Mathematician Terence Tao argues in a new essay that AI could plunge mathematics into a crisis reminiscent of the foundational upheaval of the early 20th century.

The math community should stop debating what AI can do, Tao writes in an essay for the 2026 International Congress of Mathematicians, and instead face a question it has largely ducked: What exactly are the goals of mathematical research, which depending on your perspective is either entering a golden age or a deep crisis?

Tao draws a parallel to the foundational crisis that rocked mathematics between 1900 and 1930, when Russell's paradox and Gödel's incompleteness theorems forced mathematicians to spell out assumptions they had always left implicit. That crisis produced a rigorous framework that has held up for a century.

Today, Tao argues, the stress test has shifted. It's no longer about mathematical truth but about "the largely implicit framework of mathematical values and practices": what counts as a contribution, what gets rewarded, what it means to understand something, and whether a machine can be said to have done the work.

His working hypothesis: "AI tools will, reasonably soon, become capable of performing a reasonable fraction of research-level mathematical tasks, with reasonable levels of success, quality, supervision, and cost."

As evidence, he points to the First-Proof Project. In the second round, ten never-published research problems were tested against four AI systems under controlled conditions. Seven of the ten received at least one passing grade from at least one system, meaning a solution judged essentially flawless or needing only minor revisions, at costs in the tens to hundreds of dollars per problem.

When metrics become goals

The many goals of mathematics have always been tightly linked, Tao writes, with solving problems, building theories, fostering community, and training the next generation all feeding into each other. AI threatens to pull them apart.

He cites Goodhart's law: "When a measure becomes a target, it ceases to be a good measure." Generative AI is especially prone to this because it chases the appearance of a good output rather than the real thing.

The financial incentives of the AI industry make things worse by rewarding exactly the kind of quotable, benchmarkable wins that mathematicians have long used as stand-ins for deeper goals.

Too many proofs, not enough understanding

If the working hypothesis holds, Tao warns, the field could shift from proof scarcity to proof abundance, with AI-generated proofs piling up faster than anyone can check, read, or absorb. The Erdős problem database already contains dozens of AI-generated submissions that no human expert has volunteered to verify.

AI-polished proofs have a different issue. In human-written proofs, Tao observes, the hard parts tend to retain natural friction: a careful lemma, a change of notation, and a paragraph that's clearly been rewritten several times. An over-polished AI proof strips away both the clutter and those useful signals, producing text that is "easy to read and hard to learn from." As Tao puts it, the "mistakes" in human exposition "can be genuinely helpful to the reader."

A proof nobody can explain is incomplete

For concrete guidance, Tao points to the Leiden Declaration on Artificial Intelligence and Mathematics, published in June 2026 and backed by the International Mathematical Union.

His own rule of thumb: "If the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published." A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

Training young mathematicians needs special care, Tao argues, because mathematicians need to protect the "irreducibly human aspect" of their work and keep AI tool use tightly restricted. The goal of training a mathematician is not achieved by producing correct homework. Tao himself discloses using AI for literature search, diagram creation, text completion, and converting his slides into paper format.

Recently, prominent mathematicians Timothy Gowers and Peter Sarnak credited large language models with real mathematical abilities but saw limits when it comes to genuinely new ideas, a finding that shows up in other research as well. Tao's essay goes beyond that debate because he's less concerned with what AI can or can't do and more with what mathematics itself actually wants.

Read on for the full picture.
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