Terence Tao tells mathematicians to stop debating AI capability and ask what math is for

Terence Tao, a UCLA mathematician and 2006 Fields medalist, has published a 12-page essay titled "Mathematics in the age of AI," based on a public lecture he gave at the 2026 International Congress of Mathematicians in July. The paper, submitted to the proceedings of that congress and posted to arXiv on August 17, takes an unusual move: instead of debating whether AI tools can do research-level mathematics, it assumes they will and asks what the mathematical community is actually optimizing for.

The capability question, parked

The essay organizes the debate around a template it calls the "AI Capability Conjecture": at some point, some AI tools will, at some cost, accomplish some research-level mathematical tasks with some non-trivial success rate. Tao notes most public evidence for or against various forms of that statement suffers from reporting bias and undisclosed costs, and he deliberately does not resolve it.

Instead he asks the reader to work under a "Working Hypothesis": that a reasonably strong version of the conjecture is true. His framing is that the mathematical community is entering an era of turbulence comparable to the crisis of foundations around 1900 to 1930, except that this time the stress test falls on implicit values and practices: what counts as a contribution, what gets rewarded, who, or what, is credited with the work.

Goodhart's law and a five-stage pipeline

Tao's central argument runs through Goodhart's law. Historically, the goals of mathematics (solving problems, building theory, training people, creating lasting works) were positively correlated, so one goal worked as a proxy for the rest. Once AI tools optimize one proxy hard, he argues, the previously aligned goals diverge. Generative systems are good at exploiting the gap between a measure and the thing it measures, and the economics of the AI industry reward exactly the kind of quotable, benchmarkable achievements built on those proxies.

He then deconstructs problem solving as a case study, iterating through five goal formulations. "Solve as many open problems as possible" fails because it produces wrong proofs of the Riemann hypothesis. Add verification, and you get a new failure mode: proofs that check out in a proof assistant like Lean or Rocq that no human understands. Tao points out this is already happening; the Erdős problems database has dozens of AI-generated proof submissions, and in several cases the submitters say they are not qualified to verify them.

Add clear exposition, and you hit a subtler problem he calls out: AI-written proofs are grammatically flawless but tend to dwell on trivialities and obscure the interesting parts, and they can even sand away the "natural friction" of human writing that signals to readers where to slow down. Add community acceptance, and then canonicalization into the definitive textbooks of a field. That last stage, Tao writes, is the slowest of all and the one least amenable to AI. The paper closes the case study by noting that AI tools themselves depend on the canonicalized theory humans built, since their training data is the output of that process.

Data points and recommendations

The essay cites the independent First Proof project, which tests frontier models on genuinely novel, unpublished problems. In its second batch, evaluated in May 2026 against four systems, seven of ten problems received at least one expert-graded passing solution, at a compute cost of tens to hundreds of dollars per problem, according to the project's results.

For how to respond, Tao points to the Leiden Declaration on AI and Mathematics, published in June 2026 and endorsed by the International Mathematical Union, which offers 23 recommendations to mathematicians, institutions, funders, and policymakers. He quotes four directed at individual mathematicians: disclose tool use in papers, make your work easier to review, keep authorship credit with humans, and invest effort in attribution. He adds his own rule of thumb: if the authors cannot give a clear, expert-level talk on their result, it should not be published, even if a machine verified the proof.

The essay ends on practical ground: education and the training of young mathematicians may need tighter restrictions on AI, while other parts of the workflow need community-defined best practices rather than vendor-defined ones. Tao also discloses that AI assistance went into literature search, diagram generation, and formatting for the paper itself.