Category Archives: charisma

Can mathematics be antiracist? Part I


If a disease like Covid-19 could push higher education to the brink of collapse, perhaps something is rotten in the system. This is what we should be addressing.        (Cinzia Arruzza, Chronicle of Higher Education)

Mathematics is deeply democratic.  You can be Black or White or any other color; male, female, or gender non-conforming; European or African or an extraterrestrial giant; ten years old or dead more than 40000 years.  As long as you know the rules, you are welcome to play; and history has shown that the rules are always flexible.

Mathematics is deeply antidemocratic.  Mathematics is not a “marketplace of ideas”; arguments are settled with QED and then they will never again be unsettled.  The rules have been established once and for all, and there is no room for dissenting opinions.

This post will not attempt to reconcile these two apparently incompatible visions of mathematics, both of which are at least implicitly invoked whenever the reasons for the field’s visible demographic imbalance are discussed.  Instead, I will take up the challenge proposed on the AMS inclusion/exclusion blog under the title #ShutDownMath:

Our goal needs to be to create an environment in which any person who WANTS to be a mathematician, can.



On second thought, I’m going to pass on this particular challenge, which, if taken literally, would necessitate a “full-blown social revolution,” to quote just different‘s comment on this blog — and the comment seems to suggest, reasonably enough, that “social revolution” talk can serve as an excuse for postponing action indefinitely.  But I do hope that whoever wrote that sentence will agree that an environment in which any person who WANTS to be gainfully employed as a mathematician — or gainfully employed at all, for that matter — has never existed; that according to Friedrich von Hayek such an environment would be impossible; and that therefore creating such an environment would require not only a thorough rejection of neoliberal thinking but more importantly a thorough reorganization of work and of the social distribution of wealth  — a social revolution, in other words.

So I will interpret the word “goal” in that sentence as “aspiration” and stick to challenges that can be accomplished, to continue quoting just different, provided “higher ed institutions radically change their curricula and admissions practices.”  This “change” may or may not be radical enough to qualify as the “change” in the sentence in #ShutDownMath that immediately follows the last one:

To change mathematics itself so that it actually serves Black and Indigenous communities.

Let me suggest as a friendly amendment, “to change North American mathematics itself…”  Leaving aside whether the rest of the world is really responsible for the aftermath of what have rightly been described as our Republic’s twin original sins, it’s wise to avoid hinting that even the wokest mathematicians in this highly militarized country are plotting to change the practice elsewhere.  Halfway through the second week of the protests following George Floyd’s murder, my French colleagues were grumbling yet again about the AMS’s soft imperialism through what they perceived as extortionate prices for MathSciNet subscriptions.   And I’m sure that they and most of my colleagues around the world would find grating the insistence of #ShutDownMath on individual rather than collective action.  (But because I don’t want to let my French colleagues totally off the hook, I advise everyone who reads French to check out this extremely timely article about French racism by filmmaker Raoul Peck (I Am Not Your NegroYoung Marx).)

But to return to the point, I don’t see any necessary connection between the last two sentences quoted.    If by “be a mathematician” #ShutDownMath means “obtain academic positions” (and not to work as quants or defense analysts or spooks or data miners), then the problem is well posed:  find the “any person[s] who WANT” to obtain such positions, and then match them through an expansion of the normal training process — unaltered — with the positions.  I’m going to pretend to share the inclusion/exclusion authors’ unfounded optimism that higher education as we know it will not collapse in the near future and there still will be stable academic positions in the numbers to which we have been accustomed.  And, just to make the speculation more lively, I will admit the next sentence from #ShutDownMath —

White and non-Black POC don’t need to recruit people as props to make us feel better, we need to get out of the way.

— and assume that the requisite number of positions for Black and Indigenous people who WANT to become mathematicians are not already occupied by White and non-Black POC.

Granting all these assumptions, I have no doubt that the people in question can actually be found by a concerted effort — something like a vastly expanded version of the Math Alliance program, together with the (possibly massive, but possibly not) funding needed to coordinate the process and to provide support as needed for Ph.D. students who presumably (because just different has taken social revolution off the table) have not enjoyed the unearned privileges of the typical Ph.D. students of the current generation.  This is probably the right place to insert an otherwise completely incongruous

White Privilege Anecdote

In January I rented a car in Manhattan for a day trip with my family to Long Island, which is home to many people (2.8 million, more or less, past Brooklyn and Queens) but for us counts as uncharted territory.  None of us had ever been to a Hampton before, and we frankly didn’t know what to make of it when we finally saw one, even after we had explored it past sundown.  On our way back, and close to the City, I stopped to refill the gas tank, then turned left down a side street, at the end of which the internet had chosen a place for our dinner.  I had hardly driven three blocks when I saw the lights of a police car flashing in my rear view mirror.  Although I had no idea what was going on, I duly pulled to the side and waited for the officer to shine a flashlight into the front seat and ask for my license (which I provided) and registration (which the rental agency had not provided).  He then explained that I had driven through a red light (which I had not seen, but which I believed was perfectly possible).  After a few minutes of cordial conversation, he told me he would not write a ticket but admonished me to stop at red lights in the future.  I thanked him (naturally) and as he left he also advised me to turn on my headlights, which I had forgotten to switch back on when I left the gas station, as I often do in a rental car.

“To change mathematics itself”

So with a good deal of attention, a commitment to funding that is (possibly, but possibly not) massive (but still negligible compared to budgets for police, not to mention the military), and considerable good will, I claim that the demographics of mathematics departments can be transformed in the space of a generation (the time to dispatch current professors to a comfortable retirement) to match those of the wider North American population, where the “environment” will be no more uncomfortable than that of society at large.  The authors of #ShutDownMath will now be justified in complaining that I either unwittingly or maliciously missed the point of the post, namely the sentences that preceded the one about “creat[ing] an environment”:

We also want to make a distinction here — the problem is systemic racism, not just underrepresentation (even “underrepresented minority” is a terrible term to use). If we continue business as usual, it is disingenuous to focus only on recruiting more Black, Indigenous, Latinx students into our programs.

My problem is interpreting the juxtaposition of “business as usual,” which presumably refers to the practices that, as we read in the next paragraph, “have marginalized many groups” and may even “cause irreparable harm,” with the image of the person who “WANTS to be a mathematician.”  No person WANTS to suffer irreparable harm, so either (a) the person doesn’t see “business as usual” as irredeemable, or (b) what the person WANTS involves “chang[ing] mathematics itself” into something about which we know primarily what it is not — not “entrenched in systems of white supremacy,” not a source of “structural and systemic oppression,” not marginalizing, not the status quo… not business as usual.

Now I have a hunch that it will be difficult to change business as usual in mathematics into something else without confronting the business as usual of higher education itself, with its hierarchies within hierarchies and a financial model whose sustainability has been visibly in question since the student debt crisis exploded in the wake of the 2008 financial crash.  But that once again leads us, if not into the treacherous landscape of social revolution which just different has taken off the table, at least into its foothills.  So let’s leave that aside for the moment and just acknowledge that there is an endless gradation between tinkering as in (a) above with hiring practices and bad attitudes and a thoroughgoing metaphysical transformation as in (b).  For example, I found this minimal list of the changes needed to “business as usual” in John Rice’s article in The Atlantic, where he calls it “third-degree racism”:

(1) acknowledging what constitutes third-degree racism so there is no hiding behind a lack of understanding or fuzzy math, (2) committing to developing and executing diversity plans that meet a carefully considered and externally defined standard of rigor, and (3) delivering outcomes in which the people of color have the same opportunities to advance.

Rice’s list is addressed to “major employers” but I suspect a majority of colleagues would agree that the changes would be welcome in mathematics departments, where the obstacles to bringing them about would nevertheless be formidable.  At least one prominent university department is already making what seems to me to be a good faith effort into working to overcome these obstacles.  And they seem perfectly compatible with Federico Ardila-Mantilla’s Four Axioms, which have been widely quoted and have been adopted by another (probably more than one) prominent university department.

However, these changes can be implemented while leaving the content and the professional practice of mathematics intact (not to mention its presumed metaphysical underpinnings).  And I worry that Rice’s talk of “delivering outcomes,” like the good faith effort mentioned above, looks dangerously like what Ibram X. Kendi calls assimilationism in his #1 New York Times bestseller How to Be an AntiracistFor Kendi assimilationism is one of the principal forms of racism.   The changes we were ready to celebrate in the last paragraph fall far short of (b), and if the “business as usual” of mathematics is already racist, none of the concrete measures I’ve seen suggested goes nearly far enough.

Is the metaphysical basis of mathematics intrinsically oppressive?  Is the mathematical hierarchy racist?  Can the metaphysics and the hierarchy even be separated?  Is the inclusion/exclusion blog’s perspective on inclusion and exclusion reformist or revolutionary — Antiracist or merely assimilationist?

It’s also worth examining whether or not this initiative to boycott police work is business as usual or a reaction against business as usual…

This was originally going to be a post about the unbelievably dreary details of the hiring process under “business as usual” in a typical pure mathematics department.  My aim was to identify stages in the process where a well-timed intervention could effectively outflank “third-degree racism,” change the delivered outcomes, and maybe even spark the beginning of a metaphysical as well as sociological transformation in the field as a whole.  But this post has gone on long enough, so those questions will have to be reserved for Part II, or for the social revolution — whichever comes first.


Villani is running for Mayor of Paris, but what does he represent?

You can read about it in English in a Reuters article signed by Elisabeth Pineau and Michel Rose, or in French in (among many other places) a Le Monde editorial (not behind a paywall).  But mostly you’ll read about how Macron’s strategy to win a symbolic victory over the established parties of the center-right and center-left has been “plunged into uncertainty.”  That doesn’t make for a very uplifting story.  You can test the argument Villani himself has put forward to motivate his candidacy

“Many complex problems will need to be worked out, which can be done by working together playing to our strengths,” Villani told supporters. “I’ve been tackling complex problems my whole life before entering politics.”

by substituting his fellow Fields medalists, one after another, for the “Lady Gaga of Mathematics”:  are you convinced?

His more relevant “strengths” can be read off the list of supporters who attended the announcement of his candidacy:

…nombre de ses soutiens sont à puiser chez les déçus de la “macronie”.

On y trouve le député Matthieu Orphelin, un proche de Nicolas Hulot qui a quitté le parti présidentiel en février, la députée Anne-Christine Lang, élue du 13e arrondissement, un ancien porte-parole de LaRem, Rayan Nezzar, et Paula Forteza, élue LaRem des Français de l’étranger.

(from the unsigned French Reuters article published on Mediapart but behind a paywall.)

The candidate Macron originally preferred has his own supporters:

Face à [Villani], Benjamin Griveaux peut compter sur l’appui du président de l’Assemblée nationale, Richard Ferrand, de Stanislas Guerini, de son ancien collègue Mounir Mahjoubi – ex-soutien de Cédric Villani – et de ministres comme Marlène Schiappa et Agnès Buzyn.

Meanwhile, the incumbent Socialist mayor Anne Hidalgo has yet to start her campaign for reelection, but her chances are looking considerably better since Macron’s movement split.  Here’s Hidalgo, looking very mayoral at a ceremony last year honoring the exiles of the Spanish Republic who played a crucial role in liberating Paris in 1944.


The Reuters article will tell you that this split

… could help Hidalgo win re-election, despite her own unpopularity due to the congested streets and polluted air of Paris.

This claim, which is typical of the English-language media, should not be taken at face value.  The bitterest attacks on Hidalgo have come from drivers, often from the wealthy western suburbs of Paris, who complain about her administration’s closure of parts of the city center to automobile traffic.  This policy has been promoted by her Green party allies (EELV — Europe Écologie les Verts) in the Paris city council.  Yannick Jadot, EELV’s candidate in the 2017 presidential election, has no problem with Hidalgo on that score:

“Le bilan d’Anne Hidalgo en matière d’écologie est celui des écologistes. Donc cette partie là, j’en suis très fier”, déclare Yannick Jadot.

Villani, meanwhile, announced that

he would be Paris’ first “truly environmentalist mayor”

But what does that mean?

Will Cédric Villani’s reputation survive his association with Macron?

Cédric Villani’s political orientation has come under increasing scrutiny since his election as deputy on the “République En Marche” (LREM) list created by French President Emmanuel Macron, whose own popularity has fallen precipitously over the first two years of his mandate.  Villani’s announcement that he would run, and occasionally dance, for the LREM slot in next year’s race to replace Paris Mayor Anne Hidalgo — Villani was president of her campaign committee in 2014 — roughly coincided with the development of the Yellow Vest movement in protest against Macron’s policies, which are perceived (quite rightly) as undemocratic and upwardly redistributive.  Macron’s approval rating has dropped to 27%, while the Yellow Vests were enjoying up to  60-75% approval, in spite of repeated attempts by the press to depict them as a Trojan Horse for the far right. Such attempts have been considerably more successful abroad than in France. No doubt this is because, for at least 25 years, every time a popular movement arose in France in opposition to the neoliberal trickle-down consensus that united the two main governing parties — the socialists (apparently fatally damaged by former President Hollande’s undistinguished record) and the right, currently called “Les républicains” — the mainstream press warned that such protests could only benefit the far right; the warning lost its force as living standards declined for much of the population, whichever party was in power, and the decline continues under Macron’s post-partisan leadership.

Each of the two major governing parties has been hit repeatedly by scandal. Macron was supposed to put an end to that, but the “Jupiterian President” quickly got caught up in a particularly sordid and growing scandal involving the unconventional and largely illegal employment of a security staffer named Alexandre Benalla (this is too complicated to explain here, but you can find an account here, not including the most recent developments).

Inevitably, Macron’s unpopularity has rubbed off on Villani.  His Tweets now routinely provoke mocking suggestions that he should stick to mathematics:  he is taken to task for failing to address the shocking consequences of police repression of Yellow Vest demonstrations — 18 eyes lost to flashballs as of this writing, 4 hands lost to dispersal grenades, and hundreds of other injuries, many of them disfiguring; for his silence on the Benalla affair; and for many of his votes in the National Assembly (his vote on the Loi Asile-immigration is the focus of an open letter by the mathematician Charles Boubel  published in the CNRS magazine Image des Mathématiques).

It didn’t have to be this way. Maurice Chiodo and Piers Bursill-Hall, in the first of an ongoing series of Ethics in Mathematics Discussion Papers, identify “four levels of ethical engagement,” in increasing order. Villani examplifies the third level, “Taking a seat at the table of power.” “[B]eing at that table,” they write, “gives a participating mathematician the potential to influence the ethical consequences of what is done with the maths.” In his insightful report on Villani in The New Yorker, Thomas Lin quotes Villani’s collaborator and former student Clément Mouhot:

Many mathematicians are glad that Villani is willing to participate in public life, Mouhot said, so that they don’t have to.

Chiodo and Bursill-Hall note with approval that a committee under Villani’s direction recently prepared a thorough and remarkably persuasive report for the French government on how to prepare France for the arrival of AI in mainstream society.  The particular strength of this report is that its authors have a deep grasp of the technology and the mathematics behind AI, and that gives them a particular political authority.
Chiodo and Lara Gordon elaborate on this in a later Discussion Paper, arguing that Villani’s report compares favorably, in both its ambition and its attention to ethical issues, with a British report on the topic.

In view of Villani’s remarkable success in persuading Macron’s administration to admit France’s responsibility in the torture and murder of the mathematician Maurice Audin — perhaps his most notable achievement to date — one would also have thought that Villani would be sensitive to the question of police repression.Villani Audin - 1

Villani at the Père Lachaise cemetery June 11, 2019, at the ceremony
in honor of Maurice and Josette Audin

As of this writing it’s not clear how far Macron’s administration will follow the suggestions of Villani’s report. One recommendation —

Doubling salaries in the early stages of their careers at the very least is a vital starting point, otherwise the pool of young graduates interested in higher education and academic research will definitively dry up. [Villani et al, p. 11]

was mentioned prominently in Le Monde‘s coverage. Not surprisingly, it was welcomed by French researchers, most of whom have seen the graph circulated in 2008 that predicts a straight-line evolution of starting research salaries to the French minimum wage, with convergence scheduled for 2025; as far as I know the evolution continues on track.  The salary recommendation was intended “to make public research careers more attractive” in the face of competition from the GAFAM tech giants. This particular suggestion disappeared without a trace, and one naturally suspects that Villani, in taking on the mission assigned to him, missed the point of neoliberalism, which is precisely its ideological commitment to market solutions to practically all problems.  Encouraging public academic research, on this reading, is perfectly inconsistent with Macron’s signature policies — lowering taxes on the rich and on corporations while raising them on the less fortunate; deep cuts in public services in health and education; elimination of favorable conditions for workers in the national railroad and plans for its future privatization; a “reform” of university admissions that has left hundreds of thousands of high school graduates with no options whatsoever; and a plan to multiply fees by 10 for students from outside the EU, including doctoral students. Attacks on the national pension system and privatization of the Paris airports are on this year’s agenda.   (See Didier Fassin’s report in the London Review of Books for a thorough analysis of Macron’s “authoritarian neoliberalism.”)
But perhaps it did have to be this way, after all.  Villani and Macron are both alumni of the French-American Foundation France, where they overlapped as “Young Leaders” in 2012. The existence of the think tank seems not to be well-known, but former “Young Leaders” include some of the French welfare state’s most prominent gravediggers, and an astonishing proportion of presidential contenders as well as leading cabinet members and political figures in both countries (included are Alain Juppé, François Hollande, both Bill and Hilary Clinton, General Wesley Clark, LA mayor Eric Garcetti, current French Prime Minister Edouard Philippe, and two directors of Le Monde, among many, many others).   Not all alumni are neoliberals — Didier Fassin’s brother Eric, a prominent leftist intellectual, is on the list, as is filmmaker Charles Ferguson — but you could do worse than to use the list to map the boundaries of mainstream political opinion.

Macron’s party is scheduled to choose their candidate this coming week for next year’s Paris mayoral election.   The French president reportedly prefers Benjamin Griveaux, but Libération reported on July 5 that it’s increasingly likely Villani will be chosen to face incumbent mayor Anne Hidalgo.  Villani was president of Hidalgo’s support committee in 2014, but he now claims to be dissatisfied with her record.  There are sensible reasons to be dissatisfied with Hidalgo, but most of the complaints I read seem to have to do with cars and bicycles.  Villani himself explained that

Quand on entre aujourd’hui dans Paris, on arrive à des portes qui sont souvent taguées, sales, dans une ambiance d’embouteillage phénoménale et ce n’est pas possible.

All true, but not more so than 20 years ago.  Many of my mathematical colleagues are convinced that Villani’s true goal is the French presidency.  How many more compromises will this “Young Leader” have to accept before he gets there?

UPDATE:  Macron’s party’s chose Benjamin Griveaux, not Villani, to run for mayor of Paris next year.  Villani is considering his next steps.  Meanwhile, yet another member of Macron’s government, François de Rugy (Minister of Ecological Transition), is caught up in a huge scandal, involving (among other things) lobsters.


Videos of the 2018 Mazur conference


The videos filmed and (barely) edited by filmmaker Oliver Ralfe were recently put online on the IHES YouTube channel.   Only about half of the events were filmed, but these include all the panel discussions (listed on the right hand side of the poster).   Here is the complete playlist:



Manjul BHARGAVA ‘Coming soon’



Poetry Panel



Persi DIACONIS – Barry Mazur as an Applied Mathematician



Philosophy/Law/Physics Panel



Jordan ELLENBERG – Heights on Stacks



History of Science Panel



Haruzo HIDA – Galois Deformation Ring and its Base Change to a Real Quadratic Field



Glenn STEVENS – Modular Symbols, K-theory, and Eisenstein Cohomology



Alexandra SHLAPENTOKH – Defining Valuation Rings and Other Definability Problems in Number Theory



Akshay VENKATESH – Derived Hecke Algebra for Weight One Forms and Stark Units



Wei ZHANG – Selmer Groups for Rankin-Selberg L-functions of GL(2)xGL(3)

Short proofs

August 2018 was this blog’s busiest month in two years.  Practically all the visits came in the first two weeks, with much of the traffic arriving from Germany (1788 of 5574 views).  The explanation, apparently, is that Peter Scholze’s Fields Medal was announced the first day of the month, and the Hausdorff Institute of Mathematics in Bonn chose my blog post  as one of three “interesting and popular articles” on his work, along with the article Erica Klarreich published in Quanta two years ago, and my chapter in the book What is a Mathematical Concept? edited by de Freitas, Sinclair, and Coles.  Quanta‘s articles on mathematics are notoriously interesting and popular; my chapter on the “perfectoid concept” may or may not be interesting, but I can’t imagine why anyone would consider it “popular”; and the blog post — which, as you may remember, is a text that did not qualify for publication in The New Scientist, is somewhere in between.

Anyway, my WordPress dashboard informs me that the Hausdorff Institute’s recommendations were picked up by (Frankfurter Allgemeine Zeitung) as well as  These two sites, together with the Hausdorff Institute, my indefatigable colleague Peter Woit’s blog, and the inevitable Google and Facebook, accounted for most of August’s referrals.

This year’s Fields Medals were widely covered by the international press, with Scholze’s story featured most consistently, along with the unexpected drama of the theft of Birkar’s medal.  Apart from Ulf von Rauchhaupt’s rather insightful article (visibly influenced by my blog post, not always with full attribution), coverage was mainly as approximate as one might expect, and was more informative about the current state of science reporting than about the priorities of contemporary mathematics.  Most entertaining for me was the article on the French website, which included this surprising bit of news:

Scholze a donc incontestablement la bosse des maths, mais il ne s’agit pas de son seul talent. En effet, il faisait partie d’un groupe de rock à 17 ans, puis a été professeur d’histoire allemande à 24 ans.

Rough translation:  “Scholze unquestionably has the math bump [a French expression that derives from phrenological notions popular in the 19th century — apparently there really is such a cranial bump, though its connection to mathematics is dubious] but it’s not his only talent: he played in a rock band at age 17, then at age 24 became professor of German history [sic!]”  Instead of a byline the article refers to three sources:  DW, El País, and Quanta.  I strongly suspect the sources were consulted and consolidated by a robot reporter which offered its own intrinsically logical interpretation of the sentence that opens the El Pais article:

Con 17 años tocaba el bajo en un grupo de rock, con 24 se convirtió en el catedrático más joven de la historia de Alemania.

The news coverage also revealed something of the network of journalists’ local contacts.  Thus the New York Times consulted Jordan Ellenberg, while El País quoted José Ignacio Burgos; Le Monde went to the trouble of finding four different mathematicians to contribute sentences about each of the four medalists:  Laurent Fargues (for Scholze), Philippe Michel (for Venkatesh), Jean-Pierre Démailly (for Birkar), and, inevitably, Cédric Villani (for Figalli).

Practically every article alluded to Scholze’s refusal of the New Horizons Prize, already discussed on this blog in 2015.   This came as no surprise to me; in fact, I had already anticipated the hypothetical reader’s fascination with this telling detail in the article I had prepared for The New Scientist, with the following sentence about his motivations:

My guess — but it’s no better than anyone else’s — is that he decided that the priorities of Silicon Valley are just not compatible with those of the mathematical community, as he sees it.

This means something very specific to me, and it may mean something to mathematicians reading this post, but to the hypothetical New Scientist reader it means exactly that Scholze refused the prize because he refused the prize, a vacuous observation embellished with the enigmatic expressions “mathematical community” and priorities.”  As we already know, this sentence never made it into the pages of The New Scientist; but, much to my surprise, it was translated into Spanish, at least twice, and at least once into German.  In each case my sentence was promoted to the status of a “speculation,” although the journalists had absolutely no reason to treat me as an authority on the matter, and besides which, as, I already explained, in the context of a newspaper article my sentence was totally devoid of content.  (Though one could always hope that a particularly attentive reader will find it surprising that not only is these such a thing as a “mathematical community” – though the word “community” disappeared from the German version — but that it even has “priorities”.   The reader may be sufficiently intrigued to wish to learn more about this, in which case:  good luck!)

Apparently Scholze’s refusal of the $100,000 prize cried out so desperately for explanation that the journalists grabbed at the only straw they found.  If they had been a little more patient, though they could have waited until August 6, when Scholze’s own answer to the question appeared in his interview with Helena Borges in O Globo:

O que posso dizer é que aquele era um prêmio e que este é outro. E é tudo que vou comentar sobre.

Rough translation, which curious readers are invited to ponder:  “What I can say is that that [the New Horizons Prize] was one prize, and that this [the Fields Medal] is a different one.  And that’s the only comment I’m going to make about that.”

The other item mentioned in practically all the press coverage recalled how Scholze distinguished himself already at age 22 when (quoting O Globo again) he “transformou uma teoria de 266 páginas em um texto sucinto de 37 folhas” — “transformed a 266-page theory into a succinct text of 37 sheets.”  Most of the other sources, starting with Erica Klarreich’s article in Quanta in 2016, identified the overstuffed “266-page theory” as none other than my book with Richard Taylor.  There is an interesting lesson hidden in that story about radical abbreviation, but that’s a silly (as well as misleading) way of putting it.  I was hoping to explain why that’s the case before I present an overview of the proof of the local Langlands conjecture to the graduate reading group that meets at Columbia tomorrow afternoon, but unfortunately I have run out of time, and I’ll have to return to the question later.

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.