Terence Tao & 25 Fields Medalists Warn: AI Math 'Breakthroughs' Cause Proof Indigestion

AI Companies Turn Math into a Marketing Stunt

Anthropic and OpenAI have recently claimed that their unreleased models can solve millennium prize problems such as the Riemann Hypothesis and the Navier–Stokes equations. These announcements are made via press releases, wrapped in marketing language, and circumvent peer review entirely.

Tao's ICM 2026 Talk: A Second Foundational Crisis

Terence Tao, responding at the 2026 International Congress of Mathematicians (ICM), drew a parallel with the 1900–1930 foundational crisis triggered by Russell's paradox and Gödel's incompleteness theorems. He argued that today we face a "crisis in the foundations of mathematical values and practices" — but that a thorough examination will ultimately make the mathematical community stronger and more resilient.

The Core Question: Not Whether AI Works, But What We Want

Tao framed the discussion around a Community Response Question : How should the mathematical community respond to modern AI technology and its real or claimed research capabilities? He broke this into sub‑questions.

3.1 AI Capability Conjecture

Most public "data points" about AI's mathematical ability are not gathered under controlled scientific conditions; they suffer from reporting bias and hidden costs. Tao stressed the distinction between the truth of a conjecture and its desirability .

3.2 Controlled Evidence: The First Proof Challenge

The only rigorous evidence Tao cited is the First Proof evaluation (1stproof.org), an independent, controlled benchmark:

Each batch contains 10 brand‑new, research‑level problems across multiple fields.

Batch 2 was tested on 4 AI harnesses under controlled conditions on 28 May 2026.

Results were judged by experts for both correctness and exposition quality .

7 out of 10 problems were solved at a "publishable" level by at least one team.

Compute cost per problem ranged from $10 to $1,000.

This, Tao said, is the scientific attitude: look at controlled experiments, not press releases.

3.3 Working Hypothesis: Condition on Strong AI Capability

Tao then made a pivotal methodological move: he declared the rest of the talk would not debate the AI Capability Conjecture but instead assume a Working Hypothesis — that a fairly strong version of the conjecture is true. This conditional analysis sidesteps the "does AI work?" quagmire and focuses on the consequences if it does.

Goals and Values: The Ghost of Goodhart's Law

Under the Working Hypothesis, the real question becomes the Goals and Values Question : What are the precise aims of the mathematical community? Tao listed goals such as solving open problems, developing new theories, understanding the world, building the community, training the next generation, contributing to the shared knowledge network, and creating works of lasting aesthetic value.

He then invoked Goodhart's Law (1975) : when a measure becomes a target, it ceases to be a good measure. Over‑optimizing for AI‑generated solutions risks decoupling these goals — solving Olympiad problems, solving Erdős problems, solving research problems, building theory, applying knowledge, building community, training mathematicians, and creating enduring aesthetic works could all diverge.

Case Study: Six Iterations of the Goal "Solve Problems"

Tao demonstrated how the definition of "solving a problem" must be refined through successive iterations, akin to a mathematical proof.

5.1 Version 1: Solve as Many Open Problems as Possible

Optimizing for sheer quantity leads to a flood of incorrect solutions to major conjectures — a danger known long before AI.

5.2 Version 2: Solve and Verify

Adding verification (e.g., via proof assistants like Lean, Coq, HOL) accelerates checking, but raises a new issue: what if the AI produces a verified proof that no human — including the prompter — can understand?

5.3 Version 3: Solve, Verify, and Express Clearly

Current AI tools produce flawless spelling and formatting but often dwell on trivial steps while glossing over or obscuring the novel, interesting ideas. They rarely connect to prior literature or provide high‑level overviews. Tao contrasted this with his own heavily annotated copy of a 1991 Bourgain paper — the "friction" of wrestling with the text is where understanding actually happens. He quoted William Thurston (1994): "We are not trying to meet some abstract production quota of definitions, theorems, and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics."

5.4 Version 4: Add Community Digestion and Acceptance

A proof must be accepted and valued by the community; other mathematicians need to digest it and incorporate it into their own work.

5.5 Version 5 (Final?): Feed into the Domain's Definitive Theory

The full pipeline becomes: open problem → unverified solution → verified solution → well‑written solution → accepted solution → definitive solution .

5.6 The Era of Proof Surplus: "Impedance Mismatches" and "Proof Indigestion"

If the Working Hypothesis holds without policy and cultural changes, the pipeline will suffer systemic impedance mismatches , which Tao calls proof indigestion :

AI‑generated proofs will pile up awaiting verification.

Many verified proofs will wait for a readable rewrite .

Even correct, well‑written proofs will overwhelm traditional peer review .

Published proofs will become so numerous that the community cannot hammer them into definitive form .

In short: we are transitioning from a "proof scarcity" era to a "proof surplus" era.

From Diagnosis to Prescription: Three Recommendations

Normalize disclosure of AI use. Avoid the worst case where authors secretly use AI and hide it to evade criticism. Tao led by example, noting in a footnote: "These slides used AI tools for text autocompletion and figure generation."

Recalibrate evaluation systems. De‑emphasize raw proof generation and being "first" to solve a problem; elevate the value of "proof digestion" — exposition, publication, and canonization.

Adopt a rule of thumb. If an author cannot convincingly demonstrate they can give a clear, expert‑level, factually correct, and properly attributed account of their own result, that result should not be published. This rule would filter out the vast majority of "AI‑generated, human‑signed, nobody‑understands" submissions.

The Joint Declaration: 25 Fields Medalists Speak

If the ICM talk was an academic analysis, the declaration on mathandai.org (https://mathandai.org) is a collective statement. Signed by 25 Fields Medalists (including 2026 laureate Yu Deng), it argues that while AI/LLMs show astonishing progress in mathematical problem‑solving, AI companies treat "solving math problems" as a benchmark and target, which is severely misaligned with the true goals of the mathematical community and may damage mathematical science itself.

The declaration calls on mathematicians, AI developers, and society to urgently address this misalignment, ensuring that as AI changes how we work, we do not forget what mathematics and intellectual work are fundamentally about. It concludes that AI does offer potential to enhance and accelerate genuine mathematical research and understanding, but whether the changes are ultimately beneficial or destructive "will depend to a large extent on the decisions of the humans who control this new technology."

https://mathlib.org/
Mathematics in the age of AI
https://arxiv.org/abs/2608.16753