Category Archives: History

The diversity statement controversy, II


From AMS Notices, January 2020, online only, p. 1

One of the minor virtues of Goodwillie’s piece, quoted in the previous post, is its clarity regarding the meaning of “diversity” in the institutional setting.  The word appears only twice, neither time with an unambiguously positive valence:

Institutional diversity is all very well, but if the “different” people do not feel truly welcome, and if mismatches between the institution and the worlds that the students are coming from are ignored, then the institution has failed them.

I’d like to think that a job applicant who meditated on Goodwillie’s post would be well-prepared to write a genuinely meaningful diversity statement.  But it would be much more than a “cuddly, feel-good” sort of diversity statement;  it might even be seen as dangerously close to the kind of commitment to social justice that the 1978 Supreme Court ruled out as grounds for affirmative action.

In contrast, no one comes off well in this latest controversy.  I had prepared a point-by-point list of some of the problematic arguments made in each of the texts, starting with Thompson’s essay and continuing through the open letters written for and against, as well as Chad Topaz’s blog post at QSIDE.  But a friend made the following comment upon reading an earlier draft:


My friend is absolutely right.  I enjoy a good quibble as much as anyone, but it’s best to keep it private.  Besides, the most serious of my points was the suggestion that the AMS Notices open its pages to an extended debate on the important topic of … inclusion and exclusion … including but not limited to the role of diversity statements.   It turns out that this debate already began in the January 2020 issue of the Notices, which arrived in my mailbox yesterday.  The for and against letters are included, and a second for letter again, with all the signatures; the total occupies a 21-page pdf file.  It’s therefore likely that more than 1400 people knew, as I did not, that my suggestion was superfluous.  This is a sign that I should perhaps be expressing myself with more humility.

I note, however, that nowhere in the 21 pages of the Notices file does anyone “take[s] the Coates point,” as my friend put it.  Lewis Powell is not identified as the author of the “diversity” opinion, and the Bakke case is only mentioned once, in passing, and in a way that, perhaps inadvertently, confirms “the Coates point.”  Xander Faber’s letter quotes this comment by Supreme Court Justice Harry Blackmun:

In order to get beyond racism, we must first take account of race. There is no other way. And in order to treat some persons equally, we must treat them differently.

The context of Blackmun’s comment, however, was his Separate Opinion,  written to clarify his agreement with the minority position in the Bakke case.   This is the position that lost out to the “cuddly, feel-good” diversity that, thanks to Powell and four other Justices, has been the limit of what the law of the land protects since 1978.  To me it is counterintuitive to rely on Powell’s vocabulary to “uphold Blackmun’s words,” as Faber writes, when the continuation of the Blackmun comment — “We cannot – we dare not – let the Equal Protection Clause perpetrate racial supremacy.” — was written as an explicit rebuke to Powell’s reasoning.

Faber’s letter has the merit of appealing to evidence, in the form of “an extensive report” produced by UC Berkeley “that documents the effect of hiring with a diversity focus in mind.”  Here is what that report had to say (on p. 49) about the effectiveness of diversity statements:

Beyond the applicant stage … no clear and consistent patterns in the data emerged that would suggest a positive statistical correlation between this practice and diversity.  We suspect there may be considerable variation in how search committees implemented this practice, and we speculate that these differences may have obscured the potential value of some forms of implementation. In addition, different institutions may use information about candidates’ commitment to diversity in different ways, and when these can be studied separately, some may emerge as considerably more promising than others. Anecdotal evidence from other UC campuses suggests that much may depend on the extent to which strong or weak “diversity statements” are used as potential deciding factors during the search deliberations. On the basis of our data and analyses to date, however, we do not think we can conclude that this is a practice showing clear promise.

This is hardly a ringing endorsement of diversity statements as a way to enhance even diversity of the “cuddly, feel-good” variety,  much less as a means of realizing the more ambitious aims of equity and inclusion to which Faber refers in his letter.  I wonder whether Faber disagrees.

Overall I have to assume that when people in this debate use the word “diversity” they have in mind something like “equity” or even “social justice” — the opposite of the meaning  Powell set out in his 1978 opinion.   Institutions like the Regents of the University of California may be confined to the legal straitjacket that Justice Powell designed for them more than 40 years ago, but there is no reason that a colleague who is genuinely committed to the values of equity or social justice should feel obliged to express their values in Powell’s vocabulary.

P.S.  I’m not sure I agree with Thompson’s judgment that “Requiring candidates to believe that people should be treated differently according to their identity is … a political test,” [my emphasis] but it is certainly political. Assuming that the US approach to identity politics has universal political validity is a symptom of the provincialism — not to say cultural imperialism — that comes too naturally to people who live in this country, wherever their opinions fall on the political spectrum.  It is particularly unwelcome as the default position of the inclusion/exclusion blog with regard to decisions that affect the very international population of candidates for jobs in the United States.  Some of these candidates come from countries where treating people “differently according to their identity” is strictly illegal.  Depending on what is meant by “treat,” this is arguably also the case in the United States — the “equal protection clause” of the 14th Amendment to the U.S. Constitution is cited 31 times in the Bakke case that is at the origin of all this talk of diversity.   I sincerely regret that Blackmun’s position did not prevail in 1978, but it doesn’t help anyone to pretend that it did.



The diversity statement controversy, I


Supreme Court Justice Lewis F. Powell, author of the legal definition of “diversity”; PD-USGov

Colleagues who are confused by the ongoing controversy surrounding Abigail Thompson’s article in the Notices of the AMS on mandatory diversity statements should reread what Ta-Nehisi Coates had to say about “diversity” in his article “The Case for Reparations“:

Affirmative action’s precise aims… have always proved elusive.  Is it meant to make amends for the crimes heaped upon black people? Not according to the Supreme Court. In its 1978 ruling in Regents of the University of California v. Bakke, the Court rejected “societal discrimination” as “an amorphous concept of injury that may be ageless in its reach into the past.” Is affirmative action meant to increase “diversity”? If so, it only tangentially relates to the specific problems of black people— the problem of what America has taken from them over several centuries. …

America was built on the preferential treatment of white people—395 years of it. Vaguely endorsing a cuddly, feel-good diversity does very little to redress this.

Thompson, a Vice-President of the AMS and Chair of the mathematics department at UC Davis, wrote her essay to object to the UC system’s use of mandatory diversity statements to “screen out [job applicants] early in the search process.”  While she compares these statements to the “loyalty oaths” that the UC Regents required during the McCarthy period, and Melissa Lutz Blouin, speaking for the UC Davis administration, retorted that

Diversity, equity and inclusion statements foster productive discussions on how current and prospective faculty can shape and improve the learning and working environment in higher education…

neither Thompson’s original article nor the subsequent controversy makes it clear whether the UC Regents favor “cuddly, feel-good” diversity statements or are willing to consider statements that relate more than tangentially to the specific problems of the communities whose concerns they are meant to address.

Coates, unlike most of the mathematicians and bloggers who have weighed in on the topic since Thompson’s essay appeared, is deeply familiar with the history of the term “diversity” within the jurisprudence that underlies UC Davis’s approach to affirmative action.   When the Bakke case to which Coates refers was decided, it was considered a defeat by those who hoped to use affirmative action as a means to remedy historical discrimination.  Alan Bakke, the plaintiff, claimed that his constitutional rights had been violated when he was rejected — by UC Davis, of all places! — because the medical school had set aside 16% of its slots for minority students.  The California Supreme Court agreed with him, and the US Supreme Court followed suit — Bakke was admitted later that year.  The Court’s judgment, written by Justice Lewis Powell, did allow affirmative action, but only as a way of “obtaining the educational benefits that flow from an ethnically diverse student body”:

An otherwise qualified medical student with a particular background — whether it be ethnic, geographic, culturally advantaged or disadvantaged — may bring to a professional school of medicine experiences, outlooks, and ideas that enrich the training of its student body and better equip its graduates to render with understanding their vital service to humanity.

(Bakke, pp. 306, 314).  Translating Powell into plain English:  an ethnically diverse student body is desirable as a bonus benefit that can “enrich” the experience of the (presumably white) majority.  Or, to quote Christopher Newfield’s Unmaking the Public University, as I already did three years ago in this post,

…in Powell’s diversity framework, diversity was the expression of an institution’s freedom to choose particularly attractive individuals, and was about ensuring this freedom for powerful institutions like… Harvard College.…Diversity acquired social influence not as a moderate mode in which to pursue racial equality but as an alternative to that pursuit.

I am suspicious of any attempt to ground a progressive approach to any question whatsoever in the ideas of the supremely sinister Powell, author of the notorious Powell Memorandum, which Wikipedia accurately calls “the blueprint for the rise of the American conservative movement.”  While the Regents of the University of California are legally bound by a jurisprudence that serves, as a friend wrote, as a means of “deflecting attention from the structural issues to the individual ones,” why is the AMS inclusion/exclusion blog so attached to the policy?  I suspect it is because its readers and authors, unlike Coates or Newfield, imagine that “diversity” can be translated into the aspirations expressed in Thomas Goodwillie’s post on that same blog.  Goodwillie’s text, which is extraordinary for its thoughtfulness and humility, should be studied before reading the second part of this post.




“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.

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.



Riemann, Cantor, and Lyndon LaRouche

Campaigner and Cantor

Facsimile of January-February 1976 issue of The Campaigner, from (itself a subdivision of

While looking for hints that Grothendieck’s influence on Alain Badiou, which is undeniable (if somewhat warped),  may have included face-to-face meetings, I came across a treasure trove of archival material documenting Lyndon LaRouche’s conviction that he is a latter-day Riemann or Cantor.   This page, for example, greets you with the message

The report you are about to experience was produced to make clear why LaRouche refers to his economic forecasting methodology as the “LaRouche-Riemann Method.” In doing so, we’ll explore the central feature of economics: our characteristic activity, as a species, is built upon mankind’s willful implementation of creative discoveries which re-define our relationship to the universe around us. To do so, we’ll enter into one crucial aspect of Cantor’s work, and the breakthroughs of Bernhard Riemann on implicit geometry and transcendentals.

Other pages introduce the unwary visitor to the thoughts of Kepler and Fermat and to the Mind of Gauss, where you can read what appears to be a complete proof of quadratic reciprocity, followed by this enigmatic comment:

Gauss found that, actually, the 4n+1 primes, because of their relationship to -1 and the Pythagorean Triples, were not really primes. They were really representations of the Complex Domain. This investigation can wait until the pedagogicals on Gauss’s work on Biquadratic Residues.

Evidence of the affinities of the LaRouche movement with mathematics are easy to find on the internet; for example I am looking forward to spending 3 hours and 20 minutes watching a YouTube video answering the question Does Mathematics Make You Stupid?  But practically all the links I’ve found lead back to the LaRouche movement itself.  There is at least one exception, however:  The Campaigner, the theoretical journal of the National Caucus of Labor Committees (NCLC), actually did publish the first English translation of Cantor’s Grundlagen einer allgemeinen Mannigfaltigkeitslehre, the first of his fundamental articles on set theory.

The cover of the issue containing Uwe Parpart’s translation is pictured above.  The translation is widely referenced — I discovered its existence in the bibliography of Peter Hallward’s 2003 book on Badiou, for example.  It’s also listed at the beginning of Joseph Dauben’s Chapter 46 of the perfectly legitimate Landmark Writings of Western Mathematics 1640-1940, edited by the late Ivor Grattan-Guinness — though Dauben adds that the translation in W. B. Ewald’s From Kant to Hilbert is “preferable.”  It seems you can even buy a copy of the NCLC translation on   It’s not listed on MathSciNet, on the other hand, and LaRouche doesn’t seem to have been mentioned on MathOverflow, so its existence may be as surprising to most readers of this blog as it was to me.

P.S.  The LaRouche people used to have a table on rue de Chevaleret in Paris, when the Institut Mathématique de Jussieu was in exile there, but I assume that was just a coincidence.

Remembering Boris Weisfeiler

aviso copia

The New York Times reports that Chilean judge Jorge Zepeda “has put an end to the 16-year investigation” into the disappearance and death of the Russian-American mathematician Boris Weisfeiler, pictured above, while hiking in Chile in 1985, at the time of the Pinochet dictatorship.  For those unfamiliar with the case, here is what Allyn Jackson wrote in January 2004 in the Notices of the AMS:

In 1985 the mathematician Boris Weisfeiler disappeared while hiking alone in a remote area of Chile. At the time, he was a professor of mathematics at Pennsylvania State University and was widely recognized for his work in algebraic groups. What happened to him remains a mystery, and to this day it is not known whether he is still alive.  Born in the Soviet Union, Weisfeiler received his Ph.D. in 1970 from the Leningrad branch of the Steklov Institute, where his adviser was E. B. Vinberg. Weisfeiler emigrated to the United States in 1975 and worked with Armand Borel at the Institute for Advanced Study. The next year he joined the faculty at Pennsylvania State University. In 1981 he became an American citizen.  Weisfeiler’s disappearance has been the subject of several newspaper articles (see, for example, “Chilean Mystery: Clues to Vanished American”, by Larry Rohter, New York Times, May 19, 2002; and “Tracing a Mystery of the Missing in Chile”, by Pascale Bonnefoy, Washington Post, January 18, 2003). Further information about media coverage, as well as the present status of the investigation into his disappearance, may be found at http://weisfeiler. com/boris
On the occasion of the publication in Chile of a book about Weisfeiler’s disappearance, the Notices decided to present a brief tribute to his life and work. What follows is a short summary of his mathematical work and a review of the book. This is not an obituary, as hope remains that Weisfeiler is still alive. Nevertheless, it seems appropriate to commemorate this lost member of the mathematical community, whose absence is keenly felt.
—Allyn Jackson
The Notices tribute consisted in two articles by Alexander Lubotzky (on Weisfeiler’s work) and a review by Neal Koblitz of the book pictured below,
published in Chile in 2002.  The book’s thesis, which Koblitz found convincing, is that Weisfeiler was arrested by a military patrol — eight of whose members were put on trial in 2012 but released by judge Zepeda’s ruling — and handed over
to nearby Colonia Dignidad, an enclave of ultrarightist German immigrants founded and at the time still led by ex-Nazi Paul Schäfer. Thinking that Weisfeiler was a “Jewish spy” working for Nazi-hunters, they imprisoned and eventually killed him.
 The website mentioned in Jackson’s introduction is still active and has a reaction to the judge’s ruling, under the title “A Travesty of Justice.”
Judge Zepeda’s ruling in this case is a direct aide-mémoire of the judicial rulings during Gen Pinochet’s dictatorship. Regrettably, today’s Chilean Justice is strongly influenced by the government as well: the Chilean political establishment continues to see the Armed Forces as a threat to political stability and prefer not to interfere in their affairs.
The website has a long list of articles about the case, mostly in Spanish.  Those who understand Spanish can read an interview with Weisfeiler’s sister Olga, dated March 16, 2016, in the Chilean website
La noticia sobre el fallo que absolvió a los culpables de este crimen fue un golpe duro para ella…. No tanto por la decisión del juez Jorge Zepeda, que descartó que fuera un crimen de lesa humanidad, sino porque, según comenta, el magistrado la engañó a ella, su familia y al gobierno de los Estados Unidos acerca de cómo finalizaría esta historia.

Programming began in Mesopotamia

Plimpton322 - 1

This is a picture of Plimpton 322, a relatively late (less than 4000 years oldOld Babylonian cuneiform tablet that just happens to be located in Columbia’s Rare Books and Manuscript Library.  It may be the most famous of all mathematical tablets, because it contains a list of Pythagorean triples, including (in sexagesimal notation) (1:22:41, 2:16:01, 1:48).  Eleanor Robson’s “reassessment” seems to be the most influential recent analysis of the tablet, but since that link is behind Elsevier’s paywall, you may prefer to read her article in the February 2002 American Mathematical Monthly.


I took that picture, and here is another picture of the tablet in my very own gloved hands,

Plimpton in gloves

taken two weeks ago by a visiting colleague.

I had been thinking about Babylonian mathematical tablets in connection with an article I’m currently writing.  The title of this post is suggested by the following quotation from an article (unpublished, I believe) by Jim Ritter, entitled Translating Babylonian Mathematical Problem Texts:

A good interpretive level would be one which would use only the cognitive tools used in Ancient Mesopotamia, which would apply to all mathematical problem texts and which would remain as close as possible to the formal structure of the text. Such an approach was first suggested for Babylonian mathematical texts as early as 1972 by the pioneer of computer algorithmic, Donald Knuth (KNUTH 1972 and 1976), an algorithmic approach based on sequences of arithmetic and control commands. Unfortunately for Assyriology, the paper was published in a computer science journal and so remained unknown there in general until this approach was independently rediscovered in the late 1980s (RITTER 1989b).19 The advantages in this approach are major. First of all, the problem texts as they have come down to us are algorithmic in nature. There is no need to move beyond what is actually written and the same approach applies to the whole corpus. Moreover the dual nature of the commands in the text—calculational (arithmetic operations) and control (initialization, storage, parallel computing)—parallel exactly the dual nature of modern programming. Perhaps most importantly, this interpretation is minimal in the sense that it does not block an algebraic or geometric further development if the reader or translator so desires.






Image from Plimpton 322, in the collection of Columbia University, via Wikimedia Commons

Beschleunigung, perfectoid or otherwise


This post will be unusual in that it will actually be concerned with the kind of mathematics I encounter in my own research, and more unusual still in that I will reveal how these encounters affect my state of mind.   Since the guiding principle of this blog, like the book, is that nothing about its author belongs in it unless it is ideal-typical, in other words exemplifies a general feature of the vocation (and this, in turn, is because there’s no reason for a reader to care about the author’s state of mind, unless the reader happens to know the author, in which case the reader should just ask in private), I begin with some background about the ideal-typical experience on which I am about to report.  My guide will be Hartmut Rosa, critical theorist and professor of sociology and political science at the Friedrich-Schiller-Universität Jena, in Germany, theoretician of Beschleunigung [acceleration]:

The time structures of modernity… stand above all under the sign of acceleration.… a grave and sharpening scarcity of time has arisen in the social reality of Western societies, a crisis of time that places in question the traditional ways in which individuals and polities could secure the possibility of shaping their own existence.

He describes the predicament of a contemporary entrepreneur, but the description applies just as well to a mathematician:

He will try to maintain control over his life…and also plan scrupulously for future developments.  However, the more dynamic his environment becomes, the more complexly and contingently its chains of events and horizons of possibility take shape, the more unfulfillable this intention will become.

These two quotations are taken from the preface to his book Social Acceleration, a translation of Beschleunigung. Die Veränderung der Zeitstrukturen in der Moderne, which was based on the Habilitationsschrift defended in 2004 in Jena at the (it seems to me) not particularly accelerated age of 39.  His two subsequent books also have Beschleunigung in the title.  I could just write “acceleration” but it seems to me that, like Durkheim’s anomie or Kierkegaard’s Angst — or Drinfel’d’s shtuka, in a different context (that is nevertheless relevant to the latter part of this post) — the German word draws attention to the singularity and novelty of the underlying concept in a way that is lost in the English equivalent.  His most recent book, pictured above [World Relations in the Era of Beschleunigung], looks particularly relevant, because the sample chapter available online quotes T.S. Kuhn, and the point of this post is that I am, and therefore all the representatives of the ideal-type to which I belong are, feeling buffeted if not battered by the waves of Beschleunigung powered by not one but two, or maybe even three, singularly accelerated Kuhnian paradigm-shifts.  This book has not been translated, but it is available from in Kindle form and I would be ready to overcome my deep misgivings about Amazon and purchase it with 1-Click but my whole point is that this morning’s Beschleunigung is so violent that I can’t even spare the time for 1-Click if (mixing metaphors) I want to have any hope of hanging on by my eyelids to the last car of the new paradigm train that is rushing by at blinding speed.

I remember earlier Beschleunigung-episodes, notably the introduction of perverse sheaves with its immediate applications to geometric representation theory, or quantum groups, or the development of motivic cohomology, not to mention various stages in the Langlands program with which I was too closely involved to be able to appreciate as a spectator.  Each episode had its distinctive contingent character but they shared a complex mix of affects, as initiates and bystanders alike experienced the euphoria of witnessing the novel methods providing unexpected solutions to longstanding problems or even more unexpected solutions to unexpected problems, the Angst that one might not be able to beschleunigen sich enough to keep up with the new developments, the anomie of realizing that questions remained, not only the old ones but even more new ones that arrived in the Beschleunigung‘s wake.  And then there is the INERTIA of those not caught up in the Beschleunigung, as described in a paper by Bart Zantvoort entitled


that cites Hartmut Rosa and that pretty much sums up the ignominious alternative that awaits me if I fail to keep my balance amidst all this buffeting and battering.

What provoked this outburst was the simultaneous appearance this morning of two preprints on the arXiv, namely

Title: Geometrization of the local Langlands correspondence: an overview
Authors: Laurent Fargues
Categories: math.NT math.AG math.RT


Title: A canonical torsion theory for pro-p Iwahori-Hecke modules
Authors: Rachel Ollivier, Peter Schneider
Categories: math.RT math.NT

The two accelerated Kuhnian paradigm shifts to which I alluded above are Scholze’s perfectoid geometry and derived algebraic geometry, and to make sense of what’s going on in the overlapping concerns of these two (and many more) papers my fellow hobos, who would just as happily sit and watch the trains pass by, need to overcome their INERTIA and beschleunigen sich enough to stay on the far side of both paradigms.  And it’s just as necessary to absorb the lessons of Vincent Lafforgue’s work on Langlands parametrization (this is where the shtukas, or chtoucas, come in), and whether or not this qualifies as a full-blown paradigm shift it certainly involves a lot of work and is likely to require a lot more as its implications hit home.

I had better get back to my personal Beschleunigung.  So I’m not going to explain why someone with my ideal-typical profile would have to overcome INERTIA when the two papers cited above appear on the arXiv; instead I’ll just mention that, in response to a request to contribute to a book that is (I think) entitled What is a mathematical concept?, edited by Elisabeth de Freitas, Nathalie Sinclair, and Alf Coles, I submitted a chapter entitled

The Perfectoid Concept: Test Case for an Absent Theory

where the absent theory is not the all-too-present perfectoid geometry but rather the theory of the kind of mathematical concept it exemplifies.  The chapter treats the euphoria and the Angst that accompanied this particular candidate for paradigmatic Kuhnian status, but not the anomie, except perhaps in this passage near the end:

Is a professional historian even allowed to believe that (some) value judgments are objective, that the notion of the right concept is in any way coherent? How can we make sweeping claims on behalf of perfectoid geometry when historical methodology compels us to admit that even complex numbers may someday be seen as a dead end? “Too soon to tell,” as Zhou En-Lai supposedly said when asked his opinion of the French revolution.



Three mathematicians, three novels, only one movie, part 3



Many followers of this blog have undoubtedly read Daniel Kehlmann’s Measuring the World [Die Vermessung der Welt].  In contrast to the books of Fonseca and Désérable, it has been a major international success, winning too many prestigious awards to list.  Yet it has also attracted the attention and admiration of numerous literary scholars, many of whom nevertheless feel compelled to characterize it as “best-selling.”  “It was on the bestseller lists for weeks on end,” writes literary scholar Nina Engelhardt (in a private e-mail), “even competing with Harry Potter and Dan Brown.  It has also received a lot of critical literary attention and is generally viewed as a successful and innovative example of combining literature and science.”  My German-speaking friends tend to describe Kehlmann as a celebrity, often to be seen on TV talk-shows; he lives in Berlin and Vienna but also holds a visiting professorship at NYU.

Measuring the World devotes alternate chapters to the historical figures of Alexander von Humboldt and our very own Carl Friedrich Gauss, familiar to every German in the widely-circulated portrait reproduced above.  Soon after Kehlmann’s novel was translated into English, Frans Oort published a review in the AMS Notices.  It’s an understatement to say that Oort was disappointed with Kehlmann’s depiction of the Prince of Mathematics.  The review begins with a report of a dream, no doubt fictional:

The young Gauss started to smile, knowing that I recognized him, and remembered this story. Then his face and and figure changed into the beautiful portrait of the young Gauss published in the Astronomische Nachrichte, 1828…. He looked at my desk, and he started to talk to me. “I see that you are reading that book! What can this man mean, slandering me in this way?”… “why does this man have so little appreciation for the deep thoughts engendered in the beautiful things that I encountered and enjoyed in my life? Do you know where I can find this Kehlmann, so that I can explain to him the beauty of my ideas, and the reasons why I set out to measure things?”

It’s one of the most elegantly written and informative reviews I’ve ever read in the Notices, but the book I had just finished left a very different and altogether more positive impression.  So I wrote to Engelhardt, whom I had already consulted in connection with the Pynchon chapter of MWA, in search of clarification.  In her lengthy reply, she agreed with Oort that readers looking for historical accuracy in Measuring the World are likely to be unsatisfied.  But, as she explained (and as already should be clear from the title), that’s precisely the point.  I quote one of the articles* she has recently published on the book:

the humorous tone of the novel, the indirect discourse continuously indicating that events and dialogues are mediated, and the characterization of the eternally grumpy Gauss and an obsessed and naïve Humboldt can leave little doubt that Measuring the World is a work of fiction.

Here Engelhardt inserts a footnote with a reference to Oort’s review, naming at least one reader who failed to detect the telltale signs that Kehlmann’s novel is a specimen of historiographic metafiction.  Her comparative study of Measuring the World and Pynchon’s Mason & Dixon actually suggests that both novels belong as well to the rather different genre of scientific metafiction:

historiographic metafiction contests the accessibility of the past, an epistemological concern that does not challenge the reality of the past, while scientific metafiction problematizes the literally “natural,” namely the nature of the physical world, and thus introduces an ontological dimension.

The telltale signs include the consistent use of indirect speech in the German original, and pointers to Kehlmann’s “epistemological concern” are pretty hard to miss, frankly.  The one on the very first pages could not be more self-referential:

Even a mind like his own, said Gauß, would have been incapable of achieving anything in early human history or on the banks of the Orinoco, whereas in another two hundred years each and every idiot [Dummkopf] would be able to make fun of him and invent the most complete nonsense about his character.

The liberties Kehlmann takes with the empirical historical record — measurable liberties, one might say — are too numerous to mention.  For example, the 11-year-old Gauß discovered the curved geometry of the earth while flying in a hot air balloon with Pilâtre de Rozier, Montgolfier’s associate.  A quick calculation shows that Pilâtre had died several years before Gauss turned 11.  Oort made the calculation, as did the literary scholar Karina von Tippelskirch; yet they draw diametrically opposite conclusions — another illustration of the indeterminacy of measurement.

Tippelskirch reads the chapter entitled The Garden as a simultaneous enactment of reversals of Kafka’s The Castle and the Grand Inquisitor scene from The Brothers Karamazov.  The Humboldt segments are, if anything, even more meta.  Engelhardt:

Humboldt forges his journal when, afraid and refusing to go back into the jungle to shoot a jaguar, he is embarrassed about his actual behavior: “He decided to describe events in his diary the way they should have happened” (90).… it is not even certain whether it is Humboldt or a hallucination who tells his travel companion Bonpland that they “had climbed the highest mountain in the world. That would remain a fact, whatever else happened in their lives.” (152) Humboldt communicates the “fact” to Europe in “two dozen letters” (153), but it is incorrect on two accounts—readers witness that Humboldt and Bonpland have to turn back before reaching the summit and that, with the discovery of the Himalayas, Chimborazo proves not to be the world’s highest mountain.

The meta-sensitive reader is not surprised that in South America Humboldt encounters magic realism — story-telling boatmen named Carlos, Gabriel, Mario, and Julio! — as well as jaguars and crocodiles.  I expect that professionally-trained readers will detect in his travels across Russia a deliberately framing in the idiom of 19th century Russian realism.  So much of literary consequence has been written about Kehlmann’s book, in fact, and so much more will be written, that I will now turn to the question of particular concern to readers of MWA, namely:  how do these three novels depict provers of theorems not as abstractions but as live flesh-and-blood beings; in other words, how do they resolve the mind-body problem that is the topic of Chapter 6 of MWA?  More urgently, how do they contribute, if at all, to the canons of mathematical nudity?
Not at all, as I remember, in Coronel Lágrimas; the Grothendieck/Quijote figure smokes and drinks (too much) but is otherwise barely material at all.  Évariste features a single mystifying nude scene, in which the author undresses the frequently apostrophized but never visible character known as mademoiselle and then has her dress up as Galois in preparation for a fictional but unfulfilling love scene with Stéphanie.
The unwary reader who treats Kehlmann’s book as reliable history, on the other hand, will remember Gauss as quite the ladies’ man.  He visits the whores in Göttingen (not forgetting to think of numbers all the while) but he truly loves his first wife Johanna.  Barely 10 pages after their wedding night she dies in childbirth, in one of the most moving scenes in the book.    It’s the earlier scene, however, that reviews invariably highlight, specifically the moment in which the lovemaking is interrupted by Gauss’s discovery of the least squares method:

er schämte sich daß ihm ausgerechnet in diesem Moment klar wurde, wie man Meßfehler der Planetenbahnen approximativ korrigieren konnte…   weil er fühlte, daß sie erschrak, wartete er einen Moment, dann schlang sie ihre Beine um seinen Körper, doch er bat eine Verzeihung, stand auf, stolperte zum Tisch, tauchte die Feder ein und schrieb, ohne Licht zu machen:  Summe d. Quadr. d. Differenz zw. beob. und berechn. -> Min.

The scene is unlikely, as Oort points out, as well as historically inaccurate; and I haven’t yet figured out the author’s cunning purpose in placing this particular discovery at this particular point of the narrative.  And apparently it wasn’t enough to redeem the movie — I did mention that there was a movie, didn’t I?  A 3-D movie in fact, directed by Detlev Buck, with a screenplay by Buck and Kehlmann, starring Florian David Fitz as Gauß and Albrecht Abraham Schuch as Humboldt, and universally panned by German critics, in spite of an estimated 10 million € budget.   I found that figure on IMDB, where the film rates a miserable 5.7.  There are no reviews at all on Rotten Tomatoes, and I don’t know whether the film was even released to English-speaking audiences.

The trailer is up on YouTube, however, and if you want to add an image of a Gaussian bunda to your private canon of mathematical nudity you will find one at 0:39 (and another bunda a few seconds later).



*‘Scientific Metafiction and Historiographic Metafiction: Measuring Nature and the Past’. Twentieth-Century Rhetorics: Metahistorical Narratives and Scientific Metafictions. Ed. Giuseppe Episcopo. Napoli: Cronopio: 2014. 145–72. In press.

Problems of a problematic vocation

Readers who (like the author) persist in wondering what I was trying to say after they have finished the book may find it useful to take the book’s subtitle more literally.  If you believe mathematics is a problematic vocation, it doesn’t necessarily follow that you believe that the problems have solution, much less that the book’s author has found them.  Just identifying the problems may help to clear up misunderstandings (for example, that certain questions necessarily have answers).  Assigning problems to appropriate categories may be even more helpful.  With this in mind, here is a short but far from exhaustive list of some of the problems examined (but not solved) in  MWA, divided among four categories:  ethical problems (taking a stance on one of these problems entails an ethical commitment, and it is difficult to avoid taking a stance); historical problems (what appears to be a question about some intrinsic feature of mathematics is better addressed by investigating the questions’ history; best left to historians); linguistic problems (the imaginative resources we can apply to understanding the problem are limited by our language); and other.

Ethical problems

1.  Is mathematics elitist and/or hierarchical, and must it be? (Mainly addressed in Chapter 2)
2.  Who should pay for mathematics?  (Mainly addressed in Chapters 3, 4, and 10)
I wrote a three-part post about this last spring after realizing that what I wrote in the book lent itself to misinterpretation.  But the misinterpretations continue, in part as a result of some of the most recent reviews so I will write another post on this topic that I hope (but don’t expect) will settle it once and for all.
3.  Are mathematicians responsible for the uses to which our work is put?  What are the implications of “Faustian bargains” with funders? (Mainly addressed in Chapters 4 and 10)
4.  How to explain number theory (or topology, or dynamical systems) at a dinner party? (Mainly addressed in the obvious place)
5.  Must mathematicians have bodies?  (Addressed briefly in Chapter 6 and even more briefly in Chapter 7)

Historical problems

1.  Does mathematics belong to high or low culture? (Mainly addressed in Chapter 8)
2.  Should mathematics be seen as an Art or a Science, or both? (Mainly addressed in Chapters 3 and 10)
3.  How are Foundations of mathematics chosen? (Mainly addressed in Chapters 3 and 7)

Linguistic problems

1.  Does mathematics have a beginning and an end?  (Mainly addressed in the short first and last chapters, and in Chapter 7)
2.  Is mathematics created or discovered?  (Mainly addressed in Chapter 7; also in Chapter 3.  See also realism vs. nominalism, etc.)


1.  Is the image of mathematics in popular culture accurate, and if not, what can be done about it? (Mainly addressed in Chapters 6 and 8, as well as in the “How to Explain” dialogues)
2.  What does mathematics have (structurally and socially) in common with the arts, especially the visual arts?  (Mainly addressed in Chapter 3 and Chapter 10 and near the end of Chapter 8)
3.  What should mathematicians write or think about the motivations of literary authors, specifically (but not exclusively) authors of fiction, who make allusion to sophisticated mathematics in their writings?  (Mainly addressed in Bonus Chapter 5, but also in the Science Wars)

I was inspired to write this list by some of Mike Shulman’s recent comments (this one, for example), and more directly after reading a recent post on the always-engaging MathTango website.  The inimitable Shecky Riemann showed excellent taste in choosing Siobhan Roberts’s biography of John Conway as his “favorite popular math book” of 2015.  And I can’t fault his taste in picking MWA for the second slot.  Shecky writes:

Harris, more than any mathematician I’ve read, has a knack for saying things that sound interesting, but that are just vague or ambiguous enough to leave one uncertain of what his exact point is. That sounds like a criticism, but in some perverse way it makes his writing all the more thought-provoking and engaging…

Perhaps the exact point is that I’m uncertain (and maybe you should be too).