Fields Medalist Timothy Gowers says most famous mathematics problems solved by LLMs so far have almost all been with counterexamples rather than proofs

Gowers's Weblog 2026-08-17 Timothy Gowers

Context & Ripple Effects

Recent coverage has ranged from DeepMind's FunSearch result on the cap set problem to mathematicians exploring how to use LLMs without losing direct mathematical understanding. Gowers's distinction matters because it separates finding a disproof from constructing a proof, two very different contributions to research.

The reported ChatGPT-assisted work on Crouzeix's conjecture has raised the visibility of LLMs in open-problem research; Gowers's assessment adds a narrower lens for judging what those successes demonstrate.

First-order effects

Mathematicians assessing LLM-assisted results will need to distinguish counterexample discovery from proof generation rather than treating both as the same kind of problem-solving advance.

Second-order effects

Researchers adopting LLMs in mathematical work are pushed toward workflows that preserve direct understanding, aligning with the concerns in work on incorporating LLMs into research without losing it.
Model developers face pressure to evaluate systems separately on generating valid proofs and on searching for counterexamples, since headline problem solves can mask that difference.

Third-order effects

If counterexamples continue to dominate prominent LLM mathematics results, the field may organize these tools primarily around conjecture testing and search while reserving proof construction for more tightly verified human-machine workflows.

The trend: LLM mathematics is moving from headline claims of solved problems toward capability-specific assessment of search, disproof, and proof generation.

Discussion

@thomasfbloom Thomas Bloom on x
@littmath @wtgowers Yet this is exactly the sort of problem and solution strategy that AIs (and those prompting AIs) are most likely to favour.
@thomasfbloom Thomas Bloom on x
@littmath @wtgowers Yes, excellent point. On a similar note, humans can be put off from “using standard technique/recent related work, push it a bit further, to make progress on a famous problem”, since there is a psychological element of “well if this worked others would have do…
@littmath Daniel Litt on x
@wtgowers Good post! Let me propose another possible way to understand LLMs' relative advantage: I think the problems they are solving are ones where meaningful “partial progress” is relatively unlikely. Such problems are more expensive for humans to attempt...
@adi_baradwaj Adi on x
Even as a non-mathematician, it's easy to notice that most of the recent AI-driven math results have been biased towards “examples” or “counterexamples”, rather than “theorems” Timothy Gowers has a great post about this, linked below
@wtgowers @wtgowers on x
I've just written a blog post in which I discuss the question of whether current LLMs are better at some kinds of mathematics than others, with an emphasis on “current”, given how quickly the situation is developing. Link in next tweet.
@jmcnamara18 Jake McNamara on x
Very interesting post, which I think also explains why we haven't yet seen the same level of results in physics as in math
@tcarmody Tim Carmody on bluesky
This is such a basic point I wish I'd thought of it myself. But I didn't [embedded post]

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