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Terence Tao says AI could trigger math's biggest crisis since Gödel
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Terence Tao, one of the world's most celebrated mathematicians, has issued a stark warning: artificial intelligence could plunge mathematics into its biggest crisis since the foundational upheaval of the early 20th century. In a provocative essay written for the 2026 International Congress of Mathematicians, Tao argues that the math community must stop debating what AI can do and instead confront a question it has largely avoided: what exactly are the goals of mathematical research? Depending on one's perspective, he writes, the field is either entering a golden age or a deep crisis.
Tao draws a direct parallel to the foundational crisis that rocked mathematics between 1900 and 1930, when Russell's paradox and Gödel's incompleteness theorems forced mathematicians to make explicit the assumptions they had always left implicit. That crisis ultimately produced a rigorous framework for mathematical truth that has held up for a century. Today, Tao contends, the stress test has shifted—no longer about truth itself, but about 'the largely implicit framework of mathematical values and practices.' This includes 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 is blunt: '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, Tao points to the First-Proof Project, where 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 ranging from tens to hundreds of dollars per problem.
The implications are profound. Tao cites Goodhart's law: 'When a measure becomes a target, it ceases to be a good measure.' Generative AI, he warns, 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 compound the problem by rewarding exactly the kind of quotable, benchmarkable wins that mathematicians have long used as stand-ins for deeper goals.
If the hypothesis holds, 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. Tao also notes a subtler issue with AI-polished proofs: over-polished text becomes 'easy to read and hard to learn from,' stripping away the natural friction—the careful lemmas, changed notations, and rewritten paragraphs—that signals genuine understanding to human readers. As he puts it, the 'mistakes' in human exposition 'can be genuinely helpful to the reader.'
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 is direct: if authors cannot convincingly demonstrate that they can give a clear, expert-level talk on their results, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if formally verified.
Training young mathematicians requires special care, Tao argues, because the field must protect the 'irreducibly human aspect' of its work. This means keeping AI tool use tightly integrated with human understanding rather than allowing automation to bypass it. The essay frames the moment as a choice: embrace AI as a collaborator while redefining mathematical values, or risk losing the very essence of the discipline. With the 2026 ICM set to debate these questions, Tao's call is likely to shape the conversation for years to come, situating mathematics alongside other fields grappling with AI's disruptive power.
