Category Archives: charisma

Time to move on

wainua               Figure 6.1 (Clairaut's diagram)

Snail image:  Creative Commons licence courtesy of Te Papa; Clairaut’s love formula from Chapter 6 of MWA

My tireless editor Vickie Kearn at Princeton University Press has brought me the welcome news that Mathematics without Apologies will be coming out in a paperback edition next spring.   I started this blog for two reasons, and one of them — to clarify my intentions in writing the book — will vanish when I add two or three pages to the preface of the new edition.   The new pages — I have already written them — will devote one paragraph or so to each of four topics, provisionally under the headings charismamemoirsutility, and ethics; each paragraph will address some of the points raised by comments on this blog as well as in some of the more negative reviews.

My other reason  for starting this blog was to find some outlet for the wealth of material that I was not able to incorporate in the book.  Most of this material has remained untapped while I composed comments on current events or new findings, and I was idly wondering when I would get around to sifting through the 7 GB  or so that is gathering nanodust on my computer’s hard drive.  My Eureka! moment came when I realized that I had already devoted a considerable amount of my free time to writing the book during the better part of three years.  Perhaps I didn’t really want to return to the old material?  With the new preface, I can finally declare the book finished and move on to something else.

Will it be another book, maybe one that will win me the mythical seven figure advance?  Or will there be another blog, or the same one under another name?   That’s for the future to decide.  Meanwhile, this one will remain visible, but with no new entries.

My thanks to the regular readers and occasional visitors who helped keep the blog from slipping into solipsism.  And my special thanks to authors of comments who, by disagreeing, often sharply, with opinions expressed here, demonstrated that the meaning of mathematics is still a matter of controversy.

This was supposed to be the last entry, but I’m now thinking I should include part of the new preface material — or all of it, if PUP allows it.  Meanwhile, in order not to let anything go to waste, here is the post on which I was working when I realized that this blog had reached the end of its natural life…

I Cunfirenti

This was originally going to be an appendix to the playlist near the end of Chapter 8:  an exploration of the attitude to mathematics in the genre of organized crime ballads.  The deeper meaning of Rick Ross’s 2009 single Mafia Music was exposed even before it was released,  but I was unable to find an interpretation of the unexpected appearance of mathematics in the middle of this rap à clef:

I thought about my future and the loops I could pin.
Walked out on a gig and I turned to da streets,
Kept my name low key, I ain’t heard from in weeks.
I came up with a strategy to come up mathematically,
I did it for da city but now everybody mad at me.

Apart from Rick Ross, Gödel is the only person Google finds who can “come up mathematically.”  My guess is that Ross’s strategy (unlike Gödel’s) involves money.  But Ross is not really a gangster, and Mafia Music is not really a mafia song at all; in fact, by naming names the song breaks what I’m told is the most fundamental of all the rules of the Italian Malavita, namely the rule of omertà, the iron law of silence.

Now it struck me when I saw this that the mathematical profession has its own version of omertà, probably not very different from other forms of academic rules of silence, having to do with forms of behavior that straddle the line that divides the unpleasant from the unethical.  The behavior protected by mathematical omertà differs from other varieties in that it tends to inspire less literary commentary.  Instead it consists in scandalous rumors whispered in corridors when they are not being shouted across barroom tables, but that must under no circumstances be mentioned in public.  (There was a scurrilous exception in a well known literary magazine a few years ago, but I will not dignify it with a link.)

I am particularly sensitive to this rule just now, because in the past few weeks I was shocked to learn of abuse of power by several colleagues I would not have believed capable of such behavior (and by a few others I can easily believe capable of anything).  Whether being the repository of such confidences is one of the perks of my charisma, or whether it’s the abusers who feel newly entitled as a result of their own charisma, the mildest punishment I could expect if I chose to betray the dark secrets of the mathematical profession is not to be privy to such secrets in the future.  Breach of Mafia omertà is treated more harshly than that.  Many of the songs on the delightful album La Musica della Mafia are devoted to the many kinds of punishment the gangster ethic  —

Laws that don’t forgive those/Who break their silence

reserves for traitors — cunfirenti, in Calabrian dialect.  For example, the song entitled I cunfirenti promises that they will find “their final resting place in concrete walls” (‘Mpastati ccu cimentu e poi murati).

The album’s title is imprecise; it’s not a collection of songs of the Sicilian mafia but rather the ballads of their Calabrian declension, the ‘Ndrangheta, who deserve to be better known, and not only for their songs:

Its success at drug smuggling catapulted the ‘Ndrangheta past its more storied Sicilian rival, the Cosa Nostra, in both wealth and power. Italian authorities now consider the ‘Ndrangheta to be Europe’s single biggest importer of cocaine.

What I find most charming about this collection is the contrast between the lively rhythms of many of the songs and the uniformly grim, often bloody, content of the lyrics.  For example:

Malavita, malavita
Appartegnu all’Onorata
Puru si c’impizzu a vita
Eu nun fazzu na sgarrata

Which means

Malavita, malavita!
I am one of the honorable society.
And even if it costs me my life,
I will never surrender.

If you’re looking for mathematical content you have to skip to the last verse:

Ed eo chi tingu sangu ´nta li vini
Su prontu d’affruntari mille infami
A chista genti ci rispunnimu
Pidi sunu pronti centu lami

Which means

And I who have blood flowing through my veins
Am ready to face 1000 traitors
As they know all too well
That 100 sharpened knives are ready for them.


The theologico-teleological apology


Comments on David Roberts’s Google+ page, May 30, 2016

David Roberts’s announcement a few months ago of his then-forthcoming review in the Gazette of the Australian Mathematical Society sounded like a warning shot, especially since I occasionally had the impression that he was trying to bait me on this blog.   The review is now out, and as far as I’m concerned it’s perfectly fair; the reviewer was even thoughtful enough to include what trade jargon calls a pull quote in the last paragraph, and you can expect to see it soon enough on the reviews page.

The review also provides (yet another!) opportunity to clear up some misconceptions, notably about charisma, as used in chapter 2.  I chose the word deliberately as a provocation, but it provokes different readers in different directions, and that’s beyond the author’s control.  The ambiguity of the word is already in Weber, it seems to me:  the charismatic leader is separated from the masses by an aura, while those possessed of routinized charisma are part of the mass of functionaries that make the community… function.  I tried to make it clear that chapter 2 was the (fictionalized) story of my acquisition of routinized charisma, in other words, of being accepted as a legitimate functioning member of the community.  So when Roberts writes

The ‘relaxed field’ that Harris discusses … is perhaps not the same for us as for those with charisma.

he is making a distinction that is quite alien to the spirit of the book; indeed, Roberts is displaying a paradigmatic form of charisma by publishing a book review in the Gazette of his learned society, and more consistently in his contributions to MathOverflow and other social media.

By the way, saying that chapter 2 was fictionalized is not the same as saying that it was made up; what I meant was, first, that it was written in acknowledgment of the narrative conventions of (a certain kind of) fiction; and that it didn’t matter for my purposes whether or not the events recounted were strictly true, as long as they were ideal-typical.

Roberts reads MWA as calling charisma a form of prestige whose acquisition is one of the motivations for doing mathematics, but this was not my intention.  No doubt mathematicians find it gratifying when our work is recognized, and much of the mass of chapter 2 is devoted to prizes and other forms of recognition, large and small, institutionalized or informal; but only André Weil is represented as actually craving prestige, and the context makes him recognizably an outlier.  An obsession with ordered lists and rosters of Giants and Supergiants is attributed to the community, rather than to individual mathematicians who hunger for recognition.  This obsession is such a visible feature of contemporary mathematics that it deserves explanation, and chapter 2 suggests an explanation that is so counter-intuitive that it seems not to have been noticed by anyone (on pp. 18-19):

The bearer of mathematical charisma… contributes to producing the objectification—the reality—of the discipline, in the process producing or imposing the objectification of his or her own position within the discipline.…The symbolic infrastructure of mathematical charisma is… the “objectification” of mathematics:  the common object to which researchers refer… In other words, it’s not just a theory’s contents that are defined by a social understanding:  so are the value judgments that organize these contents.

This brings me to Urs Schreiber’s instructive misreading of MWA‘s intentions, quoted above.  Most likely it’s a misreading based on no reading at all of MWA, because he seems not to be aware that the words “meaning” and “reality” that he cites as the aims of a self-aware mathematician are examined repeatedly in MWA, especially in chapters 2, 3, and 7.

Chapter 3 refers to three main forms of “apologies” for mathematics, labelled in keeping with the western philosophical tradition as “good, true, and beautiful.”  The word “tradition” is fundamental.  The one thing I find unforgivable when mathematicians make general comments about the values and aims of mathematics is the suggestion that they are saying something original.  Talk of values and aims is necessarily embedded in a philosophical and literary and social tradition; a failure to acknowledge this is merely a sign of ignorance, not of intellectual independence.  THAT is why MWA has nearly 70 pages of endnotes and more than 20 pages of references:  in order to record the author’s efforts to purge himself of the notion that his ideas are his own — and, no doubt, to encourage others to take the same path.

MWA cites those three main forms of “apologies” because they are the ones actually on offer; writing about them is my way of grappling with “reality.”  I attended the meetings described in chapter 10 not out of masochism (the champagne receptions were not bad at all) but because they were really happening, they were organized and attended by real decision-makers (“Powerful Beings”) whose decisions have real consequences for the future of the discipline; and the representations of mathematics (and of scientific research more generally) presented at those meetings were the real attempt of the community to procure the external goods necessary for its survival in its present form.  (I procured no pleasure, not even Schadenfreude, when I read the documents listed in the bibliography under “European commission”; but they are terribly important for anyone who is concerned about the future of mathematics.)

Anyway, Schreiber’s speculations cited above are irrelevant to MWA, but they are instructive nevertheless, because they exemplify what might be considered a fourth kind of apology that might be called Theologico-teleological.  One doesn’t need to believe in a supreme being to be a seeker of “answers to deep questions” or “meaning” or “reality,” but one has to believe in something.  I don’t know how to attach consistent meanings to the terms in quotation marks in the last sentence, and I don’t think Schreiber does either.  But I do know one name that has been given to the process by which meanings accumulate around terms like that:  tradition-based practice, specifically in the writings of Alasdair MacIntyre.  Two separate texts, both cited in the bibliography, led me to MacIntyre:  David Corfield’s article Narrative and the Rationality of Mathematics Practice and Robert Bellah’s book Religion in Human Evolution, which I read at the suggestion of Yang Xiao.  Both texts propose ethical readings of important human social phenomena, and this is important to me, because I have found that most arguments about the nature of mathematics, including Schreiber’s comments, turn out to be ethical arguments in disguise.

(Like “beauty,” the “answers to deep questions” or “meaning” or “reality” that Schreiber appears to be seeking can also be interpreted as euphemisms for “pleasure,” but I will leave this for another occasion.)

Leçon inaugurale


Few of the 420 seats of the magnificent Amphithéâtre Marguerite de Navarre were empty when Claire Voisin performed the ceremony marking her entry to the “most prestigious institution of the French university” system, the 486-year old Collège de France.  Jean-Pierre Serre, whose work was amply cited by Voisin in her hour-long account of the branches of complex geometry — analytic, Kähler, and algebraic — sat in the middle of the front row, together with past and present Professors of the Collège.  The talk was systematic, organized, and comprehensive, like Voisin’s introductory two-part book on Hodge Theory and Complex Algebraic Geometry:  not a dinner party explanation by any means, it was “a rather complete tour of the subject from the beginning to the present” in the words of the review in the Bulletin of the AMS by Herbert Clemens, who noted “the break-neck pace of Voisin’s clear, complete, but ‘take no prisoners’ exposition.” Serre, who turns 90 (!) this September, was alert as always; colleagues to my left and right in the tenth row were happy with the pace and the content but speculated that the distinguished medieval historians and classicists in Serre’s row were already dozing off by the time Voisin defined complex structures in her second slide (my apologies for the blurry photos)

nombres complexes

and in any case long before she concluded her lecture with an allusion to her work on the generalized Hodge conjecture.

lastslide - 1

Readers of MWA will not be surprised to learn that the lecture was followed by a sumptuous champagne reception.  The new Professor has so many friends and admirers that the petits fours ran out well ahead of schedule, but there was (just barely) enough champagne for the jubilant crowd.  For Paris mathematics, the inauguration was undoubtedly the social event of this (very rainy) season.

I was wondering who would replace Don Zagier when he vacated his Chair in Number Theory a few years ago, so — as always in these situations — I consulted the best-informed of my colleagues.  He told me that no one had yet been named, but that it had been decided to create a Chair in Algebraic Geometry.  It was obvious for whom such a Chair was intended, but the formal announcement took some time to appear, and I was surprised to see that the press took no notice of what in the English-speaking world would certainly be considered an event of historic magnitude:  the naming of the first woman professor of mathematics to the most prestigious position in the French academy.  Journalists had no doubt chosen to heed Voisin’s own preemptive and scathing critique of this approach to “diversity”:

L’idée qu’augmenter le nombre de femmes à l’Académie des sciences aurait un impact sur la désaffection des femmes pour certains domaines des sciences est tout simplement grotesque. D’abord parce que l’Académie des sciences n’intéresse personne et ensuite parce que le choix de faire une carrière scientifique ne repose que sur les aspirations intellectuelles et le talent, et non sur des considérations mondaines. Personnellement, je supporte de moins en moins d’être passée en quelques années du statut de mathématicienne à celui de femme-mathématicienne, et de subir l’oppression grandissante de l’obsession paritaire, transportée à grand bruit par les médias.

Je souhaite que mon statut de femme, qui me plaît beaucoup, reste du domaine privé, et que l’évaluation et la reconnaissance de mon travail ne se trouvent pas polluées par la prise en compte de ce statut (ce qui est insultant en général : être une femme n’est pas un handicap !).

Je souhaite aussi ne jamais devenir Madame Quota, et surtout que cela n’arrive pas à mes filles. A supposer qu’on ne puisse pas parler d’autre chose que de la fameuse parité, serait-il possible de mentionner que les quotas sont à différents égards (dont certains non mentionnés ci-dessus) une menace pour les femmes scientifiques ?

Voisin’s point of view is rarely expressed so forcefully, but it is widely shared in France.  When I arrived from the United States in the early 1990s, it was disorienting, to say the least, to hear male mathematicians routinely making comments about the physical appearance of their female colleagues, behind their backs; but I also heard female mathematicians commenting (appreciatively or not) about the looks of their male colleagues.  There’s much more to be said, but I learned quickly enough that if I opened my mouth on this (or any other) subject I would be accused of being an “anglo-saxon” — and it was futile to brandish my Beowulf and point out that this status would be denied me in any actual English-speaking country.

After an hour of catching up with long-lost acquaintances I managed to push my way through the crowd of well-wishers to congratulate the newly-named Professor.  I hope she will not hold it against me that I have briefly extracted her “status of woman” from the private sphere.  She told me that she has a copy of MWA but that she hadn’t yet found the time to read it.  I advised her to skip ahead to Part II, and maybe chapter 4, which is where she would find the best jokes, and by all means avoid reading the boring chapter 3!  That’s the advice I give everyone, but after taking a look yesterday at his comment on David Roberts’s Google+ blog I feel I ought to make an exception for Urs Schreiber, who is specifically advised to reread the discussion of tradition-based practices on pp. 74-77 and to decide whether that discussion didn’t anticipate his objections.  I will have more to say on this topic when I comment on Roberts’s review for the Australian Mathematical Society.







“Je m’en f…,” he wrote

Pierre Colmez has pointed out a few passages published in the Serre-Tate Correspondence where Serre and Tate express their opinions about the correct way to identify that conjecture about which so much ink has been spilled.  The date is October 21, 1995, the papers of Wiles and Taylor-Wiles have been published, and Tate is confiding that

I am tired of the Sh-Ta-We question. But it doesn’t go away.

Joe Silverman had just written to him about the correct nomenclature for the Japanese translation of his book (presumably one of his books about elliptic curves):

Springer-Tokyo wondered if we still wanted to call it the  “Taniyama-Weil Conjecture,” since they say that everyone in Japan now calls it the “Shimura-Taniyama Conjecture.” I certainly agree that Shimura’s name should be added to conjecture (Serge’s file is quite convincing), but I don’t feel strongly about whether Weil’s name should be omitted. I hope you won’t mind that I told Ina that for the Japanese edition it would be all right to call it « Shimura-Taniyama », although I suggested that they add a phrase « now proven in large part by Andrew Wiles ».

Tate is finished for the moment with Sh-Ta-We but wonders whether the new theorem should be named after Wiles or Taylor-Wiles, at which point he wrote the censored passage above.

Before we move on to the substance of the question, let’s speculate as to the reason for Tate’s ellipsis.  We know that Tate was born in Minneapolis, and we heard a lot about Minnesota nice earlier in this election cycle; could it  just be the way of Minnesotans to talk in ellipses?  Or maybe the passage was censored by the Société Mathématique de France, publisher of the Serre-Tate Correspondence.

The significance of Tate’s expletival indifference is easier to ascertain.  If anyone occupies the pinnacle of charisma in contemporary number theory, it’s John Tate; but here he is choosing not to exercise his charisma to manipulate public opinion in favor of one historical label or another.  There’s no doubt in my mind that his survey article on elliptic curves was largely responsible for popularizing the conjecture and associating it with Weil’s name (“Weil [82] has the following precise conjecture…”) — but there’s no reason to think this was his intention; he had called it Weil’s-Shimura’s conjecture in a letter to Serre dated August 4, 1965 (on p. 262 of the first volume of the Serre-Tate Correspondence — it’s August and Tate writes “Bonnes Vacances” at the end of his letter).

Colmez adds a footnote to the 1995 letter.  I quote verbatim:

Il me semble qu’une bonne part de la dispute vient de ce que l’on s’obstine à considérer deux conjectures bien distinctes comme une et une seule conjecture. Si E est une courbe elliptique définie sur Q, considérons les deux énoncés suivants :

two statements

La théorie d’Eichler-Shimura [Ei 54, Sh 58], complétée par des travaux d’Igusa [Ig 59] et de Carayol [Cara 86], permet de prouver que le second énoncé implique le premier. Réciproquement, le second est une conséquence de la conjonction du premier, de la théorie d’Eichler-Shimura, et de la conjecture de Tate pour les courbes elliptiques sur Q. Comme la véracité de la conjecture de Tate n’est pas vraiment une trivialité, cela donne une indication de la différence entre les deux énoncés. Une différence encore plus nette apparaît quand on essaie de généraliser les deux énoncés à un motif. Le premier se généralise sans problème en : la fonction L d’un motif est une fonction L automorphe – une incarnation de la correspondance de Langlands globale. Généraliser le second énoncé est plus problématique, et il n’est pas clair que ce soit vraiment possible.

Colmez’s mathematical gloss is impeccable, but he provides no evidence that anyone actually confused the two statements.  Shimura certainly did not; when I was in Princeton the year after Faltings proved the Tate conjecture for abelian varieties, he (unwisely) asked me whether I thought the proof was correct, instead of consulting any of the numerous experts on hand.

Only a professional historian of mathematics can determine the accuracy of Colmez’s explanation for the disagreement.  Serre, in any case, was never convinced by Lang’s file.  In his message to Tate on October 22, 1995, he wrote

La contribution de Weil (rôle des constantes d’équations fonctionnelles + conducteur) me parait bien supérieure à celle de Shimura (qui se réduit à des conversations privées plus ou moins discutables).

Serre clarified his position in a letter to David Goss, dated March 30, 2000, and reprinted two years later in the Gazette of the Société Mathématique de France.  I alluded to this letter in the text quoted in the earlier post.  To my mind, the most interesting part of this letter is his explanation of what he sees as Weil’s contribution to the problem.

b) He suggests that, not only every elliptic curve over Q should be modular, but its “level” (in the modular sense) should coincide with its “conductor” (defined in terms of the local Néron models, say).

Part b) was a beautiful new idea ; it was not in Taniyama, nor in Shimura (as Shimura himself wrote to me after Weil’s paper had appeared). Its importance comes from the fact that it made the conjecture checkable numerically (while Taniyama’s statement was not). I remember vividly when Weil explained it to me, in the summer of 1966, in some Quartier Latin coffee house. Now things really began to make sense. Why no elliptic curve with conductor 1 (i.e. good reduction everywhere)? Because the modular curve X0(1) of level 1 has genus 0, that’s why!

One can agree or disagree with Serre’s criterion — who apart from Serre has proposed numerical verifiability (or falsifiability!) as the boundary between meaningful and questionable (“discutable”) conjectures? — but at least it has the merit of being stated clearly enough to serve as the starting point for a philosophical consideration of authorship of a conjecture.  It’s not merely rhetorical.

They’re here!

Grothendieck on Amazon

One published in 2015, two new ones this year.  Plus last month’s addition to the list:



Douroux is a journalist at Libération, who, according to the jacket, “spent four years tracking down the brilliant hermit.  He has scoured [épluché] the archives and flushed out its last secrets.”  Fat chance!  Still, his book is at 5851 on’s best seller list, the others somewhat lower.

None of these books has been translated into any language other than French, as far as I can tell, and the definitive biography — the one on which the definitive movie will be based — has yet to be written.

The Taniyama, Shimura, Weil controversy in Herts.


Hatfield Galleria,  by Cmglee (own work) CC BY-SA 3.0 via Wikimedia Commons


In honor of the Abel Prize Committee’s decision to award credit for the celebrated conjecture on modularity of elliptic curves to Shimura, Taniyama, and Weil (in that order) in the course of awarding the Abel Prize to Andrew Wiles, I am publishing here for the first time an excerpt from the text of my talk, entitled Mathematical Conjectures in the Light of Reincarnation, at the conference Two Streams in the Philosophy of Mathematics that took place in 2009.  (The remainder of the talk was reworked and expanded into Chapter 7 of MWA.)  The conference was organized by David Corfield and Brendan Larvor and was held on the campus of the University of Hertfordshire in Hatfield, England.  There are no photographic records of the conference, which left practically no internet trace whatsoever, apart from the program posted on the FOM website, so I have included a Wikimedia commons photo of the Hatfield Galleria Shopping Mall, which is where the conference dinner was held, across a traffic circle (roundabout) from the university campus.

For years I have wanted to write a comprehensive article about the controversy over the name of the conjecture for a philosophy of mathematics journal.  But I have never had the patience to organize the themes of the controversy, and instead, as a way of relieving my persistent irritation with the way the controversy has been addressed, I have been inserting cranky fragments of arguments into articles and presentations where they don’t necessarily belong.   Here is an example from an October 2009 draft that is actually called PHILOSOPHICAL IRRITATION.  The original reason for my irritation is explained in the second paragraph, and it goes back to Serge Lang’s earliest interventions on behalf of Shimura, before Wiles proved Fermat’s Last Theorem; the words “chagrin” and “avalanche” allude to an incident that took place some five years after Wiles’s announcement, about which, perhaps, more will be written later.

Even before the Science Wars erupted, I had observed with increasing distress a bitter debate over the appropriate nomenclature for a conjecture of fundamental importance for my own work in number theory. As a graduate student I had been taught to refer to the conjecture as the “Weil conjecture” but a few years later, after a series of consultations I have not attempted to reconstruct among senior colleagues, the name was changed to “Taniyama-Weil conjecture.”  By the beginning of my career, which fortuitously coincided with the official consecration, in the form of the four-week Corvallis summer school, of the Langlands program, this name designated this program’s iconic prediction and unattainable horizon, though even at the time it was understood to hold this position only as an effect of convergence after the fact, since Langlands had found his way to his program by another route, and the conjecture was primarily iconic for number theorists.[1] About ten years later the conjecture underwent another promotion when it was discovered[2] that it implied Fermat’s Last Theorem as a consequence. Serge Lang then began an energetic campaign (some details are recorded in the AMS Notices) to change the name on the grounds that, after an initial hesitant formulation by Taniyama, it had been proposed in a more precise form by Shimura, the first to suggest the idea to Weil who, after a period of skepticism, not only published the first paper on the conjecture but wrote both Taniyama and Shimura out of the story. At this point the name of the conjecture underwent several bifurcations: Shimura-Taniyama-Weil for those inclined to generosity (and alphabetical order), Taniyama-Shimura-Weil for those with a certain view of history, Taniyama-Shimura or Shimura-Taniyama for those in Lang’s camp (including Shimura himself, as I later learned to my chagrin) who saw Weil as a treacherous interloper, and occasionally Taniyama-Weil for those who took pleasure in baiting Shimura.  An often acrimonious exchange of opinions on the question, which enjoyed a brief revival after Wiles proved the conjecture in sufficient generality to imply Fermat’s Last Theorem and Taylor and his collaborators proved it in complete generality, has led to the current impasse where there is still no consensus on what the conjecture (now a theorem) should be called: French sources generally include Weil’s name, whereas many if not most American authors do not[3].

What I found and continue to find most disheartening is that none of the quarrel’s protagonists saw fit to provide any guidance to resolving similar conflicts in the future. Indeed, with the exception of a relatively late[4] contribution by Serre, to which I return below, no one acknowledged that it might be of interest to consider the dispute as other than sui generis, and the discussion remained largely in the forensic mode initiated by Lang. How and to whom to attribute ideas — a more or less isolated conjecture, a research program (such as the Langlands program), or a key step in a proof — is the question of most moment in the development of individual careers, the distribution of power and resources, or the evolution of the self-consciousness of a branch of mathematics.  It can be fruitfully analyzed by historical or sociological methods, specifically by science studies in one or another of its incarnations. It can also be given the status of a philosophical question. Indeed, the claim that such a question has philosophical content beyond what is accessible by history or sociology — that it can in some sense be analyzed in terms of principles whose nature remains to be determined — is itself a philosophical claim, and one that is likely to be contested. When such a question arises I would like to be able to answer it on the basis of principled arguments and not by joining a transitory alliance or actor network. It is not the sort of problem that typically appeals to philosophers.

The problems about mathematics that do appeal to philosophers, according to the Oxford Handbook[5], include (for example)

  1. What, if anything, is mathematics about?
  2. …how do we know mathematics [if we do]?
  3. To what extent are the principles of mathematics objective and independent of…?

and so on. I see no room on this list for an account of how to attribute authorship to a conjecture. The word “conjecture” does not even appear in the index of this 800-page handbook (“theorem” occurs, but sparsely; more popular index entries are “truth,” “proof,” “proposition,” and “sentence,” as well as topics like “arithmetic,” “geometry,” and “number”). No guidance is forthcoming from the handbook as to whether a conjecture is ontological rather than epistemological or methodological. A conjecture must be a matter of importance to mathematicians, though, if so many of them, and not only the rival claimants and their friends, are willing to sacrifice valuable working time for fruitless belligerence in order to arrive at an accurate attribution.

It may just be that the conjecture is something the working mathematician is forced to cite by name, and that one attempts to trace it back to its original source in order to avoid irritating short-tempered colleagues. I briefly thought I had found a way to evade responsibility by referring to it as “the conjecture associated with the names of” followed by three names, a formulation that is undoubtedly objectively true (and admitting of ostensive proof in this very text); and at least one colleague followed me[6] along this deceptively innocuous path until the day we walked into an avalanche. Don’t follow my lead in suggesting that the problem can be evacuated into a matter of typographical convenience:  do we know what kinds of “something” we can be “forced” to treat in this manner, and just what is this “force” that holds us in its grip? As of this writing many colleagues have given up on nominal attribution altogether and refer to the conjecture by what it says (the “Modularity Conjecture for elliptic curves,” for example). But how can a conjecture “say” anything?[7]

[1] Langlands’s own priorities were elsewhere, as he has frequently pointed out. His insights have been so influential in so many branches of mathematics that he can hardly be said to own the Langlands program any more. But he certainly has been clear and consistent about his own reasons for formulating the program that bears his name.

[2] By Frey, Serre, and Ribet. The word “discovered” is used here in the colloquial sense and expresses no philosophical commitment.

[3] I haven’t checked Japanese practice.

[4] Its publication came late in the story, but it’s clear from discussions with French colleagues that Serre had expressed his opinions on the matter quite early on.

[5] Stewart Shapiro, Chapter 1, p. 5; letters mine.

[6] Langlands used a similar formulation in a recent article in Pour la Science but I have no reason to think he didn’t come up with it on his own.

[7] This at least looks more like a Handbook question — but which one?

By the end of this text my irritation has expanded from the “short-tempered colleagues” who had promoted the controversy to include, uncharitably, the contributors to the Handbook who didn’t think to include material that would help to clarify what, if anything is really at stake (beyond personal antipathies, which should not be underestimated) in controversies of this kind.

Serre’s “relatively late” contribution to the debate is quite interesting, and has influenced a number of colleagues, but it looks like I never got around to revising the draft to keep the promise to explain what Serre added to the discussion.  Maybe I will on this blog, or maybe I’ll actually write that philosophical article.



How MWA promotes the (oppressive?) hierarchy


From Langlands, Is there beauty in mathematical theories?, Lectures at Notre Dame University, January 2010, published in The Many Faces of Beauty, Vittorio Hösle, ed.

Finally I can begin to fulfill the promise I made last August to point out a few of the things that are really wrong with Mathematics without Apologies.   The diagram reproduced above is taken from an article Langlands wrote when the editor invited him to contribute an article on mathematical beauty to the volume cited above, with the title The Many Faces of Beauty.  (In the summer of 2011 I was invited to contribute an article on mathematical beauty to an issue of the Portland-based literary journal Tin House with the title Beauty.  So Langlands and I have at least that in common.)   The diagram lists  “the names of some of the better-known creators of the concepts” that contributed to the solution by Andrew Wiles of Fermat’s Last Theorem, which Langlands chooses as the starting point of his account of the theory of algebraic equations and, ultimately, automorphic forms.

One will have noticed that all the names belong to men, practically all of them European.  This is a problematic feature of the hierarchical nature of mathematics, but it’s not my topic today.  The question instead is Who speaks for mathematics? which (it seems to me) is at least implicit in Piper Harron’s identification of the oppressiveness of mathematical hierarchy.  Much to my regret, MWA did not break with the convention of quoting the reflections of Giants and Supergiants and those most visible among our contemporary colleagues, the people whose names appear in lists and diagrams like the one copied above.   Thus the question Who speaks for mathematics? is answered by pointing to those who occupy the most prominent positions in the (oppressive?) hierarchy.

This is in part due to the unsystematic nature of my research and in part due to structural features of the hierarchy, which I emphasize on p. 39, in what I have already identified as the key passage in Chapter 2:

We’ll see throughout the book  quotations by Giants and Supergiants in which they conflate their own private  opinions and feelings with the norms and values of mathematical research,  seemingly unaware that the latter might benefit from more systematic  examination.  One of the premises of this chapter is that the generous licence  granted hieratic figures is of epistemological as well as ethical import.

My own experiments with the expression of what appear to be my private opinions resemble this model only superficially and only because they conform  to the prevailing model for writing about mathematics.  My friend’s point was  that even my modest level of charisma entitles me not only to say in public  whatever nonsense comes into my head…

In other words, it’s much easier to be quoted if you have published your thoughts in the first place, and it’s much easier to get your thoughts published if you are identified as a consequential mathematician.  I don’t know how to overcome this (possibly oppressive) characteristic of the mathematical hierarchy, and it’s one of the main reasons I am hoping sociologists will pay closer attention to mathematics.

Having said that, I should dispel any notion that Langlands took advantage of his mathematical eminence, in the article from which Diagram B is taken, to write “whatever nonsense” came into his head.  On the contrary, while the modesty of his intentions is evident throughout, there is no nonsense but rather a good deal of profound and unconventional thinking about the nature of our vocation.  So I will take the risk of promoting the (oppressive?) hierarchy once again and encourage you all to read the article, if you have not done so already; and I will quote a few of its more memorable passages.

On his own limitations and uncertainties:

I learned, as I became a mathematician, too many of the wrong things and too few of the right things. Only slowly and inadequately, over the years, have I understood, in any meaningful sense, what the penetrating insights of the past were. Even less frequently have I discovered anything serious on my own. Although I certainly have reflected often, and with all the resources at my disposal, on the possibilities for the future, I am still full of uncertainties.

On the effects of (what MWA calls) charisma:

Because of the often fortuitous composition of the faculty of the more popular graduate schools, some extremely technical aspects are familiar to many people, others known to almost none. This is inevitable.

On Proposition 78 of Book X of Euclid’s Elements:

This is a complicated statement that needs explanation. Even after its meaning is clear, one is at first astonished that any rational individual could find the statement of interest. This was also the response of the eminent sixteenth century Flemish mathematician Simon Stevin…

On the beauty of Galois theory:

What cannot be sufficiently emphasized in a conference on aesthetics and in a lecture on mathematics and beauty is that whatever beauty the symmetries expressed by these correspondences have, it is not visual. The examples described in the context of cyclotomy will have revealed this.

On cooperation — and charisma! — in mathematics:

Like the Church, but in contrast to the arts, mathematics is a joint effort. The joint effort may be, as with the influence of one mathematician on those who follow, realized over time and between different generations — and it is this that seems to me the more edifying — but it may also be simultaneous, a result, for better or worse, of competition or cooperation. Both are instinctive and not always pernicious but they are also given at present too much encouragement: cooperation by the nature of the current financial support; competition by prizes and other attempts of mathematicians to draw attention to themselves and to mathematics.