notes

CHAPTER 1 INTRODUCTION: THE VEIL

1. (Hilbert 1900).

2. See Gillies (1995). Although Kuhn’s work has been criticized by professionals, I cite him because his vocabulary is so familiar and lends itself to the picture imagined in the previous paragraph.

3. (Kuhn, 1996, p. 33).

4. mit der reicheren Entfaltung einer Wissenschaft die Notwendigkeit auftritt, die ihr zu Grunde liegenden Begriffe und Prinzipien zu verändern. Es geht auch in diesem Punkte der Mathematik nicht anders wie den Naturwissenschaften : neue Erscheinungen stürzen die alten Hypothesen um und setzen andere an ihre Stelle (Kronecker 2001, p. 233). Subsequent chapters, especially chapter 7, will return to Cantor’s set theory and the Foundations Crisis.

5. Siegel was sympathizing with his contemporary, Louis J. Mordell, who had just written a famously savage review of Serge Lang’s book Diophantine Geometry. Mordell and Siegel make brief appearances in the next chapter (Lang 1995).

6. Gray (2010, p. 37) refrains from using Kuhn’s framework, undoubtedly for good reasons.

7. Mumford, whose career is evenly divided between the purest branches of pure mathematics and the most challenging questions of artificial pattern recognition, has been awarded most of the highest distinctions in the field; he taught at Harvard for many years before moving to Brown. The quotation is from Casazza (2011). Frank Quinn (2012, p. 33) reports that Paul Halmos claimed to be in “roofing and siding.” Humanists see things differently. Alex Csiszar quotes Bruno Latour’s claim that “[i]t is hard to popularize science because it is designed to force out most people in the first place,” and concludes that “the particular difficulty that mathematicians have in talking to nonmathematicians about what they do, in fact even in talking to fellow mathematicians who are not quite a part of the same subdiscipline, is a telling indicator of the discipline’s tendency to exclude all but the very few” (Csiszar 2003).

CHAPTER 2 HOW I ACQUIRED CHARISMA

1. That the quest for certainty can awaken a vocation is a staple cliché of mathematical reminiscences (which this and similar paragraphs in this chapter exemplify only by way of pastiche). Thus, Helmut Hasse chose mathematics “because even in my earliest youth the impulse toward knowledge of objective, irrefutably valid truths was so strong that it overshadowed every other interest” (Hasse 1952, my translation). It’s tempting to claim that the certainty of mathematics appealed to me as well, as a response to the turbulence of 1968, but it didn’t really happen that way. It was only after comparing the biochemistry and mathematics programs during my first year in Princeton that I opted for the latter. Certainty, in the philosophical sense, was not an issue for me at the time. The stability of the hierarchy set out in the IBM poster was appealing, but not more so than the apparently endless string of Nobel Prizes in molecular biology. There are many paths to mathematics and just as many mathematical personalities.

2. The Eames couple was and still is famous for the lounge chair they designed in 1956. Through the Eames Studio, Redheffer produced a series of portraits of more than a hundred distinguished mathematicians, many of whom were still alive when I first saw the IBM poster. The portraits are posted in the corridors in the heart of the UCLA Mathematics Department (Gamelin 2005). Other noteworthy collections include the portraits of academicians on the top floor of the Steklov Institute of Mathematicians in Moscow, the Michael and Lily Atiyah Portrait Gallery at the University of Edinburgh (a mix of historical figures and contemporaries and collaborators of Sir Michael Atiyah), and Faces of Mathematics at Herriot-Watt University in Edinburgh, whose mission is to “present […] the human side of this most austere and challenging area of modern science.” The book (Cook 2009) pairs portraits of contemporary mathematicians with one-page interviews.

3. Here E. T. Bell is notoriously inaccurate. Caroline Ehrhardt (2011, p. 196) calls his book a “biographical novel.” See also the discussion of Amir Alexander’s (2011) book in chapter 6.

4. On April 5, 2012, IBM released a digitized version as a free iPad app called “Minds of Modern Mathematics” (still including only one female mind), expressing the hope that “classes and students will use the app, provoking more people to pursue math, science or technology-related educations and jobs.” Large-scale versions of the IBM timeline are on display at the New York Hall of Science in Queens and at the Museum of Science in Boston. (http://www.wired.com/gadgetlab/2012/04/new-ibm-app-presents-nearly-1000-years-of-math-history/). The word giants was inadvertently suggested by one of the Princeton University Press readers of my book proposal. Many future mathematicians are still initiated into the field at high school summer programs like the one I attended at Temple. The PROMYS program at Boston University is one of the most successful and influential; 100 of its graduates are currently professors, 70% of them in mathematics (Clay Mathematics Institute 2013). It is much more mathematically challenging than the Temple program, devotes less attention to the “romantic” chapters in the field’s history, and is wary of promoting a “great man” version of history of mathematics. Still, the Giants and Supergiants are inevitably part of the story. Glenn Stevens, who has been leading PROMYS since 1989, explained (in a private communication) that “the point is not to see the pioneers as the ‘great masters’ so much as to have an internal understanding of how they might have been motivated to develop these theories in the first place. We hope the students will begin to identify personally with the ‘masters.’ ”

5. Neil Chriss, in (Lindsey and Schachter 2007, p. 111).

6. (MacIntyre 1981, p. 181). Also, “insofar as the virtues sustain the relationships required for practices, they have to sustain relationships to the past—and to the future—as well as in the present” (p. 206).

7. (Smith 1759). Smith adds: “It is not always the same case with poets, or with those who value themselves upon what is called fine writing.”

8. (Krantz 2002, p. 186). Weil remained a snob even when he was apparently being generous: “It is also necessary not to yield to the temptation (a natural one to the mathematician) of concentrating upon the greatest among past mathematicians and neglecting work of only subsidiary value. Even from the point of view of esthetic enjoyment one stands to lose a great deal by such an attitude, as every art-lover knows … ” (Weil 1978). This is the published version of a lecture I attended in which Weil explained that what “one stands to lose” is precisely a feeling for the gulf separating genius from “subsidiary” talent. I hasten to add that Weil’s vanity was remarkable because it was atypical—humility is still expected and can in fact be read as a sign of prestige, since the mathematicians who know the most are also those who realize how much more there is to know. And I shouldn’t be too hard on Weil, tempting target that he is, not only because his name is on many top-ten lists, but specifically because of his overwhelming influence on number theory, in particular on the kind of number theory to which this book devotes much of its attention.

C. L. Siegel was last seen in chapter 1 complaining about a pig in a beautiful garden.

9. (Weber 1922).

10. (Menand 2001, pp. 153-54). Benjamin Peirce has given his name to the coveted mathematics instructorships (“BPs”) at Harvard; he is better known as the father of philosopher C. S. Peirce.

11. See (Zarca 2012, p. 318) and the surrounding discussion. Maybe this is typical of “high-consensus fields.” “Physicists often describe how they are able to rank-order the achievements of fellow physicists” (Hermanowicz 2007).

12. http://mathoverflow.net/questions/10103/great-mathematicians-born-1850-1920-et-bells-book-x-fields-medalists. One of the lists granted Kolmogorov only an “honorable mention.” It would be interesting to compare the list to Redheffer’s UCLA portrait gallery, for which I was unable to find a catalogue online.

13. No less prestigious are the Nemmers Prize (since 1994), the Kyoto Prize (every four years since 1985), and the Crafoord Prize (since 1982), the latter only occasionally granted to mathematicians. The monetary value of three of the four IMU prizes—the Fields Medal, Nevanlinna Prize, and Gauss Medal—is modest, but the Chern Medal, created in 2010, is worth $250,000, and some of the other prizes mentioned bring close to $1 million. And this year (2014), billionaire Yuri Milner and his Silicon Valley friends will start to “transform [mathematicians] into rockstars” with the first of the new $3 million Breakthrough Prizes in mathematics. Milner thinks scientists “should make at least a fraction of what some Wall Street trader makes” (Walker 2013).

14. For Kovalevskaia, see (Ann Hibner Koblitz 1983) and (Audin 2008). For Schwartz, the best source is his autobiography (Schwartz 2001). Several complete biographies of Grothendieck have long been in the works; the most complete is (Scharlau 2010) and its two companion volumes. There’s also Grothendieck’s own 900+-page Récoltes et Sémailles, as well as the two-part report (Jackson 2004). Perelman is the subject of the modestly titled (Gessen 2009). Two trade books were written about Erdős shortly after his death [in this book I cite (Hoffman 1998)], and there is also a sixty-minute documentary.

15. In hindsight I realize he was spaced out rather than distant, focusing his inner eyes on his research rather than his visible eyes on the person to whom he was speaking. This characteristic trait, familiar to anyone who has lived with a mathematician, was already well known in antiquity (see chapter 6).

16. For example, the polynomial equation x² + y² = 1 in two variables defines a circle; the equation x² + 2y² = 5 defines an ellipse; the equation x² + y² = z² defines a cone in three dimensions. There are pictures in chapters β and δ for those who don’t remember the x-and y-axes of high school algebra. Once there are more than three variables, the figure defined by the equation can no longer be represented by a picture, but it can be represented in many other ways, which together make up the subject of algebraic geometry.

17. See (Rashed 2005). The IBM chart informs us that al-Khayyam is best known for his Persian poetry cycle, the Rubaiyat, and Wikipedia says the same thing; but (Rashed and Vahabzadeh 1999), the definitive edition, in Arabic and French, of al-Khayyam’s mathematical works claim (on p. 5) that no contemporary documents prove that al-Khayyam the poet and al-Khayyam the mathematician (and philosopher) were, in fact, the same individual.

18. More precisely, what Weil counts are solutions to congruences, where the number of solutions is always finite: see chapter γ. Tim Gowers (2013) has given a very readable account of Weil’s topological idea in the presentation of Pierre Deligne’s work that accompanies the announcement that Deligne was awarded the 2013 Abel Prize, most notably for his proof of Weil’s conjectures.

19. Take a swimming pool (any shape you like as long as there are no islands in the middle), cover it completely with dandelion fuzz, and set up a smooth current on its surface. After one hour (or any other lapse of time you choose), at least one piece of fuzz will be back where it started. This is an example of Brouwer’s fixed point theorem, which, of course, is not about swimming pools and currents but about flat surfaces—specifically flat surfaces without holes, in the technical sense discussed in chapter 7—that are subject to an internal motion that is continuous in the sense that nothing is allowed to be torn. (More precisely, the theorem deals with continuous mappings—using the surface as a map of itself that may be distorted but is not torn.) For more complicated surfaces, or configurations of higher dimension, there is the more elaborate Lefschetz fixed point theorem that tells you how many points are likely to be left in place after a continuous motion. Mathematical economists have used versions of this theorem to convince themselves that economic systems tend to equilibrium, in spite of considerable evidence to the contrary, but that’s another story.

20. Henry Brougham, Lives of the Men of Letters and Science, who Flourished in the Time of George III, quoted in (Shapin 2008, p. 8). Shapin explains that Brougham was including “natural philosophers”—scientists, in other words—among philosophers.

21. (Weber 1978, section 10).

22. The term research program is associated with Lakatos, especially (Lakatos 1978). In this book I use the term more informally, as in (Mazur 1997, especially section 6).

23. B. H. Gustin uses the word in much the same way as I do: “Much as religious charisma must be routinized … the charisma of science is regularized, attenuated, and embodied in the institution of publication, as well as, of course, in prizes and prestigious professorships” (Gustin 1973, p. 1131). Gustin does not stress the symbolic importance of mathematical charisma. An anonymous reader suggested that I was using the word charisma to designate “an emergent property of one’s structural position” rather than an intrinsic (and magical) aspect of one’s character, and that what I mean by charisma corresponds rather to “what sociologists mean by the term ‘status,’ which is to say the prestige accorded to individuals because of the abstract positions they occupy rather than because of immediately observable behavior” (Gould 2002). I thank the same reader for bringing to my attention the following quotation from (John Levi Martin 2009): “The very charisma that makes this person appear extraordinary may better be seen as a property of the pattern of attributions of charisma, reversing cause and effect.”

<The Martin quotation certainly corresponds to my application of the term to my own situation; but I wouldn’t necessarily characterize the charisma of the Giants on the wall in this way. One feature that distinguishes them—and also initiators of research programs, like Grothendieck or Langlands—is that they are remembered for creating the “structural positions” to which their names are subsequently attached, rather than for fitting into preexisting positions—including the positions that preexist in association with features of the natural world (like elementary particles, or aromatic hydrocarbons, or DNA) and that often have no counterparts in mathematics. It is common to use words like “visionary” and “prophetic” when speaking of their work, and the reader will recall from the preface my observation that the way we talk about value in mathematics borrows heavily from the discourses associated with religion. It therefore doesn’t seem at all inappropriate to adapt Weber’s terminology of charisma in connection with such Giants, nor to suggest that this charisma rubs off, as in Weber’s model of “routinization of charisma,” on those of us who make even modest contributions to their research programs.

24. (Grothendieck 1988, p. 50).

25. (Shils 1968; Max Weber 1978, Part one, III, p. 242). For “seriousness” see chapter 8.

26. Felix Browder (1989) cites Weber’s distinction between “rational-bureaucratic” and “charismatic leadership” to explain how M. H. Stone “transformed” the University of Chicago mathematics department “into the strongest mathematics department in the U.S. (and at that point probably in the world).” Stone’s charismatic leadership was organizational and maybe personal, but not scientific in the sense I have in mind; he transformed the department but not the discipline.

27. The Matthew Effect consists “in the accruing of greater increments of recognition for particular scientific contributions to scientists of considerable repute and the withholding of such recognition from scientists who have not yet made their mark” (Merton 1968), quoted in (Cole 1970). The name is an allusion to Matthew 25:29: “For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken even that which he hath.” Merton also introduced the expression “role model” into sociology. The present chapter attempts to demonstrate that the Giants and Supergiants of the IBM poster are much more than role models in Merton’s sense.

28. See the text that accompanies Serre’s photo in (Cook 2009, p. 144), or the article (Huet 2003) for samples of his charm.

29. (Max Weber 1922). Analyzing the Grothendieck-Serre correspondence, Leila Schneps observes (following Grothendieck) that “self-analysis in any form strikes Serre as a pursuit fraught with the danger of involuntarily expressing a self-love which to him appears in the poorest of taste” (Schneps 2014b).

30. This position was expressed with memorable violence by G. H. Hardy (2012), as we’ll see in chapter 10, and this may account in part for the suspicion greeting mathematical public intellectuals in the English-speaking world. The situation is rather different in France, where Fields medalists regularly participate in public debates. Attitudes may be changing, in part thanks to the success of Gowers’s blog and similar Web-based platforms.

31. Three well-known incidents hint at a different model—the growth of the Moscow mathematical school in the twentieth century (Graham and Kantor 2009), the creation of the Bourbaki movement in Paris in the 1930s, and the self-education of G. Shimura and his contemporaries in Tokyo in the 1950s. Kuhn’s model of paradigm shift is as artificial as class struggle in explaining these developments. In each case a new generation, inspired by foreign practices, created a new and original approach. For Bourbaki it was Göttingen, especially the abstract algebra of Emmy Noether, the one woman on the IBM poster. Grothendieck’s program was mainly built on the Bourbaki model, whereas Langlands was more eclectic, incorporating important insights of the Bourbakist Séminaire Cartan, the novel ideas of Shimura and Taniyama in Japan, and the branch of the Moscow school around Gelfand, as well as many others. The Frye quotation is from (Frye 1957, p. 186).

32. (Shils 1968).

33. (Bourdieu 1984, p. 208). Nor am I thinking of passages like this one: “The privileged classes find in what one can call charismatic ideology … a legitimation of their cultural privileges … ” (Bourdieu and Passeron 1964, 1985, p. 106). This recent quotation from (Clerval 2013), also in the Bourdieu spirit, is more appropriate. “The intellectual petite bourgeoisie as a whole is in a paradoxical social position … dominated by the bourgeoisie in being exploited in its work and in suffering the degradation of working conditions … but it also receives … a kind of partial retrocession of capitalist surplus value that bears witness to its participation in the exploitation of popular classes.” Exploitation by mathematicians is primarily symbolic and inadvertent; a good example is the subject of chapter 4.

34. (Joyal 2008, p. 154). Joyal is a mathematician, not a sociologist, and the quotation is from a recent textbook. Reuben Hersh, also a mathematician, has promoted a social constructivist approach to mathematics in a series of books (sometimes with Philip Davis), notably (Hersh 1997). But his primary interest is the social nature of mathematical truth. Truth is important, but it’s only one of the values that motivates mathematicians.

35. (Ehrhardt 2011).

36. “The advent of ‘league tables’ of university excellence, first produced in 2003 by Shanghai Jiaotung University, was perhaps the inevitable consequence of the convergence during the 1990s of liberalisation of international markets, enabled by new communications technologies, and the shift of the global economy towards one based on information and knowledge…. universities, as sources of innovative ideas and highly skilled manpower, have come to be seen as vital agents in maintaining national competitiveness…. Amongst governments, the rankings indicated the extent to which their universities were achieving the excellence presumed to be needed to drive and support national economic prowess” (Boulton 2010).

37. (Boulton 2010), Times Higher Education Supplement, June 24, 2010.

38. (Cole and Cole, 1967; 1968). See also (McCain 2010) for some of the history of citation indices. In 1980, sociologists could write that “it is not logically necessary that … bibliometric centrality is equivalent to intellectual centrality in the development of a research area …” while claiming that it was legitimate to use bibliometric indices to analyze “eminence hierarchies” in the case under study (Hargens et al. 1980).

39. (Hirsch 2005).

40. (Hagstrom 1964).

41. “… status honor is normally expressed by the fact that above all else a specific style of life is expected from all those who wish to belong to the circle” (Weber 1978, vol. II, section IX, p. 932). Val Burris proposes in (Burris 2004) to explain the persistence of status rankings of university departments on the basis of a Weberian model of social exchange, which of course is not inconsistent with authentic merit. Burris studies reciprocal hiring practices of leading sociology departments, but I have no doubt that mathematics departments in the United States, United Kingdom, and France behave similarly. Mathematics departments are ranked by U.S. News and World Report and by the Times Higher Education Supplement, for example; the lists are not consistent.

42. (Cole and Cole 1973).

43. Langlands (2013), expressed with characteristically understated acerbity. Mario Biagioli (2003, p. 266) writes more generally that “The [scientific] author is the producer of the work, but he or she is also ‘produced’ (i.e., recognized and rewarded as such) by his or her peers.

44. (Foucault 1984, p. 112).

45. The topic is explored at length in (Mazur 1997) and from a different point of view in chapter 7.

46. The lectures presented during the seven installments of SGA were published in 12 volumes that come to a total of 6209 pages. There are very few pictures.

47. (Shapin 2008, pp. 266–267).

48. Langlands’s fairy tale article is (Langlands 1979a). The analogy of motives or automorphic forms with elementary particles should not be taken literally; and it should be understood that both Langlands’s “speculation” and Grothendieck’s “imagining” are rooted in the kind of formally rigorous reasoning that characterizes mathematics.

49. Formulated in the early 1960s by the English number theorists Bryan Birch and Sir Peter Swinnerton-Dyer, now emeritus professors at Oxford and Cambridge, respectively. See (Ash and Gross, 2011). I discuss some implications of the conjecture at length in chapter δ.

<As for the other Clay problems related to number theory: the third of Weil’s conjectures, solved by Deligne on the basis of Grothendieck’s program, was a geometric version of the Riemann Hypothesis, and it is widely believed that the completion of Langlands’ program is intimately connected with the solution of Riemann’s original version, the first of the Clay problems (see chapter α for a brief description). The Hodge conjecture predates Grothendieck’s work—though early in his career Grothendieck corrected Hodge’s original prediction—but its most satisfying interpretation is as a precursor (or avatar, see chapter 7) of Grothendieck’s theory of motives, which, in turn, is the starting point for Langlands’s Märchen.

50. “Seen as” is conceptual shorthand. “Seen as an avatar of an automorphic form,” in the sense to be explored in chapter 7, is more accurate. The existence of an automorphic form connected to an elliptic curve was already conjectured at the time—it’s a long and complex story, described in (Ash and Gross 2011)—and had long been known for equations (E1) and (E5) but was proved in general only in 2000 by C. Breuil, B. Conrad, F. Diamond, and R. Taylor, using the methods introduced by Wiles in his proof of Fermat’s Last Theorem.

51. (Mulkay 1976, pp 448, 454).

52. The articles were J. Coates and A. Wiles, “On the conjecture of Birch and Swinnerton-Dyer,” Inventiones Math 39 (1977), 223-251; B. H. Gross and D. B. Zagier, “Heegner points and derivatives of L-series,” Inventiones Math. 84 (1986), 225–320; K. Rubin, “Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication,” Inventiones Math. 89 (1987) 527–559; B. Perrin-Riou, “Points de Heegner et dérivées de fonctions L p-adiques,” Inventiones Math. 89 (1987) 455–510; V. A. Kolyvagin, “Finiteness of E(Q) and ш(E,Q) for a subclass of Weil curves,” Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522–540; K. Kato, “p-adic Hodge theory and values of zeta functions of modular forms,” Astérisque 295 (2004): 117–290. The reader will have guessed that Inventiones Math. is a “great journal.” A few of the authors were a bit older and already had “great jobs” at the time of their breakthrough, but most belong roughly to my age cohort. If this chapter were autobiographical, I would let the reader know how it felt to enter professional life at the same time as these colleagues, especially before I acquired tenure.

53. “Research scientists gain in professional repute to the extent to which they are seen to have contributed intellectually to areas of scientific consensus” (Mulkay 1976, p 448). On mentoring in science, see (Hooker et al. 2003).

54. In Truth and Truthfulness, Bernard Williams, who describes himself as a genealogist, refers to genealogists who talk this way as “deniers.” The Persian scholar al-Biruni had already met such colleagues in the eleventh century:

“[I]f he is shown that arithmetic and geometry are impossible to understand unless one proceeds systematically from first principles, unlike other sciences in which he may be acquainted with something of their middle (parts) or their ends without knowledge of their beginnings, he thinks that this is intended to [turn him away] from his appreciation and to confuse him. This, he imagines, is similar to the ignorance into which (non-initiate) members of (secret) sects (are led) with regard to the doctrines of their sects until they had taken the oaths, entered into the covenants, and made a long practise and training. This adds to his revulsion, so that the stopping of his ears with his fingers becomes his most potent recourse, and the raising of his voice in shouts his most powerful equipment.”

From al-Biruni’s preface to (al-Biruni 1976, p. 6). Other references here and later are to (Corfield 2012).

55. (Bloor 1991, p. 86). The strong programme in the Sociology of Scientific Knowledge is the genealogical approach to sociology of science pioneered by Bloor, in the book just cited, and Barry Barnes.

A recent article by Barany and MacKenzie (2014), which takes as its starting point the central role of chalk and blackboard (or their substitutes) in mathematical communication, is to my mind an unusually successful and insightful example of a “critical and naturalistic” study within the genealogical approach to sociology of mathematics. Its examination of the practice of “following the argument” at seminar talks is one of a number of fruitful contributions to a future tradition-based sociology of mathematics. For my purposes, its focus is nevertheless excessively epistemological, as the concluding sentences suggest: “Mathematical ideas are not pre-given as the universal entities they typically appear to be. The most important features of mathematics can be as ephemeral as dust on a blackboard.” When the authors write, “Given the stereotype of the lone mathematician and the importance of breakthrough stories in post-facto accounts of mathematical innovation … the predominance of project-work in mathematics is … surprising,” I’m surprised to learn this is surprising. The view of mathematics as a “science of ideals” to which the authors refer may be widely shared by mainstream sociologists of science, less so by mathematicians—unless the “ideals” are “naturalized” in connection with the values explored in the present chapter.

56. Exceptions are (Hagstrom 1964) and (Stern 1978). Mathematics is frequently considered along with other sciences by sociologists of both tendencies, for example, in (Hagstrom and Hargens 1982).

(Restivo 2001) was not quite, as the author claimed, the first book by a sociologist on the sociology of mathematics; it was preceded by (Heintz 2000) and followed by (MacKenzie 2001) and (Rosental 2003). All four books, while very different, are genealogical in inspiration, though Heintz makes some gestures in the tradition-based direction.

A much more recent study, closer to the encyclopedist tendency than to the other two, is (Zarca 2012). Zarca’s is the only study I’ve seen based on an extensive survey of values, practices, and social conditions of mathematicians and (for comparative purposes) scientists in related fields. The survey, conducted anonymously by Internet, included responses of roughly 1000 to 2000 individuals (some responses were more complete than others). The author restricted attention to scientists working in France (I responded to the survey) but quotes other studies and biographical material to supplement his data. The book appeared after I had completed a first draft of this chapter, and I could not incorporate Zarca’s conclusions at length, but I note that he and I share many of the same concerns. In particular, he devotes a subsection of twenty pages to “l’élitisme des mathématiciens” (in which he quotes Weil extensively!) but does not treat the hierarchy as a constitutive factor in mathematics.

57. The statement deserves to be qualified: most of the encyclopedist texts I cite in sociology are empirical studies (like Zarca’s) and (unlike Zarca’s) have very little to say about philosophical questions, whereas the Strong Programme seems more interested in philosophy of Mathematics than in mathematics; see note 60. Here I refer primarily to English-language philosophy. For continental philosophy, the subjective experience of the mathematician is often central; this was especially true of the phenomenological approach initiated by Husserl and still very much a factor in France (but, interestingly, totally absent from Zarca’s discussion).

58. If philosophers of Mathematics truly monopolized the certification of mathematical legitimacy, in the way that psychiatrists monopolize the dispensing of anxiety medicine, then logicians, working on behalf of philosophers of Mathematics, would have to be viewed as mathematics’ “puppet government.” Most mathematicians, however, would disregard the latters’ directives even if most logicians hadn’t ceased issuing them long ago, and most philosophers know it. For example, Marcus Giaquinto’s philosophical treatment of the Foundations Crisis—see chapter 3—is entitled The Search for Certainty (New York: The Clarendon Press, Oxford University Press, 2002), but it concludes with the reminder that “[m]athematics is definitely not just logic.” Pointing to the variety of modes of typical mathematical reasoning, “reasonings that involve modalities (contingency, necessity, possibility), or that are affected by the passage of time, or that concern individuals, or that require iconic displays, or that link heterogeneous acts of thought or speech or kinds of things.,” Emily Grosholz concludes that “it is no wonder that logic all by itself cannot express mathematics” (Grosholz 2007, p. 284).

59. The distinction, which overlaps my classification, between mainstream and mavericks in philosophy of mathematics, goes back to (Kitcher 1984). What I call philosophy of mathematics is often called the philosophy of mathematical practice; I will avoid this expression, because it is long-winded and because it implies that what deserves to be called “philosophy of mathematics” is concerned with something other than what mathematicians actually do.

I’ve always thought it peculiar that the sociologist Bloor attempted to account for “The Standard Experience of Mathematics”—the title of the chapter from which the preceding quotation is taken—without reference to any sociological data whatsoever. (He describes a game invented by mathematical educator Z. Dienes but not the people playing the game.) Such a priorism is typical of the philosophy of Mathematics, and I agree that sociology cannot be merely empirical, but the notion of a sociology without people challenges the imagination.

60. (Greiffenhagen 2008). In her review of (MacKenzie 2001), Bettina Heintz (2003) makes a similar claim: “The few [sociologists] who have dealt with mathematics did it in a rather programmatic way and with reference to sometimes quite arbitrary examples from the history of mathematics. The empirical reality of modern mathematics, however, was hardly touched.” One of the sociologists Heintz had in mind was Sal Restivo, who claimed in 1999 that sociology of mathematics—by which he apparently also means sociology in the SSK mold—“has attracted very few practitioners and seems to be a citation orphan in core STS.” See also (Greiffenhagen, in press) and (Greiffenhagen and Sharrock 2011), a helpful antidote to the exclusive focus on epistemic consensus—on Mathematics rather than mathematics—that, as the authors demonstrate, led to a misreading of the relation between public and private aspects of mathematical research. Greiffenhagen and Sharrock find that even Reuben Hersh (see note 34), who for thirty years has taken the lead in promoting a humanistic alternative to the philosophy of Mathematics, exaggerates the contrast between the (formalist) “front” and (constructivist) “back” of mathematics in (Hersh 1988; 1991).

It bears mentioning that, as with most contemporary work in philosophy of Mathematics, Greiffenhagen’s examples are taken from logic, whose organization and practices are far from representative of mathematics as a whole; but the phenomena to which his video analysis draws attention are familiar in more central branches as well. See also (Greiffenhagen and Sharrock 2009a, 2009b).

61. (Corfield 2012, pp. 253, 255). This is in line with the frequently repeated observation that what matters in mathematics is not truth as much as interest; see the discussion of heuristics in the next chapter. (C. S. Fisher 1973) claims that mathematicians’ “methods of attack are not rationally chosen,” that “They are derived from a combination of training, taste, and personal predilection.”

62. The preferred technique, the Arthur-Selberg trace formula, is a close relative of the fixed point theorem mentioned in note 19. Langlands’ unpublished letters are magnificently preserved at http://publications.ias.edu/rpl/, along with his published articles. For Act 1, see especially the letters in sections 5 and 6 that record the evolution of Langlands’ ideas during the crucial early years; the goals of Act 2 are described in the articles in sections 8 and 10, especially those published in the 1970s.

I have repeatedly been invited to contribute a chapter (after paying a hefty publication fee) to a book project on endoscopic surgery; however, this has nothing to do with endoscopy in Langlands’ sense.

63. His first work on (FL1) was in collaboration with G. Laumon; his complete solution was published in (Ngô 2010). Objects G and H are examples of what are called perverse sheaves, a notion developed by a number of mathematicians around 1980 to reduce the study of complicated geometric objects to a more familiar geometry. Grothendieck has written in Récoltes et Sémailles that he dislikes the terminology.

64. (Sinclair and Pimm 2010). Membership in the tradition is in principle not class-based. In his Vie de M. Descartes, Adrien Baillet tells the story of the Dutch peasant and shoemaker Dirck Rembrantsz van Nierop, who, not satisfied with the mathematics he had found in books written in the vernacular, left his village in the hope of meeting Descartes. The first time he showed up at the philosopher’s door, he was sent away. The second time, he was offered money, which he refused. Finally, on the fourth visit, he was admitted to the presence of Descartes, who “immediately recognized his talent and merit” and soon counted him among his friends. Rembrantsz is remembered as “one of the leading astronomers of his century” as well as a cartographer and published a considerable number of books in mathematics. (Baillet 1691, Tome II, pp. 555–57).

65. For Jevons, see note 18 to chapter 10. For mass culture, see chapter 8, especially the comments of F. R. Leavis and Pierre Boulez.

66. Respectively, the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the National Science Foundation.

67. See the list near the end of the chapter and in note 75.

68. Gowers’s announcement was at http://gowers.wordpress.com/2012/01/21/elsevier-my-partin-its-downfall/. The signatures were collected at http://www.thecostofknowledge.com/, which also hosts the Statement of Purpose.

69. (Clarke 2010) lists “five traditional functions of journals: Dissemination, Registration, Validation, Filtration, and Designation.” Henry Cohn, a signatory to the Statement of Purpose, added the function of “archiving … particularly crucial in mathematics, because the half-life of math papers is much greater than in other scientific fields….” Merton, of course, was a functionalist. For the record, my own position is that a scientific journal is not reducible to a collection of functionalist functions but is rather an expression of a social relation. For more of this kind of talk, see (Marx 1858). Tao’s and Cohn’s comments were at http://publishing.mathforge.org/discussion/7/whats-the-point-of-journals/#Item_0.

70. (Gustin 1973, p. 1128).

71. Many of the “greatest journals” are in fact published at relatively low cost by university presses or professional societies, but there are apparently not enough of them to fill a CV, especially in certain branches of mathematics.

72. (Bellah 2011, p. 633, note 54).

73. http://publishing.mathforge.org/discussions/?CategoryID=0. Mathematical physicist John Baez, whose “This Week’s Finds in Mathematical Physics” was one of the first successful mathematics blogs, opened a Math 2.0 discussion with a proposal to add a simple 1-to 5-point rating system to existing preprint servers. Baez emphasized that his goal was not to replace the current peer review system; others have made more radical proposals (for example, at www.papercritic.com).

At lunch one day in Paris, a colleague who had signed the SoP described a less radical vision of a future without commercial publishing, of journal editorial boards directly vetting articles that had been posted on dedicated internet file servers, Tao’s skepticism notwithstanding; maybe a single file server, like arxiv.org, would suffice. “What about the journal hierarchy?” gasped a publisher’s representative. No problem; editorial boards would sort that out among themselves.

74. For an academic mathematician, these duties include but are not limited to publication in peer-reviewed journals, speaking at national and international conferences, training PhDs, and applying for research grants. Not all mathematicians are academics, of course; but most professional pure mathematicians do teach at universities.

75. Also the Research Institute for Mathematical Sciences (RIMS) in Kyoto, the Isaac Newton Institute near Cambridge University, institutes in Bonn and Münster, Moscow and St. Petersburg, Barcelona, Vienna, Montreal, and Beijing; as well as the even longer list of institutes I haven’t yet visited, notably IMPA (Instituto de Matemática Pura e Aplicada) in Rio de Janeiro. Weekly conference centers on the Oberwolfach model include the Centre International de Rencontres Mathématiques (CIRM) on the limestone cliffs overlooking the Mediterranean coast near Marseille and the spectacular Banff International Research Station (BIRS) in the middle of Banff National Park in Canada.

76. See also note 43. Grigori Perelman reportedly turned down his 2006 Fields Medal and the $1,000,000 prize awarded by the Clay Mathematics Institute in 2010 for his solution to the Poincaré Conjecture, in part because he felt the award overlooked the fundamental earlier contributions of Columbia University professor Richard Hamilton. Perelman’s unusual attentiveness to what is ostensibly one of the guiding ideals of science—disinterestedness, third on Merton’s list of norms—is discussed in (Zarca 2012, pp. 20–21). See also chapter 6, note 36.

77. (Bourdieu and de Saint Martin. 1996, p. 19).

CHAPTER α HOW TO EXPLAIN NUMBER THEORY (PRIMES)

1. Debbie Epstein, Heather Mendick, and Marie-Pierre Moreau, “Imagining the mathematician: young people talking about popular representations of maths,” Discourse: Studies in the Cultural Politics of Education 31, No. 1 (February 2010): 45–60. The Eilenberg Lectures mentioned in this chapter are named after the late Samuel Eilenberg, professor for many years at Columbia, who had an impressive gray beard.

2. (Gowers 2008).

3. (Hasse 1952).

4. At least since (Hardy 2012), which included brief proofs of the irrationality of the square root of 2 and the infinity of primes to illustrate his mathematical aesthetic. See chapter 10.

5. This is also a recurring theme in Thomas Pynchon’s novel Against the Day; see bonus chapter 5. Every number theorist knows, by the way, that the Riemann hypothesis doesn’t quite claim that N/log(N) is statistically optimal; but I don’t advise you to aim for a more accurate statement at your next dinner party. I also recommend that you avoid mentioning, as I have up to now, that log(N) stands for the natural logarithm of N; this year (2014) marks the 400th anniversary of John Napier’s publication of his Description of the Wonderful Canon of Logarithms [Mirifici Logarithmorum Canonis Descriptio], and their wonders could fill the dinner hour all by themselves.

6. (Doxiadis 1992). Highly recommended. Although the twin primes conjecture still seems a long way from being settled, a breakthrough in that direction, due to the previously obscure (but now famous) Yitang Zhang, gave number theorists around the world an unexpected reason to celebrate during the spring of 2013. At about the same time, Harald Helfgott announced the solution of a problem closely akin to the Goldbach conjecture: that every odd number greater than 5 is the sum of three primes.

7. Frege was the first to define the natural numbers in this way: in (Frege 1884), the number 1 is defined (in section 77) as the concept (Begriff) “equal to 0,” where 0 has been defined (in section 74) as the concept “not equal to itself.” (Rotman 1987) places the discovery of zero at the heart of several fundamental transformations in painting and money as well as arithmetic.

8. P. A. is mistaken: Kronecker’s considerations are metaphysical, not necessarily theological. But the mistake is understandable.

9. This is what the logicians don’t understand. It’s perfectly legitimate for them to want to post rules, like traffic signs, to keep the explanations from crashing into one another. The problem is when they convince themselves that they were the ones who built the roads.

P. A.: Please don’t count on my support if you’re trying to drum up sympathy for some obscure and tedious feud with logicians.

N. T. (Crestfallen): As you wish.

10. Both of these ideal types were encountered briefly in chapter 1; we will see more of them in the next chapter.

CHAPTER 3 NOT MERELY GOOD, TRUE, AND BEAUTIFUL

1. (Collini 2011).

2. Quotations from the CBC TV program “Power and Politics,” June 28, 2011, was still online two years later at http://podcast.cbc.ca/mp3/podcasts/powerandpolitics_20110628_18370 mp3, but the podcast has since been taken down. I thank Vladimir Tasic for transcribing the sentences quoted.

3. Quoted in (Donoghue 2008, p. 5) and by Joan Wallach Scott in her March 30, 2011, lecture on academic freedom at the IAS.

4. (Scott 2009, pp. 452–453). Compare (Fischer 2007): “From the point of view of system theory, the politically intended penetration of research by social, economic and political goals and standards … is pathological because science [is] called upon or even forced to make values and codes foreign to its system into its primary ones…. only such research as “pays off” is socially “relevant” or politically correct is to be financed by society.” Wagner (2007) finds “this diagnosis … of special relevance, since the contemporary reform of the European science system” requires that “science itself … be organized according to market-economic principles.”

5. The titles speak for themselves: Science Mart (P. Mirowski); Lowering Higher Education: The Rise of Corporate Universities and the Fall of Liberal Education (J. Cote and A. Allahar); A vos marques, prêts … cherchez! (I. Bruno).

6. Here is a recent French version of this cliché, published as the conclusion of the conference Maths à Venir in December 2009, to which we return in chapter 10. “[Les mathématiques] interviennent de manière cruciale dans de nombreuses sciences naturelles, humaines ou sociales, dans la technologie moderne, et dans la vie de tous les jours, même si on n’en a pas toujours conscience. Elles sont utilisées pour l’imagerie médicale, les jeux vidéo, les moteurs de recherche sur internet, la téléphonie mobile, dans les modèles climatiques, dans la finance, pour ne citer que quelques applications. La vitalité et la santé de l’école mathématique française sont donc devenues un enjeu stratégique.”

7. Quotation from M. J. Nye, Michael Polanyi and His Generation: Origins of the Social Construction of Science, reviewed in (Shapin 2011).

8. (Flexner 1939). Among other contributions, Flexner mentioned electricity, radio transmission, relativity theory, insurance, and bacteriology. We return to this article, to Flexner, founder of the Institute for Advanced Study, and to Maths à Venir, in chapter 10.

9. See for example the Mathematics and Climate Research Network, http://www mathclimate.org/.

10. “Houses of Refuge and Entertainment: Why are Institutes for Advanced Study Proliferating?” an after-hours lecture by Peter Goddard, February 17, 2011.

11. The four color theorem is the claim that it is possible to color any map with a maximum of four colors so that no two contiguous countries have the same color. For a social science perspective, see (MacKenzie 1999).

12. The Kepler conjecture asserts that the “grocer’s” stacking of identical balls (or-anges) in three-dimensional space is the most efficient. Like the four color theorem, Kepler’s conjecture has now been proved with the help of computers. The classification of finite simple groups is a basic problem in algebra: it asks for a complete list of all the distinct finite groups—like the Galois groups we will encounter in chapter β—that are simple in the sense that they are not put together from smaller groups.

13. For all this, see http://code.google.com/p/flyspeck/wiki/FlyspeckFactSheet

14. (Gray 2004, quotations on pp. 27, 37).

15. See www math.ethz.ch/news_events/bombieri_math_truth.pdf.

16. Mentioned in (Ruelle 1991, pp. 3–4); emphasis added.

17. Transcribed from (Voevodsky 2010), quotations from 8′50″ and 43′30″. Gödel’s second incompleteness theorem denies the possibility of proving the foundations of the interesting parts of mathematics consistent. I note in passing that the “proscience” side of the “Science Wars” of the 1990s explicitly discounted the possibility of a reaction like Voevodsky’s to Gödel’s theorems.

18. Still the most prestigious honor in mathematics; see chapter 2. The IAS has long had the world’s largest concentration of Fields Medalists. Among the mathematicians mentioned in this chapter, Bombieri, Deligne, Atiyah, and Tao, as well as Voevodsky, have all been Fields Medalists; all but Tao either are or have been IAS professors as well.

19. As does “the knowledge that material things exist” and “all evident reasoning about material things” (Descartes 1991, Part 4, paragraph 206).

20. (Heidegger 1962, paragraph 21).

21. Gödel’s more famous first incompleteness theorem put an end to another goal of Hilbert’s program: to show that every true theorem in the system admits a proof based on the rules. See note 25.

22. From the announcement of a workshop held in the Netherlands in late 2011, at Lorentz Center (2011).

23. See note 15. Bombieri adds that these considerations “may change with time.”

24. (Frege 1879, Preface, p. 5). quoted at http://plato.stanford.edu/entries/frege-logic/.

25. The most sustained attempt to mechanize the verification of mathematics is associated with David Hilbert, whose goal was to read off the provability of a proposition from its syntax. Gödel’s incompleteness theorems are conventionally said to have put an end to Hilbert’s program and to have ended the Foundations Crisis by declaring it incurable. Frege himself insisted in his Grundlagen der Arithmetik that, although “It is possible … to operate with figures mechanically, just as it is possible to speak like a parrot, that hardly deserves the name of thought. It only becomes possible at all after the mathematical notation has, as a result of genuine thought, been so developed that it does the thinking for us, so to speak.” So is Frege a mechanizer? Edward Kanterian (in a private communication) quotes from Frege’s article on Boole: “‘our thinking as a whole can never be coped with by a machine or replaced by purely mechanical activity.’ But he immediately adds: ‘It is true that the syllogism can be cast in the form of a computation’. Overall, he seems undecided about the relation between inferring as a ‘live’ and as a mechanical process.” Thanks to David Corfield for forwarding Kanterian’s message.

26. Blog post March 11, 2011, on the n-Category Café, http://golem.ph.utexas.edu/category/.

27. See (MacKenzie 2001) for a review of these arguments.

28. (Lanier 2010, p. 10).

29. The question opening this paragraph is not merely rhetorical and is one of the main themes of (MacKenzie 2001). On pages 326–327, MacKenzie quotes a computer scientist who argued that mathematicians’ failure to perform proofs formally, “by manipulating uninterpreted formulae accordingly [sic] to explicitly stated rules” proves that “mathematics today is still a discipline with a sizeable pre-scientific component, in which the spirit of the Middle Ages is allowed to linger on” (Dijkstra 1988); a philosopher (Peter Nidditch) who claimed that “in the whole literature of mathematics there is not a single valid proof in the logical sense”; and another philosopher (Paul Teller) who doubts “that mathematics is an essentially human activity.” Already in 1829, Thomas Carlyle could complain of “the intellectual bias of our time,” that “what cannot be investigated and understood mechanically, cannot be investigated and understood at all” and that “Intellect, the power man has of knowing and believing, is now nearly synonymous with Logic, or the mere power of arranging and communicating” (Carlyle 1829).

30. Examples can be found in (Daston-Galison 2007) and (Alexander 2011).

31. The meaning of this passage, of fundamental importance in understanding the role of mathematics in Western culture, is obscured by modern translations. In this 1917 translation by Bernadotte Perrin of chapter 14 of Plutarch’s Life of Marcellus, as in Dryden’s seventeenth-century translation, the first two occurrences of “mechanics” correspond to the word organikos, the third to mekhanikos. We return to the distinction between mekhanikos and geometry in chapter 8.

32. (Rashed and Vahebzadeh 2000, p. 112).

33. (Cziszar 2003).

34. In “Science as Vocation,” Weber, writing before Hardy’s Apology, asserts that “academic man … maintains that he engages in ‘science for science’s sake.’ ” Many mathematicians must feel this way, but it’s not so easy to defend this position in public. Weber himself goes on to ask “what … does science actually and positively contribute to practical and personal ‘life’?” In addition to technological applications and “tools and … training for thought,” Weber adds a third objective: “to gain clarity.” Weber’s own discipline of sociology contributes in exemplary fashion to this objective. So, I would like to say, does mathematics. The so-called A-G requirements for admission to the University of California seem to support this contention: mathematics is item c, and the A-G guide expects prospective students to arrive with “awareness of special goals of mathematics, such as clarity and brevity … parsimony … universality … and objectivity.” But very few students graduating from the UC system will devote their lives to research in pure mathematics. It takes a book at least the length of this one to explain what kind of clarity is to be gained from that particular choice.

35. (Tymoczko 1993).

36. “Cela semble si naturel aux mathématiciens, tant l’aspect artistique de notre discipline, plus encore que des autres sciences, est évident, que nous ne voyons pas où est le problème. … ce qui fait généralement avancer un mathématicien, c’est le désir de produire quelque chose de beau. … quand nous prenons connaissance d’un résultat ou d’un théorème, notre premier souci est de juger de sa beauté.” [Comptes Rendus 2010].

37. (Muntadas 2010, pp. 47–48, 58, 82). Muntadas interviewed dealers, collectors, gallery owners, museum directors and curators, “docents,” critics, media representatives, and a few artists.

38. His article, renamed “Narrative and the Rationality of Mathematical Practice” is (Corfield 2012).

39. The first quotation is from (Hardy 1929), the second, due to Atiyah, is reproduced on page 256 of (Corfield 2012). Thanks to David Corfield for reminding me of the latter quotation. Proof also plays an important role, not to be explored here, in institutional validation. For Bourbaki, while logic is “extremely useful,” it is but one aspect of their “axiomatic method … indeed the least interesting one” (Bourbaki 1950, p 223).

40. Atiyah is hardly the only mathematician to see proofs in this way. Compare “The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it” (Pólya 1962, p. 127).

Ian Hacking’s new book (2014) makes an illuminating distinction between Cartesian and Leibnizian mathematicians. The Atiyah quotation would be representative of the former attitude, while Voevodsky’s project is typically Leibnizian insofar as it aims at mechanization. Hacking’s consistency in treating the two attitudes in parallel, if pursued by other philosophers, could go a long way toward reconciling the philosophies of Mathematics and mathematics.

41. (Huizinga 1950, p. 49).

42. Thus, Ursula Martin defines mathematical practice descriptively as “ … producing conjectural knowledge by means of speculation, heuristic arguments, examples and experiments, which may then be confirmed as theorems by producing proofs in accordance with a community standard of rigour” (Martin 1998). Compare the following quotation from G. Cantor’s 1867 doctoral thesis: “In mathematics, the art of proposing a question must be held of higher value than solving it.” For epistemology and heuristics in the work of (philosopher and mathematician) Bernhard Bolzano, see (Konzelmann Ziv 2009).

43. http://www math.ias.edu/~vladimir/Site3/Univalent_Foundations html, http://ncatlab.org/nlab/show/homotopy+type+theory#introductions_31.

44. Urs Schreiber (2012) offers an example from one of his own papers on the border between topology and string theory. “[A] certain kind of problem that poses itself in the context of string theory, which … was generally regarded to be among the more subtle problems in a field rich in subtle mathematical effects … finds an elegant and simple solution once you regard it from the perspective of homotopy type theory.” In technical language: “homotopy type theory … automatically produces the correct answer, the ‘E 8-moduli 3-stack of the supergravity C-field in M-theory.’ A solution that looks subtle to the eye of classical logic becomes self-evident from the point of view of homotopy logic/homotopy type theory.” “It is one of those cases,” Schreiber adds, “where a simple change of perspective leads with great ease to a solution of what seemed to be a difficult technical problem.” A new mathematical formalism—new foundations in Manin’s sense—is adopted, at least by the affected subculture, when there is an accumulation of similar examples.

45. (Manin 2004). In (Garfield 2010), Jay Garfield points out a second distinction between foundationalism of content—“certain sentences or cognitive episodes are taken to be self-warranting and to serve as the foundation for all other knowledge” and foundationalism of method—“certain faculties or methods of knowing are taken to be self-warranting and foundational”. Garfield mentions Descartes’ method of clear and distinct perceptions as an example of the latter and argues that the early Buddhist philosopher Nāgārjuna (see chapter 7) is specifically opposed to this kind of foundationalism. In the epistemology of mathematics the two—axioms and rules of argument—tend to be considered part of one system of foundations.

46. (Weyl 2009, p. 188; Benacerraf 1973; Grosholz 2007).

47. (Kreisel 1967; Gray 2010, p. 203, p. 273 for the Enriques quotation).

48. Mehrtens (1990, p. 12) is describing objections to the set-theoretic axiom of choice and alludes to Brouwer’s account of the difference between formalists and intuitionists (p. 178).

49. (Bloor 1991, p. 155).

50. J. Avigad, “Understanding Proofs,” in (Mancosu 2008, 317–353).

51. (Pimm and Sinclair 2006, chapter ω).

52. (Zarca 2012, Tableau 17, pp 278–279 and surrounding discussion). More precisely, respondents were asked to rate “components of the socioprofessional and psychosocial dimensions of the professional ethos.” Teaching came next after pleasure on the scale of importance; at the very bottom were applications to science and the “social world” (20% and 8%, respectively, for pure mathematicians), along with administration of research (12%), concern for France’s international standing (17%), and “the glory of establishing an important result” (18%).

Zarca does not indicate whether or not his respondents explained what they meant by plaisir, but he does cite Weil’s comparison of mathematical to sexual pleasure, noting that the former typically lasts longer. None of the items on the list looked at all like a Golden Goose, but this may just have been an oversight on Zarca’s part.

53. (Zarca 2012, pp. 259–66, especially tableau 14). Some readers may be wondering whether or not the mathematicians who chose to accept Zarca’s e-mail invitation to participate in his Internet survey, rather than to ignore it, form a representative sample of the profession. Zarca addresses just this question in an appendix (pp. 343–34). Information identifying the respondents was sealed off from their responses, and Zarca was thus able to ascertain that his sample is indeed representative as far as age, position, and sex are concerned. Conversations after the survey was closed revealed that a higher than average proportion of “high-level” mathematicians did not bother to fill out the questionnaire. By including the opinions expressed in these conversations in his discussion and in autobiographical writings, Zarca feels he managed to compensate for their relatively low level of participation. There remains the question of whether the sample might be “psychologically unrepresentative” of the profession precisely because it gives excessive weight to those who were eager to respond when there was no obligation to do so. (For example, this might imply that the high proportion of respondents for whom pleasure is a primary motivation is merely an artefact of the methodology.) Zarca admits there is no way to know whether or not this was the case but concludes that to disqualify the results on such grounds would amount to denying the possibility of “any survey that aims to establish [statistics] for relatively concentrated populations.”

54. A handful of amateurs have left their mark on mathematics, including the French jurist Pierre de Fermat, Supergiant of the seventeenth century, when the status of professional mathematicians had not yet been defined, and (in the mid-twentieth century) the German Kurt Heegner, whose ideas continue to be used in work on the Birch-Swinnerton-Dyer conjecture. But research in pure mathematics is overwhelmingly reserved for professionals.

55. (Beebee 2010). In the mid-1930s, the British journal Philosophy published an series of letters on the topic “The Present Need of a Philosophy.” The responses covered much the same utilitarian ground as the BPA letter (no computers, of course) but were more carefully argued. I thank Brendon Larvor and David Corfield for these references.

56. (Zarca 2012, p. 274; Huizinga 1950, p. 132). Quotations in the next two paragraphs are from pp. 160–61 and p. 203; the Aristotle quotation is from Politics, viii 1339 A, 29.

57. Of course Huizinga did not use Kuhn’s language of paradigms. The Bellah quotation is from (Bellah 2011, p. 594).

58. (Freud 1958), translation slightly modified.

59. (Lindsay and Schachter 2007, pp. 108–109). The complex relations between mathematicians and the reality of “the real world” is explored in chapter 7. For the comparison of formalized mathematics to chess in the writings of Weyl and Heinrich Behmann, see Mancosu (1999).

60. Der Meister der Moderne… bestimmt sich als “freier Mathematiker,” als “Schöpfer,” und die Gegenmoderne wirft ihm “Schöpferwillkür” vor” (Mehrtens 1990, p. 10).

61. (Huizinga 1950, p. 3, p. 2; Netz 2009, p. x).

62. Hermann Weyl also saw mathematics as “schöpferish,” but (as usual) his version of the argument was idiosyncratic. “Mathematics plays a central role in the construction of the world of the mind. Mathematizing, like myth, speech, or music is one of the primary human creative attitudes….” And again: “Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man […] Theoretical creation is something different from intuitive insight; its aim is no less problematic than that of artistic creation.” (Weyl 1968, Vol III, p. 293, my translation; Vol. II, p. 533, translation in (Mancosu 1998, p. 136)).

63. (Burghardt 2005, p. 77–78). Burghardt attributes the term “relaxed field” to G. Bally (entspannten Feld) and K. Lorenz. Bachem (2012) mentions that “bluethroats and blackbirds sing the most artful songs when they have no objective”—here “artful” (kunstvoll) refers to “songs of greater complexity and subtlety.”

64. (Dewey 1987, p. 283).

65. The European Research Council asks reviewers to assign objective grades on a scale of 1–4 when answering the following questions:

Does the proposed research address important challenges at the frontiers of the field(s) addressed? Does it have suitably ambitious objectives, which go substantially beyond the current state of the art (e.g. including inter-and trans-disciplinary developments and novel or unconventional concepts and/or approaches)? … Does the proposed research involve highly novel and/or unconventional methodologies, whose high risk is justified by the possibility of a major breakthrough with an impact beyond a specific research domain/discipline?

66. (MacIntyre 1981, p. 175).

67. (Muntadas 2011), pp. 38, 66 et passim.

68. (MacIntyre 1981, p. 207). Corfield (2012) also cites these passages.

69. (Wright 2011). Wright goes on to say that “the full value of philosophy—as both activity and research discipline—does depend upon the existence of the equivalent of a ‘grass roots,’ a culture of philosophical education and awareness…. So of course philosophy should be funded, but only because it belongs to the kind of value that the subject has that the style of thinking it involves, and its preoccupations, is capable of communication to, and beneficial impact upon, the lives of the population in general. In essence, the reasons why philosophy should be funded are more or less the same as those (non-instrumental) reasons why education generally should be funded.”

The same holds for mathematics. The chapters labeled by Greek letters are meant to be an experiment in communication. If I had space to explain the structure of the experiment, I would devote it to talk about the creation of ways of thinking. But the premise of this book is that the “style of thinking” as well as the “preoccupations” of mathematics are grounded in a practice that is the book’s proper subject.

70. Jacques Rancière writes that “Art exists as an autonomous sphere of production and experience since History exists as a concept of collective life” and dates this existence back to the mid-eighteenth century (see also note 20 in chapter 10). Replacing Art by (small-m) mathematics in the preceding sentence, it says that the existence of mathematics as a self-conscious tradition-based practice is tied up with its projection in history, which is consistent with the themes of chapter 2. The timing for mathematics may be different.

Some differences between mathematics and the arts also have bearing on the purpose of this book. Those involved with the arts exhibit much more cultural confidence—you’re not likely to read a book called An Artist’s Apology unless the artist did something genuinely and specifically awful; artists tend to have better parties than mathematicians; and art, unlike mathematics, is a vehicle for large-scale individual and institutional investment. These three differences may be linked.

71. (Bellah 2011, p. 200). Preceding quotations from pp. 139–44, 188. Capitalization of “Powerful Beings” is my addition.

CHAPTER 4 MEGALOPREPEIA

1. “It is difficult to live in a world whose productive medium is money.” Nicole El Karoui, French mathematician, specialist in Mathematical Finance, speaking at the Académie des Sciences, quoted in Les Echos, November 20, 2008.

2. Gertrude Stein, Dr. Faustus Lights the Lights.

3. That is, ethics made quantitative, as opposed to the ethics of mathematics, which is the guiding theme of this book. Quotations are from (Boulding 1969).

4. (My[confined]space 2008; Fisher et al. 2009; Ellerson 2009).

5. The rebellion was crushed by eight Roman legions under the command of Crassus (played by Laurence Olivier in the 1960 Stanley Kubrick film). “A top quant farm” is from (Patersson 2010, p. 140).

6. (Rocard 2008).

7. (Legge 2013; Ferguson 2012, pp. 3, 206; Parliamentary Commission 2013, Vol. 1, p. 7).

8. At the Hague, where he would have liked to see Milton Friedman in the dock: see (Besson 2008). In (Jouve 2011), a colleague asks “Devra-t-on un jour traduire messieurs Black et Scholes devant le tribunal pénal international de La Haye?”

9. (Kahane et al. 2009). Le Monde chose not to print the response, which appeared instead in the online mathematics journal of the CNRS, the French national research agency.

10. Both the SMAI (Société de Mathématiques Appliquées et Industrielles) and the SMF (Société Mathématique de France) published special dossiers on mathematics and finance in the wake of the crash: Matapli, 86, 87 (juin, novembre 2008) and Gazette des Mathématiciens, 119 (janvier 2009). The Yor quotation is on pp. 75–76; Karoui’s is on p. 24 of Matapli 86.

11. (Lambert-Mazliak 2009; Rogalski 2010; Guedj 2008).

12. (OMI 2012).

13. (G. Cohen 2012). Addressing the French Senate, Gauss Prize winner Yves Meyer reported that “when Nicole El Karoui was teaching there, 70 % of École Polytechnique graduates went to work for banks” (Comptes Rendus 2010). Meyer may have been referring to mathematics students; in his opening speech at Maths à Venir 2009 (see chapter 10), Philippe Camus, quoting the president of EADS, claimed the figure was 25% of all graduates (Roy 2010, p. 55).

14. Quoted in (Halimi 2012).

15. See (Taylor 2007, pp. 176ff) for the history of “The economy as objectified reality.”

16. Writers about quants seem to have been issued identical stocks of clichés. Bridge metaphors abound, alongside allusions to Faust and the Devil. Just after Lehman Brothers collapsed, an article entitled “Don’t blame the quants” protested that “[w]hen a bridge collapses, no one demands the abolition of civil engineering” (Shreve 2008). A quant quoted anonymously by the New York Times in 2009 rejected the analogy: her work is “not like building a bridge. If you’re right more than half the time you’re winning the game.” In his fine review of Patterson (2010])’ for the Notices of the American Mathematical Society, David Steinsaltz (2011) writes “one can hardly imagine top civil engineering academics reacting with nonchalance if their best graduates were not building bridges but finding bridges prone to collapse so they can cash in buying insurance on them.”

17. (Smith 1776, chapter VI). Interestingly enough, this is the sole use of the word derivative in The Wealth of Nations.

18. (Braudel 1981, p. 471). Brian Rotman links these innovations to the semiotic opportunities and perils created by the introduction of the number zero: “If the xeno part of xenomoney threatened collapse of the world money system from the past, from unsupportable debt … the money part of it threatens … collapse from the future, from an unsustainable mutability of money signs created by the financial futures markets.” “Xenomoney” is Rotman’s (1987, pp. 96, 93) term for a certain kind of fictitious capital, supplanting paper money and viewed as “a sign able to signify its own future.” Writing in 1987, Rotman predicted the collapse of the options market on purely semiotic grounds; was anyone paying attention?

19. Mephistopheles:

Ein solch Papier, an Gold und Perlen Statt,
Ist so bequem, man weiß doch, was man hat;…
Und das Papier, sogleich amortisiert,
Beschämt den Zweifler, der uns frech verhöhnt.
Man will nichts anders, ist daran gewöhnt.
…

Kaiser:

Das hohe Wohl verdankt euch unser Reich;
Wo möglich sei der Lohn dem Dienste gleich.
Vertraut sei euch des Reiches innrer Boden,
Ihr seid der Schätze würdigste Kustoden.…
Wo mit der obern sich die Unterwelt,
In Einigkeit beglückt, zusammenstellt.

20. From the Benjamin Jowett translation of Politics, Book 1, 1259a, online at classics mit.edu. Thales was traditionally considered to be one of the founders of Greek mathematics as well as philosophy.

21. (Compétences 2005).

22. (Patterson 2010, p. 38). Black became ineligible for the Nobel Prize when he died in 1995. Estimates for total losses due to the collapses depend on sources and timelines, but they tend to be in the low billions of dollars for LTCM and in the trillions for the 2008 crisis. See, for example, FCIC (2011) for losses just within the United States.

23. MacKenzie’s (2003, p. 835) article is unsurpassed for the clarity of its account of the Black-Scholes equation. In its emphasis on “performativity”—roughly, how the mathematical model brings its own reality into being—it is a strikingly successful example of what I called the “genealogical” approach to the sociology of mathematics. But it touches only briefly on questions of mathematical practice.

24. (Wilmott 2000) was written nearly a decade before the 2008 crash!

25. Maybe not so many after all. Columbia doesn’t list its students online, but NYU’s comparable Masters in Mathematical Finance graduates only 30–35 students per year.

26. Such equations also provide one class of avatars, one théorie cohomologique, for Grothendieck’s hypothetical motives. This is the avatar relevant to the Hodge conjecture—one of the six remaining million dollar Clay Millennium Problems. See chapter 2 for the Clay problems and chapter 7 for Grothendieck avatars.

27. Compare the Galois symmetry of chapter β. Much more general equations of this kind, with no special symmetry, are treated in Chung and Berenstein (2005), whose bibliography refers to a paper Langlands wrote on this topic by with F.R.K. Chung.

Although the financial data themselves are assumed to vary stochastically, figure 4.1 expresses a deterministic relation between the option price and the price of the underlying security. MacKenzie (2003) includes an account of the authors’ initial difficulty in finding its solution. Black wrote that he “had never spent much time on differential equations, so I didn’t know the standard methods used to solve problems like that.” Ultimately, they obtained the solution by financial reasoning. A mathematician—not necessarily a specialist in differential equations—would recognize that the equation in figure 4.1 is a slightly modified version of the heat equation whose discrete form underlies the Stable Equilibrium game. Compared to the usual form of the heat equation, the arrow of time is reversed, in the sense that instead of initial conditions, the solution is determined by terminal conditions. I thank Ivar Ekeland for this explanation.

28. This requires qualification. Deterministic systems are not always predictable; probabilistic methods were introduced in physics to study deterministic systems, like the movement of an ideal gas, or the Brownian motion of particles in a liquid suspension, with too many parameters to calculate deterministically. The equations of Brownian motion provide the model for the evolution of stock prices and, thus, for the Black-Scholes equation. Even when the number of parameters is small, initial conditions cannot, in general, be known with sufficient accuracy to permit reliable prediction; this is the insight at the root of what is popularly known as chaos theory. Probabilistic methods can also be used to provide qualitative (and quantitative) information about chaotic dynamical systems.

29. The attempts of Rudolf Carnap and Ian Hacking have been particularly influential. For the sociologist Elena Esposito, “the numbers calculated on the basis of stochastic models represent a kind of doubling of reality, a fictive reality, that does not compete with real reality but forms an alternative description, the increases the available complexity” (Esposito 2007, p. 30).

30. (Thorp 1966, p. 182). This book, which explains Ed Thorp’s “winning strategy” for blackjack—Wikipedia considers Thorp “the father of card counting”—was followed by Beat the Market, written with Sheen Kassouf, which applies similar probabilistic methods to stock options. In The Quants, Patterson calls Thorp “The Godfather.”

31. (Steinsaltz 2011; Ekeland 2010; Wilmott 2000). The Notices of the AMS is the American Mathematical Society’s house journal, and is undoubtedly the most widely-read of all general-interest professional journals for mathematicians.

32. (Tao 2008a), posted on Tao’s blog shortly before the Lehman collapse, lists the following (not very plausible) assumptions, without which the Black-Scholes model is unrealistic: infinite liquidity, infinite depth, no transaction costs, no arbitrage, infinite credit, infinite divisibility, (unlimited) short selling, no storage costs. Most of these assumptions need no explanation, and most of Tao’s account is accessible to a reader who doesn’t know what a differential equation is. At the end it becomes technical: Tao actually derives the Black-Scholes differential equation using a discrete (difference equation) stochastic model not so different from the probabilistic hybrid between Stable Equilibrium and Matthew Effect described previously.

33. (Derman and Willmott 2009). “As more and more complex securities … were downgraded, banks experienced … losses and write-downs which reached US$700 billion in November 2008 … far in excess of what pricing models, rating models and risk models would have predicted” (Crouhy 2009). Hélyette Geman, a specialist in financial mathematics, was quoted in the Swiss Le Temps on November 12, 2008: “Our reputation is now tarnished because mathematical models have made the risks of toxic investments opaque … From 1986 to 1996 the contribution of probability was very positive” but there followed “a phase of excessive mathematization where the beauty of the results” was distorted by confusing the model with the real world.

34. (Shreve 2008; Chemillier-Gendreau and Jouini 2008).

35. (Krugman and Wells 2011; J. Macdonald, “Elephant Tears,” review of Money and Power: How Goldman Sachs Came to Rule the World, London Review of Books, 3 November 3, 2011).

36. More precisely, “by 2007 the trade in derivatives worldwide was one quadrillion (thousand million million) US dollars—this is 10 times the total production of goods on the planet over its entire history,” says Stewart. “OK, we’re talking about the totals in a two-way trade, people are buying and people are selling and you’re adding it all up as if it doesn’t cancel out, but it was a huge trade” (Harford 2012). See also (BIS 2013).

37. (El Karoui and Pagès 2009).

38. (Chance and Brooks 2012). Quotations are from pp. 17–18, exercises are below p. 25.

39. The speaker is Neil Chriss, from Lindsay and Schachter (2007, p. 134); see chapter 2, note 5. Academia was not for Chriss—recall what he said about Hesse’s Glass Bead Game. Not everyone who moved from academia to Wall Street had Chriss’ options: there were more mathematics and physics PhDs than jobs at the time. Admittedly, starting pay on Wall Street even then was often higher than a full professor’s salary. But who cared about money?

40. (O’Neil 2008).

41. (Harkinson 2011).

42. (Patterson 2010, p. 295).

43. (Jouve 2011, Overbye 2009). To be fair, God sometimes gets into the act as well, for example, in this sentence from Georg Simmel’s Psychology of Money: “Just as God in the form of belief, so money in the form of the concrete is the highest abstraction to which practical reason has attained.” Simmel also quotes the Meistersinger Hans Sachs: “Money is the secular God of the world.” [Quoted on page 238 of (Simmel 1978).]

44. (Lindsay and Schachter 2007, p. 202).

45. If you don’t know how much you need to qualify as an “Ultra High Net Worth Individual” (UHNWI), then you probably are not one, which would explain why you have not been approached by “[b]rands in various sectors, such as Bentley, Maybach, and Rolls-Royce in motoring, … to sell their products. Figures gathered by Rolls-Royce suggest there are 80,000 people around the world with disposable income of more than $20 million. They have, on average, eight cars and three or four homes. Three-quarters own a jet aircraft and most have a yacht.” (Wikipedia 2013). More precise information can be found in Capgemini and RBC Wealth Management (2013).

46. The Rand quotation is unsourced and perhaps apocryphal but is often attributed to her on the Internet and, as such, deserves to count as part of her influence. For Rand’s influence on Greenspan, see (Greenspan 2007, pp. 51–53).

47. This is from the translation by H. Rackham, online at http://www.perseus.tufts.edu. To translate epistemon as “scientist” might be more appropriate—see chapter 10—and certainly more in line with the objectivist view of wealth creation—or the neoliberal view of Baroin or Thatcher’s TINA, which come to much the same conclusions.

48. (Smith 1759); the second quotation is in (Bowles and Gintis 2011).

49. (Alvey 2000).

50. (Stephenson 1999, p. 29).

51. (Higgins 2001, pp. 168–69; Patterson 2010, pp 82–85; Nocera 2009).

52. A study by Olivier Godechot (2011, p. 14) determined that “finance constitute[d] 37% of the headcount of the top 0.01%” of salaries in France in 2007. On p. 2, he cites an earlier British study estimating that “70% of the recent increase of the share of the top 1% in the United Kingdom was captured by workers of the financial industry.” In 2013, HNWIs account for roughly 1% of the U.S. population and about .6% of the population of France and the United Kingdom.

53. Quotation from (Lindsay and Schachter 2007, p. 200). Goldman Sachs has changed since 1994. Greg Smith, explaining “Why I Am Leaving Goldman Sachs” in the New York Times in March 2012, caused something of a sensation when he wrote that “five different managing directors refer to their own clients as ‘muppets,’ ” and “These days, the most common question I get from junior analysts about derivatives is, ‘How much money did we make off the client?’ ”

54. Quotation from (Lindsay and Schachter 2007, p. 46). Some slivers of my grand-parents’ lifestyle are still on display at the Lower East Side Tenement Museum.

55. http://www.urbandictionary.com defines fuck-you money as “An amount of wealth that enables an individual to reject traditional social behavior and niceties of conduct without fear of consequences.” The quotation is from (Stephenson 1999, p. 26).

56. (MacIntyre 1981, p. 149). MacIntyre also mentions megalopsuchia, “magnanimity.”

57. (Guerzoni 1999, p. 338).

58. Crassus, who “acquire[d] wealth dishonestly, and then … squander[ed] it uselessly” was “the most covetous man in the world” according to Dryden’s translation of Plutarch’s Life of Crassus. Plutarch explains that Crassus made his money in real estate rather than finance: “making advantage of the public calamities,” by buying houses on fire “and those in the neighbourhood” at bargain prices. Plutarch uses the words philoploutia and philokerdes rather than pleonexia.

59. “Since kings and princes have a lot of property and wealth, they should give bigger recompenses and they should spend with more pleasure and promptness.” From Romanus Aegidius’s De regimine principum, written between 1277 and 1279; quoted in Guerzoni (1999, pp. 355–356). Guerzoni traces the cult of what Latin authors called magnificentia from Plato and Aristotle through Cicero and Aquinas to the Italian sixteenth century, when “aristocratic magnificence … became a heroic princely attribute” and “every action or aristocratic consumption could be morally justified by the desire to pursue full magnificence.”

60. [Muntadas 2011], quotations on pp. 18, 102, 32, 31.

61. (MacIntyre 1981, pp. 181, 178).

62. This is (painter) Allan Kaprow, quoted in (Muntadas 2011, p. 102).

63. (Cook 2009, p. 46; Ferguson 2012, p. 230; Simons 2010, starting at 48′20).

Krugman (2009) also questions the “social value” of high-speed trading. Patterson (2010, p. 116) mentions an allegation of fraud directed at Simons’s Renaissance Technologies but gives no reason to think it is credible.

Ferguson’s opinion doesn’t sound especially flattering, but if you’ve read his book, you’ll know that Simons, almost uniquely among Ferguson’s headliners, is accused neither of committing nor conniving with nor even facilitating criminal behavior. At worst he is “a pure drag on the economy, like spam e-mail,” “providing no social benefit”—but hasn’t the same been said (albeit on a much more modest scale) of pure mathematics?

64. (Patterson 2012, pp. ii, 234).

65. Quoted in (Overbye 2009).

66. The Clay Mathematics Institute, already mentioned in chapter 3 in connection with the million dollar Millenium Prize Problems, was founded in 1998 by Landon T. Clay III, who, according to the CMI Web site, has “had a distinguished career as a successful businessman and in finance and science-based venture capital funding.” CMI supports many research and educational initiatives, among them the PROMYS program for high school students mentioned in note 4 to chapter 2. AIM was created by John Fry, of Fry’s Electronics, the chain of big-box electronics stores based in California, and is (temporarily) located in the Palo Alto big box; they receive funding from the NSF as well as the Fry family.

Eric Weinstein imagines a “Nobel Prize for Backers”—HNWI individuals who have experience with risk management. See http://streamer.perimeterinstitute.ca/Flash/47c17555-ac4e-4143-aa10-b3d4ad499b40/viewer.html (around 15′).

67. (Pieper 2007).

68. (Mehrtens 1990, p. 377–385; following quotations from p. 385). For the IHP, see http://www.ihp fr/fr/presentation/histoire.

69. In this respect mathematicians are not alone. A recent New York Times article entitled “Billionaires With Big Ideas Are Privatizing American Science” described “a cottage industry … offering workshops, personal coaching, role-playing exercises and the production of video appeals” in order “to help scientists bond quickly with potential benefactors.” “Today, federal funding of basic research is on the decline… The best hope for near-term change lies with American philanthropy.”

The author of this book feels obliged to disclose that he has personally benefited from the generosity of several of the philanthropic foundations mentioned in this section.

70. (Lafforgue 2008), my translation.

71. (Skovsmose 2010). “The good” and “the true” converge when responsibility is mechanized.

72. (Lave 1988, pp. 125–126).

73. (Steinsaltz 2011, p. 703). In (Mirowski 2004, p. 6), we read that “Politically pugnacious economists … threw their weight behind the supposed ‘positive/normative distinction’ in their theories…. Philosophers, together with statisticians and economists, began to pretend that elaborate statistical algorithms … perhaps fortified with game theory … could somehow provide solutions to the value relativity of measurement and quantification…. All … tended to treat some generic thing called ‘mathematics’ as if it were capable by itself of cutting the knot binding science and the economy.”

74. (Einstein 1949).

CHAPTER β HOW TO EXPLAIN NUMBER THEORY (EQUATIONS)

1. Ignoring degenerate linear equations, like x = x, that are tautologically true for all numbers x, and inconsistent linear equations, like x = x + 1, that have no solutions.

2. This is the celebrated quadratic formula, known since antiquity but which only gradually took on this standard form, starting with Christoff Rudolff’s introduction of the radical sign in the sixteenth century. At present count, YouTube features 1290 “quadratic formula song” videos, in which high school students and their teachers set B² − 4AC and company to tunes by the likes of Justin Bieber and Lady Gaga as well as to “Pop Goes the Weasel.” German students instead study the equation x² + px + q—the”pq-Formel”—and if you want to see the solution set to music you should check out the rap Die Lösungsformel, http://www.youtube.com/watch?v=tRblwTsX6hQ. There are (in 2013) videos but no songs about the formula in French and neither videos nor songs in Dutch. A colleague has alerted me to efforts to eliminate the quadratic formula from the British curriculum, jeopardizing future dinner-party explanations of number theory to British Performing Artists.

3. This is a manner of speaking. As a general rule, the problems of finding integral solutions and finding rational solutions are both interesting. It’s the latter that drives the plot at this point in the story.

4. This was the tenth problem on Hilbert’s list from the 1900 Paris Congress. The (negative) solution was completed by Yuri Matiasevich in 1970 on the basis of earlier work by Martin Davis, Hilary Putnam, and Julia Robinson. See Putnam’s forthcoming Intellectual Autobiography.

5. The graphs of quadratic equations are the conic sections that we will meet again in the next chapter.

6. The complete version, for even numbers or for odd numbers that are not primes, is just a little more complicated to state: any positive integer n can be written as the sum of two squares if and only if each of its prime factors of the form 4k + 3 occurs an even number of times in its prime factorization.

7. It would be more accurate to call it Brahmagupta’s equation, since Brahmagupta, working in Rajasthan, analyzed such equations in the seventh century; and Bhaskara II solved them completely by the twelfth century—500 years before John Pell.

8. You can find it in any elementary book on number theory—for example, in (Stark 1987, pp. 239–45).

9. Posterior Analytics, Book I, 7. The author is indebted to Marwan Rashed for this and all subsequent discussions of ancient Greek and Islamic philosophy of mathematics.

10. Quoted in (Fisch 1999, p. 145).

11. In his commentary on Euclid, al-Khayyam “argues that a geometrically obtained irrational ratio can be understood only by accepting irrationals as numbers—and in so doing he emerges as the first mathematician to admit irrationals to the status of numbers” (Özdural 1998). For al-Khwarizmi’s ontology of numbers, see chapter 10.

In Hamilton’s quaternions, i² = j² = k² = −1 and ij = k but ji = −k. They make an appearance in chapter 5 also in number theory, but not in this book.

12. (MacIntyre 1981, p. 196).

13. (Cook 2009, p. 7).

14. Suppose we label the four solutions:

There are 24 ways to permute the solutions, but only 4 of them are in the Galois group: these are

A ↔ B, C ↔ D (exchange A and B, exchange C and D; in other words permute 3i and –3i);

A ↔ C, B ↔ D (in other words, permute and );

A ↔ D, B ↔ C (in other words, permute the pairs 3i and –3i and and );

The other 20 permutations fail to preserve the rules of arithmetic. For example, we see that A + D = 0. If we switch A with B but leave C and D unchanged, then A + D is replaced by B + D, which is not equal to 0. Thus this permutation is not allowed in the Galois group. In the Hamlet analogy, say A is Gertrude, B is Claudius, D is Hamlet, and the equation A + D = 0 corresponds to the sentence “Gertrude is Hamlet’s mother.” Exchanging Gertrude with Claudius while leaving Hamlet alone gives the sentence “Claudius is Hamlet’s mother,” which belongs to a different play.

15. In the version emphasized by John Tate.

16. For example, pp. 184–186, pp. 200–202.

CHAPTER 5: AN AUTOMORPHIC READING OF THOMAS PYNCHON’S AGAINST THE DAY

1. (Dalsgaard 2012, p. 165).

2. From the review by Martin Paul Eve at http://www.berfrois.com/2012/02/pynchonite-generosity-martin-eve/.

3. On December 7, 2008, it gave one, which is still there and is still the only (real) one, as you can check for yourself.

4. (Sante 2007; Menand 2006). Menand is being too modest; his book The Metaphysical Club (Menand 2001) includes a perfectly creditable account of the history and philosophy of probability and statistics in its chapters on C. S. Peirce.

5. Compare http://quarterlyconversation.com/the-christophe-claro-interview.

6. So does Wikipedia: “The paths of Stencil and Profane through the novel form a sort of metaphorical V.” Also see also the next note.

7. In his 1990 review of Vineland, Rushdie wrote, “His novel ‘V.’ was actually V-shaped, two narratives zeroing in on a point, and “Gravity’s Rainbow’ was the flight path of a V-2 rocket, a deadly parabola that could also be described as an inverted V.” Dalsgaard (2012) alludes to the “difference in propulsion between the ballistic-rocket and narrative parabolas in Gravity’s Rainbow.”

More meticulous analyses point to a morally charged dialectic in Gravity’s Rainbow between deterministic (meta)physics, symbolized by the parabola, represented by the Pavlovian Ned Pointsman, and pointing at death; and statistical or stochastic (meta) physics, personified by the sensitive Roger Mexico and associated in the text with ellipses and the richness and unpredictability of life: the “ellipse of uncertainty” of the rocket’s landing spot, an ellipse of blood left after delivery of a child, an ellipse of light illuminating an anarchist encampment. The scene of a particularly convoluted inter-twining of dynamics of life and death is illuminated, literally, by a flashlight beam alternating between elliptical and parabolic outline. It’s nevertheless clear that Gravity’s Rainbow is written under the sign of the parabola.

8. But compare with Wikipedia: “ … the line approximates a segment of a small circle upon the surface of the (also approximately) spherical Earth.”

9. Like a Keplerian orbit, rather than a Copernican circular orbit—so it is worse than Copernicus after all….

10. In late 2013 Roberto Natalini, an applied mathematician in Rome, sent me a copy of (Natalini 2013), a chapter on the role of mathematics in the work of the late writer David Foster Wallace, in a book devoted to the latter. Wallace had an avowed interest in mathematics; he was a great admirer of G. H. Hardy’s Apology (see chapter 10) and wrote a full-length book on Cantor’s set theory. Natalini finds several mathematical structures in Wallace’s Infinite Jest, so precisely calibrated with the (fragmented) plot that it is impossible to believe that their presence is merely incidental. Here I just want to mention that, on Natalini’s analysis, the trajectories of the two main characters of Infinite Jest follow the two arcs of a hyperbola that “are close to merging in the middle of the novel, when [the characters] live a few hundred meters apart.”

Natalini and his colleague Lucia Caporaso coordinate the Italian Web site MaddMaths, dedicated to popularizing mathematics and guided by two principles: that “mathematics is more entertaining and less frightening than what is usually believed” and that “mathematics is not only the basis of innovation and technological development but also of the modern idea of citizenship” (Caporaso and Natalini 2013).

11. (Engelhardt 2013). Real and imaginary axes are present both within Pynchon’s narrative and outside it in our reading of it, yet another doubling or bilocation, just as the quaternions are a doubling of the original complex numbers. The Joyce quotation is from (Ellmann 1983, p. 573).

12. Pynchon chose the twenty-fifth anniversary of C. P. Snow’s “Two Cultures” lectures for one of his rare departures from public silence: (Pynchon 1984).

13. (Dalsgaard 2012, p. 157).

14. This MH, not the author of (Harris 2003).

15. (Harris 2003).

16. (Slade 1990). I have no idea what he means by this. Slade thanks the mathematician John S. Lew for bringing the parallels between Atman and Galois to his attention. See also (A.-G. Weber and Albrecht 2011).

CHAPTER 6 FURTHER INVESTIGATIONS OF THE MIND-BODY PROBLEM

1. Quotation from Has translated from French subtitles; the translation of Plato is from (Nightingale 2004, p. 115), with slight modifications suggested by Marwan Rashed.

2. The aesthetics and psychology of the cinema, trans. Christopher King.

3. (Lévi-Strauss 1992, 99–100).

4. “[A]n immense musical accelerator, built for the sole purpose of detecting the Higgs boson of the universe of love” (Spice 2013). See especially (Dreyfus 2010).

5. Actually an authentic equation in the theory of vertex algebras, reproduced in figure 6.8 from the article (Frenkel el al. 2011).

6. See http://videos.tf1.fr/infos/2010/mathematiques-le-genie-russe-refuse-1-million-de-dollars-5787192.html.

7. (Spinney 2010; Mazzucato 2010).

8. “It’s worth noting that almost all films involving mathematicians depict them as clever, whether they are also crazy, schizophrenic, or ruthless murderers.” referring to Robert Connolly’s The Bank, from (Emmer 2003). The first of these mathematical psychopaths may well have been Sherlock Holmes’s nemesis Professor Moriarty who, after writing a “treatise upon the Binomial Theorem which has had a European vogue” gave up a “most brilliant career” in pure mathematics in order to pursue a life as the “Napoleon of crime.” Quotations are from The Final Problem by Arthur Conan Doyle.

9. (Alexander 2011, p. 3). The next quotation is from p. 161.

10. It’s not true, for example, that Galois first wrote up his main discoveries the night before his fatal duel; nor was he ignored and rejected by the mathematical establishment; and there is no reason to believe his duel over Stéphanie was arranged by his political enemies. Alexander was not the first to bring these facts to our attention—see (Rothman 1982)—but to my knowledge he was the first to theorize why the fabrication was and remains so successful.

11. Or Thales, who, in his cameo in Plato’s Theaeteus, falls in a well while studying the stars.

12. The quotation under the entry for wrangler dates from 1751. Toby Gee, whom we will be meeting in chapter 8, was Senior Wrangler his year.

13 (Diderot and Grimm 1765). Diderot’s 1748 novel Les bijoux indiscrets is dubious about the allure of mathematics. The novel concerns a magic ring that endows a woman’s bijou with the gift of speech, and the secrets revealed in this way. The bijou of a woman who had studied geometry spoke incomprehensibly: “Ce n’était que lignes droites, surfaces concaves, quantités données, longueur, largeur, profondeur, solides, forces vives, forces mortes, cône, cylindre, sections coniques, courbes, courbes élastiques, courbe rentrant en elle-même, avec son point conjugué …”

14. (Clairaut 1745), reprinted and placed in context in (Boudine and Courcelle 1999). My thanks to Olivier Courcelle and Jean-Pierre Boudine for this reference. See also (Boudine 2000), quoted in (Justens, 2010). Diagram from books.google.com.

15. Maupertuis was a friend of the Marquis d’Argens, author of the novel Thérèse philosophe, which features a chapter entitled Thérèse se procure machinalement des plaisirs charnels. The quotation is from Mauperthuis’ La Vénus Physique.

16. With respect to the sixty-six-day version of the book, no longer considered reliable. The quotations can be found in the authoritative 1804 version, edited by François Rosset and Dominique Triaire. In my edition the flirtation begins on page 231 and the marriage is announced on page 607.

17. (Alexander 2011, p. 74; Brown 1997). The French were hardly the first to acknowledge the erotic potential of mathematics. Medieval Hindu exercise manuals included problems like this one from the Manoranjana, a commentary on Bhaskara’s Lilavati written by one Ramakrishna Deva (Brahmagupta and Colebrook 1817):

The third part of a necklace of pearls, broken in an amorous struggle, fell to the ground: its fifth part rested on the couch; the sixth part was saved by the wench; and the tenth part was taken up by her lover: six pearls remained strung. Say, of how many pearls the necklace was composed.

18. Byron wrote in his diary that “Lady B. would have made an excellent wrangler at Cambridge” (see note 12).

19. {Zamyatin, 1924, Record Twenty-seven; Edgeworth 1881), my emphasis. Edgeworth continues: “but we seem to be capable of observing that there is here a greater, there a less, multitude of pleasure-units, mass of happiness; and that is enough.” See also chapter 10, note 61.

20. (Jung 1969, p. 59).

21. Bourbaki’s structuralism is treated at length in the next chapter. The catastrophe theory of (the non-Bourbakist) René Thom, like structuralism, is a twentieth-century revival of Leibniz’ mathesis universalis, but arising within mathematics itself. At one time or another, catastrophe theory, like the structuralist double binary opposition scheme illustrated here, has been applied to everything. For a catastrophe-theoretic approach to sex, see (Hubey 1991). Hubey’s abstract begins “A nonlinear differential equation model and its associated catastrophe is shown to model the simplest version of the sexual response of humans. The mathematical model is derived via well-known and non-controversial aspects of sexual orgasm as can be found in the literature.” Hubey’s love equation is

where “Ψ represents ‘sexual tension,’, Ω is the frequency of the ‘excitation,’ and µ is some nonlinearity parameter.”

22. From his letter to Simone Weil, 1940. The same imagery, culminating in an allusion to the Bhagavad-Gita, reappears in Weil’s 1960 article quoted in chapter 7. Both of Weil’s texts derive pleasure from witnessing an intimate coupling between two branches of mathematics; a text by Grothendieck from the 1980s, quoted less frequently, evokes the “deep kinship between the two passions that had dominated my adult life” by casting the mathematician and la mathématique as the erotic partners:

It’s surely no accident if in French as well as in German, the word that designates [mathematics] is of the female gender, as is “la science” which encompasses it, or the still vaster term “la connaissance” [knowledge] or again “la substance.” For the genuine mathematician, by which I mean the one who “makes mathematics” (as he would “make love”) there is indeed no ambiguity regarding the distribution of roles in his relation to la mathématique, to the unknown substance, then, of which he acquires knowledge, which he knows by penetrating her. Mathematics is then no less “woman” than any woman he has known or merely desired—of whom he has sensed the mysterious power, attracting him into her, with that force at once very sweet, and unanswerable (Grothendieck 1988, p 496).

Frenkel, at two generations’ distance from Grothendieck, would never express himself in this way, but some mathematicians in the United States—men as well as women—found the imagery of Rites’s trailer uncomfortably reminiscent of the “distribution of roles” Grothendieck found so natural and protested when Berkeley’s MSRI announced it was cosponsoring a special showing of Rites in a downtown Berkeley theater. The protestors worried that spectators viewing Rites would get the message that the “genuine mathematician” is a man and convinced the MSRI to withdraw its sponsorship (the showing was nevertheless sold out, with more than 100 left waiting on the pavement and many standing in the back of the theater). Some of the most interesting comments were recorded at http://www math.columbia.edu/~woit/wordpress/?p=3318 on Peter Woit’s always stimulating blog “Not Even Wrong.”

It’s funny to note that the only mainstream film to have represented the mathematician’s partner unequivocally as a sex object is the avowedly feminist Conceiving Ada, whose “computer genius” protagonist shares her private moments with a husky bearded boyfriend, who divides his on-screen time between (a) watching adoringly while she interacts with phantoms behind the computer screen, (b) getting her pregnant, and (c) making coffee.

23. http://aejcppfreefr/lacan/1969-12-03 htm, reproduced from Magazine Litté-raire Spécial Lacan, number 121 de Février 1977. See also http://www.psychanalyse-paris.com/Il-eusse-phallus html for a discussion of Lacan’s analysis of the castration complex in terms of the golden ratio.

24. Elisabeth Roudinesco, quoted in French Wikipedia. More recently, French philosopher Alain Badiou has attempted to place Lacan’s theory of love on a sound theoretical footing. I am indebted to Vladimir Tasić for drawing my attention to this excerpt from a discussion of Badiou’s La Scene du Deux. UC Irvine professor Juliet Flower MacCannell writes that “Badiou has succeeded. He has formalized—to an unparalleled degree—the work of the [Lacanian] object a’ [the upper-right corner of Figure 6.5, MH] in structuring the nonrelation of the sexes and opening the way to the relation that is in excess over them: the supplement of Love.” MacCannell quotes Badiou (her translation) in (MacCannell 2005):

An amorous encounter is that which attributes eventmentally to the intersection—atomic and unanalyzable—of the two sexed positions a double function. That of the object, where a desire finds its cause, and that of a point from which the two is counted, thereby initiating a shared investigation of the universe. Everything depends, at bottom, on this u ≤ M and u ≤ W being read in a double fashion: either that one is assembling there the inaugural non-relation of M and W, inasmuch it affects the non-analyzable u with being only what circulates in the non-relation. The positions M and W are then only in this misunderstanding over the atom u, cause of their common desire, a misunderstanding that nothing sayable can lift off, since u is unanalyzable. This the first reading. Or one reads it in the other direction: starting from u, either (W − u) and (M − u), two positions like those the atom u supports by subtracting itself from it (pp 186–87 of Badiou’s original text).

Tasić made the following attempt at an interpretation:

The sexes (M and W) are not in a relation but both dominate the same atom (u) in some poset. Apparently the successful solution of Lacan’s riddle (“there is no sexual relation”) is a love formula having something to do with the notion of a partially (but not totally) ordered set.

25. However, see http://www.cambridge-news.co.uk/Whats-on-leisure/Art/Cambridge-don-Dr-Victoria-Bateman-dares-to-bare-in-nude-portrait-20140515065433 htm. Rites was in competition at the June 2010 Sexy International Paris Film Festival (SIPFF) but was not listed among the prize winners. See www.sexyfilmfestparis fr.

26. She has instead been consecrated, like her contemporary Galois or her father, as a romantic martyr; see the film Conceiving Ada mentioned before, as well as William Gibson’s novel The Difference Engine and several books for a general audience.

27. Quotations from (Alexander 2011, pp. 86–89).

28. (Alexander 2011, p. 13). This may not be unrelated to their poor reputation as lovers: “ … it’s maybe because of math’s absolute, wholly abstract Truth that so many people still view the discipline as dry or passionless and its practitioners as asocial dweebs” (Wallace 2000).

29. Rites of Love and Math, opening frames.

30. (Alexander 2011, p. 49). After reading this paragraph, Alexander pointed out a third possible archetype, see following text.

31. Чем глубже мы погружаемся в материальный мир, тем дальше мы от него отдаляемся в направлении мира идеального. Light and Word (Свет и Слово), in [Parshin 2002].

32. From Proof, by David Auburn; cited in (Hofmann 2002).

33. All quotes from (Ikeda and Bonnet 2008). Gross, a number theorist, is especially famous for his work with Don Zagier on the Birch-Swinnerton-Dyer conjecture; see chapter 9 and endnote 52 in chapter 2.

34. (Menkes 2010).

35. http://abcnews.go.com/Entertainment/wirestory?id=10017982&page=2.

36. http://www.youtube.com/watch?v=YZXLLNbi76o.

37. Comments and quotation at http://www.youtube.com/watch?v=eQAuSGvQjN0&feature=related.

38. See http://www.claymath.org/poincare/laudations html. Mikhail Gromov, also on hand at the Perelman ceremony in Paris, had been quoted a year earlier to the same effect: Perelman’s “main peculiarity is that he acts decently. He follows ideals that are tacitly accepted in science” (Gessen 2009, p. 111). Gessen herself decided while writing her book that Perelman “has an internally consistent view of the world that is entirely different from the view most people consider normal”—her definition of “crazy” (on the book’s amazon.com page).

39. No coincidence: there’s always a champagne reception.

40. Two films, actually, including one still in production in collaboration with my colleague Pierre Schapira.

41. (Alexander 2011, pp. 253–255).

42. Much less that she anticipated Kepler’s theory of planetary orbits by a good twelve centuries.

43. (Aczel 2001).

44. (Metz 1977, p 10, p. 68).

45. We will encounter the very different figure of naked truth, Horace’s nuda veritas, in chapter 10.

46. From http://en.allexperts.com/q/Russian-Language-2985/truth htm. I thank Yuri Tschinkel for this link. A. N. Parshin, on the other hand, points out an old Russian proverb where the emotional content of the two terms seems to be reversed. The proverb is based on Psalm 85:11, which reads “Truth shall spring out of the earth; and righteousness shall look down from heaven” in the King James version. In the Russian Bible, where the verse is numbered 84:12, “Truth” and “righteousness” are translated istina and pravda, respectively, and so the proverb becomes Истина от земли, а правда с небес: istina from the earth, and pravda from the heavens. The Hebrew words are emeth and tsedek.

47. Among the Chinese characters composing Mariko’s name is one meaning “real” or “true.”

48. (Florensky 1996). The original text can be found online: … истинный художник хочет не своего во что бы то ни стало, а прекрасного, объективно-прекрасного, то есть художественно воплощенной истины вещей. … Лишь бы это была истина—и тогда ценность произведения сама собою установится.

49. http://www.thomasfarber.org/b_body.php.

50. In which a police inspector screams, “Did you fucking know about this? You and your fuckin’ equations?”

51. To forestall possible misunderstandings, I remind readers that the equation in question was printed in an article having nothing to do with love or film several years before Frenkel began his collaboration with Reine Graves. It might also be mentioned that the algorithm for the online dating service OK Cupid, created by four Harvard math majors in 2004, was sold in 2011 for $50 million, making it by some measure the most successful love formula in history—unless you count Facebook, created at Harvard that same year.

52. Точно узнать продолжительность жизни, по мысли Френкеля, вполне реально, если иметь в распоряжении статистику половых актов конкретного человека. http://woman-todayru/themes/world/9942/. No one knows where they got this idea.

53. As a teenager, he may have seen this movie on Soviet TV: http://www.imdb.com/title/tt0216755/.

54. A. N. Parshin points out that the name-worshippers’ postulate continues “but God is not His name.”

55. Graham and Kantor (2009) give a detailed account of Florensky’s martyrdom. Florensky has been repeatedly accused of antisemitism, especially in connection with the Mendel Beilis blood libel of 1913. Pyman (2010) examines the evidence for this accusation and finds it inconclusive.

56. From Light and Word in (Parshin 2002) and Staircase of Reflections (Parshin 2007), respectively.

57. Erwin Panofsky claims Suger wrote this verse under the influence of Pseudo-Dionysius the Areopagite, author of On the Divine Names and The Celestial Hierarchy, among other works. According to Parshin, Pseudo-D. insists in The Celestial Hierarchy on the fact that “every angel is a mirror: he accepts light in order to reflect it.” (Parshin 2005). I found this translation at http://www.esoteric.msu.edu/VolumeII/CelestialHierarchy.html:

[The Hierarchy] moulds and perfects its participants in the holy image of God like bright and spotless mirrors which receive the Ray of the Supreme Deity—which is the Source of Light; and being mystically filled with the Gift of Light, it pours it forth again abundantly, according to the Divine Law, upon those below itself.

58. Florensky is quoted in (Parshin 2007) in the Russian original: Тело есть наиболее одухотворенное вещество и наименее деятельный дух, […], но лишь в первом приближении; Тело есть осуществленный порог сознания; Тело есть пленка, отделяющее область феноменов от области ноуменов. The translations are mine. The Russian word пленка, which traditionally designates a very thin boundary between two media, is also the contemporary term for the material on which the images of a film are recorded. Note that for Florensky, the body is a film separating two regions of the mind.

59. То, что за телом, по ту сторону кожи, есть то же самое стремление само-обнаружиться, но сознанию сокрытое; то, что по сю сторону кожи есть непосредственная данность духа, а потому не выдвинутая вне его. Сознавая, мы облачаем, и переставая сознавать-разоблачаем себя самих. (The English version is my translation, with Parshin’s help.)

60. In (Parshin 2008). The concept of a mirror that exchanges the images on either side is paradoxical. The reader is encouraged to try to draw a diagram, which may or may not resemble that drawn by Florensky in his Limits of Knowledge; for Florensky, there is an infinite staircase of steps between the two worlds. I thank A. N. Parshin for this reference.

61. See Rig Veda, 10:129.

62. Ведь любовь возможна к лицу, а вожделение—к вещи; рационалистическое же жизнепонимание решительно не различает, да и не способно различить лицо и вещь, или, точнее говоря, оно владеет только одною категорией, категорией вещности, и потому все, что ни есть, включая сюда и лицо, овеществляется им и берется как вещь, как res (Florensky 1997).

63. (Zinoman 2010).

64. Faust, Part I, lines 1856–59; translation http://www.online-literature.com/goethe/faust-part-1/5/; part II, lines 11804–11807, translation with the help of Yuri Tschinkel).

65. This is also the theme of Frenkel’s (2013) book, entitled Love and Math. Neuroscientist Arjan Chatterjee is not convinced: in his analysis of the neurological basis of aesthetic response, he distinguishes “liking” from “desire” and is inclined to classify the response to mathematical beauty under the former heading (Chatterjee 2014, p. 62).

66. This may or may not disqualify an artificial theorem prover as a future subject of mathematical biography in the vein of Minds of Mathematics.

67. (Levinas 1987, p. 56).

CHAPTER β.5.

1. Taking their lead from (Hardy 2012), to which we return in chapter 10. For the use of the word trick in mathematics see chapter 8. The diagonalization trick is sketched in note 25 to chapter 7.

2. (Kontsevich and Zagier 2001).

3. (Turing 1936-1937). In the appendix, Turing shows that his computable numbers are equivalent to the “effectively calculable numbers” introduced by A. Church.

4. Rafael Bombelli, “the first mathematician to accept as valid the solutions of third-degree equations … containing imaginary numbers” called irrational numbers “impossible to name” in his Algebra, written in 1550 (La Nave 2012). The numbers I 2 am calling indescribable are much more radically impossible to name than π or . See, however, (Hamkins et al. 2013) for models of set theory in which every real number is definable in the sense of being “specified as the unique object with some first-order property.” Hamkins explains in Math Overflow (where he is the reigning champion) that the Wikipedia article on definable numbers is mistaken in arguing that this is not the case.

5. See video.ias.edu/voevodsky-80th, around 33′30″.

6. For the purposes of this dialogue I am considering only numbers that can be defined or described in the setting of a dinner party, where I think the subtle considerations of the article of Hamkins et al. do not apply.

7. (Weyl 1968, p. 177), quoted in (Schappacher 2010).

CHAPTER 7 THE HABIT OF CLINGING TO AN ULTIMATE GROUND

1. The Vimalakīrti Nirdésa Sūtra mentions three aspects of Dharma worship, among others: “attaining the tolerance of ultimate birthlessness and nonoccurrence of all things … realizing the ultimate absence of any fundamental consciousness; and overcoming the habit of clinging to an ultimate ground.” See (Thurman 1976).

2. The reasons for this belief deserve to be explored. In what sense is such a belief rational?

3. For example, see (Pettie 2011). Erdős wasn’t the first to define mathematicians in this way; the quotation is reportedly due to his fellow Hungarian and collaborator Alfréd Rényi: see http://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi. Erdős is, however, famous for having said it.

4. Fragment I, translated in (Kahn 1981, p. 29).

5. (Hofmann 1998). For Dr. Gonzo and his medicine cabinet, see Hunter S. Thompson, Fear and Loathing in Las Vegas, passim.

6. Quoted in (Jackson 2004, p. 1196). In his unpublished writings, Grothendieck frequently refers to his internal rêveur—the dreamer—which in one text he identifies with Dieu.

7. Godement’s comments were reported by Leila Schneps. For the milk and bananas see (Jackson 2004, p. 1044). Godement told Schneps that, as a doctoral student in Nancy in the early 1950s, “Grothendieck would periodically suddenly decide that he was never going to eat anything again except for milk, cheese and bread. So if they invited him to lunch, he would come, but he would scold them all for eating too much, and demand to be served only milk, cheese and bread while everyone else ate whatever Godement’s wife had prepared.” The forty-five-day fast is recounted in (Scharlau 2010, pp. 250–251): “Vielleicht wollte er … durch diesen Akt Gott zwingen, sich ihm zu offenbaren.”

8. “Mirror Symmetry” in (Gowers et al. 2008, p. 528).

9. I. Grojnowski, “Representation Theory,” in (Gowers et al. 2008, p. 421).

10. Compare this verbatim quotation: “I don’t want to be too precise [about the definition] because it would take a quarter of an hour.”

11. Translated in (Mazur 2004a), with Grothendieck’s boldface restored. Here is the original text:

Contrairement à ce qui se passait en topologie ordinaire, on se trouve donc placé là devant une abondance déconcertante de théories cohomologiques différentes. On avait l’impression très nette qu’en un sens qui restait d’abord assez flou, toutes ces théories devaient “revenir au même”, qu’elles “donnaient les mêmes résultats”. C’est pour parvenir à exprimer cette intuition de “parenté” entre théories cohomologiques différentes, que j’ai dégagé la notion de “motif” associé à une variété algébrique. Par ce terme, j’entends suggérer qu’il s’agit du “motif commun” (ou de la “raison commune”) sous-jacent à cette multitude d’invariants cohomologiques différents associés à la variété, à l’aide de la multitude des toutes les théories cohomologiques possibles à priori.

12. For example, (Corry 2004, especially chapter 7; McLarty 2006).

13. (Drinfel’d 2012).

14. (Cook 2009, p. 156). Deligne, a Belgian, studied with Grothendieck in France and remained there for two decades before moving to the IAS. The Weil quotation is from an article entitled “De la métaphysique aux mathématiques,” reproduced in volume II of his collected works; the English translation is from B. Mazur’s (1977) article. Weil’s title refers to the eighteenth-century mathematicians who used the word metaphysics to designate what Deligne is calling “philosophies” or “yogas.” Grothendieck’s allusion to “yoga” in the next paragraph is from (Colmez et al. 2003, p. 347).

15. (Krömer 2007, p. 9).

16. Since this chapter was originally part of a presentation at a philosophy conference, I was thinking of its use by philosophers like Brentano and Husserl. Compare also H. Weyl, “existence is only given and can only be given as intentional content of the conscious experience of a pure, sense-creating ego,” quoted in (Schappacher 2010, p. 3274).

17. Worrying to what extent this is also the case in the natural sciences leads to another abyss that is not the topic of this paper.

18. Ian Hacking observes that Wittgenstein first used übersichtlich and its cognate übersehbar in connection with mathematics in his Remarks on the Foundations of Mathematics that dates from the 1930s. Translated “perspicuous” and “surveyable,” respectively, these terms remain highly influential in contemporary philosophy of mathematics. Hacking points out that Wittgenstein’s subsequent allusions to these words are comments on their first appearance in his work and advises philosophers to be more attentive to this context. See (Hacking 2014, chapter I, paragraph 24).

19. The dates and illustrations in English and French are taken from the Oxford English Dictionary and Le Robert or the Dictionnaire de l’Académie, respectively.

20. (Deligne 1971; Coleman et al. 1969). I thank IAS director Peter Goddard for drawing my attention to the latter article. The word had already appeared in 1929 in the very first recorded adventure of Deligne’s most famous compatriot, in the Land of the Soviets: “Le « Petit XXe », toujours désireux de satisfaire ses lecteurs et de les tenir au courant de ce qui se passe à l’étranger, vient d’envoyer en Russie Soviétique un de ses meilleurs reporters: Tintin! Ce sont ses multiples avatars que vous verrez défiler sous vos yeux chaque semaine” (Hergé 1930).

21. This is indeed the spirit of the Beilinson conjectures, which strongly influenced Voevodsky’s work.

22. The unifying principle was identified by the Milesians as water (Thales), air (Anaximenes), “the infinite” (to apeiron; Anaximander), or “being” (to eon; Parmenides). Plato frequently thematizes identity and difference, notably in the Timaeus, Sophist, and Parmenides; Aristotle treats the question in particular in the Topics and the Metaphysics. The Upanishads contain a monistic theory later summarized by Sankara’s identification of atman with brahman; even in the Rig Veda one can read “The wise speak of what is One in many ways; they call it Agni, Yama, Matarishvan” (1.164.46); see (Doniger 2010). The logic of difference was analyzed by the Nyaya and Vaisesika schools as early as 2000 years ago: see (Ganeri 2009).

23. (Gowers 2000). Party planners are warned that Erdős’s estimate is certainly far short of the real value of R(k) [often written R(k, k)]; determining the exact values of R(k) even for small k is one of the outstanding unsolved problems in combinatorics.

24. Ten years after the article’s publication, the case no longer needs to be made. Thanks in large part to the successes of Gowers and a few of his colleagues and students, Erdős-type mathematics is now a reliable source of charisma at the highest levels: Erdős’s collaborator Endre Szemerédi was awarded the Abel Prize in 2012 (and received the rarer honor of being compared to Leonardo, Einstein, and the Buddha in the pages of the Guardian by his compatriot, the novelist László Krasznahorkai) for his work in theoretical computer science as well as combinatorics; a July 2012 conference on additive combinatorics at the Institut Henri Poincaré held its official banquet at the Louvre.

25. Quotations from (Dauben 1979, pp. 290, 245–246, 144–146). The final quotation is from a letter to Cantor by Cardinal Johannes Franzelin.

The set denoted P(S) is the power set of S, the set of all subsets of S. If S = {a, b} has two elements, P(S) has the four elements {a}, {b}, {a, b}, and the empty set. If S has n elements, then P(S) has 2ⁿ elements, more than S. The diagonalization trick shows that P(S) is bigger than S even if S is infinite. Here is the trick in a few words. Suppose S and P(S) had the same cardinality; in other words, for each s in S, there is a subset f(s) in P(S), and every member of P(S) is labeled in this way. Now let T be the set of s such that s is not in f(s). Suppose there were a t such that T = f(t). Now if t were in T, then t would be in f(t) and, hence, by definition, is not in T. Thus t must be in T; but then t fails the criterion to be in T. Hence there is no such t, and P(S) cannot be indexed by the elements of S. The reader will note the resemblance to Russell’s paradox.

The paradox described here is related to but not identical to the Burali-Forti Paradox regarding the endless sequence of ordinal numbers. See also (Garfield and Priest 2003, especially the discussion on pp. 17_19) for similar paradoxes in Nāgārjuna’s philosophy (roughly AD 200).

26. Compare (Meillassoux 2006, pp. 115–118).

27. (Vallicella 2010).

28 MMK, chapters I, XXV, translations (Garfield 1995, pp. 3, 74; Yadav 1977, p. 463). See also (Nāgārjuna 2010). Compare the entry on Mādhyamika in the Stanford Encyclopedia of Philosophy: “There is … broad agreement [among Mādhyamika thinkers] that language is limited to the everyday level of understanding and that the truth of nirvana is beyond the reach of language and of the conceptualization that makes language possible.”

29. (Stcherbatsky 1978, p. 215). These translations may be peculiar to Stcherbatsky.

30. “The Principle of Identity,” in (Heidegger 1969). Heidegger adds on page 26 that “what is successful and fruitful about scientific knowledge is everywhere based on something useless,” namely, the identity of an object with itself.

31. (Nāgārjuna 2010, IX, 16).

32. (Corry 2008).

33. (Mazur 2008b), the most readable account I’ve seen of how category theory treats equality.

34. http://ncatlab.org/nlab/show/identity+type. This is actually characteristic of intensional type theories, of which the homotopy type theory favored by Voevodsky is one example.

35. Quoted in (Garfield and Priest 2003, p. 15).

36. golem.ph.utexas.edu/category; ncatlab.org/nlab/show/nPOV.

37. (Langlands 2013).

38. (Grosholz 2005, p. 262).

39. (Azzouni 2010, p. 91). Azzouni writes that “the word ‘exist’ isn’t amenable to an analysis that yields this, or any other, criterion”; that the word is “isolated and criterion-transcendent.” Other philosophers naturally disagree. (Burgess and Rosen 2000) is a book-length analysis of the role of nominalism in philosophy of mathematics.

40. For Udayana, see (Yadav 1977). Udayana also appealed to the cosmological (or Uncaused Cause) argument, familiar from Aristotle and Aquinas. The Descartes and Kant arguments, in the Fifth Meditation and the Critique of Pure Reason, respectively, are summarized in plato.stanford.edu/entries/descartes-ontological/#1, where the connection is made to Russell’s theory of descriptions.

41. Matters are not so simple in fluid mechanics. The Navier-Stokes equation describes the behavior—more precisely, the pressure and velocity—of an incompressible fluid (like the Mississippi River) in two-or three-dimensional space (within or sometimes overflowing the river’s banks). “Existence” in this setting comes with a price tag: existence of smooth solutions to the three-dimensional Navier-Stokes equation is another Clay Millennium Problem. The two-dimensional case was settled long ago. See (Fefferman 2000).

42. Grothendieck, letter to D. Quillen, February 19, 1983; distributed in Pursuing Stacks (original in English). Over the next few days, he continued this letter, having found a “simple guiding principle” after all; but the general point remains valid.

43. (Hoffman 1998, p. 54).

44. Proving or investigating special cases or consequences of the Langlands conjectures is a full-time research career for (at least) hundreds of mathematicians and increasing numbers of physicists around the world. This could serve as a textbook example of the successful research program in mathematics; but neither philosophers nor sociologists nor historians have begun writing the textbook.

45. “Human spontaneous non-demonstrative inference is not, overall, a logical process. Hypothesis formation involves the use of deductive rules, but is not totally governed by them; hypothesis confirmation is a non-logical cognitive phenomenon: it is a by-product of the way assumptions are processed, deductively or otherwise.” (Sperber and Wilson 1995, p. 69).

46. Langlands drew profound conclusions from this particular avatar relation—which was also at the heart of Wiles’s proof of Fermat’s Last Theorem—in his “fairy tale” paper (Langlands 1979a). Drinfel’d, one of the great mathematical prodigies of the last century, announced his first results in this direction at the very Galois-appropriate age of 20; forty years later, he is not only still alive but fantastically creative.

47. A capsule version of the ontological conundrum at the heart of this chapter: did virtual reality exist before people like Lanier invented it? Since it’s essentially a collection of computer programs, I believe a mathematical Platonist would have to answer that it did exist and that it was discovered rather than invented. I find that hard to accept.

Quotations from (Lanier 2010, pp. 185, 187). For VR vs. LSD, see (Rheingold 1992, 354–55).

48. L’ “évidence” de Grothendieck n’est pas liée à la proximité de deux termes dans une chaine déductive, mais à l’effet de “naturel” lié à l’abolition de l’espace entre le symbole qui capte et le geste qui est capté” (Châtelet 1997, p. 16).

49. (Velleman 2008, pp. 421–422).

50. Mehrtens (1990, p. 8) continues: For what he calls the Moderns, “Wahrheit und Sinn der Texte bestimmen sich in der Arbeit an ihnen, nicht mehr in der Repräsentation der gegebnene physischen Welt, auch nicht im Bezug auf eine transzendente Ordnung. Dagegen protestiert die Gegenmoderne und sucht den ‘Ur-Grund,’ in dem die Mathematik ‘wurzelt’ und aus dem sie ihre Wahrheit und ihre Ordnung bezieht.”

51. Grothendieck left Bourbaki after a few years because “he lacked humor and had difficulty accepting Bourbaki’s criticism” according to Laurent Schwartz—but also because of his conflicts with Weil, especially over the latter’s refusal to admit the categorical perspective into Bourbaki. See (Corry 2008). The term “programmatic manifesto,” referring to The Architecture of Mathematics (written by Bourbaki founder Jean Dieudonné) is in Corry (2004, p. 303).

52. ReS, p. 48. Grothendieck’s emphasis, my translation. Bourbaki was usually not a Platonist. The full quotation is La structure d’une chose n’est nullement une chose que nous puissions “inventer”. Nous pouvons seulement la mettre à jour patiemment, humblement en faire connaissance, la “découvrir”. S’il y a inventivité dans [notre] travail, et s’il nous arrive de faire oeuvre de forgeron ou d’infatigable bâtisseur, ce n’est nullement pour “façonner”, ou pour “bâtir”, des “structures”. Celles-ci ne nous ont nullement attendues pour être, et pour être exactement ce qu’elles sont! Mais c’est pour exprimer, le plus fidèlement que nous le pouvons, ces choses que nous sommes en train de découvrir et de sonder, et cette structure réticente à se livrer, que nous es-sayons à tâtons, et par un langage encore balbutiant peut-être, à cerner.

Compare Pynchon’s Vineland: “I thought [theorems] sat around, like planets, and … well, every now and then somebody just, you know … discovered one.”

53. Private communication with Leila Schneps, based on her conversations with Godement. Schneps is a founder of the Grothendieck Circle: http://www.Grothendieckcircle.org/.

54. See (Friedlander and Suslin 2003) for an account of Voevodsky’s construction. Others are due to Marc Levine, M. Hanamura, and Madhav Nori. The first attempt to define mixed motives rigorously was due to Annette Huber, who followed Deligne’s lead and worked with their avatars (called realizations, though they could just as well have been incarnations) since the ulterior items could not be defined.

55. Phaedo, 79c (translation from http://www.perseus.tufts.edu/hopper/text).

56. (McLarty 2007). See Serre’s comments in chapter 8, note 80.

57. (Cartier 2014, p. 18). Grothendieck’s elimination of dependence on choice should not be confused with working in the paradoxical “set of all sets.” Mazur (2008b) also uses a maritime metaphor: “If we are of the make-up of Frege, who relentlessly strove to rid mathematical foundations of subjectivism, we look to universal quantification as a possible method of effacing the contingent—drowning it, one might say, in the sea of all contingencies. But this doesn’t work [because of Russell’s paradox].”

Of the mathematicians responding to Zarca’s questionnaire, 38% defined themselves as atheists, 15%, as believers, 6%, as “having faith,” with the rest offering a variety of answers (Zarca 2012, p. 252).

58. (Yadav 1977, p. 452). I have been unable to find an equivalent quotation in any translation of the Prasannapada. For infinite regress, see MMK VII, 3 and the commentary in the Prasannapada.

59. (Graves and De Morgan 1882).

60. The Abraham interview is on page 101 of (Kirn 1991); his photo is captioned “Prof Ralph Abraham believes in mixing math and acid.” Mullis’s drug use is well known; his Wikipedia page attributes the comment quoted here to “BBC Horizon—Psychedelic Science—DMT, LSD, Ibogaine—Part 5.”

61. (Kaiser 2012, p. vii)

62. (Cook 2009, pp. 148, 138, 58, 32; Grothendieck 1988, p. 675). I use “hooked” to translate Grothendieck’s “accrocher.” The French word has the connotations of addiction and appears quite often in ReS; for example, he describes how in 1977 he got “strongly ‘hooked’ on a substance of exceptional richness” (p. 319).

63. (Scharlau 2010, p. 144). Scharlau writes, and I certainly agree, that “Grothendieck’s life is so unusual and singular that it does not belong only to him.” The Molinos quotation is from Guía espiritual, Capítulo XX.

64. (Hoffman 1998, pp. 21, 22, 243).

65. More precisely, an (∞, 1)-category, for which the standard text is Lurie (2009); but it’s only the beginning of the story, because an endless chain of (∞, n)-categories await their founding documents. Identity becomes problematic in ∞-categories, in the sense that the usual identity has many equivalent ways to be an identity; and the equivalences have many equivalent ways to be equivalences; and so on to ∞. There’s no way to say this in a few words: the definitions are “too long,” never mind the proofs; Lurie’s book is 925 pages long.

66. Grothendieck (1988), letter to Quillen.

67. After writing this sentence, I discovered Manin’s article “Foundations as Super-structure” and Reuben Hersh’s “Wings, not Foundations!” which subject the foundations metaphor to a “deconstruction” (Hersh’s word) similar to that carried out in the present chapter: (Manin 2012; Hersh 2005).

68. On page 39 of Récoltes et Sémailles, Grothendieck places his foundations at the base of a house, but it’s the only instance of its kind in the book and it’s clear he’s just using the conventional metaphor in order to talk about something else. He has built a house from the foundations up but he belongs in the open air … Je me sens faire partie, quant à moi, de la lignée des mathématiciens dont la vocation spontanée et la joie est de construire sans cesse des maisons nouvelles. Chemin faisant, ils ne peuvent s’empêcher d’inventer aussi et de^⋄façonner au fur et à mesure tous les outils, ustensiles, meubles et instruments requis, tant pour construire la maison depuis les fondations jusqu’au faîte, que pour pourvoir en abondance les futures cuisines et les futurs ateliers, et installer la maison pour y vivre et y être à l’aise. Pourtant, une fois tout posé jusqu’au dernier chêneau et au dernier tabouret, c’est rare que l’ouvrier s’attarde longuement dans ces lieux, où chaque pierre et chaque chevron porte la trace de la main qui l’a travaillé et posé. Sa place n’est pas dans la quiétude des univers tout faits, si accueillants et si harmonieux soient-ils …. D’autres tâches déjà l’appelant sur de nouveaux chantiers, …. Sa place est au grand air [my emphasis].

69. Quoted in Gray (2010, p. 240).

70. In the spirit of the theory of actants of sociologist Bruno Latour. We return to Krishna and Arjuna in the next chapter.

71. See Minhyong Kim’s comment at (Mathoverflow 2012; Lurie, 2009, p. 50; Mazur 2008b).

72. (Carter 2004; Grosholz 2007). For Voevodsky’s inaccessible cardinals, see (Shulman 2011).

73. (Kronecker 2001, p. 233, 228; Azzouni 2010, pp. 98–99).

74. For Bourbaki’s manifesto, see note 51. Compare “The pathway of hypotheses sometimes snakes between, sometimes bridges, various domains of mathematical research” (Grosholz 2007, p. 60), reviewing Carlo Cellucci’s theory of mathematical discovery.

75. Grattan-Guinness illustrates the problem by referring to one of André Weil’s historical articles: “ … the algebraic reading of Euclid has been discredited by specialist historians in recent decades. By contrast, it is still advocated by mathematicians, such as Weil … who even claimed that group theory is necessary to understand Book 5 … and Book 7 (introducing basic properties of positive integers)!” (Grattan-Guinness 2004).

76. (Tappenden 2008). See, specifically, Tappenden’s discussion of quadratic reciprocity as a special case of class field theory. In algebra, prime numbers become prime ideals. Quotations from pages 269–70.

77. (Tappenden 2006). Nowadays the complementary approaches of Weierstrass and Riemann are taught side by side in every mathematics department in the world.

78. The homological mirror symmetry conjecture is an active research program, rooted in part in Grothendieck’s work and overlapping with a geometric version of the Langlands program. Kontsevich, one of the most influential living mathematicians (1998 Fields Medal, 2008 Crafoord Prize, 2012 Shaw Prize, 2015 Breakthrough Prize in Mathematics….), is currently a professor at IHES.

79. http://en.wikipedia.org/wiki/Talk%3ABarrington_Hall_%28Berkeley,_California%29.

80. (Garfield 1995, p. 91).

81. G. G. Joseph (2011, p. 345) hints cautiously at such a link in the chapter on ancient Indian mathematics; he suggests that “zero as a concept probably predated zero as a number by hundreds of years” and speculates that the link between the counting and spiritual senses of shunya may be traced as far back as the Rig Veda. F. Gironi (2012) addresses the theme from the standpoint of continental philosophy, especially Derrida and Badiou. See also (Goppold 2012), especially section 3.

82. So much for Hans Reichenbach’s distinction between “context of discovery” and “context of justification.”

83. Example (for mathematically sophisticated readers): one categorifies a ring A by devising a category whose objects correspond to generators of A and morphisms to additive or multiplicative relations.

84. (Bellah 2011, p. 9).

85. Grothendieck’s Wikipedia pages in most of the languages I can read describe him as stateless; however, Cartier (2014) writes that he “consented to apply for French citizenship” after 1980. As for Erdős, “There was even an interval when he was nearly stateless: after a trip abroad, U.S. officials refused to readmit him—on the grounds that he was a security risk! Israel came to the rescue with a passport” (Hayes 1998). Quotation from Hoffman (1998, p. 29).

86. (Pinker 1997, p. 21).

87. A topos is one of the notions Grothendieck introduced to look at space; in the hands of F. W. Lawvere and others, topos theory has developed into an alternative to set-theoretic logic.

88. http://mathoverflow.net/questions/59520/how-true-are-theorems-proved-by-coq.

89. (Stcherbatsky 1978, p. 44; Wittgenstein 1922). Juliette Kennedy (in press) quotes a private communication by the eminent logician S. Shelah expressing much the same sentiment in a much more technical context: “Considering classical model theory as a tower, the lower floors disappear—compactness, formulas, etc…. the higher floors do not have formulas or anything syntactical at all.”

CHAPTER 8 THE SCIENCE OF TRICKS

1. This is from the the 1979 Penguin translation by Dim Cheuk Lau; however, I have replaced Lau’s “plausible men” by “clever talkers,” as in the 1938 Arthur Waley vintage translation (Confucius 1979, 1938). The term corresponds to the Chinese ning ren, elsewhere translated as “persuaders” or “specious talkers.” “Wanton” is also translated “licentious,” “obscène,” etc.

2. Compare with Tao (2008b) in math.CA,tricks: “ … Tim Gowers … has begun another mathematical initiative, namely a ‘Tricks Wiki’ to act as a repository for mathematical tricks and techniques…. today I’d like to start by extracting some material from an old post of mine on ‘Amplification, arbitrage, and the tensor power trick’ … ” Note the seepage of financial metaphors into mainstream mathematics. I’m told the “Tricki” is currently more or less dormant.

3. See, for example, “The Mythological Trickster: A Study in Psychology and Character Theory” in (Manin 2007). For “mathematical wit,” see (Hoffman 1998, p. 49).

4. A tensor product trick goes unmentioned in chapter 9 but was of crucial importance nonetheless.

5. Here is the incriminating sentence, from my article (Harris 2009): “The principal innovation is a tensor product trick that converts an odd-dimensional representation to an even-dimensional representation.” Calling it “Harris’ tensor product trick” was the idea of (Barnet-Lamb et al. 2014).

6. I’m still waiting.

7. For the trickster as creator and crosser of boundaries, figure at the crossroads, see (Hyde 1998, 2010, pp. 6, 119, 220–225; Gates 1988). Compare “Kunstgriffe verblüffen, indem sie scheinbar verschiedene Aspekte miteinander verbinden und daraus Nutzen ziehen” from (Rump 2011). The article mentions the famous “Kunstgriff von Rabinowitsch” that appeared in van der Waerden’s classic 1930 algebra text.

8. See (Dante 1981). I follow Barry Mazur’s translation in (Mazur 2004a) but added my emphasis.

9. (Langlands 1979b). Grothendieck referred to himself as a builder (bâtisseur). The description in Récoltes et Sémailles, Prelude, section 2.5, deserves to be quoted in full. “Il n’a que deux mains comme tout le monde, c’est sûr—mais deux mains qui à chaque moment devinent ce qu’elles ont à faire, qui ne répugnent ni aux plus grosses besognes, ni aux plus délicates, et qui jamais ne se lassent de faire et de refaire connaissance de ces choses innombrables qui les appellent sans cesse à les connaître. Deux mains c’est peu, peut-être, car le Monde est infini. Jamais elles ne l’épuiseront! Et pourtant, deux mains, c’est beaucoup. … ”

10. Bas Edixhoven and Frans Oort, personal communications.

11. According to Wikipedia, the word is also used for acrobatic stunts and secondarily for magic tricks, and a tryuk is typically “dangerous or unfeasible for the untrained.” http://ru.wikipedia.org/wiki/трюк. The primary word for magic tricks is fokus.

12. Compare tour, derived from “to turn,” with the epithet polutropos, “of many turns,” applied to Odysseus. Odysseus is also called polumekhanos, usually translated “of many devices” but sometimes “of many tricks.”

13. “Kunstgriffe gehen somit einer Theorie voraus, sie greifen ins noch Unbekannte, verbinden das scheinbar Getrennte, damit dieses durch weiteres Nachdenken seinen natürlichen Platz in der allgemeinen Theorie erhält und damit bekannt wird” (Rump 2011).

Michael Atiyah sees trick as the first way station, followed by technique and method, along the path “from problem to theory.” “[I]n order to solve a problem you have got to have a clever idea, some kind of trick. If it is a sufficiently good trick and there are enough problems of a similar type you go on and develop this trick into a technique. If there is a large number of problems of this kind, you then have a method and finally if you have a very wide range you have a theory; this is the process of evolution from problem to theory” (Atiyah 1989, pp. 21116). In this connection, I note that Sanjoy Mahajan paraphrases George Pólya to the effect that “A tool … is a trick I use twice,” in (Mahajan 2010). Tools and techniques are discussed later. See also the long discussion at http://mathoverflow net/questions/48248/what-do-named-tricks-share.

14. (Hurston 1943, p. 452).

15. A complete translation is available online at http://ebooks.adelaide.edu.au/s/schopenhauer/arthur/controversy/chapter3.html. In Plato’s Cratylus, 409 D, Socrates says “Now watch out for my special trick (mekhane) which I have for everything I can’t solve!” The translation is from (Huizinga 1950, p. 150).

16. (Gheury 1909). Quotation marks are as in the original.

17. (O’Toole 1940). The Mathematical Association of America is the professional organization of mathematics teachers, whereas the American Mathematical Society is the professional association of mathematical researchers. Of course, there is a significant overlap.

18. (AMS 1962, p. 2).

19. In his second-most highly rated MathOverflow post, entitled “Proofs that require fundamentally new ways of thinking,” Tim Gowers list six examples of what might be called tricks that have since been incorporated into mainstream mathematics; he finds them interesting because “they are the kinds of arguments where it is tempting to say that human intelligence was necessary for them to have been discovered.” http://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking.

20. (Restivo 1988; G. A. Miller 1908).

21. Unless the first computing machine was constructed by Wilhelm Schickard, as he described in his letter to Kepler dated September 20, 1623: “I constructed a machine [machinam extruxi], which includes eleven full and six partial pinion-wheels, which can calculate automatically, to add, subtract, multiply and divide.” Text at http://history-computer.com/MechanicalCalculators/Pioneers/Schickard html, in the original Latin and in English translation.

22. “Peter Sarnak, Professor in the School of Mathematics, compares the fundamental lemma [proved by B. C. Ngô; see chapter 2] to a screwdriver, functoriality to opening a screw, and the Langlands program to the big machine working to reveal the underlying structure of automorphic forms” (Kottwitz 2010).

23. Plutarch, Table Talk, VIII 2.1 [718e]; also Life of Marcellus, XIV, 5–6 (Bernadotte Perrin translation). Plato complained that the mathematicians in question had “descended to the things of sense” rather than “abstract thought” and moreover used “objects which required much mean and manual labour.” This objection would in principle not apply to the canonical delooping machine or any other construction that is only metaphorically mechanical.

24. From a letter to Schumacher translated in (Ferreirós 2007, p. 216).

25. William Casselman, private communication, September 3, 2003. Casselman, a professor at the University of British Columbia, was one of the earliest collaborators on the Langlands program. The blurb for a planned semester program at the MSRI in Berkeley illustrates the continued appeal of the term Langlands philosophy (www msri.org/web/msri/scientific/programs/show/-/event/Pm8951).

26. See Deligne’s talk “L’influence de la philosophie des motifs,” January 15, 2009, announced at http://www.ihes fr/jsp/site/Portal.jsp?document_id=1661&portlet_id=999.

27. “By “yoga” he meant a unifying point of view, a lead in the search for concepts and proofs, a method one could use repeatedly.” Pierre Cartier, February 25, 2011 (private communication).

28. (E. T. Bell 1944).

29. (Dumézil 1992, p. 101). In this passage, Dumézil also insists that tripartite organization is absent in the mythology of non-Indo-European cultures.

30. (Langlands 2007). Langlands cautions that “still unforeseeable difficulties … will make great demands on the inventive powers of analytic number theorists.”

31. Or “of many tricks”—see note 12.

32. To save the Pandava brothers’ army from slaughter at the hands of the invincible warrior Drona, Krishna devised a ruse to convince Drona that his beloved son Aśvatthāma had been killed. Bhima (the brother associated with brute force, a kshatriya in Dumézil’s reading) killed an elephant named Aśvatthāma and shouted that Aśvatthāma was dead. Drona asked Yudishthira (the brother associated with the brahmin function) whether this was indeed true. Yudishthira, as Drona knew, was incapable of lying, but at Krishna’s instigation, he replied “Aśvatthāma the elephant has been killed,” pronouncing the word “elephant” inaudibly (or drowned out by Krishna’s conch, in some versions). Brokenhearted at the news, Drona left the battlefield and went straight to heaven, as Krishna had anticipated.

Deceit is also a recurring feature in Ferdowsi’s tenth-century epic of the Persian kings.

33. (Cook 2009, p. 138). Compare (Littlewood 1986, p. 195): a mathematician “has to be completely honest in his work, not from any superior morality, but because he simply cannot get away with a fake.”

34. The word Kunstgriff appears on page 4 of Die letzten Gründe der Hohern Analysis by Joseph Nürnberger, 1815. It seems to be used in a slightly different sense in Anfangsgründe der Mathematik: Anfangsgründe der Analysis …, Volume 3, Issue 1, published by Abraham Gotthelf Kästner in 1767. . For the use of the word trick in English, see also Sadler (1773).

35. For ta-mar, see Friberg (2007). For Hellenistic mathematics, see Netz (2009, pp. 78, 67, 17ff, 54, 77–78). The Lilavati is (Brahmagupta and Colebrook 1817).

Netz identifies three characteristic “ludic” features of Hellenistic mathematics: “mosaic composition”—“the juxtaposition of apparently unrelated threads that, put together, delight with the surprise of a fruitful combination, or startle with the shock of incongruity” (p. 117)—“narrative surprise,” and “generic experimentation” (including the so called carnival of calculation). We have seen that the first two (but not the third) of these features also help to identify contemporary mathematical tricks, but it seems to me there is an important difference: the latter don’t call attention to themselves as tricks—as vehicles for entertainment or display—but primarily function as shortcuts on the way to a defined goal, possibly destined to be integrated in normal mathematics. The parallels and differences between Hellenistic and contemporary mathematics deserve further analysis.

36. From the verb root ḥ-w-l. Hans Wehr’s dictionary lists the following meanings in Form VIII: “to employ artful means, resort to tricks, use stratagems….” Also “to deceive, beguile, dupe, cheat, outwit, outsmart … or achieve by artful means, by tricks.”

Jens Høyrup translates ‘ilm al-ḥiyal by “science of ingenuities” (following Gherard de Cremona’s scientia ingeniorum—see below—and writes “the term ingenio can be read in its double Latin sense, as ‘cleverness’ and as ‘instrument’ (in agreement also with the Arabic text)—we might speak of the discipline as ‘engineering’ or as ‘applied theoretical mathematics’ ” (Høyrup 1988). It’s possible to write the tricks out of medieval Islamic mathematics. But why would one want to?

37. From the MacTutor History of Mathematics Archive, http://www-history.mcs.st-andrews.ac.uk/Biographies/Banu_Musa html. Al-Khwarizmi’s algebra also placed magnitudes and quantities on an equal footing: see chapter β.

38. Al-Fârâbî actually referred to al-jabr wa al-muqabala, from the name of al-Khwarizmi’s ninth-century treatise. And it should be noted that al-jabr originally referred to the (mechanical?) operation of bonesetting. For Ibn Sina, see (Rashed 2011, p. 753). For mekhane see note 12. Most reproductions of the al-Khayyam quotation refer to the translation by Amir-Moez (1963); I have instead used the English translation by (Rashed and Vahabzadeh 2000, p. 171) but follow Amir-Moez in writing “trick” for ḥila. In their French translation, Rashed and Vahabzadeh write “artifice.” This is, in any case, the only place the word ḥila appears in al-Khayyam’s mathematical papers.

39. See http://plato.stanford.edu/entries/archytas/. Pierre Hadot, in his Veil of Isis (see chapter 11), points out that “[t]he idea of trickery—and ultimately, of violence—appears in the word ‘mechanics,’ since mekhane signifies ‘trick.’” (Hadot 2006, p. 101)

40. G. de Cremona, De Scientiis, reprinted in (al-Fârâbî 2005; de Gandt 2000); for Kunstgriff as translation of ingenium, see (Glowatzki and Göttsche 1990, p. 23; Kausler 1840, p. 546).

Descartes defines ingenium in this passage from Rule XII: “In accordance with these diverse functions the same power is called now pure intellect, now imagination, now memory, now sense; and it is properly called mind (ingenium) when it is either forming new ideas in the phantasy or attending to those already formed.” The roots of Descartes’ use of the term are explored in (Robinet 1996) and (Sepper 1996). In constrast to de Gandt, neither Robinet nor Sepper mentions that ingenium is a translation of mekhane. The word ingenium has so many meanings in postcartesian philosophy that its association with Kunstgriff can hardly be considered primary.

The questions of where (or whether) to draw the line between art and technique or between the artist and the artisan dominate many of the aesthetic debates of the nineteenth and twentieth centuries, as reconstructed in (Rancière 2011).

41. Private communications of Kai-Wen Lan and T. Yoshida, September 2011.

42. J. reine angew. Mathematik 10 (1833): 154–166. The original text is “Bei allen im Vorigen gezeigten Verwandlungen der Kettenbrüche in andere, wurde der Kunstgriff angewandt, dass man einen Theil des Kettenbruchs als summirt betrachtete, diese Summe durch einen Buchstaben ausdrückte, und durch die Verbindung dieses Buchsta-bens mit den übrigen Gliedern des Kettenbruchs neue Ausdrücke fand.”

Apart from publishing the earliest Kunstgriff in Crelle’s Journal, M. A. Stern has the distinction of being the first unconverted Jew to be named to a professorship at any German university, in any subject (Göttingen, in 1859). The reader may have noticed that a number of the tricks mentioned in this chapter are associated with Jewish authors, including the author of these lines. No doubt this is a coincidence.

43. One of the most apparently deceitful of mathematical tricks is called, depending on the context, either the Eilenberg swindle (after Samuel Eilenberg, one of the creators of category theory) or the Mazur swindle (after my thesis adviser Barry Mazur). Though highly counterintuitive, the trick is logically perfectly legitimate.

44. Oresme’s comment is on page 188 of his translation of Aristotle. I thank Jean-Benoît Bost for directing my attention to Rey’s dictionary and for providing these references, from 1967 and 1908 respectively: “Astucieux est en train de perdre la nuance de malice et de duplicité qui y était attachée. Il signifie ingénieux, habile, perspicace” [Astucieux is losing the nuance of malice and duplicity with which it was associated. It means ingenious, clever, perspicacious.] “On dit: avec astuce pour « avec ingéniosité,” et aussi: dire une astuce, faire une astuce, où astuce désigne « quelque chose d’ingénieux,” [astuce designates “something ingenious”] References online at http://www.cnrtl fr/lexicographie/astucieux; http://www.cnrtl fr/lexicographie/astuce.

45. The Greek word kataskeue for geometric construction, also used in connection with building, is not in Euclid but was used by later geometers, including Pappus.

46. On page 7 of (Lipschitz 1870), my translation. At the turn of the nineteenth century, the poet Novalis had already written “Am Ende ist die ganze Mathemat[ik] gar keine besondre Wissenschaft—sondern nur ein allgem[ein] wissenschaftliches Werck-zeug” [in the end the whole of mathematics is not at all a special science—but rather a general scientific tool].

47. For Technik, see the 1835 book Die darstellende Geometrie:(La géometrie déscriptive) by S. Haindl (six results on Google books).

48. The Jahrbuch published abstracts and reviews of mathematical articles from 1868 through 1942; it was succeeded by Zentralblatt für Mathematik und ihre Grenzgebiete (1931—), which has made the Jahrbuch’s archives available on its online Web site, as well as Mathematical Reviews, published by the AMS since 1940, and the Russian Referativnyi Zhurnal, which began publication in the 1950s. Dieudonné’s review is MR0179805 (31 #4047a), b.

49. (De Sigorgne, 1739–1741; Darboux 1866, the word attaqué is on p. 117; Dixon 1882). The word had already appeared in The Analyst—predecessor of the extremely distinguished Annals of Mathematics—in an 1876 article purporting to solve the equation of the fifth degree, which Abel and Galois had proved impossible.

50. (Bourbaki 1948, esp. pp. 40–41; 1950).

51. (Bourbaki 1950, p. 223).

52. (Gates 1988, p. 6). Analyzing Transylvanian trickster stories, L. Piasere sees the figure mediating a transition between potential and actual infinity, in Aristotle’s sense: “Ora, il trickster sembra proporsi come una figura dalle possibilità indefinitamente aperte … e quindi sembra essere più dal lato dell’infinito potenziale, dell’infinito aperto; ma il fatto di riunire su di sé, cioè nella propria identità, le infinite alterità concepibili, lo pone contemporaneamente dal lato dell’infinito attuale, dell’infinito chiuso” (Piasere 2009). Dean A. Miller (2000, p. 243) sees the trickster-hero as “intermediary between [the supernatural] zone and the zone of heroic human action.” Lewis Hyde (1998, 2010) points out that the trickster is not the Devil; but neither is Mephistopheles.

53. I’ve repeatedly drawn attention to clichés—you’ll notice I just slipped one in myself—and this is probably the right place to explain that the abundant use of clichés in books about mathematics for the general public is not a response to contracts in which authors are paid by the word but rather express a judgment about what the audience can be expected to know. In a book about classical music pitched at the level of this one, you will not read that Bach had a lot of children or that Mozart was a prodigy or that Beethoven was deaf when he composed his Ninth Symphony, because that information belongs to general culture in a way that the irrationality of the square root of 2 or Cantor’s diagonalization argument does not.

54. See “Aesthetics of Music” in W. Apel, Harvard Dictionary of Music, Harvard University Press (1972), p. 14. Just sticking to number theorists, Harvard’s Noam Elkies is a composer of (serious) classical music; Princeton’s Manjul Bhargava (Fields Medal 2014) plays tabla at (close to) professional level.

The affinity mathematicians feel for music is not a myth. B. Zarca’s sociological survey (cited in chapter 2) finds that, asked to choose from nine “worlds” those to which they felt close, mathematicians in France systematically placed the “world of music” at the top of their list; pure mathematicians were even more categorical. Scientists in allied fields felt closest to social sciences and “industry and finance.” The “world of media” was at the bottom of everyone’s list but was especially severely rejected by pure mathematicians (Zarca 2012, pp. 294–95). Honesty compels me to mention that the Zarca study did detect a significant preference for classical music over other genres among mathematicians in France, and this preference is more marked among mathematicians than among other scientists (p. 261).

55. (Christensen 1987).

56. “As the study of axioms eliminates the idea that axioms are something absolute, conceiving them instead as free propositions of the human mind, just so would this musical theory free us from the concept of major/minor tonality […] as an irrevocable law of nature … Whereas geometric axioms are sufficiently justified if their combinations prove them to be both independent of and compatible with each other, the accuracy of musical axioms can be proved exclusively by their fitness for musical practical use” (Krenek 1939, pp. 206–7).

57. (Xenakis 1963).

58. David Lewin, Dmitri Tymoczko. http://dmitri.tymoczko.com/geometry-of-music html, Guerino Mazzola. Since the year 2000, there has also been a regular seminar called MAMUPHI (Mathématiques, Musique, Philosophie) at the École Normale Supérieure.

59. (Dreyfus 1996, p. 9). C.P.E. Bach was famously quoted by his contemporary J. P. Kirnberger: “You can say out loud that my principles and those of my late father are anti-Rameau” (Kirnberger 1771).

60. (Berlioz 1836; Saint-Saëns 1900; Blackburn 1900). I thank Laurence Dreyfus for the Brahms reference.

61. (Schenker 1997).

62. (Richardson 2011; Hewitt 2011; Born 1995, p. 96).

63. (Lippman 1990, p. 92).

64. The book (Assayag et al. 2002) is also overwhelmingly concerned with classical music. I neglect jazz in this chapter because I have been unable to find any striking quotations linking jazz with mathematics. See also note 97.

65. Joan Wallach Scott notes that some of the founders of the American Association of University Professors considered “a faculty member’s demeanor” important for establishing and maintaining legitimacy. She quotes the AAUP Committee on Academic Freedom and Academic Tenure, which argued in 1915 that academic responsibility presupposed “dignity, courtesy, and temperateness of language,” “a peculiar obligation to avoid hasty or unverified or exaggerated statements and to refrain from intemperate or sensational modes of expression” (Scott 2009).

66. Pierre Bourdieu’s study of distinction emphasizes the role of shared cultural values, including tastes in music and the arts, necessarily inclined to seriousness, in qualifying Homo Academicus as the personification of legitimate authority: “nothing more clearly affirms one’s ‘class,’ nothing more infallibly classifies, than tastes in music.” See the long discussion of music’s special role in defining the “aristocracy of culture” in Distinction (trans. Richard Nice), Harvard University Press (1984), pp. 18–19. Bourdieu also examines the relations between “cultural capital” and “academic capital” in Homo Academicus and The State Nobility.

67. The very word culture with its present associations only came into use in the English language in the mid–nineteenth century. Raymond Williams points out that what Coleridge called “cultivation, the harmonious development of those qualities and faculties that characterize our humanity” had previously been “an ideal of personality—a personal qualification for participation in polite society”; in the mid-nineteenth century, it had “to be redefined, as a condition on which society as a whole depended.” For Matthew Arnold, culture was “the study of perfection” and the guardians of the standards of perfection, the condition of “general cultivation,” were what Coleridge called the Clerisy, including “the sages and professors of … all the so-called liberal arts and sciences” (Williams 1958, pp. 61–64, 115). The Coleridge quotation is from On the Constitution of Church and State, V. The place of scientists and mathematicians in culture varied—Arnold wrote in his Philistinism in England and America, in connection with the mathematician Joseph Sylvester, that “for the majority of mankind a little of mathematics, even, goes a long way.” Reprinted in volume X of The Complete Prose Works of Matthew Arnold, University of Michigan Press (1974).

68. Stephen Shapin argues that the credibility of research in early modern times presupposed the conformity of the researcher to the ideal of the gentleman whose word could be trusted. Thus, in England, Henry Peachum wrote in 1622 that “we ought to give credit to a noble or gentleman before any of the inferior sort.” “Categories of people—women and the vulgar in particular—in whom [the higher intellectual] faculties were poorly developed might be … constitutionally prone to undisciplined and inaccurate perceptions.” “The relevant maxim … in the civil conversations of early modern gentlemanly society, was ‘believe those whose manner inspires confidence.’ ” Although the gentlemanly repertoire was alien to enlightenment France, manner mattered there as well: in the éloges composed by Fontenelle, perpetual secretary of the French Académie des Sciences, and by his successors, the “eighteenth century men of science were described as embodiments of Stoic fortitude and self-denial.” The Encyclopédie characterized the ideal “philosopher” as “an honnête homme, polite, civil, and autonomous” who “serves Truth alone.” (Shapin 1994, pp. 69, 77, 221; 2008, pp. 36–37).

69. “… a character who attracts our attention by his balanced pose and his ugliness.” See http://www.diariodecultura.com.ar/web/news!get.action;jsessionid=B23A495DF6437ED023AA643237DC57D9?news.id=7119. In the catalogue raisonné of Luca Giordano’s works (Ferrari and Scavizzi 1992), this painting is called Il filosofo. For the painting, on display at the Museo Nacional de Bellas Artes in Buenos Aires, see http://www.mnba.gob.ar/coleccion/obra/2858. The commentary at the Buenos Aires Web site suggests that the inscription may be a representation of the “proto-Hebrew” believed to have been the universal language before Babel.

70. (Thorndike, 1923–58, Chapter LXI).

71. For Padova, (see Favaro 1888); for Galileo’s astrology, see (Favaro 1881, p. 4). Kepler’s De Fundamentis Astrologiae Certioribus expresses some skepticism regarding astrology—“If [astrologers] at times do tell the truth, it ought to be attributed to luck, yet more frequently and commonly it is thought that this comes from some higher and occult instinct”—but like Brahe, he derived much of his income from horoscopes. The IBM timeline entries for Galileo and Kepler do not mention horoscopes.

72. From the Faustbuch 1580 (my emphasis). The English translation of this passage [online at (Wulfman 2001)] makes Faustus, incorrectly in my opinion, into a mathematician: “… being delighted … so well [in those diuelish arts] that he studied day and night therein: in so much that hee could not abide to be called Doctor of Diuinitie, but waxed a wordly man, and named himselfe an Astrologian, and a Mathematician …”

73. (Saiber 2003).

74. (Evenden and Freeman 2011, pp. 265–66). The Dee quotation, dated 1570, is taken from (Pimm and Sinclair 2006, chapter α). For his alchemical exploits, see the introduction to (Josten 1964, p. 90). John Dee is the hero of the semirock opera Dr. Dee and was also the alter ego of Doctor Destiny, a supervillain from the Justice League of America, known for creating the Materioptikon, “a device which allowed him to create reality from the fabric of dreams.” (Source: Wikipedia entry on Doctor Destiny.)

75. (Josten 1964, p. 85). Huizinga, perhaps thinking of characters like Dee or Cardano, writes that “modern science, so long as it adheres to the strict demands of accuracy and veracity, is far less liable to fall into play as we have defined it, than was the case … right up to the Renaissance, when scientific thought and method showed unmistakable play-characteristics” (Huizinga 1950, p. 204).

76. (Weyl 1935, 1939; Hardy 2012, Chapter 11). “Very serious man” is from Peter Pesic’s introduction to Weyl (2009), an edition of Weyl’s writings on philosophy, mathematics and physics; the quotation on Newton’s trick is on page 45 of the same book. See note 13 for similar comments by Pólya and Atiyah.

Høyrup, writing with Al-Fârâbî in mind, sees Jordanus de Nemore’s thirteenth-century work as a stage in the legitimation of algebra, along very similar lines: “the De numeris datis transforms a mathematically dubious ingenium into a genuine piece of mathematical theory” (Høyrup 1988, unpublished, Høyrup’s emphasis; see note 36).

77. (Hyde 1998, p. 100).

78. (Jackson 2004, p. 1203). Comparing Serre and Grothendieck, Leila Schneps finds that “Serre was the more open-minded of the two; any proof of a good theorem, whatever the style, was liable to enchant him, whereas obtaining even good results ‘the wrong way’—using clever tricks to get around deep theoretical obstacles—could infuriate Grothendieck.” [L. Schneps, “A biographical reading of the Grothendieck-Serre correspondence,” p. 18; a short version was published in Mathematical Intelligencer, 29 (2007). See note 80].

Grothendieck banned tricks from his own work as well. Armand Borel mentioned in a lecture in Mumbai that Grothendieck refused to publish his proof of what is now known as the Grothendieck-Riemann-Roch theorem because (in positive characteristic) he made use of a trick; the proof was instead published by Borel and Serre. (I thank T. N. Venkataramana, who attended Borel’s lecture, for this information.)

79. “Proof, in its best instances, increases understanding by revealing the heart of the matter” (Davis and Hersh 1981, p. 151).

80. (Langlands 2007). Serre briefly made a similar point in a 1985 letter to Grothendieck: “I know well that the very idea of “getting around a difficulty” [contourner une difficulté] is alien to you—and perhaps that’s what you find the most shocking in Deligne’s work ([an] example: in his proof of the Weil conjectures, he “gets around” the “standard conjectures”—that shocks you, but it delights me [cela me ravit]).” (Colmez and Schneps 2003, letter of July 23, 1985).

Zarca found that “19% of mathematicians associate the idea of a trick [ruse] with the notion of an elegant proof (and none the opposite).” Zarca’s findings on elegant versus natural proofs, analyzed on pages 267–72 of (Zarca 2012), deserve a longer discussion than we can provide here.

81. “Was gute Kunst ist, läßt man sich nicht vom Kaiser und dem in ihm repräsentierten Volk sagen, auch wenn er die Ankäufe der großen Museen kontrolliert. Was gute Mathematik ist, läßt man sich nicht von Ingenieuren, Lehrern und Philosophen vorschreiben, die sich auf Prinzipien nach Art des Kaisertums berufen. Als Besonderung geht in die Gemeinsamkeit der Beziehung Profession-Klientel hier das “Schöpferische” ein, mit dem für die Mathematik in Anlehnung an die Kunst der Standort im kulturellen System markiert, der Wille zum Neuen bezeichnet und der Wert der eigenen Arbeit beschrieben wird.” All quotations from (Mehrtens 1990, pp. 515, 543). In his best-selling novel Die Vermessung der Welt [Measuring the World[, Daniel Kehlmann has C. F. Gauss complaining (anachronistically?) 100 years before Liebmann about the obligation to respect court etiquette in order to obtain funding.

82. I thank Vladimir Tasic for this observation and for the reference to (Eagleton 1996, pp. 24–28), from which all quotations in this paragraph are drawn.

83. “Hijo, esto de ser ladrón no es arte mecánica sino liberal.” The Gauss character in Kehlmann (2005) has in common with Till Eulenspiegel a disdain for hypocrisy and a rejection of social decorum, sometimes taking extreme forms. Since Gauss and Eulenspiegel are the two most famous individuals associated with the city of Braunschweig (Brunswick), this may not be a coincidence. The Forsyth quotation is reproduced in (Heard 2004, p. 250).

On the other hand, one female colleague reminded me after reading this section that the freedom not to aspire to being taken seriously is a luxury not available in equal measure to all members of the profession.

For the transition from “rude and tumultuous mechanic persons” to the “mechanical philosophy” of the Royal Society, see (Hill 1975, pp. 255–60). The relation to tricks and magic, if any, is not a simple one.

84. Regarding finance, D. Steinsaltz, reviewing Patterson’s The Quants, writes that an investment bank’s “team of MIT Ph.D.s is a token of seriousness, like the Picasso in the lobby, the marble columns, and the expensive wristwatch” (Steinsaltz 2011, p. 703).

But I should make it clear that mathematics is hard work and that the value system of mathematics, if not that of Powerful Beings, rates this very highly. This must be why “work” occurs in G. H. Hardy’s Apology as frequently as “beauty” and “beautiful” combined, though it’s for the latter that his book is known. Tricks thus conflict with seriousness as conceived by Powerful Beings as well as by the value system of mathematics. But finding the right trick is also hard work!

85. In fact I did. From http://www math.utah.edu/~ptrapa/finalreport/linernotes html:

[I]n 1998 … several bored IAS members concocted a radical plan to break the grip of academic tedium. What the staid Institute needed, they believed, was a red-lining, full-blown, hard-driving Rock band … The band conspirators thought that … “music in the popular idiom” would complement [musician-in-residence Robert] Taub’s classical menu … So the Institute’s first (and, no doubt, last) band was born. It took the name “Do Not Erase”….

At least three of the band members were mathematicians—representation theorists, in fact. See also note 90.

The lyrics are from I Ain’t No Joke by Eric B. and Rakim.

86. Space limitations prevent me from exploring mathematical music in other languages—for example, the ranchera Las Matemáticas by Los Alteños de la Sierra, the tango El Algebrista, Matemáticas by the Spanish MC ToteKing, Per ogni matematico by the Italian folk singer Angelo Branduardi, and so on.

87. La Tribu, “the tribe,” was and perhaps still is the name of the Bourbaki association’s internal bulletin. See (Corry 2004, p. 293).

88. http://genealogy math ndsu nodak.edu/.

89. The two kinship systems are not quite independent. There is not exactly an incest prohibition: it is perfectly acceptable to collaborate with your offspring, since the normal result is a published article rather than another descendent, and indeed the shortest path between me and Toby Gee includes an article he wrote with his “grandfather,” Richard Taylor. But it is considered poor form for an adviser to cosign a student’s first published article, since that might create an impression of dependence, harming the student’s career prospects.

90. http://www.spinner.com/2011/06/06/battles-gloss-drop/. Atlas is online at http://www.youtube.com/watch?v=IpGp-22t0lU. Another example: When at 24 the “cool guy,” Peter Scholze, became the youngest professor in Germany with a Lehrstuhl (named chair) at the University of Bonn—in part for his work on the Langlands program—his high school teacher reminisced in the press: “Peter is not a typical mathematician … He is popular, involved, and played bass guitar in a band” (Harmsen 2012).

91. The Patti and Fred Smith lyrics are from It Takes Time on the soundtrack album from Until the End of the World by Wim Wenders. The Grebenshchikov quotation is from http://www.aquarium.ru/misc/aerostat/aerostat137.html, downloaded February 17, 2012, and translated (from a different site) on September 15, 2011. For Lynch, see (Nochimson 1997). Other rock numbers expressing a skeptical attitude to mathematics include Algebra by Soul Hooligan, Algernon’s Awfully Good at Algebra by Malcolm McLaren, and In Real Life There Is No Algebra by Beardfish. More recently, Lynch and Smith claimed an abiding interest in mathematics when they teamed up with geometer Misha Gromov for the exhibition Mathematics: A Beautiful Elsewhere at the Fondation Cartier in Paris.

92. And another track entitled “Who taught you Math?” under his own name. Hood once reminded “the younger generation” to “remember it’s all about the rhythm … and not to get so serious and so caught up in the hype of minimalism” (2Sheep4Coke 2009).

93. “Because when I grew up in the Bronx in the Eighties … there was a part of rap music where guys would battle each other on the streets, and sometimes the battling … would be talking about science, for example, and they even called it droppin’ science, like Eric B and Rakim” (Stephon Alexander 2008).

The term “dropping mathematics” was also common, according to Alexander (private communication). See (Enders 2009, p. 84): “to this day, a lot of us old Gees still be dropping mathematics.” No doubt this is due in part to the influence of the Nation of Gods and Earths, also known as the Five Percenters, an offshoot of the Nation of Islam. The Five Percent Nation’s cosmological system centers on Supreme Mathematics and the Supreme Alphabet. Rakim was only one of many hip-hop leaders to be a Five Percenter. For an early report on Five Percenter influence on hip-hop, see (Ahearn 1991); see also (Gibbs 2003). This is not intended as an explanation of the phenomenon under discussion: it should be obvious that a creative community predisposed to see mathematics as antithetical to their art would have rejected the message of Supreme Mathematics.

94. Lyrics copied from http://rapgenius.com/Mos-def-mathematics-lyrics, where you can also read that “Rap—like physics—is fundamentally a mathematical discipline.”

95. Also known as Ronald Maurice Bean. Allah Mathematics, like other members of the Wu-Tang Clan, has been associated with the Five Percent Nation. The quotation is online at http://wu-international.com/Killabeez/wuelements_pro htm.

96. Mos Def’s Mathematics lyrics mentioned in the previous note are mainly devoted to quantifying this marginalization. A longer quotation from the same song opens the article (Terry 2011).

Mathieu Guillien, whose book (Guillien 2014) focuses on the minimal techno of Detroit and specifically the music of Robert Hood, agrees (private communication): “Rock is the music of a white middle class that can permit itself to criticize an education to which it had access. The politics of rap have identified education as one of the things the white establishment has denied the Afro-American population, and it seems to me that for that reason mathematics is still perceived with respect.” It’s probably no accident that a Washington Post staff writer, wondering why Hollywood has yet to give the civil rights movement the treatment it deserves, proposed Mos Def for the role of Robert Moses, legendary organizer of the 1964 Mississippi Freedom Summer. Moses used his 1982 MacArthur Fellowship to found the Algebra Project, whose motto is “Math literacy is the key to 21st century citizenship” (Hornaday 2007).

97. Quotations respectively from (Gates 2010) and from Run-DMC, “It’s Tricky.” Readers who suspect that the point of this chapter is to suggest that improvisation, rather than rule following, is the essence of mathematics, may be surprised that jazz has been omitted from the discussion. Here is an inadequate answer. On the one hand, no one these days (Theodor Adorno notwithstanding) denies that jazz is serious music, and the exclusion of jazz from the classical canon looks increasingly arbitrary and untenable. On the other hand, the little I have read suggests that jazz musicians and classical musicians feel much the same way about mathematics. There are exceptions, however: mathematical physicist and jazz saxophonist Stephon Alexander draws astonishing parallels between John Coltrane’s chord progressions in Giant Steps and the principles of galaxy formation (Stephon Alexander 2012).

98. (Williams 1958, p. 306).

99. Boulez, in (Born 1995, p. 139); Leavis, quoted in (Williams 1958, p. 253).

100. Arthur Waley explains in the notes to his Vintage translation that “ordinary as opposed to serious-minded people had the same feelings” to the music Confucius approved “as they have towards our own classical music today. Waley cites the following passage from the Liji (Book of Rites): “ ‘How is it,’ [asked] the Prince of Wei … ‘that when I sit listening to old music … I am all the time in terror of dropping off asleep; whereas when I listen to the tunes of Cheng and Wei, I never feel the least tired?’ ” (Confucius 1938, p. 250).

101. (Ferreirós 2007, p. 220); Russell, Mysticism and Logic; (Hardy 2012).

102. See Meno, 86b and note 55 to the previous chapter.

103. I once heard Serre claim at the beginning of a lecture that nothing is easier than to make an auditorium full of mathematicians laugh; within two minutes he had proved his claim. For Serre, see (Huet 2003); for Villani, see (Delesalle 2011) and http://math.univ-lyon1 fr/~villani/cv html.

Chapter 28 of (Villani 2012), with nearly 10 pages detailing Villani’s eclectic listening habits, should put to rest any naive assumption that mathematicians are naturally inclined to prefer classical to popular music, or vice versa. French pop singer Catherine Ribeiro is a special favorite of Villani.

104. (Dreyfus 1996, p. 31).

105. F. Schiller, On the Aesthetic Education of Man (Gopnik 2009, p. 14; Pinker 1997, p. 525). Now, cognitivists use the word spandrel instead of cheesecake, referring to an architectural detail that appears not by design but through the interaction of two or more structural features and, by extension (following a well-known article by S. J. Gould and R. Lewontin), an indirect result of natural selection that is not itself selected by the environment.

To say that mathematics is the structure of music is only a way of saying that mathematics is never far when we talk about structure. If mathematical structure is more visible in music than in other fine arts it’s because there is no other distracting source of structure; this is what it means to say that music and mathematics are both abstract.

106. (Oort 2014). Oort is translating Serre’s “Cela me ravit” (see note 80).

107. (Levi-Strauss 1969, pp. 15–16; Elias 1993, p. 131).

108. Bruce Bethke, who invented the word cyberpunk, traces it to “cross-pollination going on between the synthesizer circuit hackers (that is, we totally cool musician types) and the homebrew computer hackers (a k.a., those terminally unhip nerds)” (Bethke 1997).

109. (Rucker 1982 (one of the first cyberpunk novels), chapter 4). The speaker in this case is an artificial intelligence. Rucker himself is a professional mathematician and computer scientist. For mathematics and anarchism in Pynchon, see (Engelhardt 2013).

110. In prehacker science fiction, the logic of the confrontation was different: “Time and again when confronted with the superior strength and knowledge of some robot enemy, Master Mechanics defeat … the machine’s ideology by forcing the logic-bound robot to attempt to process information which is fundamentally irreducible to true/false resolutions. Thus the ultimate superiority of human intelligence is asserted, and that superiority is fugured as based on our emotions and their theorized ability to accept contradictory assertions …” (Allen 2009). Still, a human hero without an understanding of the machine’s ideology had no hope of prevailing.

111. A decisive (?) confrontation of the Counterforce and the Firm (one avatar of They) in Gravity’s Rainbow takes the form of a banquet scene in a bad taste that is extraordinary, even for a novel that contains many scenes in bad taste, in which chance triumphs, provisionally, over determinism.

CHAPTER γ HOW TO EXPLAIN NUMBER THEORY (CONGRUENCES)

1. Equations (E1) and (E5) are examples of elliptic curves; there are graphic representations in chapter δ. At the launching in 2007 of the Fondation des Sciences Mathématiques de Paris, a grouping of mathematics research laboratories including the one where I work, one could read that “The most abstract mathematical theories are intimately linked to technologies omnipresent in our daily life. This is true of elliptic curves” (FSMP 2007). This was an allusion to the applications of the theory of approximate solutions to equations like (E1) and (E5) to modern cryptography and data security; some less precise versions of the same claim are quoted at the beginning of chapter 10. The story starts (in one version) with the patenting in 1983 by Ron Rivest, Adi Shamir, and Leonard Adelman of a public-key cryptosystem, the RSA algorithm, based on Fermat’s little theorem. Elliptic curves came into use a few years later: see chapter 14 of Koblitz (2007), written by one of the inventors of elliptic curve cryptography, quoted in chapter 10.

The National Security Agency has approved elliptic curve cryptography in its “Suite B” for “the secure sharing of information among Department of Defense, coalition forces, and first responders” (NSA 2009).

CHAPTER 9 A MATHEMATICAL DREAM AND ITS INTERPRETATION

1. The idea recently resurfaced in the work of Wei Zhang, presently at Columbia. Both of the ideas needed to carry out this program—presented, respectively, by Rallis and Rapoport in 1992—have matured in the intervening years. Zhang has a thorough understanding of all the main questions and has been making rapid progress, in part in collaboration with Rapoport—but I’m getting ahead of my story.

2. I will need to refer to the letter and have reproduced the mathematical argument in its entirety in this note, though I don’t think it will be of much interest to most readers.

   We start with a representation ρ of D* (D the division algebra with invariant 1/n) and use it to construct a local system L(ρ) on the Drinfeld upper half space Ω. We assume ρ is associated to a supercuspidal representation of GL(n) (I’m thinking in terms of mixed characteristic, by the way) and we make the following two hypotheses:

(a) H*(Ω,L(ρ)) is concentrated in degree n–1;

(b) H*(Ω,L(ρ)) is discrete series.

Verifying these hypotheses may turn out to be the hard part. In any case, (b) guarantees that the representation of GL(n) on H*(Ω,L(ρ)) is determined by the values of its character on the elliptic regular set. Now any elliptic regular element g in GL(n) has exactly n fixed points in Ω. The following hypotheses involve the development of technology but the technology shouldn’t depend on the specific situation.

(c) The character of g on H*(Ω,L(ρ)), i.e. the value of the function rep-resenting the distribution character at g, is given by the Lefschetz fixed point formula. (This is actually a series of hypotheses, including something about the summability of the character and something about l-adic cohomology − or whatever cohomology − of rigid analytic spaces not of finite type.)

Assuming (c), one can calculate the character of g as the sum over the fixed points x of Tr(g;L(ρ)_x). Assuming (a), we get

Note that the (−1)n−1 (I lost the − sign somehow) is exactly the sign relating the characters of ρ and the corresponding representation of GL(n). Sorry about the missing parentheses. We need another hypothesis:

(d) At each fixed point x of g, Tr(g;L(ρ)_x) = Tr(ρ(g′)), where g′ is an element of D× corresponding to g.

Assuming all four hypotheses, we then conclude that the representation of GL(n) on H^n-1(Ω,L(ρ)) is isomorphic to n copies of the representation corresponding to ρ; alternatively, is isomorphic to (representation corresponding to ρ)⊗σ where σ is some n-dimensional representation of the Weil group …

Hypothesis (d) would follow from the identification of the action of g on the fiber of the Drinfeld cover of Ω with the action of g′, a sort of reciprocity law at the fixed points, which ought to extend to the Weil group as well. I suggested that the fixed points ought to be obtained as follows: let K = field generated by g (or by its eigenvalues), and let LT be the Lubin-Tate formal group with CM by K. For some embedding of K in D, we can construct

and this should be a special formal O_D-module (the tensor should involve rings of integers). This should also provide an alternative proof of Genestier’s theorem on the irreducibility of the Drinfeld cover (by letting the elliptic torus vary). Genestier has been working on this and he seems to have verified what is needed (it isn’t clear whether the indicated method would also imply geometric irreducibility; Genestier thinks it doesn’t). Of course the association of g′ to g depends on some choices, but it seems that these choices are already implicit in the moduli problem, just as in the classical theory of elliptic curves, the relation to the upper half-plane depends on the choice of a basis for the homology.

I have some vague ideas about (a) and (b) involving Poincare duality and a cohomological interpretation of the Jacquet module, but I realize that I can’t go anywhere until I actually learn the theory. The article of Boutot-Carayol is clear but it isn’t easy. Drinfeld’s original article is not so helpful. But of course if I want to think about intersection theory I will have to read everything. Laumon has made a number of suggestions regarding (c); he doesn’t like the shape of the cohomology (inverse limits vs. direct limits) but he thinks it may be possible to rewrite everything in terms of vanishing cycles on the special fiber, where the cohomology theory really exists. Problems (a) and (b) are closer to things I have thought about in the past, but as Laumon points out, they are both false when the coefficients are trivial (Schneider-Stuhler) so they can’t be easy to prove in the supercuspidal case. On the other hand, any proof would have to find a characterization of the supercuspidals in terms of the representations ρ of D*.

3. They still are. Although I eventually went on to write several articles on the local Langlands correspondence for GL(n) and, with Richard Taylor, wrote a book containing the first proof of this correspondence, it is significant that our work used only the most formal properties of representations of GL(n). The representation theory of the group J appears only in a minor role in a counting argument due to Henniart and quoted in a crucial way, until 2010, in every proof of a local Langlands correspondence, including the one found by Henniart soon after my work with Taylor. Henniart’s proof is certainly the most natural one, but it also uses very little of the detailed representation theory developed by Bushnell, Henniart, and Kutzko. In 2010, Rapoport’s student Peter Scholze found a new proof of the local Langlands correspondence that does not make use of Henniart’s counting argument. The relation between the local Langlands correspondence and representation theory, in the sense of Bushnell, Henniart, and Kutzko, remains an important open question, on which Jared Weinstein has recently made progress.

4. A counting argument of the kind developed by Henniart had first been used in this context by Jerry Tunnell to give a nearly complete proof of the local Langlands correspondence for GL(2); the first complete proof in this case was discovered by Kutzko. Tunnell studied with me at Harvard and will reappear near the end of this story.

5. (Harris 2012).

6. (Max Weber 1922).

7. Most of my time with Rapoport in Wuppertal was spent trying to understand the Gross-Zagier formula in the framework of his book with Zink. On this we made no progress whatsoever, but Rapoport continued to think about the question. A few months later, he began a collaboration with Steve Kudla, who had been developing a much more detailed and flexible framework for understanding the Gross-Zagier formula in a very general setting. This collaboration continues to this day. More recently Rapoport has written a paper with Wei Zhang (see note 1) and Ulrich Terstiege that revisits some of the ideas we discussed in Wuppertal.

8. More precisely, a version of (b) at the level of group characters, especially that the character of the cohomology of coverings of Ω on the Euler characteristic was concentrated on elliptic elements.

9. The letter continues with the following details: “ … since the spectral sequence on page 30 of your article degenerates for supercuspidals. Moreover, by using a (global) argument of Taylor, one obtains a Galois representation of the right dimention in the cohomology of the Shimura variety (but independent of the global realization; more precisely, its restriction to the decomposition group at p depends only on the original supercuspidal representation).”

10. In reality, the local and global approaches were not so different. The global approach was based on the Arthur-Selberg trace formula rather than the Lefschetz trace formula, but it is well known that the two are closely related and in many cases are different ways of saying the same thing. As I had already hinted in my December message to Rapoport, one of the principal sources of my intuition in this unfamiliar subject was Wilfried Schmid’s work, some twenty years earlier, on the representation theory of real Lie groups. During the period described in this essay I was quite conscious that Schmid had made use of global as well as local trace formulas, perfectly analogous to the ones to which I allude here, in his articles of the 1970s. More recently, he had discovered a new geometric approach to the local trace formula in joint work with my Brandeis colleague Kari Vilonen that I cited explicitly in the first paper written on the basis of my dream. More about Schmid shortly.

11. This refers to Henniart’s counting argument, mentioned in note 3.

12. The dream gave rise directly to the publication of (Harris 1997). The local Langlands conjecture was proved in (Harris and Taylor 2001). Not only our own ideas were involved: an observation made by Pascal Boyer in his thesis was crucial, as was the generous support of Vladimir Berkovich, already mentioned previously.

13. (Schmid 1971).

14. I can be even more specific. My five-year plan was to find a p-adic analogue of Schmid’s calculation of the distribution characters of discrete series of real Lie groups as “Lefschetz numbers of certain complexes.” This is in an obvious sense absurd, because Schmid’s calculation is apparently dependent on the use of Hilbert space methods, and there is nothing of the sort available in the setting of the Berkovich cohomology theory. But my letter to Rapoport already invokes “Zuckerman’s algebraic version of Schmid”—meaning Schmid’s PhD thesis as well as the article (Schmid 1971)—and I had strong hopes that I could define an algebraic version of the Berkovich cohomology of the Drinfel’d coverings to which Schmid’s methods could be applied. I still suspect this is possible, and in a more general setting. But I have no idea how to carry this out.

CHAPTER 10 NO APOLOGIES

1. “ … or relativity,” the sentence continues. All Hardy quotations not otherwise attributed are from (Hardy 2012). On page 120, Hardy was quoted defining utility as accentuation of inequalities or destruction of human life. In the original quotation from 1915, he continues, “The theory of prime numbers satisfies no such criteria.”

2. (Wolfe 2009: Singh 2000). Singh adds, perhaps to spite Hardy’s pacifism, that “military and government encryption is also based on prime number theory.” Also: “Hardy’s claim that real mathematics is almost wholly useless has been over-played and, to my mind, is now very dated, given the importance of cryptography and other pieces of algebra and number theory devolving from very pure study.” J. Borwein, in (Pimm and Sinclair 2006, p. 24).

3. Tom Waits, quoted in (Adams 2011). Also: “Everyone’s afraid of Amazon,” said Richard Curtis, a longtime agent who is also an e-book publisher. “If you’re a bookstore, Amazon has been in competition with you for some time. If you’re a publisher, one day you wake up and Amazon is competing with you too. And if you’re an agent, Amazon may be stealing your lunch because it is offering authors the opportunity to publish directly and cut you out” (Streitfeld 2011).

4. This thesis was tested empirically and found to be erroneous: see (Stern 1978).

5. (Wallace 2003, p. 25).

6. This is taken from Jacobi’s letter to Legendre, responding to criticism by the late Fourier. The full quotation is “Il est vrai que M. Fourier avait l’opinion que le but principal des mathématiques était l’utilité publique et l’explication des phénomènes naturels; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est l’honneur de l’esprit humain, et que sous ce titre, une question des nombres vaut autant qu’une question du système du monde.”

7. (Gauss 1808). Gauss spoke on taking over the direction of the Göttingen Astronomical Observatory and astronomy was the Wissenschaft he felt obliged to justify, but he made it clear that he also had in mind “the more beautiful parts” of mathematics, including what is now called number theory—a field largely shaped by Gauss’s Disquisitiones Arithmeticae. See the extended discussion in (Ferreirós 2007), from which much of the translation is taken. I thank Ian Hacking for bringing this reference to my attention.

8. (Pieper 2007, pp. 211–12).

9. “zu einem wichtigen Wettbewerbsfaktor für die Wirtschaft geworden” in the original (MFO undated; (Greuel et al. 2008, p. ix).

10. Neunzert’s list is on p. 111 of (Greuel et al. 2008). SIAM’s careers page is http://www.siam.org/careers/sinews.phpf.

11. (Deloitte 2013). In view of these conclusions, one would expect the coalition government to be throwing money at mathematical research, but there is no sign of this happening.

12. Quotations here and following are from (Roy and Bonami 2010) and from the texts and videos available online at the Web site (Maths à Venir 2009). The quotations from the décideurs on the closing panel are transcribed from the video at http://www maths-a-venir.org/2009/les-math%C3%A9matiques-ressource-strat%C3%A9gique-pour-lavenir, at approximately 1:00:00, 1:06:00, and 1:20:10, respectively.

13. (Zinoman 2006), a review of A First Class Man.

14. Hardy’s aphorism serves as epigraph to the preface of a book by Manfred Schroeder entitled Number Theory in Science and Communication, whose (treacherous) purpose is to show just how useful number theory is after all.

15. (Mehrtens 1990, p. 538). The history is far from exhausted. A few months after he received his Fields Medal, Cédric Villani, already well on his way to becoming the national celebrity he is today, was invited (along with fellow Fields Medalist Ngô Bảo Chau and Gauss Prize winner Yves Meyer) to speak about mathematics before the Office parlementaire d’évaluation des choix scientifiques et technologiques. Asked to explain his comparison between mathematics and “certain aspects of art,” Villani began by remarking that the comparison “seems so natural to mathematicians, because the artistic aspect of our discipline is so obvious, even more than for other sciences”—note that for Villani a science can also be an art. The rest of his answer covered familiar ground (“ce qui fait généralement avancer un mathématicien, c’est le désir de produire quelque chose de beau” [what generally makes a mathematician progress is the desire to produce something beautiful]), but, after claiming that, “sociologically speaking,” mathematicians and artists work in much the same way, he justified his claim by pointing out that “in mathematics there are tendencies and tastes,” concluding that “[t]he world of mathematics like that of art” is “parcouru par des modes et des groupes” [crisscrossed by fashions and groups]. Villani was definitely not recycling the mathematical romanticism, symbolized by Galois, which we encountered in chapter 6. See also (Hasse 1952).

16. For example, (Lockhart 2009), which is subtitled How School Cheats Us out of Our Most Fascinating and Imaginative Art Form, includes this statement on page 34: “Mathematics should be taught as art for art’s sake. [The] mundane ‘useful’ aspects would follow naturally as a trivial by-product. Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful.”

17. (Pater 1873, preface and conclusion).

18. (Ruskin 2009), W. S. Jevons, Methods of Social Reform, London: Macmillan (1883), quoted in (Maas 1999). See also (Mosselmans and Mathijs, 1999).

For the educational benefits of the arts in France, see (Rancière 2011, chapter 8, especially pp. 173–75). The aesthetic theorists Rancière treats in this chapter, which covers a period stretching from Ruskin through the Paris Exposition Universelle of 1900 to the Bauhaus, have in common a vision of art as “the power to order the forms of individual life and those in which the community expresses itself as community in a single spiritual unity” (p. 178). This ethic of art is much more familiar than the model on which Hardy draws and it is hard to imagine its application to mathematics in any way.

19. (Heard 2004). The extended quotation, reproduced on page 238 of Heard’s thesis, is from (Salmon 1883). Regarding Pater, Heard argues that “in Pater’s philosophy the value of a beautiful thing lay entirely in its ability to give pleasure to the individual, rather than in some moral truth that it conveyed, and from which the individual and society would benefit. And in the realisation of this philosophy by the aesthetes, the individuals who gained pleasure were their own elite group.”

20. (Williams 1976, p. 42). Williams’s book is about the use of words in English, but the timing was not so different in other European languages. Jacques Rancière claims that art or Art, as we now understand it, is a notion that dates back at most to the eighteenth century and, specifically, to the publication in 1764 of Johann Joachim Winckelmann’s Geschichte der Kunst des Alterthums [The History of Ancient Art]. With Winckelmann, art is “no longer the competence of makers of paintings, statues, or poems, but as a sensible medium of coexistence of their works … Art [with a capital A] becomes an autonomous reality with the idea of history as a relation between a medium, a form of individual life, and the possibility of individual invention.” (Rancière 2011, pp. 31, 33). Pater (1873, preface), on the other hand, writing a century and a half before Rancière, sees Winckelmann as belonging “in spirit to another age … the last fruit of the Renaissance.” Undoubtedly both are right.

21. This sort of talk was not uncommon in those days: in the 1850s, for example, William Stanley Jevons had entitled an unpublished manuscript On the Science and Art of Music. The explanation of Kronecker’s defense is taken from André Weil’s (1976) review of Eisenstein’s collected works. According to Borel, an oral communication of Kronecker, reported by Paul Du Bois-Reymond, is the only source for the claim about Eisenstein (Borel 1983; Du Bois-Reymond 1910). No less tragic a figure than Abel and Galois, Eisenstein died at age 29 and was largely forgotten, according to Weil, for over a century.

22. Nicomachean Ethics, III.3, VI.3, VI.4, respectively.

23. Marwan Rashed, private communication. The grounds for the claim that a science applies to its specific ontological domain can be found in Aristotle’s Posterior Analytics, section 7 (from which this quote is taken) and section12.

24. (Al-Khwarizmi 1831, pp. 5, 87). Contemporary algebra replaces “thing” by the variable x and rewrites the first sentence as the equation 3 1/2 + 2/5 x = x and the second sentence as the instruction to subtract 2/5 x from both sides. Al-Khwarizmi devotes several long chapters to applications of his new methods to financial questions.

25. (Høyrup 1988). Høyrup writes that the De numeris datis of the medieval European mathematician Jordanus de Nemore “transforms a mathematically dubious ingenium”—algebra—“into a genuine piece of mathematical theory.”

Just for completeness, here is how the late antique pseudo-Aristotelian text Problemata mechanica distinguishes tricks from art (or from the other “parts” of art):

When we have to produce an effect contrary to nature, we are at a loss, because of the difficulty, and require skill [techne, art]. Therefore we call that part of techne which assists such difficulties, a device [mekhane].

26. The interpretation of medieval Islamic Aristotelianism is based on conversations with Marwan Rashed.

27. The first quotation is from (Fried and Unguru 2001, p. 26). The al-Khayyām quotations are from Al-Khayyam’s Treatise on Algebra (1070) in (Rashed and Vahabzadeh 2000, pp. 111–13). In the translations I have twice replaced “mathematical” by “scientific” to represent the Arabic word att’alimiya in order to emphasize the derivation from ‘ilm (science, epistêmê). This is the only appearance of the term “scientific art” (sina‘a ‘ilmiya) in Khayyām’s extant work. The reader should not assume, however, that the boundaries of the domains identified as sina‘a and ‘ilm were less fluid than those of art and science today.

One century after al-Khayyām, al-Jazari combined ‘ilm, sina‘a, and ḥiyal in the title of his best-known work: Al-Jami’ bayn al-‘ilm wa ‘l-‘amal al-nafi’ fi sina‘at al-ḥiyal, translated A Compendium on the Theory [Science?] and Useful Practice of the Mechanical Arts. Al-Jazari would now be called a mechanical engineer.

The dispute about the nature of algebra was still lively in nineteenth century, when William Rowan Hamilton contrasted three schools of thought, “the Practical, the Philo-logical, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation.” “Instrument” comes closest to art, “Contemplation” to science. See (Fisch 1999)

28. (Cardano 1968, Chapter XXXVII, p. 220). Cardano was specifically alluding to his use of the square roots of the negative number −15, what we would now call imaginary numbers, but to which he could assign no meaning.

29. (Cottingham 1976, pp 47–48). I thank Olivia Chevalier-Chandeigne for this reference.

30. Among philosophers, Hegel, Heidegger, and Collingwood, among others, have drawn very different conclusions from the fact that the Greek word tekhne refers to arts as well as crafts. More recently, we have also to contend with the fashionable notion of technoscience.

31. (Max Weber 1922, p. 7).

32. (Sinclair and Pimm 2010).

33. (Lavin and Lavin 2012).

34. Quotations are from a pamphlet entitled “Our Debt to Mathematics,” prepared by Veblen and Julian Coolidge because they were “pessimistic … about the chances of getting a substantial endowment grant from the Rockefeller Foundation” and therefore “focused on potential industrial patrons.” The pamphlet also “explained how mathematics develops (‘by the devoted labors of highly trained scientists who pursue their studies with indefatigable patience, unnoticed by the world, undeterred by the supercilious pity of those who are unable to appreciate their work’) [and] asserted that abstract mathematical principles often evolve into practical results” (Feffer 1998).

35. The quotations, in order: (Max Weber 1922); quoted in (Mulkay 1976, p. 457), but Mulkay adds on p. 459 that “in general it appears that there is a pronounced bias in Britain in favour of fields with high scientific status but low economic relevance.” (Galison et al. 1992, pp. 54–55): Webster worried that “physicists would occupy their time producing knowledge for litigation rather than for innovation); (Shapin 2008, p. 19): from a speech by Robert C. Dynes, chancellor of UCSD at the time (and later president of the UC system), in his State of the Campus address, March 22, 2001; from Whyte’s The Organization Man, 226–27, quoted on page 176 of (Shapin 2008; Conference des presidents d’université 2011).

36. (Shapin 2008, pp. 125, 104).

37. (Gelfand 2009; SIAM 2012). Manin stressed that his remarks apply only to mathematicians; the SIAM report is concerned with mathematical research in the corporate setting.

38. (Mehrtens 1990, p. 57).

39. Blair’s government listed the thirteen industies in 1998: film, TV and radio, publishing, music, performing arts, arts and antiques, crafts, video and computer games, architecture, design, fashion, software and computer services, and advertising. See (Ross 2009, pp. 28, 25).

40. Hardy’s function theory is applied throughout mathematics—in particular, to differential equations like the Black-Scholes equation—for example, in (Arendt and De Pagter 2002). The art that contributes to that 7.3% of the British economy naturally differs from the kind of art Hardy had in mind, whose economy is examined in (Muntadas 2011) and (Stallabrass 2006), especially chapter 4 of the latter; the point is rather that pure mathematics and what we might call “pure art” engender similar Golden Geese.

41. (Smith 1759).

42. Or so claimed the minister. Those who spoke up at the roundtable I attended were either business executives or representatives of various levels of government; the only researchers visible were those who had created either partnerships with industry or their own startups.

43. The scope of the Loan, as initially planned, was not limited to Madame Pécresse’s ministry (10 billion euros for higher education, 3.5 billion for “valorisation” of public research, of a total of 35 billion); and it’s hard to tell to what extent the Loan’s initial priorities survived the realities of the economic crisis and the transition to Hollande’s socialist government. Some information is available at the Web site (progfrance 2009–2013). The contribution to the operating budget of my university, at least, seems to be much less than anticipated.

44. My handwriting here is blurry and I can hardly believe anyone really said this, but Google finds the expression “quête de rente” in 101,000 Web pages, so who knows?

45. No enlightenment on this score seemed likely at the other three roundtables, on health/well-being/food (biotechnology), l’urgence environmentale (ecotechnology, especially energy and transport), or information/communication (nanotechnology and telecommunications).

46. This is an old idea. John Dewey already wrote in 1902 that “Teaching … is something of a protected industry; it is sheltered…. There is always the danger of a teacher’s losing something of the virility that comes from having to face and wrestle with economic and political problems on equal terms with competitors.” Dewey adds that “Specialization … leads the individual … into bypaths still further off from the highway where men, struggling together, develop strength.” Reprinted in (Dewey 1976).

47. Also, for the record: (3) There is concern about the condition of the researcher, especially in the creation of spinoffs; and how to manage the return of the researcher to the university if the spinoff is sold; (4) Universities have to agree to finance their offices of transfer/innovation on their overhead budgets; “then they will be viable”; (5) Mao’s “hundred flowers” image is still alive among French décideurs.

48. Respectively, 20%, 15%, and 15% (AFP 2006).

49. (Koblitz 2007). In a private communication, Koblitz clarified his position: “My own personal feeling is that it is unseemly for academic mathematicians to go lusting after money. We receive decent salaries as full-time professors and have near-perfect job security. That’s enough, it seems to me.”

50. (Biagioli 2003, p. 255); European Commission 2008; European Commission 2009, Section 9, “Communication on University Business Dialogue”). The three keywords in 2008 were “governance, commercialization, entrepreneurial mindset.” Commitment to these values crosses party lines: Geneviève Fioraso, the socialist party Minister of Higher Education and Research, called for instilling the entrepreneurial mindset as early as preschool—and for décloisonnement of research and business (Talmon 2014).

51. (SIAM 2012). Neunzert is hearing those same voices: “Auf einer breiten und soliden Basis allgemeiner Mathematik muss, zumindest für die Mehrheit der Studenten, die später ihr Auskommen in eben dieser Praxis finden müssen, ein vertieftes Wissen in mathematischer Modellierung und in numerischen Methoden gesetzt werden. Das geht nicht ohne Kenntnisse in reiner Mathematik, aber es geht auch nicht ohne Kenntnisse in angewandter Mathematik, in Modellierung und Numerik. Außerdem sollte in den jungen Mathematikern das Interesse für die Anwendungswissenschaften und die Bereitschaft der Zu-sammenarbeit mit deren Vertretern geweckt werden” (Greuel et al. 2008, p. 119 [emphasis added]).

And in France, the socialist government has drafted a bill to include “transfer of results [of scientific research] to the socio-economic world” among the official “missions” of higher education and research, alongside the six “missions” defined in the law of 2007 (listed in the Code de l’éducation—Article L123-3, at http://www.legifrance.gouv fr). See http://www.enseignementsup-recherche.gouv.fr/cid70881/une-nouvelle-ambition-pour-la-recherche-mesure-15.html.

52. It is quoted in The Beginning of the End (2004) by Peter Hershey, p. 109.

53. Neunzert in (Greuel et al 2008, p. 118; ADEC 2011).

54. It is understood—and understandable—that social scientists would want to avoid “going native” at all costs and are subject to a professional requirement not to accept the beliefs of their informants uncritically. But reflexivity works both ways: mathematicians are (mostly) no fools and know what it means to act as informants. There’s no simple way out of this conundrum.

The Deligne, Sally, and MacPherson quotations are from (Simons Foundation, undated). In (Cook 2009, p. 156). Deligne said much the same thing: “It was a wonderful surprise to learn one could at the same time play and earn one’s living.” And one finds a similar statement in the biography that accompanied the announcement, posted by the Norwegian Academy of Science and Letters, that Deligne had been awarded the 2013 Abel Prize.

55. (Gopnik 2009).

56. Philebus, 66a–c. First on the list is “measure, moderation, fitness.” We have seen that Aristotle had little patience with those who identified pleasure with “the good,” but he acknowledged that “each activity has a proper pleasure” (oikeia hedone; Nicomachean Ethics X, 5)—for example, the “proper pleasure” of tragedy, which “is the pleasure that comes from pity and fear by imitation,” (Poetics, XIV). In his Metaphysics, on the other hand, we read that ‘The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful” (Metaphysics XIII, 3.107b).

57. Quotations from (Cook 2009, pp. 68, 160, 188). This is probably the place to mention that Belgian mathematicians can aspire to honors not available elsewhere: Deligne is a Vicomte and Daubechies is a Baroness.

58. Transcribed from BBC Horizon/PBS Nova video entitled The Pleasure of Finding Things Out, at around 22:30–23:30. Some of this was published in a book by the same name, but the word relaxed was omitted.

59. (Hume 1739, Part III, Book II, Section X).

60. (Greuel et al. 2008, p. 115; SIAM 2012, p. 37). Do these quotations contradict Hardy’s claim that only the “dull and elementary parts of applied mathematics” are useful?

61. (Edgeworth 1881, p. 72; Cohen, undated). The second postulate is perhaps more familiar: “that making social utility as great as possible is the ultimate moral imperative.”

62. (European Commission 2009, Section 9). Isabelle Bruno, a specialist in EU research and innovation policy, wrote an 832-page thesis on “benchmarking” in EU policymaking; the word pleasure [plaisir] appears only in the acknowledgments and inconspicuously in a footnote, whereas the word happiness [bonheur] is only used in connection with her historical review of utilitarianism.

Compare these thoughts of Harvard President Drew Gilpin Faust (Faust 2009), reacting to the economic crisis that started in 2008: “As the world indulged in a bubble of false prosperity and excessive materialism, should universities—in their research, teaching and writing—have made greater efforts to expose the patterns of risk and denial? Should universities have presented a firmer counterweight to economic irresponsibility? Have universities become captive to the immediate and worldly purposes they serve? Has the market model become the fundamental and defining identity of higher education?”

63. (Burke 1757; Hume 1739, part II, section V; Kant 1790).

64. (O’Doherty et al. 2003; Ishizu and Zeki, 2011). The author of a recent survey (Chatterjee 2014) writes that he is “not aware of any studies that have examined the neuroaesthetics of math.” Here and in what follows, the mOFC stands in for any “ensemble of neural subsystems” that might some day be found responsible for the human aesthetic response.

65. It is! At least that’s what Zeki and his collaborators—including no less an authority than Fields Medalist Sir Michael Atiyah—announced in February 2014, after this chapter was written. More precisely, “the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely, field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.” See (Zeki et al. 2014).

I have to say I’m a little skeptical of the research protocol, which consisted in asking subjects to rate equations as beautiful, neutral, or ugly (before and after MRI scans), given that Euler’s identity 1+ e^iπ = 0 won the beauty contest, while an formula of Ramanujan—much less elaborate than the Frenkel et al. Love Equation—came in dead last.

66. Quotations from An Essay in Aesthetics, in (Fry 1920).

67. This chapter suggests that Hardy’s Apology is aligned with Bloomsbury values; but these can be read in more than one way. Regarding the relation of Bloomsbury ethics to utilitarianism, Craufurd Goodwin writes “The Bloomsburys were convinced that the imaginative life could not be understood simply as responses to utilitarian stimuli, coordinated through the market mechanism…. On the supply side of the labour market in the imaginative life, artists, writers, and scientists did not typically perceive their activities as onerous and a source of disutility that had to be compensated by utility derived from expenditures of income. Quite the contrary, they were driven to action by an irrepressible urge to communicate … ” (Goodwin 2001). On the other hand, according to Raymond Williams, Bloomsbury “was against cant, superstition, hypocrisy, pretension and public show. It was also against ignorance, poverty, sexual and racial discrimination, militarism and imperialism. But … [w]hat it appealed to, against all these evils, was not any alternative idea of a whole society. Instead it appealed to the supreme value of the civilized individual … The profoundly representative character of this perspective and commitment … is today the central definition of bourgeois ideology” (Williams 1980).

68. (Borel 1989).

69. (Grothendieck 1988, p. 202, text and footnote), my (rough) translation.

70. (Langlands 2013). The “division of elliptic integrals” to which Langlands refers is an early stage in the study of cubic equations (see chapters γ and δ) and thus represents part of the historical background to the Birch-Swinnerton-Dyer conjecture.

71. Psychologically, this is very different from Aristotle’s list: order, symmetry, and limitation, “the greatest forms of the beautiful” (Metaphysics, XIII, 3.107b); or, in another vein, the triplet “coldness, tedium and irrelevance” of “cultural perceptions of mathematics” cited by Pimm and Sinclair (The Many and the Few). Michael Atiyah, in his video interview at [Simons Foundation undated], opts for “elegance, simplicity, structure, and form … all sorts of things,” which sounds more aristotelian than Hardy’s list. Alain Badiou suggested a somewhat different three-term list—economy, rational totalization, and fruitfulness—in a public lecture in Paris on June 16, 2011. He did not cite Hardy.

72. Mathematical tricks, as we characterized them in chapter 8, qualify as beautiful on all three of Hardy’s counts: economy is represented by what we called a “short cut” while unexpectedness and inevitability come together in a trick to create an Aha! experience. Nevertheless, Hardy himself used the word trick infrequently and pragmatically rather than aesthetically. Google Scholar returns 457 articles authored by G. H. Hardy—there are many repetitions—and only two of them include the word trick. In “Prolegomena to a chapter on inequalities” [J. London Math. Soc. (1929): 61–78], he stressed the need for those working in function theory to master both the “main results and the tricks of the trade”; in “Note on the theory of series (XIX)” with Littlewood [Quarterly J. Math. (1935) 304–15] it is said that “some special trick” is needed to prove a certain statement.

73. Much of the following discussion inevitably overlaps with the exhaustive work on mathematical aesthetics by Nathalie Sinclair and her collaborators. See, in particular, (Sinclair 2011).

74. (Fry 1956, p. 8).

75. For example, see (Fry 1920, p. 54).

76. This reading of the Apology is from (Kanigel 1991). The quotation is from (Moore 1903, section 113). In his biography of Paul Dirac, Graham Farmelo suggests that “Moore’s common-sense approach to beauty probably influenced his scientific colleagues at Trinity, including Rutherford and … G. H. Hardy” as well as Dirac: (Farmelo 2010, p. 74).

77. All Bell quotations from (Clive Bell 1914).

CHAPTER δ HOW TO EXPLAIN NUMBER THEORY (ORDER AND RANDOMNESS)

1. The actual condition is that the equation define a curve of genus 0. Equations in two variables of degree higher than 2 can define curves of genus 0, but this is the exception rather than the rule. Wiles’s description of the Birch-Swinnerton-Dyer conjecture on the Clay Millenium Problem Web site (Wiles 2000) gives (Hilbert and Hurwitz 1890) as the source for the complete characterization of curves of genus 0 with infinitely many rational points.

2. Tunnell’s insight was to reinterpret the congruent number problem in terms of the Birch-Swinnerton-Dyer conjecture and to use the latter to provide a criterion that can easily be tested (but remains, for the moment, purely conjectural).

For this and for much more about elliptic curves and the BSD conjecture, at a level of mathematical sophistication that may not be appropriate for all dinner parties, see Elliptic Tales (Ash and Gross 2011).

3. Actually, it plots the scaled difference (p − S(p))/, so that the result always fits between −2 and 2. The histogram was produced by William Stein and was published in (Mazur 2008a).

4. Quotations here and following are from (Gray 2006).

5. P. A. The Bhagavad-Gita is unambiguous:

When your intelligence has passed out of the dense forest of delusion, you shall become indifferent to all that has been heard and all that is to be heard. (P.A.’s emphasis)

N. T. Is that what the Gita says? In that case, the author should have been more careful and not have trusted Weil blindly …

P. A. Weil is using the Gita to say very clearly that the indifference applies to all knowledge, past or future.

N. T. But a few lines later he says the opposite:

Heureusement pour les chercheurs, à mesure que les brouillards se dissipent, sur un point, c’est pour se reformer sur un autre. [Fortunately for researchers, insofar as the fog dissipates around one point, it is only to reappear around another.]

(Translation of verse 52, chapter 2 of the Bhagavad-Gita from a lecture of Barry Mazur in February 2014, available on his home page at http://www math.harvard.edu/~mazur/.)

6. (Plassard 1992, p. 35; Rancière 2011, p. 207).