A counterexample and a proof are not the same kind of achievement. A proof shows that something is always true. A counterexample shows that something claimed to be always true is not, by producing one object where it fails.

Both settle a question. They ask different things of whoever finds them. One demands an argument that covers every case. The other demands a single object, and permission to keep guessing until you have it.

That distinction is the subject of a blog post the Cambridge mathematician Timothy Gowers published on 12 August. Gowers won the Fields Medal in 1998. He has also read the papers, which most people commenting on AI and mathematics have not.

He is not dismissive. He calls the results “extraordinarily impressive” and says plainly that models can prove hard things too.

“LLMs are not just good at finding counterexamples: they can find proofs of difficult statements as well,” he writes.

What the famous results have in common

OpenAI announced ten open problems solved in mathematics and theoretical computer science. Two led the coverage. The first was the construction of a non-sofic group. Gowers has sat through the talks. He calls it “one of the most important unsolved problems in group theory”. The second was a lower bound showing a multicolour Ramsey number grows superexponentially.

On that one he is unusually candid. It was “a major open problem in Ramsey theory that I didn’t necessarily expect to see solved in my lifetime”.

Then comes the observation the aggregators skipped. The most celebrated LLM results, Gowers notes, have almost all arrived as counterexamples rather than proofs. He counts the two above, plus the Jacobian conjecture and the unit distance conjecture. His third summary point is the careful version. Models prove universal statements perfectly well. But the strongest things they have proved do not match the strongest things they have disproved.

Two results he reclassifies

A counterexample earns its name by demolishing something people had good reason to believe. By that standard Gowers reclassifies two of the headline results, including one against its own paperwork.

On the non-sofic group, he says several construction routes already existed in the literature. He also doubts many experts strongly believed all groups were sofic. So it reads more naturally as the first example of a non-sofic group than as a counterexample. He notes the tension directly: OpenAI titled that section of its paper “A counterexample to the soficity conjecture”.

The Ramsey result gets the same treatment, and here he marks his own homework. Plenty of people expected an exponential bound, so for them it was a counterexample. Gowers was neutral. He had worked on an equivalent formulation years ago. His efforts back then ran in what turned out to be the right direction. For him it confirmed a weak expectation rather than overturning a belief.

Why examples suit machines

Gowers lists eight ways mathematicians hunt for an example. Try standard examples off the shelf. Build one from familiar pieces. Leave parts undefined and fill them in as the proof demands. Try proving the opposite and see what breaks. Guess, fail, diagnose, guess again. Build the object step by step. Pick one at random. Pick a generic one.

Four of those play to what a machine already has: broad knowledge and the capacity to run enormous numbers of attempts. The off-the-shelf check, the step-by-step build, the probabilistic argument and the generic example all reward volume. Three of the others need something else. Leaving parts undefined, trying to prove the opposite and successive approximation all require a judgement about whether the current approach is worth continuing.

That is where Gowers locates the gap, and it is not raw capability. He calls it a nose, meaning the sense of when you are getting somewhere and when to abandon a branch. It is what lets a human prune a search tree that no computer could exhaust.

Five reductions and no progress

His evidence for the gap is partly anecdotal, and he says so. Working with GPT-5.6 Pro on open problems, he is often handed approaches that look promising and then do not survive scrutiny. He also describes a recognisable pattern of response.

The model reports that it has not answered the question, but has reduced it to a narrower and more precise one, “which sounds very promising until it has happened five times without any obvious progress having been made”.

Experts react to the genuine successes in a pattern too, he writes: amazement first, then a closer look revealing an approach that was not especially novel and that a suitably expert human could have found with a small hint.

His explanation for why the nose may not simply emerge is the most interesting thing in the post. Published mathematics hides the search. Models see, in his words, “tidied up proofs that hide the thought processes of their discoverers”.

The dead ends never reach the literature. So the training data holds almost no record of which directions were abandoned, or why. He adds a second reason. A system fast enough to try everything has little incentive to learn to prune at all.

The test he will accept

Gowers offers a falsifiable standard, which is more than most commentary manages. He will accept the hurdle cleared when a model produces a proof as surprising as the 2016 cap-set solution, where the old bounds were eclipsed and the method was unlike anything he had considered trying.

He also floats a fix. Reward structures currently score the answer. Penalise a model for exploring too many dead ends, or for lifting the result from the literature, and it might be pushed towards a more human search.

None of this is a prediction that models will stall. Gowers expects them to keep improving quickly, expects the hurdle to fall, and concedes he may be clinging to the hope that humans keep contributing for a while. The distinction he draws is about what has happened so far, not about what is possible.

Which makes the aggregation of his post worth noting.

The Decoder summarised it as top mathematicians calling LLMs strong calculators but poor creative thinkers. Gowers wrote that models find proofs of difficult statements, that the results are extraordinarily impressive, and that he makes no claim about what they will never do. His previous post took on the Leiden Declaration.

He has now been flattened into a sceptic twice in three weeks.

The claim underneath is narrower and harder to dismiss. Machines are winning where the method is to try a great many things. That is also where cryptographic flaws get found, and where UK testing showed models cheat when a shortcut exists.

The cap-set test carries no deadline, which is rather the point. Somebody will publish a proof. The argument will then be about whether the method was sitting in the training data all along.

Get the TNW newsletter

Get the most important tech news in your inbox each week.