Category Archives: Philosophy

Is it common knowledge that anyone is fit to be US President?


A few weeks ago, Terry Tao used Donald Trump’s perceived lack of qualification for the presidency to illustrate the difference between mutual knowledge and common knowledge, in a blog post with the normative title It ought to be common knowledge that Donald Trump is not fit for the presidency of the United States of America.  It’s common knowledge that Terry Tao, in addition to being one of the Mozarts of mathematics, is a very sensible person, and like every sensible person he is appalled by the prospect of Trump’s election as president.  As an attempt to account for this unwelcome prospect, Tao suggested that the correctness of Proposition 1 above is a matter of mutual knowledge  —

information that everyone (or almost everyone) knows

but not (or not yet) common knowledge

something that (almost) everyone knows that everyone else knows (and that everyone knows that everyone else knows that everyone else knows, and so forth).

It seems to me, though, that Tao’s formulation of the question — whether Trump is “fit for the presidency” or, in the words of Proposition 1, is “even remotely qualified” — is ambiguous.  The only axiomatic answer is the one provided by Article II, Section 1 of the U.S. Constitution, which implies unequivocally that Trump, like me but (unfortunately) unlike Tao, is indeed “eligible to the office of President” — though I admit I haven’t seen his birth certificate — and eligible is here the only word that is unambiguous and legally binding.

Now I realize that, even if you are a mathematician and therefore legally or at least professionally bound to respect the axiomatic method, you will object (at least I hope you will) that Tao did not mean to suggest that Trump’s bare eligibility was in question, but rather that Trump did not meet the more stringent criteria of fitness or even remote qualification.  By analogy, no one would deny that  ø (the empty set) is eligible to be a set, according to the usual axioms of set theory, but rather that

  1. ø is hardly anyone’s favorite set;
  2. ø is in no sense a paradigmatic set; and
  3. ø is not the kind of set for which set theory was designed.

Thus, even if it were mutual or even common knowledge that Trump is, so to speak, the empty set of American politics, that would hardly count as a consensus on his fitness or even remote qualification.  I’m naturally sympathetic to this kind of argument, but Tao made it clear that only comments that

directly address the validity or epistemological status of Proposition 1

were eligible for consideration on his blog.  While I’m hardly a strict constructionist, I don’t see how to avoid interpreting the word epistemological in terms of the maximal epistemological framework I share with Tao, which in this case can only be Article II, Section 1 (together with the Zermelo-Fraenkel axioms, but I doubt they are of much help here).

I was already leaning to a different explanation of the Trump phenomenon before offered this helpful but depressing roster of the worst (and best) presidents in the history of the United States, according to (unspecified) “scholars.”  Running down the list, one sees that, although Barack Obama is undoubtedly one of the most fit of all the presidents, intellectually as well as academically speaking, he only shows up near the middle of the ranking.  Presumably this is because he has been less effective as a politician than the presidents at the top of the list.  Judging by his words, I would like to say that Obama is one of the most morally fit of the presidents on the list; judging by his deeds, on the other hand — these, for example, or these — the record is much less appealing.  Jimmy Carter has proved to be both intellectually and morally admirable since leaving the presidency, but he made two of the biggest foreign policy blunders in recent history while in office (he ranks quite poorly on the list, probably for different reasons).

It is clearly mutual knowledge that the notion of fitness to lead a modern democracy, in particular fitness for the presidency of the USA ,correlates strongly with a shocking disdain for the notion that elections are designed to reflect the popular will.   My sense is that Trump’s supporters, and their counterparts across Europe, would like this to be common knowledge.  Fortunately, they are not the only ones.

This will be the next-to-last post for the summer; the next post will explain why it may be time to put this blog to rest permanently.


“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.



The belief that there are natural laws of finance

I continue the discussion of Frédéric Lordon’s Jusqu’à quand, which contains an explanation of the practices that made the 2008 crash inevitable that has yet to be translated into English although it is more specific and complete than Margot Robbie in a bubble bath or celebrity chef Anthony Bourdain making fish stew.  Here is a quotation that grapples directly with the most dangerous of the many illusions promoted by the manufacturers of mathematical models, namely that the objectivity of such models can be separated not only from the empirical observations needed to confirm and perfect them but also from need for a conceptual framework in which the question of the model’s objectivity can be raised.   Lordon argues that quantitative (probabilistic) models of finance are meaningless if there is no reason to believe that finance is governed by natural laws:

Obviously the most devoted partisans of quantitative finance will argue that any imperfection is transitory and remind us how the development of science is progressive but irresistible; even if today’s models still make a few mistakes, there is no problem that won’t eventually give way to hard work and research.  There are nevertheless reasons to think that this optimism will come up against a fundamental and insurmountable obstacle, rooted in the very question of knowing whether it is possible, and if so how far, to grasp financial risk through the calculus of probabilities.    Transcribing risk in the language of probability is such a common practice that it is never called into question.  The modelers, who consider the question trivial, are thus barely aware of the — absolutely non-trivial — theoretical choices they undertake when they make the hypothesis that the prices of financial assets are governed by this or that probabilistic law. … Of all the tools offered by the calculus of probability, the so-called “Gaussian” law is the one most commonly used … because it’s the simplest to manipulate.  But Gaussian laws have the unfortunate property of considering large price variations as events of minuscule probability… even though they are frequently observed in financial reality.  [Thus there is a competition to find the most realistic law…]  But the more frenetic the search for the “right hypothesis” becomes, the more one loses sight of the essential point…:  the belief that one will someday find the “right probability density” amounts to the belief that there are natural laws of finance.  This belief can be given the name of “objective probabilism” because it presupposes that there objectively exists a certain law of probability — “we’ll find it in the end” — that governs the price of assets.

Lordon finds more credible an “alternative approach,” which he associates with the name of André Orléan, according to which

“the” probability density doesn’t fall from the sky of “objective natures” but is rather the endogenous product of the interaction of financial operatives.  … It’s the radical modification of the configuration of interactions between operatives, expressed notably by the brutal variation of the degree of heterogeneity (or homogeneity) of behaviors, that produces the regime shift, and the qualitative transformation of the relevant probability density.

Students of dynamical systems will recognize the similarity to the language of René Thom’s catastrophe theory.  Lordon continues:

If one is absolutely set on maintaining the notion of law to describe the phenomena of finance — and the observation can be extended to all the phenomena of economics — one must bear in mind that the laws in question are not natural and invariant but rather temporary, variable, and contingent.   If one wishes, one can therefore preserve the general probabilistic framework but only after amending it substantially… where quantitative finance believes firmly in an objective probabilism, what one could even call a transcendent probabilism, there is in reality only an immanent probabilism:  the laws of probability do not fall to earth from a heaven of ideas, they emerge from below, shaped by the concrete interactions of agents — another way to rediscover that God doesn’t exist.

The last few words notwithstanding, here we see Lordon on the road to Spinozism.  We will not follow him there, but rather draw the conclusion that, if “There is no alternative,” the formula associated with Margaret Thatcher, it’s because the decision-makers, the Powerful Beings, as they are called in MWA, have contrived to make alternatives impossible.  It’s certainly not because alternatives are mathematically unthinkable.


Department of Euro-American mathematics?


Soviet 4 kopeck stamp in honor of Muhammad al-Khwarizmi

Jay L. Garfield and Bryan W. Van Norden have just published an opinion piece in the New York Times decrying the “systematic neglect” of non-European traditions in philosophy departments in the US and Canada.

…of the 118 doctoral programs in philosophy in the United States and Canada, only 10 percent have a specialist in Chinese philosophy as part of their regular faculty. Most philosophy departments also offer no courses on Africana, Indian, Islamic, Jewish, Latin American, Native American or other non-European traditions. Indeed, of the top 50 philosophy doctoral programs in the English-speaking world, only 15 percent have any regular faculty members who teach any non-Western philosophy.

Since they find it unlikely that the situation will change any time soon, they propose that most philosophy departments rename themselves Department of European and American Philosophy to reflect their “true intellectual commitments.”  I have a lot of sympathy with their position.  MWA quotes Garfield extensively as an authority on ancient Indian philosophy, specifically in regard to philosophy of mathematics.  I would only add Russian philosophy to the traditions that are generally absent from departments in North America; and I worry that the lack of diversity in philosophy departments appears to be much more severe than Garfield and Van Norden suggest.  For example, a colleague has told me that you can’t get a job in a Scandinavian philosophy department if you are a specialist on Kant, never mind Hegel or Derrida.  Whether or not this is an exaggeration, Department of European and American Philosophical Logic would be an accurate title for a great many departments on both sides of the Atlantic.

Garfield and Van Norden are not quite right, though, when they attempt to refute what they claim is a typical retort to their complaints about eurocentricity:

Others might argue against renaming on the grounds that it is unfair to single out philosophy: We do not have departments of Euro-American Mathematics or Physics. This is nothing but shabby sophistry. Non-European philosophical traditions offer distinctive solutions to problems discussed within European and American philosophy, raise or frame problems not addressed in the American and European tradition, or emphasize and discuss more deeply philosophical problems that are marginalized in Anglo-European philosophy. There are no comparable differences in how mathematics or physics are practiced in other contemporary cultures.

What is or is not “comparable” is in the eyes of the comparer, of course, and it’s no doubt true that cultural differences are no barrier to communication between contemporary mathematical practitioners in Asia and the rest of the world.  Historically, however, mathematics developed around the world in conjunction with a variety of metaphysical traditions, and this has inevitably affected the approaches to foundational matters.  I continue to believe what I have already written on this blog, namely that

I’m convinced that the most interesting problem currently facing philosophy of mathematics is to clarify how or whether Chinese and European mathematics differ and how or whether these differences reflect differences in the respective metaphysical traditions.

See also this earlier post for a discussion of how an abstract universalism tends to mask the persistence of privilege that very strongly aligns with the eurocentrism that Garfield and Van Norden rightly find objectionable.

How MWA promotes the (oppressive?) hierarchy


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

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

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

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

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

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

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

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

On his own limitations and uncertainties:

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

On the effects of (what MWA calls) charisma:

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

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

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

On the beauty of Galois theory:

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

On cooperation — and charisma! — in mathematics:

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


Alain Badiou bows down to mathematics


S’il existe une authentique Internationale, aujourd’hui, c’est bien celle des mathématiciens.

Alain Badiou, Éloge des mathématiques

When Alain Badiou, who is proud to call himself a Communist, claims that mathematicians represent the only authentic International (with a capital “I”), you know that, whatever the problem, he sees mathematics as part of the solution.   Badiou has written in Éloge des mathématiques that “the fundamental relation between philosophy and mathematics is a relation of reverence, if I may say so.  Something in philosophy bows down to mathematics.”

What, according to Badiou, is the purpose of philosophy?

I believe that philosophy has no other goal than this:  to allow anyone to apprehend, in the field of [his or her] vital experience, what is a happy orientation.  One could also say this:  to place at everyone’s disposal the certainty that the true life [vraie vie], that of a Subject freely oriented according to a true idea, is possible.

Mathematics contributes to this goal because the mathematical Subject’s orientation is free but disciplined:

…by virtue of their aesthetic force and of the invention they require, mathematics forces one to become a Subject whose freedom, far from being in opposition to discipline, demands it.  Indeed, when you work on a mathematical problem, the invention of the solution — and thus the creative freedom of the spirit — is not some sort of blind wandering, but is rather determined like a path always bordered in a way by the obligations of global coherence and of the rules of proof.  You accomplish your desire to find the solution not against rational law, but together with its help and its prohibitions at the same time.  Now that is what I began to think, in the first place with Lacan:  desire and law are not in opposition, they are dialectically identical.

Badiou’s secret mathematical love formula:  Desire = Law.  (Maybe someone can suggest a symbol for identity that is more dialectical than =?)  In a different formulation:

 Is there a happiness greater than [the pleasures that one finds in commerce]?  There’s the great question of philosophy.  Our societies, domesticated by Capital and fetishism of goods, answer:  no.  But philosophy, tenaciously and since the beginning, has labored to make us think that it exists.

For Badiou, although mathematics is not for everyone, it does offer a “model, limited but convincing, of the possible dialectical relation between the finiteness of the individual who works and strays, and the infinity of the Subject who has understood a universal truth.”

I have underlined many more sentences in my copy of  Éloge des mathématiques and could go on quoting them, explaining both my agreements and disagreements.  But the few passages translated above should give a sense of his aims.  Readers who are only familiar with philosophy of mathematics in the analytic tradition will probably find this all baffling, but you should bear in mind that Badiou sees philosophy not as a series of footnotes to science but rather, in the spirit of the ancient Greeks, as an accompaniment in the search for happiness and the vraie vie.   Mathematics plays a central role because, as Badiou sees it, philosophy only became possible with the development of systematic mathematical reasoning.

Here is the list of Badiou’s New York appearances last December (and a few from previous years).  Can you find the word “mathematics?”

Alain Badiou, “The Fundamental Contradictions of the Contemporary World”

Monday, December 15, 2014 – 6:00pm

Location: East Gallery, Buell Hall (Maison Francaise)

Event Category: Talks

Eminent French philosopher Alain Badiou will deliver a public lecture on the topic of the fundamental contradictions of the contemporary world.
Alain Badiou is a philosopher, playwright, novelist and political activist. Heis professor emeritus at the École Normale Supérieure in Paris and continues to teach seminars at the Collège International de Philosophie and the European Graduate School. Trained as a mathematician, Alain Badiou is one of the most original French philosophers today. His philosophy seeks to expose and make sense of the potential of radical innovation (revolution, invention, transfiguration) in every situation. In addition to several novels, plays and political essays, he has published a number of major philosophical works, including Theory of the Subject, Being and Event, Being and Event II: Logics of Worlds and Being and Event III: The Immanence of Truths (forthcoming).

Below is his December program in NYC:

December 13th, at 6pm

He will be at the Miguel Abreu Gallery for the book launch of the English translation of Gilles Châtelet’s To Live and Think Like Pigs, for which he wrote the foreword.

Miguel Abreu Gallery, 36 Orchard Street, New York, NY 10002.

(Gilles Châtelet obtained a Ph.D. in mathematics before he went on to a career in philosophy; more about him in future posts.)


December 13th, at 7 pm:

He will participate in a lecture-performance entitled “A Dialogue Between A Chinese Philosopher And A French Philosopher” at The Educational Alliance.

The Educational Alliance, Manny Cantor Center, 197 East Broadway, New York, NY 10002


December 15th, at 6 pm: 

Badiou will give a presentation entitled “The fundamental contradictions of the contemporary world”, at the Maison Française of Columbia University.

Maison Française of Columbia University, 515 West 116th Street, New York, NY 10027

December 16th, at 6 pm:

On the occasion of the US publication of his book, The Age of the Poets: And Other Writings on Twentieth-Century Poetry and Prose, he will give a talk about “Literature and Onthology” at the Maison Française of NYU.

Maison Française of NYU, 16 Washington Mews, New York, NY 10003


December 17th, at 7 pm:

He will be at the Jack Tilton Gallery to lead a discussion entitled “Some considerations about contemporary art”.


Literature and Philosophy: Roman/Romanesque

Alain Badiou in dialogue
with Emily Apter

Thursday, December 10, 2015, 7:00 p.m.

A New French Philosophy Event, co-sponsored by the departments of French and Comparative Literature

ALAIN BADIOU is a philosopher, playwright, and author. His books include Logics of Worlds; Being and Event; Theory of the Subject; The Century and a host of treatises and manifestos on aesthetics, Arab Spring, love, and, most recently, mathematics.

EMILY APTER teaches in the departments of French and Comparative Literature at NYU. She co-edited a collection of Badiou’s essays on literature with Bruno Bosteels (Verso, 2014), and is currently completing a book on “unexceptional politics.”