Roundtable video, incorrect proofs, true theorems

The Helix Center has now posted the video of Saturday’s round table on “Mechanization of Mathematics”.  The discussion was lively and everyone agreed that we should meet again, or even that we should organize a conference on the theme.

Since concern about correctness of proofs is one of the primary motivations of mathematicians who are active in automated proof verification, it was interesting to hear several colleagues at the IAS quote the following remark about Solomon Lefschetz:

He had marvelous intuition, and so far as I know, all of the results he claimed in algebraic geometry have now been proved. When I was a graduate student at Princeton, it was frequently said that “Lefschetz never stated a false theorem nor gave a correct proof.”

This is Philip Griffiths reminiscing, in his contribution to the biographical memoirs of Lefschetz (on p. 289).  The Helix Center discussion did raise the question of mechanizing mathematical intuition, but didn’t reach any conclusions.  The mathematicians I know would prefer to have correct proofs of correct theorems, but if our choice were between mechanical generators of false proofs of false theorems and false proofs of true theorems I guess we would pick the latter — especially if they were as consequential as the hard Lefschetz theorem.

And indeed, I was surprised to learn — but perhaps I should not have been — that Griffiths’s description of Lefschetz fits quite of few of the mathematicians I have most admired (I won’t name names).

Several people I admire were in the audience and others were watching the livestream.  Kevin Buzzard congratulated me for finding a way to quote William Burroughs (at 53′).  I repeat the quotation for the reader’s convenience:

[The] junk merchant does not sell his product to the consumer, he sells the consumer to his product. He does not improve and simplify his merchandise. He degrades and simplifies the client.

—William Burroughs, Naked Lunch

UPDATE:  A point I was trying to make at the roundtable, and also in the middle of this article, and in this post, about the inevitability of mechanization of mathematics (and of everything else, and of the monetization of the resulting data by tech companies), is made much more clearly and forcefully by Rose Eveleth in an article published today on Vox.


Roundtable on Mechanization of Mathematics

From the announcement:

Proof, in the form of step by step deduction, following the rules of logical reasoning, is the ultimate test of validity in mathematics. Some proofs, however, are so long or complex, or both, that they cannot be checked for errors by human experts. In response, a small but growing community of mathematicians, collaborating with computer scientists, have designed systems that allow proofs to be verified by machine. The success in certifying proofs of some prestigious theorems has led some mathematicians to propose a complete rethinking of the profession, requiring future proofs to be written in computer readable code. A few mathematicians have gone so far as to predict that artificial intelligence will replace humans in mathematical research, as in so many other activities.

One’s position on the possible future mechanization of proof is a function of one’s view of mathematics itself. Is it a means to an end that can be achieved as well, or better, by a competent machine as by a human being? If so, what is that end, and why are machines seen as more reliable than humans? Or is mathematics rather an end in itself, a human practice that is pursued for its intrinsic value? If so, what could that value be, and can it ever be shared with machines?

With Stephanie Dick, Brendan Fitelson, Thomas Hales, Michael Harris (who will largely follow the script already presented here), and Francesca Rossi.  At the Helix Center, 247 East 82nd St.

Mathematics, music, philosophy, and Alain Badiou


Panel at IRCAM, June 7, 2019.  Left to right:  François Nicolas, Yves André, Fernando Zalamea.  Alain Badiou is seated in the audience on the left.

To celebrate the publication of the third and final volume of Alain Badiou’s Being and Event trilogy, the organizers of the Paris MAMUPHI seminar — MAthématiques, MUsique, PHIlosophie — devoted a two-day conference at IRCAM (Institut de Recherche et Coordination Acoustique/Musique), under the title L’hypothèse du contemporain.

For the 20th anniversary of the Mamuphi seminar (mathematics-music-philosophy), these encounters are dedicated to L’Immanence des vérités, the latest work by the philosopher Alain Badiou, and, more particularly, to his theory of “works-in-truth”. How are works distinguished from “waste” and, incidentally, “archives”? The final part of the work by Badiou formalizes a limitless alternative to the oppression of finality. These days in June gather together mathematicians, musicians, philosophers, and the author to formulate their own hypothesis in the shadow of their reading of the contemporary in the 21st century.

Yves André invited me as one of the mathematicians, and because of my deep respect for André’s writings about mathematics — and of course for his mathematical work — I was pleased to accept the invitation.

Badiou’s three-volume system is heavily based on set theory and much of the third volume is devoted to the theory of large cardinals, with chapters on ultrafilters, theorems of Scott, Jensen, and Kunen, 0#, and much more.  I have no idea what the upcoming Columbia graduate workshop will make of all this.  My own presentation had nothing to do with set theory; my aim was to explain why Badiou was wrong to hint in his book, in passing, that the mathematics of Andrew Wiles belonged with the “waste,” or at best the “archives.”  You can watch my talk or you can read it (preceded by a couple of pages explaining my misgivings about the theme of the conference).

I have to confess a less highbrow motivation, though.  Here is an excerpt from a review by Stanley Chang of Mathematics without Apologies that appeared in Society, dated June 25, 2018.

Other reviewers, both academics and nonacademics, have quite forcefully deprecated his use of ideas without context, the irrelevancy of various sections, an unreadably poor organization, and a purposely opaque stream-of-consciousness that prohibits understand [sic] rather than encourages it. One of my own friends, an anthropologist in academia, laughingly said that his treatment of Badiou is something that you would expect from a bad first-year philosophy essay from a bad student at a bad university.

Although Chang gave excellent reasons for his evident dislike of the book, he went out of his way to give it a fair reading, and I have no problem with his review.  But why did he make up this part about Badiou?  MWA contains no “treatment of Badiou.”  According to the index, Badiou’s name appears three times, and only in endnotes.  Two of the references are direct quotations, without anything that can be construed as a “treatment,” and the third quotes Juliet Flower MacCannell’s comments on a quotation by Badiou regarding Lacan’s theory of love, along with Vladimir Tasić’s gloss on the quotation and the comments.

In the Q&A following my talk in Paris I got a laugh from Badiou by suggesting that the mere mention of his name would provoke the laughter of many American philosophers, not to mention anthropologists.  But I don’t think that explains Chang’s sentence.  Maybe he was confusing Badiou with Bourdieu?  Or maybe the treatment in question was on this blog, for example here?

I would fault the editors of Society for allowing the publication of that last sentence, or any sentence, on any subject whatsoever, that quotes an anonymous anthropologist — laughing no less — for the sole purpose of taking a cheap shot.  But in fact I have no idea what the sentence is about.

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?

The Ted Hill affair and the new Science Wars

the gap is so vast between whatever such studies measure and anything resembling an appreciation of the difficulties of coming to grips with the conceptual content of mathematics… that the label of science simply doesn’t apply.

The full article is at Politics/Letters.  See my article in Science for the People for an idiosyncratic comparison of the New Science Wars with the vintage version.

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.