When the machine has the idea

Graphic composition made with AI. Photo: Vitaly Gariev/Unsplash

Raphaël Cerf has spent more than thirty years on a single problem. Last August the French mathematician, former head of the probability group at the École Normale Supérieure in Paris, posted online a 708-page text gathering everything he had learned along the way: failed attempts, a significant partial result, dozens of illustrations. He hoped it would draw other researchers to what he calls "this beautiful problem". Within days the problem had been solved – not by a colleague, but by Claude, the artificial intelligence developed by Anthropic.

"On one hand, this is certainly a great achievement," Cerf says. "On the other hand, it will bring the research around this problem to a halt, at least until the proof is understood and made accessible to the human mind."

Coffee, forests and a threshold

Percolation studies how something spreads, or fails to spread, through a medium: water through ground coffee, fire through a forest, an infection through a population. Picture an infinite grid in which each link is open with a certain probability. Below a critical value, open links form only finite islands; above it, an infinite connected cluster appears. The question that resisted for decades is what happens exactly at the threshold. Since the 1960s the expectation has been that no infinite cluster survives there – hence the name "dying percolation" conjecture.

It had been proved in two dimensions and, thanks to sophisticated techniques, in eleven dimensions or more. The cases from three to ten, including the three-dimensional space we live in, remained open. "It was the main open problem in percolation," Cerf says, "the main obstacle to understanding the model in dimension three."

The way in came from Gady Kozma and Shahaf Nitzan, who in 2024 showed that the conjecture would follow from a family of very general inequalities on finite networks – so general that, as the mathematician Gil Kalai recalled, many experts expected a counterexample to turn up. At the end of August, a proof of the weakest of these inequalities, generated by Claude and entirely written in Lean, a language in which a computer checks every logical step, appeared in a code repository published by Anthropic. Days earlier the Fields Medal winner Hugo Duminil-Copin had predicted, in an essay on the blog Proofs and Prompts, that a machine would get there before humans.

Formal verification guarantees that the proof is correct. It does not guarantee that anyone understands it: "So far, nobody has understood how this part of the proof works," Cerf says. "The output of the computer is not intelligible to the human mind. Now there is difficult, painful work to do to understand what the computer has done and why these inequalities are true."

Closer to home

Carlo Mariconda, professor of mathematical analysis at the University of Padua and the university's Digital Learning Advisor, had never used generative AI in his research until last April: the results had always seemed disappointing. In May, stuck on a passage in a paper he was trying to decipher, he uploaded it to the version of ChatGPT provided by his university and asked for an explanation. The model simplified the notation and made the argument accessible: "That allowed me to finish a piece of research that had been stalled for a couple of years," he says.

The real surprise came later. For about ten years Mariconda has worked with Giulia Treu, his colleague in Padua, and Pierre Bousquet of the University of Toulouse on the Lavrentiev phenomenon, a question in the calculus of variations: roughly, whether the best value of certain optimisation problems changes when the search is restricted to better-behaved functions. "On some aspects of it we are, without false modesty, probably among the world's leading experts – not so much because we are particularly clever, but because it is a very niche field." Their latest work revolved around a conjecture they could neither prove nor disprove: that any solution with finite energy can be approximated by bounded functions without losing anything in energy. "It was as if the conjecture said that all healthy cats have four legs, and we had shown that ginger cats have four legs, that black ones do too, and so on – without ever finding a cat with fewer legs, or proving it for every cat."

Half in jest, over dinner, the three wondered what ChatGPT would say. The next morning Mariconda put the question to it, uploading the latest draft of their paper. The model found no counterexample, judged the conjecture plausible and suggested it might be proved by generalising one of their results and combining it with homogenisation techniques the authors knew little about. Asked to sketch a proof, it did. "The scheme, which the three of us examined carefully, seemed to work," Mariconda says. With further back-and-forth, they reached a proof they consider "correct and elegant". The paper is now on arXiv. A second one, which proves the conjecture, openly acknowledges the role of ChatGPT, and the authors reworked ChatGPT's argument into a shorter one of their own. "It is an early example in which ChatGPT was not used only to rewrite arguments, improve exposition, or assemble ideas already selected by the authors, but also to suggest a genuine hint for the proof strategy of a new result," Mariconda says. The paper is careful to add that AI "does not replace mathematical authorship; rather, it changes part of the exploratory process".

Two details matter. The choice of model was decisive: the same question submitted to the institutional chatbot Lucrez-IA and to the free versions of other chatbots – Claude, Gemini and DeepSeek – produced nothing useful, and some even claimed the conjecture was false. And context counted as much as capability: the same ChatGPT model, used on an account that had not been given the team's latest paper, failed to reach any result.

Satisfaction and dismay

The emotions in Padua mirror those in Paris: "On one side, great satisfaction at having settled a question we had been asking ourselves for years," Mariconda says. "On the other, a certain dismay, almost as if we had been robbed, at least in part, of the pleasure of discovery."

The two cases also show two different ways of working with the machine. In Padua, AI acted as a brainstorming partner: it proposed an idea, and humans checked, understood and rewrote it. In the percolation case, it produced a certified result that people now have to decipher after the fact. Both raise questions that have been building for months – since an OpenAI model settled a problem posed by Paul Erdős, and as mathematicians ask whether AI will take their jobs or whether mathematics is AI's next victim.

The first concerns training. "We have spent our lives writing papers by our own hand," Mariconda says. "How will new generations develop the ability to recognise the elegance of a proof and judge whether it is correct, having necessarily had less practice?" Then there is the volume of research, now that a paper can be written and rewritten in hours, and the delicate matter of credit: how should a correct, new result be published if its decisive idea came mainly from a machine?

Cerf adds a question of access: "This kind of verification cannot be done by someone who does not have access to enormous computing power," he says. "The romantic era when a mathematician alone could attack any problem, with paper and pencil, is definitely over."

The part that stays human

Neither scholar foresees the end of mathematics, or of mathematicians. Mariconda believes that what happened to his team "will become normal" and amounts to "a turning point that will profoundly change the way research is done, at least in our field". Cerf, for his part, points to what three decades on a single problem gave him: "By working on a challenging and fascinating question, we learn a lot, and it generates a lot of interesting problems."

Even among Mariconda's co-authors, opinions differ. "When it comes to the use of AI in mathematics, I often change my mind," says Pierre Bousquet. "At the moment, I think that the journey is more important than the destination, that trying to establish a result is more important than the result itself. And therefore, I think it is preferable not to use AI." He also remains convinced that its environmental, social and societal impact "is too significant in relation to the mathematical results we obtain with AI".

Twenty-five Fields Medal winners have published a joint declaration warning about what they see as a severe misalignment between AI companies and the mathematics community. https://t.co/pFwHDX4Lq2

— Andrew Curran (@AndrewCurran_) September 11, 2026

Giulia Treu sees both sides. "It is hard not to use a technology when it works and, personally, I also find it fascinating to test it and observe 'how' it works," she says. The other side, though, raises serious problems, starting with evaluation: "In Italy, where recruitment and career progression rely largely on quantitative indicators – number of papers, h-index and so on – the situation can become truly critical. How do you tell a young researcher who works seriously and makes a thoughtful use of AI from someone who uses it indiscriminately?" In recent months, she notes, cases of colleagues quickly producing many results with AI, without declaring it, have multiplied: "For now it is still relatively easy for referees to notice, but I think that will soon no longer be the case."

These concerns were also given voice in September 2026 by an appeal promoted and signed by numerous Fields Medal winners. The document, entitled A Severe Misalignment of AI in Mathematics, warns against the risk of confusing the ability to solve problems with the ultimate goal of mathematics: to understand, develop new ideas, and communicate them. If artificial intelligence makes it increasingly easy to produce results, the signatories argue, the challenge will be to ensure that this acceleration does not impoverish the very ground from which new mathematics grows.

And perhaps that is the real lesson. A machine can now tell us that something is true, and sometimes suggest how to prove it. Understanding why – the part that has always given meaning to every kind of human inquiry, science included – remains, for now, an exclusively human prerogative.