross misreading of the true state of affairs). Or I could revert to the Golden Goose argument. But if the question is taken at face value, it answers itself. Indeed, if the notion of general (or public) interest means anything at all, it should be a matter of general interest that work be a source of pleasure for as many people as possible.
Play, as analyzed in Homo Ludens, shares many characteristic features with the practice of pure mathematics. Zarca asked his subjects to choose two from a list of eight descriptions of mathematical activity; 62% of pure mathematicians chose “a very pleasurable sort of game, with the appeal of the unknown,” nearly twice as many as the next most popular choice (“an inner necessity”). Huizinga defines play as “an activity which proceeds within certain limits of time and space, in a visible order, according to rules freely accepted, and outside the sphere of necessity or material utility.”56 The condition on time and space may be irrelevant, but the other three items on Huizinga’s list are unmistakable characteristics of pure mathematics.
On ethical grounds, Aristotle’s account of music can serve as a model (if debatable) justification for mathematics: “Nowadays most people practice music for pleasure, but the ancients gave it a place in education, because Nature requires us not only to be able to work well but also to idle well” (my emphasis). Huizinga comments that “for Aristotle [idleness] is preferable to work; indeed, it is the aim (telos) of all work.” Aristotle’s “demarcation between play and seriousness is very different from ours….” “Idleness” for Aristotle means the freedom to pursue “such intellectual and aesthetic preoccupations as are becoming to free men …. The enjoyment of music comes near to being … a final aim of action because it is sought not for the sake of future good but for itself. This conception of music sets it midway between a noble game and ‘art for art’s sake.’ ” Bellah adds that, for Aristotle “Theoria”—contemplation, the search for knowledge—“is useless. It is a good internal to itself, but it has no consequences for the world.”
In trying to distinguish science from play, Huizinga argues that the two differ in that the rules of science are subject to paradigm shifts, “whereas the rules of a game cannot be altered without spoiling the game itself.”57 What does this say about mathematics? We have seen that the rules of mathematics—the “foundations” in Manin’s epistemologically neutral sense—are regularly modified in response to changed priorities; Voevodsky’s univalent foundations may be adopted as new axioms. But the perception on the part of those who adopt new rules is that they represent an improvement over the earlier rules, and I think Huizinga would have been wrong to argue that a proposed improvement of the rules would spoil a game.
Freud claimed that “the opposite of play is not seriousness, but—reality.”58 I am tempted to say that, whereas Mathematics aspires to be Real as well as Serious, mathematics may well be serious but is, at best, virtually real: not at all opposed to play. But play, like Aristotle’s idleness, is not only hard to sell to the Gradgrinds of Britain’s Department of Business, Innovation, and Skills; it also has awkward associations for mathematicians. Most mathematicians agree with Voevodsky that Hilbert’s formalism, his reading of mathematics as a rule-based game, failed on technical grounds to achieve its primary goals. But it was scarcely more successful as an ideal. Long before Gödel announced his incompleteness theorems, Weyl wrote that mathematics “is not the sort of arbitrary game in the void proposed by some of the more extreme branches of modern art.” More recently, Neil Chriss, who chose to forgo a promising future in the Langlands program to work for a series of wildly successful hedge funds, explained that he gave up mathematics in part because “it is so disconnected with the rest of the world.”
“[Hermann Hesse’s] The Glass Bead Game [Magister Ludi—literally “master of the game”] is a favorite novel among my mathematician friends.” In it “a monastic order of intellectuals has split apart from [a future] society. They engage in a complicated, all consuming game—the Glass Bead Game—and strive to achieve mastery of it … to members of the order it represents everything that is important in the world…. it is hard to believe that Hesse wasn’t describing research mathematics.”59
“The master of the moderns defines himself as a ‘free mathematician,’ as a ‘creator,’ ” writes Mehrtens, “and the countermoderns accuse him of creator’s arbitrariness.”60 Even when we grant that mathematical play is not sterile, in the way that the Glass Bead Game strikes most readers as sterile, it’s risky to identify mathematics as a form of creative play. Computerization may have stripped chess of much of its pathos, of the sense that it engages the players on the deepest strata of their existence (imagine Bergman’s Seventh Seal with HAL the computer in the role of Death); and a mathematics that sees itself as a game played on an endless chessboard leaves itself open to a similar devaluation.
So let’s leave play to the side for the moment and retain from Homo Ludens a theoretical principle—“We may well call play a ‘totality’ in the modern sense of the word, and it is as a totality that we must try to understand and evaluate it” and an ethical stance: “Most [hypotheses] only deal incidentally with the question of what play is in itself and what it means for the player…. ‘So far so good, but what exactly is the fun of playing?’ ” Reviel Netz echoes this stance in a book entitled Ludic Proof: “people do the things they enjoy doing.”61 What Netz calls a “modest” assumption seems unremarkable; but we have seen that the dominant justifications of mathematics all take for granted that mathematics is a “thing” people do with some other goal—goodness, truth, or beauty—in mind. That it should be a matter of general interest that work be a source of pleasure for as many people as possible, as I wrote before, ought to go without saying. But apparently it doesn’t where pure mathematics is concerned; otherwise there’s no explaining why mathematical research is consistently, and implausibly, presented as a means rather than an end.
When the German mathematician Heinrich Liebmann characterized mathematics as a “free, creative art” [freie, schöpferische Kunst] in his 1905 Leipzig inaugural address, the stress was on the adjectives, the word art being no more than the noun required by syntax. The advantage of using art—rather than that-which-is-free-and-creative, which is what is really meant here—isn’t just that “art” is shorter but that it conjures up a picture. It’s a misleading picture: when mathematicians say art, we’ve seen that they generally mean beauty. Other pictures are available.62 Ethologist Gordon Burghardt proposes five criteria to distinguish play from other forms of animal behavior; the fifth is that play takes place “in a ‘relaxed field.’ ” “One of the prime characteristics of play” for Burghardt “is that the animal is not strongly motivated to perform other behaviors. The animal is not starving (or even very hungry), it is not at the moment preoccupied with mating, setting up territories, or otherwise competing for essential resources or escaping predators.”63
Burghardt’s relaxed field is Eden (or Flexner’s paradise) before the fall. The deer and the antelope play in their relaxed field, but they don’t apply for National Science Foundation research grants. The theory of the relaxed field posits play as the opposite of stress. Play, for John Dewey, is not the opposite of work: “Play remains as an attitude of freedom from subordination to an end imposed by external necessity, that is, to labor; but it is transformed into work in that activity is subordinated to production of an objective result.”64 Reviewers asked65 to assign objective grades to mathematics grant proposals are also mathematicians and interpret their instructions according to the values impressed upon them by their socialization; they seek objectivity in the locus of Liebmann’s free creativity, which is another name for the relaxed field.
If we take the values themselves as the starting point, rather than the relaxed field in which they are free to express themselves, then we find ourselves in the sphere of ethics. Crispin Wright, a prominent philosopher of Mathematics, was more candid and more credible than the British Philosophical Association when asked “Should philosophy be funded, even if funding it holds forth almost no prospect of improving the lives of ordinary people? If so, why?”
[C]ontrary to what one might first assume, philosophical process is a very large part of the value of philosophy. Suppose it became possible to program computers to take over most of the projects of philosophical research currently being pursued in academia … Few would feel, “Well good, we can leave all that to the machines now, and get on with other things.” … it is crucial that [the products of philosophical research] be attained by human beings, and strongly preferable that they be attained by a shared process in which there is conversation and mutual understanding of why what results results, of the conceptual pressures and constraints that shape it (Wright 2011).
In other words, philosophy is what Alasdair MacIntyre calls a tradition-based practice—though Wright might not use this vocabulary to describe it—and this is responsible for much of its value.
MacIntyre uses the term practice to mean
any coherent and complex form of socially established cooperative human activity through which goods internal to that form of activity are realized in the course of trying to achieve those standards of excellence which are appropriate to, and partially definitive of, that form of activity, with the result that human powers to achieve excellence, and human conceptions of the ends and goods involved, are systematically extended.66
This isn’t really meant as a definition; the part from “standards of excellence” to “form of activity” looks temptingly circular, for example, and I wouldn’t want to pin down what excellence or complex means to MacIntyre any more than what important or ambitious or major might mean to the authors of the ERC guidelines (or what visual perfection, quality, and aesthetic mean in the interviews recorded in Muntadas’s book67). But I don’t see that as a problem. The tradition-based approach is circular: like the Mehrtens quotation with which I ended the first chapter, it asserts that the point of doing mathematics is to do mathematics. And the tradition-based approach is not circular: it invites us to attend to description rather than prescription—to address the “why” question by returning to the “what” question.
Or to put it another way: there is no need to seek the meaning of mathematics elsewhere than in the practice constituted by tradition; and the telos of mathematics is to develop this meaning as a way of expanding the relaxed field. MacIntyre is at pains to distinguish his notion of tradition from the conservative version associated with Edmund Burke. “[A]n adequate sense of tradition” for MacIntyre “manifests itself in a grasp of those future possibilities which the past has made available to the present.” “[T]he history of a practice in our time is generally and characteristically embedded in and made intelligible in terms of the larger and longer history of the tradition through which the practice in its present form was conveyed to us.”68 MacIntyre’s nondefinition of practice draws attention to just what it is that matters about the “socially established cooperative human activity” of pure mathematics; and this focused attention is what makes mathematics work in Dewey’s sense, “production of an objective result,” while remaining relaxed, free “from subordination to an end imposed by external necessity.”
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Crispin Wright had this to say about “improving the lives of ordinary people”:
A good philosophical process will be one which, necessarily, is appreciated as such by its participants, as interesting, eye-opening, inspiring, and perhaps importantly revisionary…. If it is a good for those who participate, then in a pluralist and civilized society, participation should be encouraged, and its scope should be widened as far as possible.69
This book does not pretend to justify mathematics; no one needs another Mathematician’s Apology. It’s goal is not so much to explain mathematics as to convey to Wright’s “ordinary people”—presumably those whose lives are not necessarily directly bound up with mathematics—what it is like to be a mathematician, freely choosing a tradition to which to adapt, not to serve the Powerful Beings of market rationality nor the metaphysical Powerful Beings of our own creation. The pleasure of the mathematical tradition is inseparable from the pathos of its relaxed field, where the poles of some familiar antinomies—mind versus body, finite versus infinite, and necessity versus contingency, just to mention some of the themes of this book’s three central chapters—stage confrontations that evolve along with the preoccupations of participants in the tradition, who don’t necessarily imagine or even care how they might eventually reach resolution.
My hope is that exposure to the peculiarity of this pathos may enrich the lives of “ordinary people.” In this book I am trying to make the mathematical tradition just real enough to make the pathos palpable. Here the presumed, but largely unsubstantiated, parallel between mathematics and the arts offers unexpected clarity. Anyone who wants to include mathematics among the arts has to accept the ambiguity that comes with that status and with the different perspectives implicit in different ways of talking about art. Six of these perspectives are particularly relevant: the changing semantic fields the word art has historically designated; the attempts by philosophers to define art, for example, by subordinating it to the (largely outdated) notion of beauty or to ground ethics in aesthetics, as in G. E. Moore’s Principia Ethica, which by way of Hardy’s Apology continues to influence mathematicians (see chapter 10); the skeptical attitude of those, like Pierre Bourdieu, who read artistic taste as a stand-in for social distinction; the institutions of the art world, whose representatives reflect upon themselves in Muntadas’s interviews; the artist’s personal creative experience within the framework of the artistic tradition; and the irreducible and (usually) material existence of the art works themselves.
Conveniently, each of these six approaches to art has a mathematical counterpart: the cognates of the word mathematics itself, derived from the Greek mathesis, which just means “learning,” and whose meaning has expanded and contracted repeatedly over the millenia and from one culture to another, including those that had no special affinity for the Greek root; the Mathematics of philosophers of “encyclopedist” schools; school mathematics in its role as social and vocational filter; the social institutions of mathematics with their internal complexity and their no-less-complex interactions with other social and political institutions; the mathematician’s personal creative experience within the framework of the tradition (the endless dialogue with the Giants and Supergiants of the IBM and similar rosters); and the irreducible and (usually) immaterial existence of theorems, definitions, and other mathematical notions. This book touches on each of these ways of talking about mathematics, but the guiding orientation is to a point somewhere between the “relaxed field” and the “tradition-based practice” in the fifth item on the list, which is where the pathos dwells in mathematics as well as in the arts.70
I would not compare mathematics to religion, but in reading Robert Bellah’s Religion in Human Evolution, I am struck by structural parallels at least as relevant to the analysis of the respective fields as the parallels with the arts. Each of the ways of talking about art and mathematics listed before has its equivalent in Bellah’s historical sociology of religion; it was in Bellah’s book that I first learned about the relaxed field; and Bellah’s allusion to MacIntyre’s notion of practice reinforced what Corfield’s approach had already taught me, namely, that many of the most persistent questions about the nature of mathematics are rooted in ethics rather than epistemology. In Bellah’s work there is also a tension, to which he draws attention only briefly, between the relaxed field of religious ritual, “a form of life not subject to the struggle for existence,” and the need to perform ritual in order to propitiate “Powerful Beings.” During the ritual dances of the Kalapalo of the Upper Xingu Basin of central Brazil, “ ‘common humanity’ … takes over from the divisions of everyday life.” Bellah quotes the anthropologist Ellen Basso: “In ritual performance … the body is an important musical instrument that helps to create … the experience—however transient—that one is indeed a Powerful Being.”
Bellah insists that the “ ‘Powerful Beings’ of the Kalapalo, who were there ‘at the beginning,’ ” are not (yet?) gods. But like other cultures, the Kalapalo believe that “Human life derives ultimately from the Powerful Beings” who “could be terribly destructive when crossed … but with whom people could identify if they followed the proper ritual, and, through identification, their power could become, at least temporarily, benign.” In more complex societies, according to Bellah, the Powerful Beings are worshipped as gods, and rituals are largely connected with maintaining the basic material conditions of life; the Hawaiian Lono, to whom the four-month carnival-like Makahiki festival is devoted, is “the nourishing god.”71
It is understood that the arts, however they are defined, entail reflection or representation of the ambient society. Reflection on this reflection is the province of critical theory and cultural studies, academic-based institutions parallel to the institutions of the art world proper, whose methods are routinely appropriated by artists themselves (like Muntadas). Mathematics has given rise to no such institution. This conspicuous difference between mathematics and the arts is more consequential than the absence, already noted, of mathematics critics; it means mathematics lacks the institutional capacity to reflect critically on the material conditions of its own creation. In pursuing their interests, today’s Powerful Beings manage to create the conditions for keeping as many people as possible in a state of stress rather than relaxation. A particularly insistent narrative of Powerful Beings in the next chapter shows how mathematicians have been innovating and applying their skills to find new ways to make this possible.
* Explanation to follow.