Terence Tao: Mathematics in the Age of AI
(August 17, 2026)
Tao's essay based on his ICM 2026 lecture. Instead of arguing about whether AI can do research math, he conditions on the assumption that it will, and asks a different question: what do mathematicians actually value?
Key ideas:
1. A second foundational crisis. 1900-1930 forced math to make its foundations explicit (set theory, formal proof). AI will force us to make our values explicit: what counts as a contribution, what we reward, who gets credit.
2. Goodhart's law hits math. Historically, all goals of mathematics (solving problems, building theory, training students, community) were correlated, so "solve open problems" worked as a proxy for everything else. AI breaks this: it optimizes for the appearance of success, and the AI industry is financially rewarded for exactly the benchmarkable metrics we used as proxies. Over-optimize one goal and the goals diverge.
3. The problem-solving pipeline is 5 stages, not 1. Generation → verification → exposition → community acceptance → canonicalization (getting into the textbooks). We only ever made the first stage explicit. AI accelerates stage 1 massively, barely touches stage 5, which Tao calls the most valuable part. Fun point: AI training data is the output of canonicalization.
4. Proof scarcity → proof abundance. Journals, priority norms, hiring, prizes were all designed for scarcity. Expect "proof indigestion": verified proofs nobody understands, correct results nobody has refereed. This is already happening on the Erdős problems database.
5. Exposition can be over-optimized too. Human proofs have "natural friction" where the author struggled, and that friction teaches you where to pay attention. AI-polished proofs sand it away, making texts easy to read but hard to learn from. He shows a Bourgain page he annotated in frustration as a grad student.
His rule of thumb: if the authors can't give a clear expert-level talk on their result, it shouldn't be published. A verified proof no human can explain is incomplete.
Also cites the First Proof project: 7 of 10 novel research problems got an essentially correct AI solution at $10-100s of compute per problem.
@abdumalik_abdukayumov
