On July 24, 2026, the International Congress of Mathematicians opened in Moscow with applause for the new Fields medalists. Within 24 hours, on the same stage, 2006 Fields medalist Terence Tao poured cold water over the celebration.
“I believe mathematics is entering a turbulent period — a foundational crisis in its values and practices.”
He chose the phrase “crisis of the century” deliberately. The last time mathematics used that word was a century ago, when Russell’s paradox (1901) and Gödel’s incompleteness theorems (1931) forced the field to inspect its own foundations — the “foundational crisis” that consumed three decades and produced today’s rigorous axiomatic frameworks. What changed, Tao argued, is the object under inspection: last time it was the foundations of logic; this time it is the foundations of values.
First, Assume AI Has Already Won
Tao’s talk was titled “Mathematics in the Age of AI,” and he refused to wade into the sterile argument over whether AI is actually any good. Instead he reached for a mathematician’s move: set a working hypothesis.
He cited First Proof, an independent benchmark testing AI on brand-new research-level problems. On May 28, 2026, its second batch of 10 problems went to four AI systems under controlled conditions with human expert review. The result: AI solved 7 of 10 at publishable quality, at per-problem compute costs of $10 to $1,000.
“Ask the audience to temporarily assume that AI will soon complete a substantial fraction of research-level mathematics at reasonable cost and quality. I’m not asking you to believe it. This is a conditional analysis.”
Granted that premise, the real questions surface: if machines can solve, what are mathematicians for? What is mathematical research actually for?
Five Revisions of the KPI: Goodhart’s Law Comes to Mathematics
Tao invoked a law every working professional knows — Goodhart’s law (1975): when a measure becomes a target, it ceases to be a good measure. Set the KPI and the metric breaks. Using just the “solve problems” stage, he walked through five forced revisions:
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Solve as many open problems as possible. Instantly rewarded with a flood of wrong proofs of the Riemann hypothesis.
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Solve, and verify correctness. Proof assistants like Lean genuinely accelerate verification — but a new problem appeared: what if AI produces a correct, machine-verified proof of enormous length that no human can read? On the website cataloguing Erdős’s problem list, dozens of AI-generated proof submissions already sit unreviewed. Most are probably right. No human expert has volunteered to check.
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Solve, verify, and communicate clearly. AI proofs are verbose in the trivial places and hand-wavy in the crucial ones. Worse, AI polishes proofs until hard steps read as smoothly as easy ones. From this Tao drew one of the talk’s most counterintuitive conclusions: the flaws in human exposition may actually serve readers.
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Solve, verify, communicate, and get the community to absorb it. A result counts when other mathematicians digest it, accept it, and build on it. Community acceptance is an external process no author or AI tool can optimize unilaterally.
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All of the above, plus entering the canon. Key results must ultimately reach textbooks and the next generation. This is the least AI-accelerable step in the pipeline — and the most valuable part of the whole process.
From Proof Scarcity to Proof Surplus
The five revisions trace a complete pipeline: generate → verify → express → publish → canonize. Once AI enters, every stage jams. Tao borrowed an electrical-engineering term — impedance mismatch: when the flow rates of adjacent stages don’t match, the whole system fails.
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AI-generated proofs pile up awaiting verification;
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verified proofs pile up awaiting human-readable exposition;
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expository manuscripts drown traditional peer review;
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even published results outpace the community’s ability to canonize them.
He named the syndrome “proof indigestion.” Then came the talk’s central judgment:
Mathematics is moving from an era of proof scarcity to an era of proof surplus.
For millennia, everything in mathematics — its culture, honor system, evaluation machinery — rested on proofs being scarce. First solver takes all the glory, fair and square. When proofs become a bulk commodity, the foundation shifts.
The Way Out: the Leiden Declaration and the Talk Test
Tao pointed to the Leiden Declaration on AI and Mathematics (June 2026, formally endorsed by the International Mathematical Union) and added his own annotations.
First, disclose AI use — to avoid the worst equilibrium, where everyone secretly uses AI while publicly pretending otherwise. Tao led by example: his own slides carry the note that AI assisted with text completion and figure generation.
Second, lower the weight of “first to solve” and raise the weight of “digesting.” Exposition, publication, canonization — historically second-class work — become the scarcest capabilities. One line of the declaration drew his special emphasis: with AI use comes full human responsibility for correctness and citation.
From there he derived the hardest rule of the talk:
“If the author cannot convincingly demonstrate that they could give a clear, expert-level, correct, and properly attributed talk on their result, then the result should not be published.”
Translated: AI may solve it for you, but if you can’t explain it, don’t publish it. He called this the “Talk test” — who can explain and teach the result will matter more than who got there first, and it is the key filter against the knowledge bubble of the proof-surplus era.
The Future: an Active Fight Over Who Defines the Rules
Tao drew a clean line. In education and talent development, emphasize the human part and strictly restrict AI — the same for hiring, grant applications, and public outreach. Everywhere else, mathematics should move proactively and define the rules on its own terms rather than being dragged by the product logic of AI companies.
Two footnotes from the “crisis talk” say more than the framework does. Next to the centennial-crisis slide, he wrote: all em dashes in these slides are human-generated. On the disclosure slide: AI assisted with text completion and figure generation. Warning of a crisis, using the tools fluently, and joking about it in the same breath — that is the posture he modeled: don’t refuse, don’t conceal, but mathematicians write the rules.
A century ago, the foundational crisis took thirty years and left mathematics with firmer ground. Handled well, this one ends the same way — except what must be secured this time is not the certainty of logic but the central place of human understanding in the production of knowledge.
References: Tao’s ICM 2026 slides, “Mathematics in the Age of AI” (PDF) · Terence Tao on AI — a living summary · leidendeclaration.ai · Simons Foundation, “AI Will Be Top of Mind at ICM”
