chapter 8

The Science of Tricks

Banish the tunes of Cheng and keep clever talkers at a distance. The tunes of Cheng are wanton and clever talkers are dangerous.

—Confucius, Analects 15:111

Individuals who never sense the contradictions of their cultural inheritance run the risk of becoming little more than host bodies for stale gestures, metaphors, and received ideas.

—Lewis Hyde, Trickster Makes This World, p. 307

MY KUNSTGRIFF

In January 2010 it was revealed that I am a trickster. Toby Gee, a young English mathematician, broke the story. Before a packed Paris auditorium, Gee explained how he and two even-younger colleagues had found a new way to exploit what he called “Harris’s tensor product trick”—implicitly allowing that it might not be the only item in my bag of tricks, that I’m not necessarily a one-trick pony—to improve on my most recent work. I had first employed the trick in a solo paper, but it had been recycled to much greater effect in my joint paper with Richard Taylor and two of his students, who in turn recycled themselves as Toby Gee’s collaborators. During his talk, he hinted that there was more to the story. At the break Gee let me know that by March, he and all three of my erstwhile collaborators would have combined a version of my trick with a few new “key ideas” to prove a big new theorem that incorporates most previous results in the field—notably all the results to which my name was attached.

The trickster, in one of many disguises—the Yoruba Esu-Elegbara or his African American equivalents, the Signifying Monkey and Br’er Rabbit; the Winnebago Wakdjunkaga, and the Coyote of the American Southwest; or Hermes, Prometheus, and Loki from European mythology—attended nearly every skirmish of the 1990s culture wars but had already tired of running laps around the postcolonialist circuit by the time the archetype’s presence as a mathematical figure was abruptly brought to my attention. And here I am barely into the second paragraph and already playing tricks on you—at least two, if you’re keeping count, with more on the way. Most blatantly, there’s no such thing as a trickster in mathematics. There are plenty of mathematical tricks, enough to fill a “Tricks Wiki” or “Tricki,” a “repository for mathematical tricks and techniques.”2 But even the rare mathematicians skilled at inventing trick after trick—the “mathematical wit” Paul Erdős comes to mind—are not known as “tricksters.” What my colleagues call a trick is a kind of mathematical gesture or speech act; and what I’m calling a trickster is a role, a persona, one of several, an attribute not of the author who invents or employs it but rather of the text in which this kind of argument or idea or style of thought appears.3

As for the trick Toby Gee mentioned in his lecture, it wasn’t really even mine, although I was the first to notice its relevance in the context of mutual interest. I learned about this particular class of tensor product tricks from a few of my contemporaries (when I was a not-yet-old dog—just around Toby Gee’s age as I write this) and I’ve applied them—we don’t “play” tricks, much less “turn” tricks, in mathematics—to solve problems in at least three completely different settings. So it would be ironic4 if “Harris’s tensor product trick” were to be the last of my marks to fade from number theory, like the Cheshire cat’s smile. And as “tricky Dick” Nixon’s career reminds us, it is a mixed blessing to be remembered primarily for one’s tricks.

You don’t really want to know the details of my trick, but you may be wondering what makes it a trick rather than some other kind of mathematical gesture. I ought to be able to answer that question if anyone can, because I’m the one who called it a tensor product trick! Whatever was I thinking? Frankly, I wish I knew. To determine when I began to view this kind of argument as a trick would require an experiment in personal intellectual archaeology or at least a time-consuming scouring of my old hard drives; but it’s too late to deny my responsibility for the terminology.5 There’s my usual false modesty, of course, labeling my new invention a (mere) trick in the hope of eliciting warm praise and protestations that “it’s more than a trick, it’s a game-changer.”6 But the hypothetical mathematical ethnographer will need to understand that not every mathematical speech act deserves to be called a trick and will want to know why.

A straightforward calculation, for example, is certainly not a trick. Nor is a syllogism, a standard estimate of magnitude, or a reference to the literature. Can I be more precise? Probably not. While capital-M Mathematics is neatly divided among axioms, definitions, theorems, and proofs, the mathematics of mathematicians blurs taxonomical boundaries. A mathematical trick, like a trickster, is a notorious crosser of conventional borders; a “lord of in-between” like the Yoruba trickster Eshu, “who dwells at the crossroads,” a mathematical trick simultaneously disturbs the settled order and “makes this world,” to quote the title of Lewis Hyde’s classic study. I would suggest that a trick involves drawing attention to an intrinsic element of a mathematical situation that appears to be, but is not, in fact, irrelevant to the problem under consideration. Alternatively, since a trick need not be subordinated to a preexisting problem, it provides an unexpected point of contact, like a play on words, between two domains not previously known to be related.7 Thus Lieberman’s trick, the first trick I saw identified as such, roughly consists in the use of multiplication in a situation when only addition seems relevant; the unitarian trick of Adolf Hurwitz, Hermann Weyl, and Issai Schur introduces a measure to solve a purely algebraic problem; my trick, like other tensor-product tricks, uses the possibility of a kind of (matrix) multiplication to reveal a structure not otherwise visible. Chapter β.5 mentioned a trick connected with Cantor’s theory of infinity. The reader (or listener) in each case undergoes a prototypical Aha! experience, the apprehension of a gestalt: the connection is obvious, but first you have to experience it.

The ambivalence of the word trick, whose associations include magic, prostitution, and deceit, neatly reflects the tension between satisfaction with a synoptic proof and disapproval of the shortcut that avoids the hard work, or the grappling with essential matters, without which recognition seems undeserved. As idealized by logical empiricist philosophers, Mathematics with a capital M is insensitive to the complex interplay of delight (a “neat trick”) and disdain (a “cheap trick”) that accompanies the revelation of a new mathematical trick and constitutes a privileged moment of pleasure, precisely like that afforded by magic tricks or like García Márquez’s reaction, nearly falling out of bed when he read the first sentence of Kafka’s Metamorphosis: “I didn’t know you were allowed to write like that.”8 The trickster is a mathematical magic realist who exclaims, “You didn’t know you were allowed to fly from peak to peak; you thought you had to trek, or at least calculate, your way across the wilderness. But look: a flying carpet!” This situates the trickster at the pole opposite to the lumberjack, who makes his one and only appearance as an incipient mathematical archetype in one of Langlands’s most oft-quoted exhortations:

We are in a forest whose trees will not fall with a few timid hatchet blows. We have to take up the double-bitted axe and the cross-cut saw, and hope that our muscles are equal to them.9

The ambivalence of tricks, the sense of getting something for nothing, persists in other languages. The Dutch word truuk (also spelled truc) is “[m]ostly used in connection with magicians and card tricks … a ‘truuk’ cannot be something very serious.”10 Russian mathematicians use the word tryuk (трюк), which in other settings can mean deceit or craftiness.11 In French a mathematical trick is called an astuce, whose primary association with “cleverness” or “astuteness” seems to connote approval, quite unlike the word tour used for magic tricks; jouer un tour means to “play a trick,” usually not a nice one.12

German mathematicians nowadays often use the English word trick for what traditionally was, and sometimes still is, called a Kunstgriff; this is how one properly refers in German to the unitarian trick of Hurwitz, Weyl, and Schur. Stüttgart professor Wolfgang Rump assigned the Kunstgriff a legitimate role in mathematics:

[T]ricks precede a theory, they reach into the as yet unknown, connect what is apparently separate, so that after further reflection the latter finds its natural place in the general theory and thereby becomes known.13

Like the African American trickster High John de Conquer, a mathematical Kunstgriff “mak[es] a way out of no-way.”14 But the German word has its own unsavory connotations. Arthur Schopenhauer’s unpublished 1830 manuscript, entitled Kunstgriffe, outlines thirty-eight rhetorical Kunstgriffe—“stratagems” in English—included in the posthumous compilation entitled The art of being right. It could be described as a list of dirty tricks for winning arguments or, alternatively, a training manual for the “clever talkers” of the Confucian epigraph. Kunstgriffe of this sort—for example, number 24, which offers advice on “stating a false syllogism”—are uniformly unwelcome in mathematics, but number four on the list may provide insight into the construction of this book and of the present chapter in particular:

Kunstgriff 4

If you want to draw a conclusion, you must not let it be foreseen, but you must get the premisses admitted one by one, unobserved, mingling them here and there in your talk…. Or, if it is doubtful whether your opponent will admit them, you must advance the premisses of these premisses…. In this way you conceal your game until you have obtained all the admissions that are necessary, and so reach your goal by making a circuit.15

THREE FUNCTIONS

Mathematicians used to air their ambivalent feelings about tricks in public. One finds educators opposing tricks to knowledge in 1909:

Much time was spent in trying to find a simpler way [to solve an examination problem] until the “trick” required was found…. The question remains: should such questions, based upon the use of special artifices, be set in examinations …? They are no test of the knowledge of candidates, and merely lead them into traps from which they emerge disheartened.16

As recently as 1940, the Mathematical Association of America (MAA) could publish an article by a college teacher who complained “that most of us proceed to teach certain sections of elementary mathematics in a way that discourages students by giving them the impression that excellence in mathematical science is a matter of trick methods and even legerdemain.”17 In contrast, in its advice to prospective authors of mathematical articles, the American Mathematical Society (AMS) gives tricks a positive valuation: “Omit any computation which is routine (i.e., does not depend on unexpected tricks). Merely indicate the starting point, describe the procedure, and state the outcome.”18 Readers are jaded by the routine but are always on the lookout for new tricks!

Voevodsky’s vision of mathematical publication cleared by automated proof checkers, if realized, will require the AMS to countermand current policy. All routine computations will have to be included in order to satisfy the expert system gatekeeper who guards the border with no orders to distinguish tricks from the routine. Ambivalence to tricks, even the sense that tricks form a separate genre, will then be seen as yet another transitory feature of the mathematics of the human period.19

By implication, the routine is what keeps mathematicians occupied when we are being lumberjacks rather than tricksters. This is not quite right; the routine is actually only one aspect of normal mathematical practice. Ignoring neutral words like method and procedure—grounded etymologically, like routine, in movement, specifically movement along the road, a meaning too remote to have retained any enchantment value as metaphor—one finds the protagonists of mathematical articles—both their actual authors and the first-person pronouns that narrate the action, so to speak—applying tools and techniques to transform ingredients into constructions. Sal Restivo, materialist sociologist of mathematics, put it this way:

Mathematical workers use tools, machines, techniques, and skills to transform raw materials into finished products. They work in mathematical “knowledge factories” as small as individuals and as large as research centers and world-wide networks. But whether the factory is an individual or a center, it is always part of a larger network of human, material, and symbolic resources and interactions … a social structure. Mathematical workers produce mathematical objects, such as theorems, points, numerals, functions and the integers.

Restivo was writing in the 1990s, but already in 1908 you could read in the MAA’s American Mathematical Monthly that

[m]athematics consists of ten thousand tools, but … Mathematics does not only develop a large number of simple tools … it especially emphasizes the putting together of these tools into powerful thought machines.20

Mathematicians actually use the word machine to refer to a procedure or, more often, a collection of procedures that yield a solution to a class of problems complete enough to be implemented mechanically. Pascal’s machine arithmétique, or Pascaline, was a literal machine that first solved this problem for basic arithmetic in 1642.21 The formulas illustrated in chapter β, which in constrast to the Pascaline are procedures rather than physical objects, can be described as “machines” for finding the roots to equations of degree 2, 3, and 4. We have already seen that Abel and Galois are famous for proving that there is no such machine for equations of degree 5 and greater. Contemporary algebraic topology has its canonical delooping machine, whereas number theory has an eigenvariety machine—blueprints courtesy of Toby Gee’s thesis adviser Kevin Buzzard—that provides a mechanical solution to a small part of the Langlands program.22

It should come as no surprise that mathematicians—as opposed to Mathematicians—lose interest in problems that can be implemented by machines. Plutarch recorded Plato’s irritation with Eudoxus, Archytas, and Menaechmus for introducing “constructions that use instruments and that are mechanical” [organikas kai mekhanikas] in the doubling of the cube.23 Gauss wrote in 1850 that in the symbolic language and terminology of his time, “we possess a level, by which the most complex arguments are reduced to a certain mechanism. Thereby the science has won infinitely in richness, but in beauty and solidity … it has lost just as much.”24 At the other end of the production process, one finds not a sneering and exploitative factory owner but rather a consensual leader, a genius, chosen according to the principles of workers’ self-management on the basis of sheer charisma, in the sense of chapter 2. At least that’s what we would like to believe. Grothendieck and Langlands provided guidance not only through the example of their work but also, as we’ve seen, through the active and energetic recruitment of students and collaborators. Although Langlands once posed as a metaphorical lumberjack and has long had a deserved reputation for not shirking hard work, his influence has been exercised not only through the Langlands program, whose ambitions and methods can be identified with precision, but more diffusely through what continues to be called the Langlands philosophy. “The word ‘philosophy’ was fashionable in 1967, no longer so by 1979. There were lots of philosophies in 1967…. It was just the way people talked.” Most participants active in the Langlands program dropped the word philosophy long ago, but beginners and nonspecialists continue to use it.25

Grothendieck is also frequently depicted as a philosopher: you can read about his “philosophy of the six operations,” or his “philosophy of anabelian geometry,” and especially his “philosophy of motives.”26 But the word more often associated with Grothendieck is yoga, as explained in Deligne’s remarks quoted in the previous chapter (the “yoga of weights,” the “yoga of six operations,” the “yoga of de Rham coefficients”). Grothendieck may not have introduced the word into the mathematical lexicon, but he seems to have given it its current meaning:

Par “yoga” il [Grothendieck] entendait un point de vue unifiant, une piste dans la recherche des concepts et des démonstrations, une méthode qu’on pouvait réutiliser.27

In between the routine and the exalted, one finds the level of normal mathematical problem solving, descriptions of which are dominated by the vocabulary of combat, as they have been since the beginning of my professional life. The metaphoric fields of mathematics become equally metaphoric battlefields, which is a little surprising given the generally pacific nature of the typical mathematician. Inspiration follows a strategy, a plan of attack, as in Piet Hein’s rhyme:

Problems worthy of attack

Prove their worth by hitting back

An attacking mathematician identifies strategic objectives and the obstacles to their attainment and mounts the attack, usually armed with concepts rather than weapons. Richard Dedekind defended his conceptualism as follows:

[I]t seems to me that … a theory based on calculation … does not offer the highest degree of perfection…. it is preferable to seek to extract proofs, not from calculation, but immediately from characteristic fundamental concepts, and to construct the theory in such a manner that … it shall be in a position to predict the results of calculation.

Though Dedekind did not express himself in martial terms, this is where E. T. Bell, writing in 1944, saw Dedekind formulating “the strategy of abstract algebra.”28 The strategy of a proof is then the step-by-step progression from hypotheses to conclusions, outlined in conceptual terms; the experienced speaker constructs a seminar talk on the basis of this outline, omitting routine details and departing from the script only when the strategy incorporates an unexpected feature—a trick, for example.

An attack in the absence of strategy is called brute force, which still follows a plan, but one based on the systematic use of routine methods. So the familiar contrast between theory builders and problem solvers is replaced by the metaphorical distinction between the strategist (who in the upper reaches shares some of the charisma of the philosopher) and the technician (who in the lower registers is scarcely more than a machine). Prestige is naturally associated with broad vision, so the latter term is typically used in a derogatory manner; in this context “technically very strong” is a double-edged compliment that often suffices to eliminate contenders for prestigious prizes. Powerful, on the other hand, applied to tools, methods, insights, or strategies, is a term of high praise, a word chosen expressly to focus the attention of hiring committees and granting agencies, as well as Powerful Beings of both public and private sectors.

It is tempting to map the descending scale of normal mathematical prestige onto the four traditional castes (varna) of Hinduism: the brahmins are the philosophers and yogis, and the kshatriyas are the problem solvers, with the techniques, constructions, and tools in the hands of vaishyas and sudras. In this way mathematical ethnography collides with Georges Dumézil’s trifunctional theory of Indo-European mythology. The three functions of the sovereign (king, priest; brahmin), the warrior (knight, soldier; kshatriya), and the producers of wealth (merchant, peasant, artisan; vaishya, sudra) correspond in Dumézil’s analysis to the “intellectual structure and mold of thought [moule de pensée]” characteristic of the social organization and mythological self-representation of all branches of Indo-European civilization.29 Not coincidentally, their responsibilities—for justice and relations with the spirit world, war, and production of goods—provide a catalogue of the matters any society must take seriously in order to assure its survival.

The three functions should not be read as a description of a caste system within the mathematical profession. Few can exercise the charisma of a Langlands or a Grothendieck, but the roles of warrior (problem solver) and worker (technician, toolmaker) do not correspond to a functional division of labor within the profession; they rather identify distinct aspects of the work of most individual mathematicians, each associated with a specific lexicon in the ritualized metalanguage. We have already seen Langlands inviting his colleagues to use the tools and the brute force of the lumberjack; but he has promoted and continues to promote specific strategies to achieve intermediate goals and, more recently, to promote his “reckless” strategy that may potentially lead to a realization of his entire program.30

Mythological wars are not particularly memorable for their strategy. The action of the Mahabharata, the Iliad, and Ferdowsi’s Shahnama is dominated by brute force and skill with weapons. When we look for hints of strategy, we instead find dirty tricks. The dirtiest trick in the western canon is undoubtedly the Trojan horse, devised by Odysseus polumekhanos [of many devices].31 At least two of the Mahabharata’s key battles were won thanks to dirty tricks suggested by Krishna, the epic’s chief trickster, over the protests of the Pandava brothers, with whom he was allied.32

There are obvious objections to the imposition of a trifunctional model on mathematicians’ apparently informal but, in fact, deeply ritualized discourse. Leaving aside the global nature of contemporary mathematics, influenced by cultural strands having little or no connection to Indo-European roots, perhaps the most cogent objection is that mathematics is hardly the only activity to which a trifunctional analysis can be applied. In business circles we hear about management philosophy, commercial strategy, and marketing tools, while politicians have a philosophy of government, a political strategy, and techniques of communication. The distinctiveness of mathematics may lie, after all, in the nature of the tricks characteristic of each activity. Unlike business, politics, or war, mathematics has no place for dirty tricks. As my colleague Marie-France Vignéras put it, “as mathematicians, we play and dream but we don’t cheat.”33 That is not because pure mathematicians are purer than everyone else but rather because cheating defeats the purpose of mathematics, however that may be construed. The pointlessness of cheating is, I submit, one of the very best clues to the peculiar appeal of mathematics as a human activity—to its similarity to play in Huizinga’s sense, for example—and poses a special challenge to science studies as well as to philosophers of Mathematics, who may consider the matter settled once mathematical machines have been programmed not to cheat.

ARCHEOLOGY

No dirty tricks, then, but tricks nonetheless. Contemporary writers frequently read tricks into early mathematics; thus several familiar arguments from Euclid’s book on arithmetic, including his proof that there are infinitely many primes, have independently received the name “Euclid’s trick.” But in my own attempts at archaeology, I’ve managed to dig down only as far as 1815 before I run out of examples of conscious use of the word, or its equivalent in another language, to designate the phenomenon described before.34 The cuneiform tablets that represent the earliest-known mathematical texts are punctuated by “you see” (ta-mar) that seem designed to force an Aha! experience. Euclid’s prose is pretty monotonous, but Reviel Netz pictures Archimedes and his “ludic” Hellenistic contemporaries setting “traps,” employing “cunning surprise,” and indulging in a “carnival of calculation,” all with the intention to “dazzle and overwhelm” the reader and to provoke “gasp[s] of delight.” The Līlāvatī written by Bhāskara II in twelfth-century India is notable for its playful tone. But none of these contains tricks in the contemporary sense.35

Like flying carpets, the habit of seeing tricks in mathematics may have come from the East. The evidence is tenuous but appealing. In the mathematical chapter of his Catalogue of the Sciences (isa’ al-‘ulum), the tenth-century Baghdad philosopher al-Fârâbî listed algebra not as a branch of mathematics like arithmetic and geometry, but rather alongside mechanical devices, in a chapter on ‘ilm al-iyal—“the science of al-iyal” (singular ila’)36—an equivalent of the Greek mekhane, variously translated as “ingenious devices,” “mechanics,” or “tricks.” A century earlier the Banū Mūsā brothers had published Kitab al-iyal, a celebrated catalogue of mechanical devices, including automata.

The Banū Mūsā were among the founders of the Arabic mathematical school, and it is perhaps not an accident that in their text Kitāb marifat masakhat al-ashkal [The Book of the Measurement of Plane and Spherical Figures], “the terminology of arithmetic is perhaps for the first time applied to the operations of geometry,” so that the ratio of the circumference to the diameter of the circle, what we now call π, was treated as a number.37 The Book of Science [Dāneš-nāma] of Ibn Sīnā (Avicenna) listed algebra among the “secondary parts of arithmetic.” Omar al-Khayyām thought otherwise: “Those who think algebra is a trick (ila’) to determine unknown numbers think the unthinkable; therefore you must not pay attention to those who judge by appearances and are of a different opinion.”38

If we dig a little deeper, we find Plutarch’s account, already mentioned, of Plato’s rejection of mechanical methods in mathematics, “a sort of foundation myth for the science of mechanics,” which must have been familiar to al-Fârâbî and Ibn Sīnā, and “which explained the separation of mechanics from philosophy as the result of a quarrel between two philosophers.”39 The Aristotelian context for the controversy, interesting in its own right, is discussed in chapter 10. For the moment, we simply note that Gherard of Cremona’s Latin version of al-Fârâbî’s catalogue translated ‘ilm al-iyal by ingeniorum scientia (the science of ingenium, the Latin equivalent of mekhane); that ingenium also figured in Latin texts on mathematics as well as in the title of Descartes’ early Rules for the Direction of the Mind [ingenium]—in more than one relevant way an exact inversion of Aristotle’s value system; and that it admits a great variety of German translations, one of which is Kunstgriff.40 The continuing associations of ingenium with machines (engineering) as well as genius, at the two ends of the trifunctional scale, neatly mirror mathematicians’ ambivalence to tricks and incidentally suggest that anything a mechanical theorem prover could invent would be assigned ipso facto the status of trick.

Back in the present, my Taiwanese colleague Kai-Wen Lan tells me “there is no convenient [Chinese] translation for the word ‘trick.’ Depending on the context, one can find more than a dozen different translations, among which many do have unsavory connotations. However, it is not natural in our culture(s?) to mix them up.” According to Teruyoshi Yoshida, the word torikku has been adopted in Japan and carries roughly the same meaning as its English homonym; but Japanese has an alternative that is especially intriguing in view of the possible medieval derivation from mekhane:

[T]he word “karakuri” (mechanism, device, strategem, trick, system), which goes back to the early Edo period (17c) and used … especially for the sophisticated puppet shows (Bunraku/Ningyo Joruri). In mathematics I hear people using the term: “I finally understood the karakuri behind this scary-looking proof” (i.e., how the proof works; idea + mechanism).” It sounds like “it was deceptively mysterious/curious but the actual mechanism is simple,” but it doesn’t have the negative implication of the English word—“trick” sounds as if the solution to (or avoiding) the profound-looking question was disappointingly easy.41

The earliest “trick” I’ve found in a mathematical journal article is a Kunstgriff in the second installment of a long article by Moritz (or Moriz) Abraham Stern, dated 1833, entitled Theorie der Kettenbrüche und ihre Anwendung [Theory of continued fractions and their application]. Since this is the first appearance of the word or one of its cognates in the oldest mathematical journal still in existence, the sentence in which it appears deserves to be quoted in full:

In all the transformations of continued fractions indicated above, one applied the trick [Kunstgriff] of treating a part of the continued fraction as if it were summed, designating this sum by a letter, and through the relation of this letter with the other members of the continued fraction finding new expressions.42

This Kunstgriff is recognizably a trick, the first of ten to be given that name during the nineteenth century in Crelle’s Journal, the oldest mathematical journal still being published.

It took centuries to dissipate the negative connotation the French word astuce acquired in 1370, when the mathematician and philosopher Nicolas Oresme wrote, “Et doncques se l’entention est malvese, tele puissance est appellée astuce ou malicieuseté* in the manuscript of his translation of Aristotle’s Ethics. In his Mer des Croniques of 1532, Pierre Desrey described an astucieux as a person “who possessed an inventive mind for finding means of deception.”43 The Dictionnaire historique de la langue française of A. Rey traces the contemporary positive usage in the sense of “ingenious invention” or “amusing joke” to the middle of the nineteenth century and specifically to the “argot” of the grandes écoles, in particular the École Polytechnique. In this case, mathematics apparently led the way to the word’s rehabilitation.44

I found these examples using online search engines. Many old journals and some old books have been converted into digitized databases that offer the possibility of searching for a word like trick or astuce or т рюк or Kunstgriff, and that’s what I did. For a few hours’ work, the results are pretty satisfying. Specimens of trifunctional vocabulary were more difficult to come by, although one might have expected the three functions to be present since the beginnings of Indo-European mathematics.45 Thus I find Sylvester writing “words are the tools of thought” in American Journal of Mathematics, 1886; and in 1870 one finds Werkzeug, the German word for “tool,” in Crelle’s Journal, used as a mathematician would today:46

img

This equation, in which the quantities ut and vt are viewed as dependent on the elements xa and xa(0), is the tool [Werkzeug] that will be used to study the normal form f(du).

“Technique,” or Technik, had acquired its contemporary meaning in mathematics by the 1830s,47 and the use was standard in the review journal Jahrbuch über die Fortschritte der Mathematik practically from its foundation in the second half of the nineteenth century. Given the intense interaction between mathematicians and philosophers over the centuries, it is probably hopeless to use databases to distinguish trifunctional from literal uses of the word philosophy in mathematical texts. On the other hand, the earliest example I found of a mathematical “strategy” is in the 1944 E. T. Bell article quoted a few pages back; the expression “strategy of the proof” makes its first appearance in Mathematical Reviews in Jean Dieudonné’s review of the Séminaire Cartan volume on the Atiyah-Singer index theorem48—and “brute force” appears at around the same time. Problems were being “attacked” at least as early as the mid-eighteenth century; the word starts to be used in mathematical journals a bit more than a hundred years later.49

A trained historian will find it easy to improve on my amateur findings, but even a superficial examination of the printed record makes it clear that the terms all became progressively much more common after World War II—really common: Google now returns nearly 5,000,000 sites for “strategy of the proof”—and reached their current configuration by 1970. Around that time, I began my own mathematical initiation, which included training in the use of the metaphoric vernacular as well as the specialized (“technical”) vocabulary in which theorems and their proofs are written. It is as if when European and North American mathematicians acquired sufficient professional autonomy to constitute themselves as a tribe, their savage minds reclaimed the three functions of their Indo-European ancestors. The following strikingly trifunctional paragraph in Bourbaki’s The Architecture of Mathematics (dated 1948, 1950 in English) is characteristic of Bourbaki’s structuralist framework (emphasis is mine):

The “structures” are tools for the mathematician; as soon as he [sic] has recognized … relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems…. Previ-ously … he was obliged to forge for himself the means of attack … their power depended on his personal talents and they were often loaded down with restrictive hypotheses…. One could say that the axiomatic method is nothing but the “Taylor system” for mathematics.

Having exhibited the mathematician as blacksmith (later to be joined by Langlands’ lumberjack) and assembly-line worker, as well as military strategist, Bourbaki reminds us in the next paragraph of the (charismatic) first function:

This is however, a very poor analogy; the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition … not the popular sense-intuition, but rather a kind of direct divination … which orients at one stroke in an unexpected direction the intuitive course of his thought, and which illumines with a new light the mathematical landscape.50

So where does the trickster fit in? Dumézil wrote a book about Loki but did not attempt to assign the trickster a consistent role in his trifunctional theory. The sector of French mathematics represented by Bourbaki had (and perhaps still has) little use for tricks. I was unable to find any trace of the word in Bourbaki’s Eléments des Mathématiques, and Bourbaki’s only allusion to tricks in The Architecture of Mathematics is dismissive:

[T]he axiomatic method has its cornerstone in the conviction that, not only is mathematics not a randomly developing concatenation of syllogisms, but neither is it a collection of more or less “astute” tricks, arrived at by lucky combinations, in which purely technical cleverness wins the day.51

My provisional hypothesis is that the trickster serves as a bridge between high and low genres. This is the role often reserved for Indo-European tricksters: think of Krishna as simultaneously the god Vishnu and his human avatar; or of Prometheus, who descended to earth with the gift of heavenly fire. Hermes was not only the messenger of the gods but also the guide who presided over the passage of the soul to the un-derworld; Mephistopheles plays a similar role in the Faust legend. Esu, the Yoruba divine trickster, limped because his legs were of different lengths: “one anchored in the realm of the gods, … the other … in … our human world.”52 In mathematical settings the trickster predates logicist or formalist idealizations and continues to offer a shortcut bypassing the (Indo-European?) idealized route from human practice to inscription of theorems in the register of the eternals.

AFFINITIES

And this brings me to the question at the origin of this chapter. How can I explain the persistent tendency to class mathematics as a high genre, along with the fine arts and specifically with classical music? Why do you find a string trio and a Steinway grand piano in the music room at the Mathematisches Forschungsinstitut Oberwolfach but no electric guitar, no drum set, no scratch mixers, no samplers?

Popular books and articles have explored the supposed affinity between mathematics and music. The authors typically start with Pythagoras,53 continue by way of the medieval quadrivium, where music was taught as a branch of mathematics (as also in Ibn Sīnā’s Book of Scientific Knowledge), through Kepler’s Harmonices Mundi and Leibniz’s claim that music was an “unconscious exercise in arithmetic,” and arrive in the present day with anecdotes about musical mathematicians.54 What is not mentioned so often is that at most times, the affinity between mathematics and “serious” music was markedly one-sided. Jean-Philippe Rameau, it is true, based his influential principle of harmony on a mathematical theory of harmonic overtones, the corps sonore; he wrote in 1722 that “only with the aid of mathematics did my ideas become clear.” Rameau corresponded with the leading mathematicians of the day, including D. Bernouilli and Euler; he sought and obtained the approval of the French Académie des Sciences for his Démonstration du principe de l’harmonie; and his good relations with d’Alembert persisted until Rameau overreached, claiming priority for the corps sonore over “the other arts and sciences,” geometry included.55 It is also true that Ernst Krenek argued in 1939 that music, like geometry, needed to be based on axioms.56 Iannis Xenakis wrote in the 1950s that

[m]usic can be defined as an organization of these operations and elementary relations [i.e., Boolean algebra and predicate calculus] between sound events […] mathematical set theory [is useful] not only for the construction of new works but also for … better understanding of works of the past. Thus even a stochastic construction or an investigation of history with the help of stochastics [is impossible] without the help of Logic or Algebra, its mathematical form, the queen of the sciences and even of the arts.57

And Milton Babbitt, who wrote a thesis in 1946 entitled “The Function of Set Structure in the 12-Tone System,” taught mathematics at Princeton before switching to the music department, the first of many professors at leading American universities who have looked to such varied branches of mathematics as group theory, orbifold geometry, and topos theory to explain musical structure.58 Even now IRCAM in Paris sponsors regular meetings on mathematical theories of music (see figure 8.1).

Outside the academy, however, such moments of convergence have been relatively rare. Long before Douglas Hofstadter wrote Gödel, Escher, Bach, C.P.E. Bach denied that his father had any mathematical leanings: “the departed [J. S. Bach] was, like myself or any true musician, no lover of dry mathematical stuff.”59 Hector Berlioz, in his 1836 eulogy of his teacher Anton Reicha, wondered whether the latter’s attachment to mathematics didn’t make his compositions “lose something in melodic or harmonic expression, in purely musical effect, what they gained (if to be sure, it was gaining) in arduous combinations, in conquered difficulties, in curious works made rather for the eye than for the ear?” Camille Saint-Saëns, referring to Berlioz, added that “One doesn’t learn art as one learns mathematics.” At the end of the century, a London music critic elaborated:

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Figure 8.1. A reconstruction of one of the less elaborate slides from Modern Algebra and the Object/Operation duality in music, a presentation by Moreno Andreatta, Music Representations Team, IRCAM/CNRS.

The Brahms Sextet … is one of those odd compositions which at times slipped from the pen of Brahms, apparently in order to prove how excellent a mathematician he might have become, but how prosaic, how hopeless, how unfeeling, how unemotional, how arid a musician he really was. You feel an undercurrent of surds, of quadratic equations, of hyperbolic curves, of the dynamics of a particle. But it must not be forgotten that music is not only a science; it is also an art. The Sextet was played with precision, and that is the only way in which you can work out a problem in musical trigonometry.60

Jazz age Paris saw arithmetic participate in the torture of the naughty child in Ravel’s L’enfant et les sortilèges, with libretto by Colette:

LE PETIT VIEILLARD :

Deux robinets coulent dans un réservoir; / Deux trains omnibus quittent une gare,

A vingt minutes d’intervalle, / Valle, valle, valle!

L’ENFANT :

Mon Dieu c’est l’arithmétique!

LE PETIT VIEILLARD :

Tique, tique, tique! / Quatre et quatre dix huit,

Onze et six vingt cinq, /Sept fois neuf trente trois.*

In 1930 music theorist Heinrich Schenker could just as well have written “mathematical” as “mechanical”—“the mechanical spreads throughout the entirety of average men like a poison gas”—when he contrasted Rameau’s “paralysis” with the living tradition represented by German musicians. “How,” he asked, “did the paralysis of the theoretical tendencies … manifest itself?” His answer was that “something mechanical … lay in the basic idea of Rameau from the start.”61

Well into the twenty-first century, mathematics is still widely seen as music’s diametrical opposite: “The pieces [Schoenberg] composed at this time were mystical, richly symbolic and emotionally charged, hardly pure mathematics,” wrote one British critic, while another adds that “once you’re inside music, all thoughts about mathematics become irrelevant.” Even Pierre Boulez, the founder of IRCAM, who elsewhere wrote that “music is a science as much as an art,” complained at one point that “what is called the ‘mathematical’ … mania … gives the illusion of [music as] an exact, irrefutable science” and referred to “number-fanatics” who seek a “form of rational reassurance.”62

Don’t you think there’s something pathetic about how certain mathematicians keep insisting that they are artists—we’ll see more of this in chapter 10—when the artists want only to keep their distance? It’s probably no accident that mathematics and classical music were not perceived as antithetical only during those periods when the notion of a “love formula” was not seen as oxymoronic. Rameau and Clairaut were contemporaries; Xenakis, Boulez, and IRCAM belong to the same world as Lacan and the structuralists, the world of an “objectivist” aesthetic of music in which “musical style … is eventually conceived in terms of statistics and music tends to lose its human significance.”63

WHY SO SERIOUS?

In his Emblems of Mind, Edward Rothstein, New York Times critic—and former mathematics graduate student at Brandeis University—explores parallels between mathematics and classical music, ignoring the historically less well worn but, in principle, no less cogent affinity of mathematics with rock and roll or rap. Catherine Nolan’s article in the Princeton Companion to Mathematics is written from the same perspective, as was a recent lecture in Paris by philosopher Alain Badiou, entitled Mathématiques/Esthétiques/Arts and sponsored by IRCAM. In the (otherwise very different) treatises of Herbert Mehrtens and Jeremy Gray on modernism in mathematics, the cultural references are Picasso, Stravinsky, Klee, or Apollinaire, not advertising, the development of a mass market, mass production, or the trade-union movement, much less the mechanically reproducible pop culture to which Walter Benjamin’s celebrated essay (indirectly) drew attention.64

“Pop music,” writes Rothstein, “fulfills a different function from art music and often has different ambitions.” Just what are these functions, and why are mathematicians reputed to find the prestige of art music irresistible, in spite of the persistent indifference of its practitioners? The relegation of popular culture to an inferior status—analogous to that of tricks in mathematics—is anthropological in nature, in that it is not generally the object of an explicit proscription, nor is it ascribed to a rational judgment (except a posteriori), but neither is it merely a matter of individual taste. Names for distinct varieties of facetiousness come readily to mind, but how can we talk about seriousness? Why do routine mathematics and classical music exemplify this virtue more convincingly than mathematical tricks or popular music? As Heath Ledger’s Joker asked in The Dark Knight, “Why so serious?”

Overlapping accounts of the origins of professorial seriousness in general stress its sources in the respect due to power,65 the respect due to respectability,66 the respect due to standards of perfection on which society depended,67 or the respect due to norms of virtuous behavior.68 These points are all valid and convincing and must be taken into consideration when trying to understand how social pressure is brought to bear upon the academic community, and upon scientists in particular, when enforcing standards of behavior. My goal in this chapter, however, is ethnographic, not historical or sociological, a description from the inside of a living culture, and therefore I want to understand how these social forces are internalized as norms within the specific culture of pure mathematics—and, no less importantly, to look for traces or tokens of these standards in the symbolic system used by members of the tribe. The ambivalent status of the trick and the supposed affinity of mathematics with classical music are my main exhibits. I am not defending the position that these characteristic attitudes originate in external social pressure, nor, on the contrary, that they arise naturally from the practice of mathematics. Though I have my opinions—musical taste looks to me more exogenous, the suspicion of tricks more endogenous—I’m not professionally qualified to make that determination. Here, then, are a few hypotheses that may deserve further study.

1. Recall the word legerdemain in the 1940 MAA article on tricks. Is the word trick perhaps an unwelcome reminder of the historic association between mathematics and magic, even witchcraft? The word mathematicus—probably the correct name for the figure in a painting by Luca Giordano (figure 8.2), pointing to what appears to be an astrological chart69—was used primarily for astrologers. The medieval scholastic Roger Bacon, who condemned the mathematics that was “a subdivision of magic,” nevertheless considered “astrology rather than astronomy … by far the most important and practical part of mathematics.”70 This state of affairs persisted until well into the seventeenth century at least: casting horoscopes was one of the duties of the chair of mathematics at the University of Padova, and Kepler and Galileo, both prominently featured on IBM’s “Men of Modern Mathematics” chart, were occasionally employed as astrologers and had at least mixed feelings about the practice.71 The matemático in figure 8.2 is thus in the same line of work as Johannes Faustus and his trickster sidekick, Mephistopheles:

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Figure 8.2. Luca Giordano, El Matemático [the Mathematicus?]. (From the collection of the Museo Nacional de Bellas Artes, Buenos Aires)

Das gefiel Doctor Fausto wol speculiert vnnd studiert Tag vnnd Nacht darjnnen / Wolt sich hernach kein Theologum mehr nennen lassen / ward ein Weltmensch / Nennt sich ein Doctor Medicinæ, ward ein Astrologus vnnd Mathematicus.72

In his work on magic, Giordano Bruno saw mathematics as a bridge between the higher and lower orders of being, “a link … between the celestial and the terrestrial”:73 just the role we assigned the trickster. John Dee, mathematician and court magus to Elizabeth I, described the bridge as follows:

A marvellous newtrality have these things Mathematicall, and also a strange participation between things supernaturall, immortall, intellectuall, simple and indivisible: and things naturall, mortall, sensible, compounded and divisible.

Dee, a possible model for Marlowe’s Doctor Faustus (and for Shakespeare’s Prospero, in The Tempest) had been arrested in 1555 for his horoscope predicting Elizabeth’s accession to the throne and was one of the most prominent mystics of his time; he had reportedly “on occasion transmuted base metals into gold.”74 He was also a leading promoter and popularizer of mathematics, a central transitional figure in the period before the elaboration of the modern scientific worldview, “one English author for whom both philosophy and mathematics figured together as vital parts of an intellectual programme.” As mathematician, Dee’s most important work was probably his Mathematicall Praeface to the 1570 English edition of Euclid; as mathematicus, a typical publication was his Monas Hieroglyphica (Antwerp, 1564; see figure 8.3), an early attempt at cosmological grand unification in the form of a list of numbered theorems, written in a arcane language apparently designed to prevent the acquisition of “an unlawful knowledge of mysteries” by “unworthy persons who”—very much like the “forces of Evil” in Rites of Love and Math—“would correctly interpret, but abuse, his all too candid exposition”:75

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Figure 8.3. John Dee. (Monas Hieroglyphica, p. 23)

2. The “very serious man” Hermann Weyl did as much as anyone to legitimize the use of the word trick in the informal language of working mathematicians. He was the first to use the English word trick in each of the two leading U.S. mathematical journals of his day, and the word Kunstgriff appears in a number of his papers in German. But there are no allusions to tricks in his highly regarded book Philosophy of Mathematics and Natural Science and only two in a separate collection of his philosophical writings; in one, he finds that a “strange scholastic trick … shows up peculiarly in the disposition of [Newton’s] Principia which is otherwise so rigorous.” So Weyl also locates the Kunstgriff on the wrong side of the border between “low” and “high.” Rump’s theoretical analysis of mathematical tricks—the only one I have been able to find—justified the Kunstgriff explicitly as a step on the way to the “general theory” with its “natural places”; this is consistent with G. H. Hardy’s attribution of what he called the “ ‘seriousness’ of a mathematical theorem” to “the significance of the mathematical ideas which it connects.”76

Hyde writes that “accidents happen in time, essences reside in eternity … [t]he [Norse] gods grow old and gray” when Loki’s intervention gives the giants a taste of the Apples of Immortality.77 The precarious status of tricks in view of Rump’s criterion is well illustrated by Grothendieck’s reported reaction to Deligne’s 1973 proof of the last of the Weil conjectures (see chapter 2), now known as Deligne’s purity theorem and considered one of the outstanding theorems of twentieth-century number theory, indeed of all time. Grothendieck made a special trip back to the IHES from his provincial exile to learn about Deligne’s shortcut solution to the problem Grothendieck had posited as the culmination and ultimate justification of his remodeling of geometry. “If I had done it using motives,” recalled Deligne, “he would have been very interested, because it would have meant the theory of motives had been developed. Since the proof used a trick, he did not care.”78 One assumes that Grothendieck felt the method had not revealed the motives at “the heart of the matter”79—just as al-Fârâbî had included algebra among the tricks. Years later, Langlands speculated that “perhaps [Grothendieck] could have drawn a different conclusion…. the last of the Weil conjectures was proven, by Deligne, in essence on the basis of a profound understanding of the étale cohomology theory accompanied by an observation arising in the theory of automorphic forms.”80

Bourbaki had no use for tricks, as we’ve seen, and among the Bourbakistes, Grothendieck found them particularly distasteful. But in this case, what Grothendieck disdained as a mere trick is now seen by the majority of mathematicians as a brilliant insight Deligne derived from his study of the Langlands program. Deligne’s proof indeed preceded a “general theory,” namely, Langlands’s theory of functoriality—but for Grothendieck it was the wrong general theory. Which goes to show that trickiness is not an intrinsic, much less quantifiable, property of a mathematical text.

3. In his Moderne Sprache, Mathematik, Mehrtens stresses that aspirations to seriousness were integral to the modernization process. “As a mathematician [German folklore figure Till] Eulenspiegel transformed himself from an anarchist fool to a theologian.” Nor did mathematics merely leave its trickster outfits behind: the German modernizers were conscious of parallels between their goals and those of their artistic contemporaries. Mehrtens explains that, when the mathematician Heinrich Liebmann described mathematics as a “freie, schöpferische Kunst” (“free creative art”) in his Leipzig inaugural address, comparable to that of the Berliner Secession painter Max Liebermann, the “freedom” Liebmann had in mind was to establish one’s own Qualitätskriterien and not to be subject to the “quality criteria” of the Kaiser and similar Powerful Beings (in the case of artists) or of engineers, teachers, and philosophers (in the case of mathematicians).81 So mathematical modernists were not only being elitist when they demanded autonomy, professional recognition, and authority comparable to that of the state; their elitism was precisely modeled on that of their fine arts counterparts. The Mehrtens book reproduces Liebermann’s 1912 portrait of Felix Klein, a central protagonist of mathematical modernization.

A parallel can be drawn with the “fashioning” of English literature as a “serious discipline”82 in the period following World War I. In 1877 it could be argued that the study of English literature “might be considered a suitable subject for ‘women … and the second-and third-rate men who … become schoolmasters.’ ” Terry Eagleton writes that even in the 1920s, “it was desperately unclear why English was worth studying at all; by the early 1930s … English was … the supremely civilizing pursuit, … immeasurably superior to law, science, politics, philosophy, or history.” This was largely the work of F. R. and Q. D. Leavis and their associates at Cambridge, whose passionate attachment to England’s language and high literature were matched only by their contempt, displayed in Leavis’ epigraph to this chapter, for “the philistine devaluing of language and traditional culture blatantly apparent in ‘mass culture.’ ”

Speaking to the British Association in 1897, Andrew Russell Forsyth fretted about a new “attitude of respect … almost … of reverence … we mathematicians are supposed to be of a different mould … breathing a rarer intellectual atmosphere, serene in impenetrable calm. It is difficult for us to maintain the gravity of demeanour proper to such superior persons; and perhaps it is best to confess at once that we are of the earth.” The adoption or reaffirmation of seriousness of demeanor as a characteristic virtue of the university professor plays itself out as a series of events in the history of the construction of the image of the professor as an authority figure. In his analysis of Liebmann’s inaugural address, Mehrtens is calling attention both to the role of seriousness in establishing status in a hierarchical society during the period of breakdown of traditional authority and to an alliance, at least de facto, between mathematicians and practitioners of the fine arts. Mehrtens may have considered it superfluous to remind his German readers that if Till Eulenspiegel dressed like a clown, it was in part because the trappings of the higher strata were forbidden to him by German sumptuary laws. (Compare the “rude mechanicals” in A Midsummer Night’s Dream or the father of Quevedo’s picaresque hero El Buscón, who reassures his son that “thievery is a liberal art, not a mechanical art.”83)

ARTISTS IN RESIDENCE

Since serious authority nowadays is vested in finance rather than royalty, it’s prudent to remember that, though Freud did claim that “the opposite of play is not seriousness, but—reality,” Huizinga—who saw seriousness as a derivative concept, defined by the absence of play—reminded us that the opposite of play can also be work. Our exploration of virtual realism suggests that Freud’s reading may be faithful to the inner life of mathematics, but work is what Powerful Beings, public or private, expect in exchange for sponsoring our research.84

Nevertheless, acceptance of recognized canons of serious demeanor is no longer necessary to establish authority. A new relation to seriousness may even now be under negotiation. Back in February 2011 at the IAS, the setting of so many of this book’s scenes, I approached the term’s first “After Hours Conversation”—artist-in-residence Derek Bermel had promised to talk about Stravinsky, Hip-hop, and the Scientific Method—with misgivings. The cultural offerings I sampled during previous Institute visits had been relentlessly lofty and Apollonian, and I could feel another Dionysiac episode coming on. Bermel, a classically trained composer and clarinet virtuoso, had brought his instrument, and he began his allotted ten minutes of remarks by illustrating metric displacement as a source of variation and development in Stravinsky’s L’histoire du soldat. He then noted that the rapper Rakim uses a similar process, moving his texts across the poetic line within a fixed meter. To illustrate the process Bermel performed this text by Rakim (note the enjambment in the first and last three lines):

Write a rhyme in graffitti in every show you see me
in Deep concentration ‘cause I’m no comedian

Jokers are wild if you wanna be tame

I treat you like a child then you’re gonna be named
Another enemy, not even a friend of me

‘cause you’ll get fried in the end if you pretend to be.

I nearly fell out of my chair. I didn’t know you were allowed to do things like that at the IAS!85

Bermel is at home in the classical repertoire of previous IAS artists-in-residence, but I can’t imagine what the latter would make of Rakim. “Performers in the classical tradition often have an extremely limited range,” Bermel told me, “and in particular have trouble with unfamiliar rhythms.” So are rockers and rappers86 the new Pythagoreans? Consider Math Rock, which, according to Toby Gee, is “characterised by a somewhat ‘angular’ sound, and maybe some tricky time signatures.” For example, 26 Is Dancier Than 4, by the Oxford Math Rock group This Town Needs Guns, is written in 26/8 time; Don Caballero’s Slice Where You Live Like Pie features guitars playing in 5/4 over a drumbeat of 7/8; Calculating Infinity by The Dillinger Escape Plan (founders of the variant of Math Rock called Mathcore) opens with a sequence of 3/8, 3/8, 3/8, 4/8, 3/8, 3/8, 3/8, 4/8, 3/8.

It’s no accident that Toby Gee has resurfaced at this point of the narrative. If I chose him as my guide to math rock it is because we are members in good standing of the same mathematical clan. The word tribe used a few pages back is not a metaphor, nor is it a sly reminder of this chapter’s anthropological theme.87 Mathematics has not one, but two, independent formally recognized kinship systems. The first is based on networks of collaboration. Mathscinet, the online version of the AMS’s Mathematical Reviews, provides a list of every collaborator with whom a given mathematician has published and calculates collaboration distance, on the model of the Erdős number mentioned in the previous chapter. Thanks to my trick, for example, my collaboration distance from Toby Gee has dropped to 2, one less than my Erdős number. The second kinship system is recorded in the Mathematics Genealogy Project (MGP).88 Genealogy refers to the ascending series of dissertation advisers. It’s common for mathematicians to refer to their advisers as “parents” and fellow students of a given adviser as “siblings.” So I am doubly linked to Toby Gee through our mutual collaborator Richard Taylor, who can claim Gee as one of his grandchildren. Gee’s lineage is particularly distinguished—Taylor was “begotten” by Andrew Wiles, whom (Cambridge professor) John Coates “begat”; before his retirement, Coates held the Sadleirian chair previously occupied by G. H. Hardy and was himself a student of Fields Medalist Alan Baker. The MGP counts me, but not Gee, among the 109,100 known descendents (as of June 2013) of Nilos Kabasilas, fourteenth-century bishop of Thessaloniki; my distant cousins include Michelangelo and Leonardo da Vinci, as well as both Grothendieck and Langlands.89

If authors of popular books about mathematics actually looked at mathematicians, they would find enthusiasm for and performance of popular music as common as among the general public. The interesting question is why they find it convenient to ignore this in favor of an exclusive insistence on the shaky alliance with classical music. Among my mathematical kin, Gee is noted for his serious interest in popular music, as is his “father,” Kevin Buzzard, not to mention his “sibling,” Dan Snaith (Manitoba, Caribou, Daphni), the most prominent indie rocker with a PhD in number theory. Skeptical, like Buzzard, about the term “math rock,” Gee advised me to listen to Atlas by Battles, “by far the best thing that gets called math rock” that Gee had heard in recent years. And so I learned that, Pynchon’s Weed Atman notwithstanding, rock’s relations to mathematics are not straightforward. In a 2011 interview, Battles’ members had this to say about the genre:

What about the term “math rock”?

[drummer John] Stanier: As a band we say we dislike it immensely, but there’s really not much you can do about it…. it’s lazy, like you can’t think of anything better to say than math rock. I hated math.

[guitarist Ian] Williams: But I loved rock.90

In Victims of Mathematics, the punk rock group Grade argued that “applying fractions to modern day living is as useful as … handing gasoline to an arsonist.” Cherry Ghost’s Mathematics is about alienation: “hear the unforgiving sounds of cold mathematics making its move on me now.” Rock’s main gripe with mathematics seems to be that—once again—love doesn’t conform to its equations. Little Boots’ Mathematics put it this way:

By the time I calculated the correct solution

The question had escaped me and so did the conclusion …
I’ll believe you ‘cause your X is equal to my Y

But equations pass me by

Patti Smith and her late husband Fred weighed in as well:

No equation to explain

Destiny’s hand

Moved, by love

Drawn by the whispering shadows Into the mathematics

Of our desire

Russian rocker Boris Grebenshchikov, who studied mathematics at Leningrad State University, placed the relation of mathematics to music in a broader context:

As one scientist to another, I’ll say to you frankly—no one, except the perfectly enlightened, can even begin to imagine how the world is really constructed…. One day, when I was studying … in the Faculty of Applied Mathematics and Control Processes, I started thinking and naively asked our professor of mathematical statistics, the marvelous Nikolai Mikhailovich Matveev: “… Statistics answers the question of the probability that some event will take place; but it seems to me much more important to know—why did this specific event happen and not some other one?” And he gently replied: “Young man, to answer that question, you will have to go to the Theological Academy.”

With that, he conclusively solved the problem of my relation to science. From that moment mathematics lost all interest for me, though I didn’t go to the Theological Academy, inasmuch as I was playing music—and probably in the depth of my soul I was certain that music would lead me to the answers to all my questions. Which is what happened.

In the same vein, filmmaker David Lynch, who recently recorded an album entitled Crazy Clown Time, once expressed “his belief that mathematics might be the only way of rationally describing what the real world is like, but he doubted that we would ever be capable of developing so complex a mathematics.”91

We turn to African American popular music with relief: there’s no stigma attached to mathematics! Thus Robert “Noise” Hood, cofounder of the minimal techno collective Underground Resistance, recorded a track entitled Calculator under the pseudonym Mathematic Assassins.92 Contrast C.P. E. Bach’s dismissive rejection of “dry mathematical stuff” with these lyrics from George Clinton’s Mathematics of Love:

Cause any percentage of you / is as good as the whole pie and any fraction thereof/brings dividends of love

Hip-hop culture, for its part, positively embraces mathematics. Kanye West’s interlocutor is meant to be flattered, not insulted, when he informs her (in Never Let Me Down) that “[y]ou don’t need a curriculum to know that you are part of the math.” Mathematical physicist and jazz saxophonist Stephon Alexander grew up in the Bronx:

I used to take the bus home from school and these guys would be on the back of the bus around, rappin’ and battling each other—droppin’ science, so to speak, and one day, you know, I heard someone say, “Yeah, man, ‘cause I got enough mathematics.”93

In hip-hop slang, “math” or “mathematics” can also mean a telephone number, as in Brooklyn Masala by Masta Ace: “She winked at me, and kinda laughed / Ripped the piece of the grocery bag and wrote her math.” But mathematics is more than just an incredibly supple idiom for urban romance. In a song entitled Mathematics—the top hit for “mathematics” on YouTube—Yasiin Bey (better known as Mos Def) advises his “working-class poor” listeners to learn the numbers to survive:

This new math is whippin motherfuckers ass

You wanna know how to rhyme you better learn how to add It’s mathematics94

Reading this entry from the online biography of Wu-Tang Clan associate, DJ/producer Allah Mathematics (also known simply as Mathematics)—

Mathematics is the universal language…. It can cross barriers of language, color, religion, ethnicity, race, and more.95

—I am tempted to speculate that the universalism that comforts the hip-hop artist speaking in the name of a marginalized population, living in a town that certainly doesn’t need more guns, is precisely what rockers like Grebenshchikov or Smith, anxious to affirm their uniqueness, find disaffecting.96

Now even those musicians—Rameau, Xenakis, or Yasiin Bey—who express admiration for mathematicians do not welcome us as fellow artists; their vision of mathematics is as exemplar of the true (or scientific) rather than of the beautiful. Nevertheless, bearing in mind that “signifying is the grandparent of rap” and that “It’s Tricky to rock a rhyme, to rock a rhyme that’s right on time,” hip-hop’s deepest affinities with mathematics may still be cultural rather than ethical or epistemological.97

UNDERGROUND RESISTANCE

Raymond Williams points to “a ready-made historical thesis” for what certain critics find disappointing in popular culture: “After the Education Act of 1870, a new mass-public came into being, literate but untrained in reading, low in taste and habit. The mass-culture followed as a matter of course.”98 The obvious corollary to this thesis has never lacked defenders. F. R. Leavis thought that being capable of a “discerning appreciation of art and literature … constitute[s] the consciousness of the race (or of a branch of it) at a given time…. Upon this minority [those capable of such discernment] depend the implicit standards that order the finer living of an age….” “[T]here’s this idea,” Boulez once said, “… that everybody should make music, every discipline is equally worthy—whether it’s rock, jazz, folk—all “les musiques,” plural. No, they’re not equal, I don’t agree. I believe in fine art, I believe in aristocracy, and I believe in elite [culture].”99

If advocates of elite culture disdain rock and rap, it’s because, like the “wanton” music of Cheng, they are said to appeal to the body’s lower sectors, unlike classical music.100 Friedrich Schiller’s poem Archimedes and the apprentice, which features the line “He who woos the goddess, seek in her not the maid,” was a favorite among nineteenth-century German mathematicians, starting with Gauss. Bertrand Russell recommended mathematics as “a means of creating and sustaining a lofty habit of mind”—“like poetry or music,” added G. H. Hardy.101 Plato found mathematics lofty to the point of bypassing or transcending the body altogether: mathematical knowledge is stored in the soul, according to the Meno; involving the body leads, as we’ve seen, to confusion and dizziness.102 This is demonstrably false. Mathematics is typically created not at the desk or blackboard, but during the strenuous Wednesday afternoon hike at the Oberwolfach institute or a leisurely stroll in the IAS woods. Well into his eighties, Jean-Pierre Serre (whom the Paris daily Libération compared to Mozart) still goes rock climbing regularly in the forest near Fontainebleau; Cédric Villani (“the Lady Gaga of mathematics,” according to the Paris weekly Télérama) lists “walking” along with music as hobbies on his CV.103

But elitist standards continue to evolve, and Bermel—who has collaborated with Mos Def/Yasiin Bey—was exercising his faculty of distinction, rather than being provocatively lowbrow, when he insisted to me that “strong and mediocre music can be found in any genre.” What matters is what Bermel (sounding very much like Bourbaki) called “coherence of structure” and what Laurence Dreyfus called the “respect for the inherent meaningfulness of the world,” a recognition of how freedom and necessity are “inextricably linked.”104 Bracketing a novel invention as a trick is a way to maintain mathematical standards while admitting that, depending on the trick’s “coherence” and “inherent meaningfulness,” its status may change in time.

Schiller, author of the Ode to Joy, thought, romantic that he was, that “Man plays only when he is in the full sense of the word a man and he is only wholly Man when he is playing.” “Plays are play,” writes Alison Gopnik, “and so are novels, paintings, and songs.” Cognitive scientist Steven Pinker has a more functional point of view. What pure mathematics and music have most clearly in common is that they both fit Pinker’s definition of “cheesecake,” “unlike anything in the natural world because it is a brew of megadoses of agreeable stimuli which we concocted for the express purpose of pressing our pleasure buttons. Pornography is another pleasure technology…. [T]he arts are a third.”105 Pure mathematics would appear to be a fourth, and what it shares with the arts, and with music in particular, is that, unlike cheesecake or even pornography, its variety is literally limitless.

Schiller’s playing, whether it is opposed to seriousness, to work, or (anticipating Freud as well as Pinker) to reality, is Burghardt’s “relaxed field,” freedom from constraint. Reflecting on his use of a truuk to prove a conjecture Grothendieck had formulated thirty years earlier, Huizinga’s compatriot Frans Oort admits that the “[tricky] aspect of mathematics” seen by some to lack elegance “has an appealing beauty to me … I find this exciting.”106 In the end the alliance between mathematics and music—and the arts more generally—is in freedom and creativity rather than beauty or formal analogy.

Some of us find Pinker’s cognitive behaviorism too … mechanical … and are drawn rather to Lévi-Strauss’s construal of myth and music as “instruments for the obliteration of time.” Mathematics, whose proofs admit the impersonal imperative and the conditional but whose conclusions are stated in the eternal present tense, is another such instrument. What need has mathematics of the formal trappings of seriousness when, to quote Norbert Elias, it is “one of the symbolic structures in whose name one may claim, as [G. H.] Hardy does, to offer eternal verities outlasting death?”107

Goethe wrote in Die Natur that “Life is [nature’s] loveliest invention, and death is her Kunstgriff for having much life.” But outwitting death, though it occasionally backfires, is a trickster specialty; outlasting death must be the finest trick of all. And the conundrum of mathematical seriousness, such as it is, may resolve itself when viewed in a broader perspective. Reading popular culture’s mathematician as a variant of the computer hacker or “wizard” who is by now a staple of thrillers of every genre, the recent fascination with mathematics is easily explained as a by-product of our deepening immersion in cyberspace as a second nature. Ed Frenkel’s fictional persona, as mathematician, occupies the same moral culture of resistance as the hackers in The Matrix, or the plugged-in primitives of Avatar, or The Girl with the Dragon Tattoo. Nearly all the apparently mathematical protagonists of popular films work in areas closely associated with computing: prime numbers for Proof, graph theory for Good Will Hunting, numerical computation for Pi—it wouldn’t be a stretch to add the rock-star mathematician’s chaos theory in Jurassic Park. In the background we hear popular rather than classical music in these films—R&B in Good Will Hunting, a blend of (black and white) electronic dance music genres in Pi, while the romantic interest in Proof plays in a rock band—and these productions are best seen as borderline variants of a much more extensive genre of which the (ideal)-typical representative is cyberpunk fiction, whose protagonist is a film noir reincarnation of the nineteenth-century mathematical romantic and whose very name is an homage to the dominant rock ethos of the time.108

Like the (often inadvertently) anarchist heroes of Pynchon’s novels, who struggle to maintain their values, their personal identities, even their lives in the face of the implacably deterministic projects of more or less precisely identified Powerful Beings, cyberpunk stages a confrontation between an artificial intelligence programmed to impose its version of certifiable seriousness and a (Neu-)romantic trickster hero whose quest is to preserve what remains of our common humanity. “I’mm goinng to do everrythinng I can to sstopp you fromm turrnning that poorr olld mman innto a piece of ssofttware in the bigg bopperrs’ memorry bannks.”109 Circumstances force the typical cyberpunk protagonist into a life of underground resistance to a political and economic order projected onto a future dominated by AI but not so different, in the end, from the instrumental market rationality with which each of us must come to terms in this life. But this faceoff mirrors the confrontation within mathematics between two facets of the same split personality: mathematics is both the condition of alienation and the skill without which community cannot be restored.110 The former is ingenium as mekhane, the relentless mechanical unfolding of an “unforgiving cold mathematics,” the Matrix, “the man,” the deterministic and Powerful Beings of Gravity’s Rainbow known only as They111—and is represented in Hollywood by mathematicians with pronounced social pathologies. The latter is ingenium as trick, the way of Br’er Rabbit and Coyote, of the cyberpunk hackers of film and fiction—and of the eternal sun-drenched present of Vineland’s group-theorist and groupie-magnet Weed Atman, avatar of Galois if not Grothendieck, and his utopian People’s Republic of Rock and Roll.


* Loose translation: “And thus if the intention is bad, such a power is called astuce, or malicieuseté.”

* LITTLE OLD MAN: Two faucets flow into a reservoir/two local trains leave a station/at 20 minutes interval / Val, val, val!

CHILD: My God, it’s arithmetic!

LITTLE OLD MAN: Tic, tic, tic / 4 and 4 make 18 / 11 and 6 make 25 / 7 times 9 make 33.