Mathematician Frank Calegari calls OpenAI's 700-paper release a watershed for mathematics

A watershed, with caveats

Mathematician Frank Calegari argues that OpenAI's release of more than 700 research papers on October 6 marks a turning point in the history of mathematics. In a guest post published on Terence Tao's blog on November 1, and featured on the Hacker News front page, Calegari writes that AI is now demonstrably capable of groundbreaking contributions across many areas of the field. His question is not whether this matters, but why so many mathematicians greeted it with dismay, and what the community should do about it.

Poorly written, sometimes wrong, still significant

Calegari does not defend the papers themselves. According to his account, they contain some strong ideas but are badly written, disregard the community's norms around attribution, and in some cases are simply mistaken. His argument is that mathematicians have almost no influence over the behaviour of trillion-dollar companies, but considerable control over how they hire, how they publish and how they train students, and that is where he thinks the effort belongs.

He also pushes back on the feeling that the discipline is being gutted. The release, he notes, barely touched the general problems of functoriality and reciprocity at the heart of the Langlands program. And even a complete resolution of Langlands would not amount to finishing number theory, any more than building the periodic table finished chemistry: headline conjectures mark the natural boundaries of current understanding, and unlocking them opens new ground rather than closing the subject.

A colleague's lament

The post highlights a reaction from Hugo Duminil-Copin, who wrote that not a single major open problem he had publicly mentioned over his career, whether in talks, lectures, articles or grant proposals, went untouched by the announcement. Everything, he said, was claimed to be proved. He had expected a few of his problems to appear on the list; instead he got all of them, at once, delivered casually. Calegari concedes he cannot say much about percolation theory, Duminil-Copin's area, but offers a historical parallel instead.

Lessons from finite group theory

In 1972, Dan Gorenstein announced a program to classify the finite simple groups. The classification was declared in 1983, prematurely in Calegari's view, since the quasithin case was only completed roughly two decades later by Aschbacher and Smith. The achievement was a success that nonetheless cooled interest in the field: once the famous problem was gone, most mathematicians turned elsewhere, even though rich questions remained. Calegari cites John Conway's fondness for asking why the Monster group exists at all.

The lesson, in his telling, is that a field whose central problems are suddenly claimed does not become finished. He argues the community now needs efforts analogous to the revised classification carried out by Gorenstein, Lyons and Solomon: consolidating and verifying what exists, refusing to pronounce subjects done, and rewarding, in hiring decisions, the people who keep exploring the older questions rather than abandoning them.

What AI actually changes

For Calegari, the practical promise of AI is that it lowers the cost of ambitious, speculative work. He describes a long-standing personal idea connecting compact arithmetic hyperbolic three-manifolds associated to imaginary quadratic fields, via quaternion algebras, into something resembling a geometric system of Heegner points bound together by an integral version of the Jacquet–Langlands correspondence. He never pursued it, he explains, because it rested on too many unproven conjectures and would have required an enormous amount of background work just to state precisely. With AI, trying such an idea no longer requires proving a theorem; the point is to explore a direction that may or may not be nonsense.

He also frames AI as a corrective to overspecialisation, writing that he has long been troubled by how often mathematicians cannot explain to anyone even slightly outside their field why their work is interesting. Fittingly, as Tao notes in an editorial remark, the post itself was converted from another file format with the help of AI.

Why it matters

The post matters less as a verdict on OpenAI's papers, whose claimed proofs Calegari himself says vary in quality and correctness, than as a proposal for how a scientific community adapts when the cost of producing research collapses. If AI can now contribute across many areas of mathematics, the binding constraints shift to verification, attribution, training and incentives: who gets hired to explore questions a model has already claimed, how students learn to check machine-generated arguments, and whether whole fields get declared closed prematurely. The classification of finite simple groups is a cautionary precedent for that last risk. Calegari's position, in short, is that mathematics has changed, and that the field's own institutions, rather than the AI companies, are the part mathematicians can actually change in response.