# If Authors Cannot Give A Clear Talk On Their Math Results, Their Proofs Shouldn’t Be Published: Terrance Tao > Source: > Published: 2026-07-26 07:51:22+00:00 Even as [AI-generated math proofs](https://officechai.com/ai/an-anthropic-researcher-says-fable-just-helped-him-disprove-the-85-year-old-jacobian-conjecture/) are [beginning to proliferate](https://officechai.com/ai/mathematician-says-gpt-5-6-disproved-the-30-year-old-dinitz-garg-goemans-conjecture-with-4-simple-prompts/) across the internet, one of the most prominent mathematicians of recent times suggests we shouldn’t be rushing into publishing them. Speaking at a [lecture](https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf) during the International Congress of Mathematicians 2026, Fields Medalist [Terence Tao](https://officechai.com/ai/ai-is-allowing-me-to-experiment-and-try-crazier-things-mathematician-terrance-tao/) laid out a rule of thumb for a field that is being flooded with machine-assisted results faster than anyone can check them. His suggestion: if the authors of a proof cannot convincingly deliver a clear, expert-level talk on their own results, correctly and with proper attribution, then that result has no business being published at all. Tao, who has spent the last several months as one of the field’s most visible commentators on AI capability, framed the talk as an attempt to separate two very different questions that keep getting tangled together. One is whether AI models can actually do research-level mathematics. The other is what mathematics is even for, once they can. Tao spent little time on the first question. He pointed to results like the First Proof initiative, an independent effort that tests frontier AI systems against ten novel research-level problems per batch under controlled conditions. In its second batch, run in May 2026, seven of ten problems were solved at publication quality by at least one of four AI harnesses tested, at compute costs ranging from 10 to 1,000 dollars per problem. Tao has also personally verified AI-assisted solutions to long-standing [Erdős problems](https://officechai.com/ai/yet-another-erdos-problem-solved-with-help-of-gpt-5-2-terrance-tao-calls-it-most-unambiguous-instance-of-ai-solving-an-open-problem/), including one open since 1980, and has said AI now lets him attempt ideas he otherwise wouldn’t have bothered with. Rather than relitigate that debate, Tao asked the audience to simply accept a working hypothesis: that AI tools will, reasonably soon, be capable of handling a reasonable share of research-level mathematical work, with reasonable success rates and reasonable supervision. Once you grant that, he argued, the far more pressing question is what the mathematical community is actually optimizing for. ## When goals stop pointing the same direction For most of the field’s history, the various reasons mathematicians do research — solving open problems, building new theory, training students, producing work of lasting value — have moved roughly in tandem. Progress on one tended to mean progress on the others, so the community never had much reason to spell out which goal mattered most. Tao warned that this alignment is a historical accident, not a law of nature, and that AI is exactly the kind of pressure that exposes it. He invoked Goodhart’s law, the old observation that any measure, once turned into a target, stops measuring what it was meant to. Mathematics, he argued, is unusually exposed to this because generative AI systems are, in his words, inherently ungrounded, and because the companies building them have every financial incentive to chase whichever metric looks best in a press release. If “solve as many open problems as possible” becomes the target, the field already has a preview of what goes wrong, since bogus claimed proofs of results like the Riemann hypothesis have circulated for decades. Tao’s response was to walk through several successive refinements of that goal, each one patching a failure mode of the last: first requiring verification, then requiring clear exposition, then requiring the result to actually be digested and accepted by the community, and finally requiring it to be folded into the field’s canonical body of knowledge. ## The trouble with proofs nobody understands It’s the middle stages of that pipeline that concerned Tao most. Autoformalization tools built on proof assistants such as Lean have already sped up both proof generation and verification substantially, and he expects that to keep accelerating. But speed on those two fronts creates a specific new problem: an AI system can now produce a proof that is technically verified as correct, yet that nobody, including the human who prompted it, actually understands. He pointed to sites like erdosproblems.com, which he noted already contains dozens of AI-generated proof submissions that are likely correct but remain unverified by any human expert willing to vouch for them, with some submitters openly admitting they aren’t qualified to judge their own AI-assisted work. Exposition, in Tao’s telling, doesn’t fully solve this either. Current models write grammatically flawless prose that tends to linger on trivial steps while gliding past or even obscuring the hardest and most interesting parts of an argument. He described how human-written proofs naturally preserve a kind of friction around the parts the author genuinely struggled with, which is often exactly what signals a reader to slow down and pay attention. An AI-polished writeup can strip that friction out entirely, presenting the routine and the genuinely difficult with the same even, confident tone, without helping anyone actually learn the underlying idea. He went so far as to suggest that mistakes in human exposition can be quietly useful for exactly this reason, since they slow the reader down at the right moment. Quoting William Thurston’s 1994 essay on the subject, Tao reminded the room that mathematics was never about hitting some quota of definitions and theorems, but about whether the work helps people understand and think more clearly. A technically correct, beautifully formatted, and completely undigested proof fails that test even if it passes every other one. ## Publication as a human bottleneck, on purpose This is where Tao’s headline suggestion landed. Since community acceptance of a result is, by design, slow and human, and can’t be optimized away by better AI tools on the authors’ side, he proposed a fairly direct gate: authors should be required to demonstrate they can give a correct, expert-level talk on their own results before publication is even considered. If they can’t explain what they claim to have proven, that’s a signal the result hasn’t actually been digested by anyone, human or otherwise. He extended the same logic to teaching and mentoring, calling for AI usage to be normalized when disclosed and treated with suspicion when hidden, and for the field to stop rewarding whoever generates a proof first at the expense of whoever helps the community actually absorb it. He also flagged the Leiden Declaration as a useful early attempt to codify some of these norms. Tao was careful throughout not to present any of this as settled. He described the current moment as a second crisis in mathematical foundations, comparable to the one triggered by Russell’s paradox and Gödel’s incompleteness theorems in the early twentieth century, this time concerning not the logical bedrock of the field but its values and incentives. That earlier crisis took roughly three decades to resolve and left mathematics with a foundation still trusted today. Whether this one resolves as cleanly, he didn’t say.