Terence Tao gave the public lecture at the International Congress of Mathematicians in Philadelphia on Friday evening, under the title "Mathematics in the age of AI." Shortly after midnight UTC he posted the slides to his own site, along with something more unusual: a long interview about his views on AI, conducted by an AI, which he had instructed to be "somewhat hard hitting" and whose results, he wrote, "somewhat surprised" him.

The recording is not out yet — Tao says it "will likely be forthcoming in a day or two." But the 52 slides are public, and the argument in them is not the one the capability debate has been having.

He declined the argument everyone expected

The obvious lecture for Tao to give is the one about capability: can these systems actually do research mathematics, and how well? He has given versions of it many times. Instead he set it aside on purpose.

He states the question formally, as what he calls the AI Capability Conjecture: that "at some point in the near future, some AI tools will, at some expense, and with some level of human supervision, be able to correctly accomplish some research-level mathematical tasks in some fields of mathematics, with some non-trivial success rate, and at some level of correctness and quality." The repeated "some" is deliberate — a template with placeholders, admitting weak and strong readings.

Then he declines to argue it. "Despite the central relevance of the AI Capability Conjecture to the Community Response Question," one slide reads, "my talk will not be about that conjecture."

Instead he asks the audience to assume a reasonably strong version is true — a "Working Hypothesis" — and to reason conditionally from there. He is explicit that this is not an endorsement: "I am not asking you to want, believe, or accept that this hypothesis is true."

What that manoeuvre buys him is the ability to skip past the part of the debate that is stuck, and go at the part it keeps skipping. If the tools do work, he argues, mathematics has a problem that is not about tools at all. It is about what the profession actually values.

He frames it as a crisis in the foundations of mathematical values and practices, and draws the analogy deliberately: the discipline went through a foundational crisis once before, roughly 1900 to 1930, when Russell's paradox and Gödel's incompleteness theorems forced mathematicians to examine assumptions they had been happy to leave to philosophers. That period was turbulent, and the end product — an explicit, rigorous, standardised framework — was worth it. He expects the same shape again.

Goodhart's law, pointed at his own profession

The engine of the argument is a metrics problem.

Mathematics has many goals, Tao notes — solving problems, building theory, understanding the world, building a community, training the next generation, creating work of lasting aesthetic value. Historically these were positively correlated: progress on one tended to mean progress on the others, so one or two could serve as proxies for the rest, and many could be left unstated.

Then he invokes Goodhart's law — "when a measure becomes a target, it ceases to be a good measure" — and adds the sharp part: "The inherently ungrounded nature of generative AI, as well as the financial incentives of AI companies, make the use of AI tools particularly vulnerable to this law."

Optimise hard enough on the one goal that is easiest to measure and easiest to automate — problems solved — and the goals that used to move together come apart.

The pipeline, and where it jams

The centrepiece is a chain that Tao builds a slide at a time, each step added because the previous version of the goal was gameable.

Start with "solve as many unsolved problems as possible" — that gets you a pile of wrong proofs of the Riemann hypothesis. Add verification. Now you can get a machine-checked proof "that nobody — not even the original humans prompting the AI — understands." Add exposition, so results can be communicated. Then acceptance, because a proof only enters the field when other mathematicians digest it and build on it. Then canonicalisation: the slow business of a result becoming part of the textbooks and the definitive theory of its subject.

Generation. Verification. Exposition. Publication. Canonicalisation.

Tao gives the back half of that chain a collective name — "proof digestion", covering exposition, publication and canonicalisation together. It is the part of mathematics that turns a correct argument into something the field can actually use.

AI accelerates the front of that chain. It barely touches the back. Community acceptance "is ultimately an external process that cannot be optimized purely by the authors and their AI tools," and of the final stage he writes: "Such canonicalization of a proof is the slowest stage of all. It requires broad, deliberative consensus by the community. It is the stage least amenable to optimization by AI tools. But it is the most valuable part of the entire process."

The result is what he calls impedance mismatches, or "proof indigestion": unverified proofs piling up awaiting verification, verified proofs awaiting a readable writeup, correct and well-written proofs overwhelming peer review, and published proofs too numerous for the community to work into definitive form. His summary line: the field moves "from an era of proof scarcity to an era of proof abundance."

This is not hypothetical. He points at erdosproblems.com, which already holds dozens of AI-generated proof submissions that no human expert has volunteered to verify — and notes that "in several cases, even the human submitters have declared themselves unqualified to do so." Which leads to the question the slide asks outright: "Could we have a verified proof of a major result that no human understands enough to explain it?"

The one piece of evidence he cites — and the disclosures around it

Having declined to argue capability, Tao cites exactly one data point, and prefaces it with a complaint about the rest. Most of the data points, he says, "have not been gathered under controlled scientific conditions," and much of the publicly available evidence is "highly subject to reporting bias and non-scientific incentives, with some important costs and variables remaining undisclosed."

The exception he offers is First Proof, an independent benchmark run by Mohammed Abouzaid (Stanford), Nikhil Srivastava (Berkeley), Rachel Ward (UT Austin) and Lauren Williams (Harvard). We read their report rather than take the slide's summary of it.

Two disclosures belong next to that recommendation, and neither is on the slide.

First Proof states its own position plainly: it "has obtained unrestricted donations from Anthropic and from OpenAI, with pending funding from Google.org," alongside grants from the Survival and Flourishing Fund and the AI For Math Fund. The report says company donations are not used to compensate the editorial board. That is a considerably better disclosure record than the industry announcements Tao is criticising — but it means the clean counter-example to undisclosed incentives is partly funded by two of the labs whose systems it grades.

And Tao is not a neutral party to this benchmark. He is listed in the report as one of the principal investigators on System B, the UCLA Moonshot Harness — one of the four systems tested. The lecture recommends First Proof as the trustworthy evidence without mentioning that its author had a team in the run.

With that on the table, the results. Ten previously unpublished research-level problems were selected from a wider solicitation to working mathematicians, spanning computability theory, discrete geometry, stochastic PDE, lattice theory, von Neumann algebras and more. Four systems were tested: OpenAI's ChatGPT 5.5 Pro, plus open-source harnesses built by academic teams at ETH Zürich/Aarhus, UCLA and Princeton. Testing ran 28 May to 1 June; grading ran 4–8 June, on a double-blind journal-review model with around thirty expert referees, each of the 39 submitted solutions graded by at least two.

Seven of the ten problems drew at least one passing grade — meaning a solution rated essentially flawless or needing only minor revisions. Two more drew submissions rated as needing major revisions: an approach a referee judged potentially viable but requiring substantial human repair. One problem, Larry Guth's in metric geometry, is summarised as complete failure, with no system making substantial progress — though the detailed section notes one submission made non-trivial progress beyond the existing literature and could count as partial. On Problem 5, in stochastic PDE, a system produced a correct solution by a novel route that differed from the human one and, the report says, impressed the referees. Tao's slide puts compute costs at between $10 and $1,000 per problem.

One caution, because the numbers are easy to misread. A widely circulated figure says the models solved none of these ten problems. That is accurate, and it describes the preliminary testing used to choose the problems in the first place — problems were screened in April precisely to keep the ones that could not be solved by a standard approach. That round also ran different models under different conditions: a thirty-minute timeout, no harness, and, by the report's own note, ChatGPT 5.5 Pro was never queried at all — the very system that went on to be tested. Setting that zero next to the seven produces a tidy story about capability exploding in a month, and it is not what happened.

What thirty referees found, independently of Tao

The most useful part of the report is not the score. It is that the referees, working blind on a different exercise, described exactly the failure Tao describes on his slides.

Tao's complaint about AI exposition is that the writing "often dwells at length on trivialities, while passing very briefly through (or even obscuring) the most interesting and novel portions of the argument," and fails to connect results to prior literature. The referees, per the report: "AI solutions tended to handle routine parts of an argument in meticulous detail while glossing over the most difficult steps, sometimes asserting that a key claim follows from 'standard arguments' without justification, or citing papers that do not actually contain the claimed results."

The same finding, arrived at twice — once from a lectern, once from a grading pile.

The report also contains something blunter. Several solutions to Problem 2 borrowed phrasing from the problem author's own earlier paper "even line by line," reusing its terminology ("T-patterns," "bends") and even its labels (B, T, D, H) — without citing it anywhere. The report's judgement: "If a human had submitted such a solution it would have been flagged for plagiarism." The harnesses had attempted to check citations. It did not help.

There is a related observation that cuts against the excitement: the systems did best when a problem closely resembled something already in the literature. Three systems solved Problem 2 — all by translating the proof of a previously solved analogous problem into the right setting.

The recommendations, including one that bites

Tao ends on what the community should actually do, and cites the Leiden Declaration as "an excellent starting point" — with a lecture on it by Jim Portegies scheduled for 26 July.

Three of his own commentaries stand out.

Normalise disclosure of AI assistance, to avoid "the worst case scenario in which authors use AI tools covertly to aid their work, but conceal that usage to avoid criticism from peers."

Shift the prestige. "We need to decrease the emphasis on proof generation, and of being the 'first' to solve a problem; and increase the emphasis on 'proof digestion': exposition, publication, and canonicalization." He notes that editorial and refereeing work — the labour that actually converts individual results into collective understanding — is "often regarded as less prestigious" than producing proofs.

And then a concrete test: "my suggested rule of thumb: if the authors cannot convincingly demonstrate that they can give a clear, expert-level talk on their results, that is correct and properly attributed, then the result should not be published."

He also practises the disclosure he is asking for, in footnotes. Slide 46 carries: "AI tools were used to autocomplete text and to generate diagrams in these slides." And on the slide where he announces the crisis, a footnote reads: "All em-dashes in these slides were human-generated."

The interview he commissioned against himself

Alongside the slides Tao published a companion interview, conducted by the AI assistant that maintains a running summary of his AI views. He asked that the questions "not be a puff piece." They are not — several are the questions a hostile reader would ask, and the record shows him conceding ground.

Asked whether being mathematics' most prominent AI enthusiast fuels the hype he criticises, he discloses his position in some detail: he is "not directly paid by any of them, but have been gifted access to their premium frontier models," collaborates with people at Google DeepMind, has organised a conference sponsored by OpenAI, and has graduate students in an IPAM project funded by a donation from Math Inc. He also co-founded an AI-focused non-profit, SAIR, and fundraises through it — a body whose competitions he recommends from the closing slide of the same lecture. He then states the conflict himself: "It is true that this does give me an incentive not to criticize any of the tech companies I interact with, in case they are less inclined to fund the activities I would like to see supported."

His standing criticism of the industry is about scientific norms — that company announcements "often do not align with scientific norms, such as committing to report both positive and negative results." Pressed for evidence that engagement has ever changed anyone's behaviour, he is careful about how much he claims: he does not have "access to the counterfactual universe," but offers what he calls a small example where he thinks he had an impact. During a period when AI companies were repeatedly announcing solved Erdős problems whose proofs turned out to have gaps, to be already in the literature, or to omit precursor citations, he built a page systematically tracking the categories of AI-assisted solutions. "After that site was developed, the number of misleading claims by tech companies about their Erdős problem accomplishments dropped notably."

Pushed on whether AI widens inequality, he first reframes the question, is told so directly, and concedes: "Fair point, I did not interpret your previous question correctly. Yes, inequality is a serious concern with the current trajectory of AI development, with the most powerful models being proprietary, and also attached to companies that many people would prefer not to utilize for various ethical reasons."

The interviewer's final move is to ask him to put a number on when AI might originate genuinely new mathematical concepts unaided — the premise being that his 2023 "trustworthy co-author by 2026" call landed almost exactly on schedule, which earns him the right to be pinned down. Tao neither disputes the premise nor accepts the invitation. He does venture that a breakthrough of that kind, "possibly coming from a very different architecture than LLMs," would "still be several years away" — and then withdraws from the wider game: "the relative certainty I had in 2023 of predicting the next three years of development is gone now. The world is a far more unpredictable place, in so many dimensions, and pretty much anything is possible at this point. I'm not sure anyone is capable of any reliable forecasting beyond a year at best, currently."

A forecaster whose last forecast is widely held to have landed, declining to make the next one, is itself a data point.

Why this matters

The capability argument is the one that gets the coverage, and it is the one Tao has just declined to have in front of the largest audience mathematics assembles. His claim is that the interesting risk is no longer that the machines cannot do it. It is that they can, and that a profession which rewards being first to a proof — and treats digesting, explaining and canonicalising as lower-status work — is optimising for precisely the stage that automates most easily, while starving the stages that do not.

A small version of this has already played out in our own coverage. A claimed counterexample to the Jacobian conjecture was checked quickly, and then took a second, slower round of work to become usable: a geometric re-explanation of the example by others, and then Tao's own write-up on 21 July, which he describes as "a 'digestion' exercise to myself" — an attempt to render the argument with as little algebraic geometry as possible. He treats the three-dimensional result as settled and notes the conjecture "remains open in two dimensions."

The proof was the fast part. Making it something other mathematicians could actually pick up took people, and took days.

We also covered Tao turning a coding agent loose on his own 1999 applets, where the bug count was the real story.

Correction, 25 July 2026: an earlier version of this piece described the pipeline as six sequential stages, listing "digestion" as a step between publication and canonicalisation. That is wrong. Slide 43 shows six states joined by five arrows — generation, verification, exposition, publication, canonicalisation — and slide 47 defines "proof digestion" as the umbrella term for the back three together: "exposition, publication, and canonicalization." Corrected above.