Is it time to sell out?

When a machine is a mathematician's best friend

Robespierre, l’Incorruptible, denying himself a delicacy, in Andrzej Wajda’s 1983 film Danton, starring Gérard Depardieu and Wojciech Pszoniak

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                                                Mathematics as modern industry

I think there’ll be different ways of doing mathematics that just don’t exist right now. I can see project manager mathematicians who can organize very complicated projects—they don’t understand all the mathematics, but they can break things up into smaller pieces and delegate them to other people, and they have good people skills. Then there are specialists who work in subfields. There are people who are good at trying to train AI on specific types of mathematics, and then there are people who can convert the AI proofs into something human-readable. It will become much more like the way almost any other modern industry works. Like, in journalism, not everyone has the same set of skills. You have editors, you have journalists, and you have businesspeople, and so forth—we’ll have similar things in mathematics eventually.

(Terence Tao, interviewed 2024¹)

When I read this sort of thing it makes me want to tear my hair out. I can’t help crying out to the cosmos, though I know I’ll receive no response: “Why didn’t the journalist ask: is mathematics an old-fashioned industry, but in need of modernization? or not an industry at all? and why are so many mathematicians now leaping at the chance to join the assembly line?”

No day goes by without a major newspaper or magazine publishing yet another story about AI destruction of jobs and entire ways of life, but the AI future mathematicians portray to journalists is invariably utopian. Tao’s imagined future is no exception, and there’s no denying that his commitment to democratization of mathematics — his blog, his participation in the polymath project, his online lectures to audiences of all levels, and much else — has been exemplary. The utopian spirit was pervasive in his Spektrum der Wissenschaft interview on AI and mathematics, up to and including the paragraph quoted above, a vision of Fordism² without Ford.

But is Fordism possible without a Ford or some contemporary VC to crack the whip? Evgeny Morozov reminds us that even American politicians were once able to dream of such a thing:

In the late 1940s, the Democratic senator Harley Kilgore saw the dangers of postwar science becoming “the handmaiden for corporate or industrial research”. He envisioned a National Science Foundation (NSF) governed by representatives from unions, consumers, agriculture and industry to ensure technology served social needs and remained in democratic control.

It gets even more enticing:

Corporations would be forced to share their intellectual property (IP) if they built on public research, and would be prevented from becoming the sole providers of “solutions” to social problems.

But we know how the story ended: “with its insistence on democratic oversight and sharing IP riches, [Kilgore’s] model was eventually defeated.”

“Democratic oversight” kills killer apps. But what will be needed to make Tao’s vision a reality? His interview starts in the present — where else would it start? — with large-scale collaborations around formalizations in Lean, but the motor of history really starts humming “in the future,” when

instead of typing up our proofs, we would explain them to some GPT. And the GPT will try to formalize it in Lean as you go along. If everything checks out, the GPT will [essentially] say, “Here’s your paper in LaTeX; here’s your Lean proof. If you like, I can press this button and submit it to a journal for you.” It could be a wonderful assistant in the future.

That’s when I started shouting at my laptop:³

Sam Altman said it cost $100 million to train GPT-4!! What angel investor is going to cough up that kind of money for the privilege of ‘exploring the space of what is true and what is false and why things are true,’ (which is how Tao explains what mathematicians do)?!?

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                                 But is incorruptibility really the only option?

If you want to be taken seriously as a cynic, you have to believe, and make known your belief, that everyone will sell out for the right price. And you also have to be consistent, which means that this cold-eyed judgment applies to you as well — or, more to today’s point, to me. And indeed, many have been the times when I have confessed⁴ that I too am capable of selling out, but that no one had ever made me the right offer.

These confessions were uniformly met with indifference. Does this mean that everyone I know is as cynical as I am, or is this just common sense? Robespierre was known as l’Incorruptible; look where that got him. Or, closer to our own time, consider the case of Grothendieck, who turned his back on his career, and ultimately on the entire mathematical community, when he learned that the IHES⁵ was (very partially) subsidized by the French defense ministry.

Cover of Survivre et Vivre, Oct-Nov 1971

Grothendieck devoted several years of his life to the ecology movement through the group Survivre et Vivre,⁶ whose journal is pictured above. I saw Grothendieck in Princeton during my first year as an undergraduate, and I can confirm that, in addition to a colloquium on topoi and a seminar on crystalline cohomology, his evening lecture on the movement Survivre — the name changed a few months later — aimed to convince his audience to quit mathematics, immediately, because of the urgency of saving the planet.⁷

And no one quit.

Grothendieck may well be remembered and admired as the “rebel legend of mathematics,”⁸ but when I was teaching in Paris, most of my French colleagues expressed regret that Grothendieck had been unwilling to compromise his principles, usually pointing out that his chair at the IHES did not long go unfilled.

I never had the opportunity to observe Robespierre up close, so I will withhold judgment as to his ultimate corruptibility. But am I so cynical as to believe that even the Grothendieck I saw in Princeton had a price? Not a price exchangeable for goods and services, certainly; as I wrote in Mathematics without Apologies,

Grothendieck's search for what his colleague Roger Godement once called "total purity" encompassed his physical existence as well: in São Paolo in 1953-54 he subsisted on milk and bananas, and years after his withdrawal from mathematics, during a 45-day fast in 1990, he went so far as to refuse to drink liquids, bringing himself to the point of death in an apparent attempt to "compel God ...to reveal himself.

But suppose the French Ministry of Defense had offered to devote itself henceforward, credibly and unflaggingly, to the pursuit of world peace. Wouldn’t that have been enough to coax Grothendieck back to his office in the tranquil Bois-Marie?

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                                              But haven't I already sold out? 

One of Grothendieck’s biographers reports this incident in May 1968:⁹

Grothendieck went to the student meetings in Orsay and was at first shocked by the wave of aggression, hatred and contempt with which he was met as a leading representative of the “establishment.” He saw scientific research as threatened, and tried in vain to present and explain his point of view in discussions.

Many academics in France had similar experiences at the time. Most of them returned to their lives and comforts when the strikes ended. Grothendieck’s response was different.

…afterwards, as if stunned, Grothendieck reflected for several weeks at home on what he had heard and experienced in the student meetings. He came to the conclusion that the students - at least "in principle" - were right.

Grothendieck had many reasons to reevaluate his career, and eventually to reject it, but the Orsay confrontation must have left its mark.

À Survivre, où il est rejoint par d’autres mathématiciens aux sensibilités plus libertaires, il prend conscience du rôle oppressif qu’il a tenu jusque-là en tant que grand savant. Intégrant la critique soixante-huitarde, il analyse la recherche comme une activité répressive, tant pour les techniciens et les « scientifiques moyens » que pour les profanes.¹⁰

Many of the students who participated in last spring’s Gaza solidarity encampments were acutely aware of their own ambiguous status of powerlessness and relative privilege, the predictable mocking of right-wing commentators notwithstanding. And while they refrained from expressing “aggression, hatred and contempt” toward professors I don’t doubt they judged them on the attitude they took to the protests.

So how do I feel as a representative of the “establishment”? Isn’t my research also a “repressive activity”? I’ll return to that question at the appropriate time. Meanwhile, you’ll have to draw your own conclusions, because I’m about to talk about my research.

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                        Dreams of an AI scribe (or, my secret plan to sell out)

727 pages; Volume 1 is another 587 pages.

For the last few years I have been working with several colleagues on the first stages of a project in the Langlands program. Our main result at the current stage is a proof of a theorem that is complete except for its dependence on the stabilization of the twisted trace formula. This is the literal translation of the title of the book pictured above, and you would think that two thick volumes would be enough to put the question to rest. But no: in 1314 pages Mœglin and Waldspurger only¹¹ complete the stabilization for groups over number fields; my colleagues and I need the same results, but for function fields.

Number fields are the familiar home (for number theorists, at least) of the solutions to polynomials with rational coefficients. If you seek the roots of a polynomial —

for example — you will find them in a number field. As for function fields, all you need to know is that their analogies with number fields were the subject of a 1939 article¹² by André Weil, the very analogies that André Weil had in mind when he called them “furtive caresses.”

I will not go so far as to propose to pry loose the theorem we need from its furtive caress with a Mœglin-Waldspurger theorem, but the analogies to which Weil refers in his article have continued to guide both terms of the analogy since the middle of the 19th century. Very recently it guided Jean-Pierre Labesse and Bertrand Lemaire in their project that solves the problems connected with one half of the formule des traces tordue, fully expecting that the other half will be completed as well and the result will eventually be subjected to stabilisation.¹³

The analogies are an imperfect guide to the function field setting. Were that not the case, Labesse and Lemaire could have simply copied the existing literature for number fields. In fact, they would not have bothered, because such an exercise would have been entirely superfluous. Instead, they write,

Le but de ce travail est d’adapter la rédaction du Friday Morning Seminar (Princeton 1983-84) de Labesse-Waldspurger [LW] au cas des corps de fonctions.¹⁴

The word “analogy” (analogue in French) is absent here but occurs 30 times in the Labesse-Lemaire paper, roughly half of them pointing to an analogous statement or formula in [LW].

Letting themselves be guided similarly through the secret passageways of the two massive green books, an expert team will immediately recognize which pages can be lifted unaltered, which pages can be maintained after a mild adaptation, and which pages contain arguments that simply break down on the far side of the analogy and can only be salvaged on the basis of entirely new ideas. If mathematics were rational, the team would focus attention exclusively on arguments of the latter sort, which I’d guess represent at most 5% of the contents of the green books. If the team manages to find substitutes valid for function fields, it would publish those and assure readers that Mœglin and Waldspurger have taken care of everything else.

But mathematics is not rational, in many ways, some of them enumerated in an earlier post. Publishing practices¹⁵ and the reward structure are among its least rational features. Briefly: no journal editor would accept a paper¹⁶ consisting of the 5% of the arguments that need to be changed to adapt the Mœglin-Waldspurger books for function fields; but nor would any editor, working within a page limit, accept a paper that reproduces the remaining 95% as well, unchanged except for notation, and therefore not of sufficient interest. And the reward structure guarantees that it would be economically irrational for an ambitious researcher to spend the time needed to rewrite the green books, knowing that colleagues who are not experts would assume that the work is a mere exercise¹⁷ in analogy.

Here’s something else irrational. My collaborators and I will soon finish a paper proving a theorem based on the stabilization of the twisted trace formula in the case of function fields. The reward structure would value the paper more highly if there were a reference in the literature to this stabilization than they will under present circumstances, even though

• Everyone knows that the stabilization will eventually be proved;

• It’s perfectly plausible that completion of the yet-to-be-stabilized twisted trace formula will solve 100% of the issues required for stabilization, and the extension of the Mœglin-Waldspurger books will simply consist in pointing this out repeatedly;

• The contents of our paper, meanwhile, are exactly the same with or without the stabilization on which some of the proofs depend.

This infuriating situation notwithstanding, I would not advise anyone to formalize the Mœglin-Waldspurger books so that Lean can decide which pages remain valid when number fields are replaced by function fields and which require new ideas. And you can be sure I’m not going to do that myself.

But I have to confess that the prospect of foisting the work off on an AI agent, even Tao’s imagined GPT, looks deeply appealing. For His Majesty’s Government money should be no object. And if that agent and I develop a durable working relationship, will we be friends? Or will the agent be more like my butler?

1. Christoph Drösser, “AI Will Become Mathematicians’ ‘Co-Pilot’” Scientific American, June 8, 2024, originally published in Spektrum der Wissenschaft.

2. The term is from the essay “Americanism and Fordism,” from the Prison Notebooks of Antonio Gramsci, who recognizes the unlikeliness of a “Fordism without Ford” in his very first pages:

The fact that a progressive initiative has been set in train by a particular social force is not without fundamental consequences: the “subaltern” forces, which have to be “manipulated” and rationalised to serve new ends, naturally put up a resistance.

The future of fraternizing with AI that Tao and other mathematicians envisage has nothing of the kind. The alert science writer should be asking: why not?

3. which doesn’t talk back any more than the cosmos does, though I suppose that could be arranged…

4. “Hinted” would be a more honest way of putting it.

5. Institut des Hautes-Etudes Scientifiques, a research center for mathematics and physics, modeled on Princeton’s Institute for Advanced Study but on a smaller scale, where Grothendieck had been a permanent professor from its founding in 1958 until he quit in 1970.

6. It became Survivre et Vivre a few months later.

7. “Je pourrais ajouter que j’ai pris l’habitude, depuis le printemps dernier [when I heard him at Princeton, in other words: M.H.], lorsque je reçois une invitation pour faire des exposés mathématiques quelque part, et lorsque je l’accepte, c’est en explicitant que cela ne m’intéresse que dans la mesure où un tel exposé me donne l’occasion de débattre de problèmes plus importants…” Alexander Grothendieck, “Allons-nous continuer la recherche scientifique?” Transcription of a lecture at CERN, January 27, 1972, available online at Écologie & politique 2016/1 (N° 52), pp. 159-169.

8. That’s the title of a five-part podcast by France Culture, which will be broadcast in August (but you can listen to it already), and which I expect to review later this year.

9. Winfried Scharlau, Wer ist Alexander Grothendieck? Teil 3: Spiritualität. Chapter 4. The Revolution, translated by Melissa Schneps.

10. Introduction to the transcription of lecture at CERN, op. cit.

11. Only!

12. Sur l’analogie entre les corps de nombres algébrique et les corps de fonctions algébrique, Revue Scient. 77, 104-106, 1939

13. For specialists: Labesse and Lemaire carried out the fine spectral expansion. It remains to carry out the fine geometric expansion, and the news from Marseille is that progress is being made, for example here.

14. From the first slide of a lecture by Labesse and Lemaire on February 4, 2021.

15. I nearly wrote “the publishing ecosystem” but caught myself in time.

16. Only a masochist would want to read such a text, but that’s a fair description of practitioners in many branches of mathematics.

17. A term of art referring to work that is not creative.