Calegari on OpenAI's 700+ math papers: a chance to broaden the field

Frank Calegari, writing as a guest on Terence Tao's blog, responds to OpenAI's October 6 release of over 700 research papers. He calls it a watershed in the history of mathematics and says AI is now undeniably capable of making groundbreaking contributions across many areas of mathematics. Tao's editorial note adds that the post was initially written in a different file format and converted using AI.

Calegari asks why some mathematicians are so unhappy. Part of the answer, he says, is the apparent disregard of AI companies for mathematical practice: the OpenAI papers, despite some tremendous ideas, are atrociously written, violate community expectations of attribution, and some are even wrong. But the community has far more control over how it hires mathematicians, publishes papers and trains students than over the behavior of trillion-dollar companies, and that is where he thinks effort is better spent. He also reads much of the reaction as a feeling that a beloved subject is under attack and being hollowed out. The release, he says, made barely a dent on the general problems of functoriality and reciprocity in the Langlands program, and even a full solution would not have solved number theory, just as the periodic table did not solve chemistry. Central problems, he argues, are the most natural boundaries of current understanding, and unlocking them is a first step toward new playgrounds.

He takes the hardest-hit case seriously. He quotes Hugo Duminil-Copin's lament that not a single major open problem he had publicly mentioned was left untouched, and that everything has been claimed to be proved. Calegari's answer is a historical precedent, the classification of finite simple groups. Dan Gorenstein announced a program in 1972, and after about a decade of intense activity announced the classification in 1983, somewhat prematurely, since the quasithin case was not completed for another two decades by Aschbacher and Smith. The classification was a success, but it also cooled interest in the subject once the famous problem was gone, even though rich questions remain (he cites John Conway's favorite: why does the Monster group exist?). He suggests the OpenAI release may feel to some fields like the classification dumped on a group theorist's floor in 1970 as smudged printouts spread like confetti. What is needed now, he says, are efforts like those of Gorenstein, Lyons and Solomon on the revised classification, and a refusal to pronounce subjects done. Those who seek out the old winding paths should be rewarded when it comes to jobs.

The positive case is personal. Calegari says that in the last thirty years human progress in mathematics has been remarkable, yet he has felt a growing unease: mathematicians often cannot explain to someone slightly outside their field why their work is interesting, and the profession has bent toward narrow technical expertise over bigger vision. He hopes AI helps people escape the habit of stopping at the first natural conjecture they cannot prove. As an example, he describes an idea he never pursued: treating different commensurability classes of compact arithmetic hyperbolic 3-manifolds as a kind of geometric Kolyvagin system, tied together by an unknown integral arithmetic version of the Jacquet-Langlands correspondence. It would have depended on many conjectures and a huge amount of background work. With AI, he says, the barriers to trying such a problem are much lower, because it is exploration of a new direction rather than an attempt to prove a theorem. He says this does not mean everyone must or must not use AI, but that priorities need readjusting. He quotes Christian Szegedy, who argues that the heroism of climbing what nobody else could will matter less, and that choosing which terrain to explore, steering fleets of airplanes and explaining findings to the world will matter more. Calegari says he is ready to board the plane. For pessimists the game is over; for optimists it has changed. Putting the existing community above mathematics itself, he says, would be a terrible mistake.

Under the heading 'The ugly truth', he criticizes the rush on the arXiv to post AI-assisted papers in order to be first. He sees a growing gulf between the ability to produce novel proofs and the ability to show understanding or push the subject forward, and says the Leiden declaration is being honored more in name than in spirit. He would give substantial credit to someone who picks a worthwhile problem, obtains a proof entirely with AI, understands it thoroughly and uses it to develop new questions, even if they supplied no new ingredient in the proof. Choosing worthwhile problems, he says, may end up mattering more than producing proofs. He warns that the field is in danger of repeating, a hundredfold, its old mistake of undervaluing exploratory work, and urges senior mathematicians with tenure to create future problems rather than just answer existing ones. A senior mathematician stressed about being scooped by AI, he says, is not being ambitious enough.

The last section asks what to do. Think pieces are pointless without a mechanism, he writes, and the most effective tools will be economic rather than rhetorical, because mathematicians want recognition, jobs and grants. The levers available are hiring and acceptance of papers at journals. His basic concrete suggestions: evaluate all work as if it is AI-assisted; make hiring much more about personal interactions than a paper file, interviewing postdocs and graduate students in person; and demand more from journals and abandon those that do not. The available text ends in the middle of this list.

Key facts

Frank Calegari, in a guest post on Terence Tao's blog, calls OpenAI's October 6 release of over 700 research papers a watershed and says AI can now make groundbreaking contributions across many areas of mathematics.
He says the OpenAI papers have tremendous ideas but are atrociously written, violate attribution norms, and some are wrong, yet he says the community should focus on what it controls: hiring, publishing and training.
He compares the moment to the classification of finite simple groups (program announced 1972, classification announced 1983, quasithin case finished two decades later), which cooled interest in the field.
He warns against the rush to post AI-assisted results first and says choosing worthwhile problems may end up mattering more than producing proofs.
His concrete suggestions include evaluating all work as if AI-assisted, hiring through in-person interviews, and demanding more from journals or abandoning those that do not.

Why it matters

This is a mathematician arguing, in a guest post on Terence Tao's blog, that OpenAI's October 6 release of over 700 papers changes the field for good, and that the right response is ambition rather than grief. Calegari does not downplay the shock: he quotes Hugo Duminil-Copin saying every major open problem he had mentioned publicly was claimed as proved. His answer is that solved central problems are boundaries, not endings, and that the profession should use the moment to broaden itself and reward exploration over quick proofs.

Who it affects

Working mathematicians at every level, especially those whose headline conjectures were covered by the release. Senior mathematicians with tenure are urged to create future problems rather than race to answer existing ones. Hiring committees, postdocs, graduate students and journals are the targets of his concrete suggestions, since he says these are the levers the community actually controls.

How to use it

The post offers practical steps rather than a tool. Evaluate all work as if it is AI-assisted. Make hiring more about personal interactions than a paper file, and interview postdocs and graduate students in person. Demand more from journals and abandon those that do not. Calegari says this is not a rule that everyone must or must not use AI. For individual researchers, he points to AI as a way to try ambitious, speculative directions that were previously too costly, such as his own unpursued idea linking hyperbolic 3-manifolds and arithmetic.

How solid is it

This is an opinion essay by one mathematician, who says he works in algebraic number theory and the Langlands program. Its claims about the OpenAI papers (poor writing, attribution problems, some errors) are his own assertions; no count of wrong papers is given, and no response from OpenAI appears. The historical account of the finite simple groups classification is his framing, not an evidence-based forecast. The visible text breaks off in the middle of his list of concrete suggestions, so the details behind them are not included here.

Risks and caveats

Calegari himself names the risks: a gulf between producing proofs and understanding them, a race to be first that leaves the Leiden declaration honored more in name than in spirit, and a possible repeat of the old habit of undervaluing exploratory work. He also acknowledges that some treasured parts of mathematical culture will not survive, and that grieving is human. His proposals are untested, and his comparison with the finite simple groups classification, which cooled interest in that area, cuts both ways.

“Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.”

— Hugo Duminil-Copin, quoted by Frank Calegari