In May, word reached Fields Medalist Timothy Gowers that an AI had resolved the Erdős unit distance conjecture — an open problem in geometry for nearly 80 years. His first reaction: if true, mathematicians may soon be out of a job. He spent the night recalibrating his worldview. The next morning, a clarifying email arrived: the AI had not proved the conjecture. It had refuted it. Gowers felt relieved. Why is a disproof more comforting than a proof? On August 12, his blog post answered that question.
The May rumour that almost ended mathematics
The backdrop: on August 1, OpenAI announced ten results in mathematics and theoretical computer science, all produced by an unreleased internal model called Astra. Per the official framing, each result solves a long-open problem or makes substantial progress on one. Developer Simon Willison compared the mood among online mathematicians to a collective "Deep Blue moment," and mathematician Kirwin Hampshire had already published a piece titled The Dark Night of Mathematics, describing a genuine crisis of confidence. Gowers conceded that the ten results are extraordinary — then asked a sharper question: if large models were stronger than every human in every branch of mathematics, their speed advantage should make results pour out like a flood. Why no flood?
Ten results from Astra — most of them demolitions
Look at the flagship results and a pattern appears. On the Erdős unit distance problem, the AI did not extend the decades-old growth bound; it imported tools from algebraic number theory into discrete geometry, constructed families of point sets over complex number fields, and showed that unit-distance pairs can grow faster than the field had long believed possible — a counterexample to a widely held estimate. On the Jacobian conjecture, mathematician Levent Alpöge, working with Claude, found a counterexample to a related question. On sofic groups, the breakthrough was constructing the first non-sofic group. On multicolor Ramsey numbers, the core move was again building highly specific combinatorial objects.
All four problems compress into one machine-friendly question: "Is there such a thing?" When the answer is yes, the model's playbook — massive search, cross-domain tool transfer, recombination of known tricks, repeated construction attempts — takes over completely. And it is cheap: the total token spend for the counterexample hunting across all ten results was on the order of a few thousand dollars at API prices.
Why "counterexample" and "theorem" are not two abilities
Mathematicians' intuition treats finding counterexamples as luck work — punch one hole and the proposition falls — and proving theorems as hard craft: no exception allowed, plus a flash of insight. Gowers argues the boundary does not hold. Consider two classical results: Vinogradov's 1937 theorem that every sufficiently large integer is a sum of three primes, and Gluskin's 1981 result that in any dimension there exist two spaces whose shapes are extremely far apart. Written in formal language, their skeletons are nearly identical — yet nobody calls Vinogradov's work a counterexample, and nobody calls Gluskin's a theorem. The difference is only where the hard step lands.
There is a subtler layer: the word "counterexample" carries psychology. It only counts when it overturns a proposition people had reason to believe. The non-sofic group is therefore better described as the first example — few experts genuinely believed all groups are sofic. And the multicolor Ramsey result, Gowers notes, is an expected example for him personally: he studied an equivalent form as a young mathematician, and the direction that proved correct was exactly the one he had worked on. The line follows the person, not the mathematics.
Eight ways to find examples; AI is strong at four
Gowers lists eight standard ways mathematicians find examples: run through the standard examples in stock; assemble from basic objects; leave blanks and fill conditions in later; prove no such object can exist; iteratively revise a previous guess; build piece by piece under constraints; pick one at random; take a sufficiently general example.
His judgment: models will be very strong at four of them — stock examples, step-by-step assembly, random picks, and general examples. These share three properties: mature routines, dense instances in training data, and low failure cost. The remaining approaches demand something else: repeatedly judging, mid-path, whether the current road is still worth walking. That is pruning — the act of cutting most branches of a search tree as soon as it starts to fork. Models know almost every standard argument and can fail thousands of times before landing on a solution, so the luckier a problem is, the more AI dominates it. The ground humans still hold is where the search tree is wide and deep.
Why AI lacks the mathematician's nose
Why can't models prune yet? Three reasons. First, training data contains no such course: published proofs are clean finished products — the dead ends, the moments of doubt, the turnarounds are erased, so models see conclusions but not process. Second, speed is itself the obstacle: humans must prune because a lifetime only allows a few paths, while models can afford search inefficient enough to crush a human — and with no scarcity, there is no pressure to prune. Third, the question is nearly impossible to experiment on.
Terence Tao, in his ICM 2026 lecture "Mathematics in the Age of AI," described the symptom from the reader's side: AI-generated proofs are painful because the emphasis is misplaced — three pages on an obvious lemma, three lines on the genuinely interesting step. Human proofs leave what he calls natural friction at hard points; AI-polished proofs smooth it away. His own example: as a graduate student he struggled through Bourgain's 1991 paper — margin note "I hate Jean Bourgain" included — and the struggle trained him to follow the author's thinking. Had that proof passed through several layers of AI polishing, he doubts he would have received the training. "Which step is hard" has never been formally recorded in the mathematical tradition, so models cannot learn it — and their polished outputs erase the last remaining traces. The result is a wrong loop: models fail to learn judgment, and their writing makes it harder for the next generation of models to learn it.
Gowers' prescription targets the reward function: penalize wandering into too many dead ends, and penalize pulling ready-made answers from the literature. Reward only results, and you get results without process. That holds for training — and, he notes, for the mathematical community itself.
The cap-set test — and what it means for your workflow
Gowers offers the strictest ruler available. The 2016 cap-set breakthrough: a new solution that threw the previous bound far behind, using an approach none of the field's expectations had anticipated — and yet, in hindsight, obviously right. A wave of researchers immediately dropped their own work to explore what the new technique could do. A genuinely great method does not just solve one problem; it changes what a community works on overnight. Only when a model produces something of that weight will Gowers consider the threshold crossed. He expects models to keep improving fast — possibly growing a stronger nose within a year or two. The human edge is rented, not owned.
Meanwhile, use models the way Gowers does: as a cheap counterexample machine. Before committing to an assumption, an estimate, or a design constraint, let a model try to break it — the entire ten-result campaign cost a few thousand dollars in tokens. But treat its confident proof sketches as proposals to be pruned, not results; and when a model says "I could not answer your question, but I reduced it to a narrower one" five times in a row, it is not converging — it is describing being lost more and more elegantly. In the same spirit as the COLM papers questioning Transformer defaults, the deepest value of AI in mathematics right now may lie less in new theorems than in new questions — and in breaking the ones everyone believed.
