Tim Gowers · 2026-08-12 · notable
Tim Gowers — LLMs crack maths problems with counterexamples, not proofs
Tim Gowers notes that the famous maths problems LLMs have cracked were almost all settled by counterexamples, not proofs. The Fields Medallist argues wide knowledge plus fast search explains the wins; pruning bad ideas is still human work.
A Fields Medallist on why LLM maths wins cluster around counterexamples rather than proofs.
What is it?
Tim Gowers picks at a pattern in recent AI maths results: the famous problems that have fallen were almost all settled with counterexamples. He points to OpenAI's announcement that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group and a proof that the multicolour Ramsey number R(3,3,...,3) with k threes grows superexponentially in k. He writes that the Jacobian conjecture and the unit distance conjecture went the same way.
How does it work?
The mechanism behind those wins, Gowers argues, is search rather than any special talent for one kind of statement. A model with wide mathematical recall can throw many candidate objects at a problem quickly, and an object that works is cheap to check once you have it. That combination of breadth and speed is what suits construction-shaped problems.
Why does it matter?
What Gowers says is still missing is judgment about which idea to drop. Working on open problems with 5.6 Pro, he reports being 'often presented with approaches that sound promising until I think about them carefully, and then seem quite a lot less promising.' He expects models to stay weakest on 'problems where the mysterious human ability to prune the proof-discovery search tree is particularly advantageous' — a practical filter for reading the next round of AI maths announcements.
Who is it for?
mathematicians and people tracking AI reasoning claims