In an op-ed published in issue 19 of ActuIA, Axel Cypel questions AI’s ability to produce theory. Recent AI contributions to mathematics give this question new scope: does producing a novel result suffice to establish that the machine thinks?
The author contrasts two ways of producing knowledge. On one hand, machine learning, which proceeds by induction, i.e., by generalization from examples. On the other, theory, which can give rise to new concepts. He draws on major physical theories, from Galilean mechanics to the standard model, which did not follow this method. Gravitational waves, predicted by Einstein’s theory of general relativity, were only sought because the theory predicted them, he notes, in what is called the hypothetico-deductive method. He also echoes physicist Étienne Klein’s expression, for whom the theoretical act functions as an ‘ontological winch.’
The text reports the question posed to him during the Roberval Prize ceremony on December 7, 2024, where his book Voyage au bout de l’IA was named the winner in the General Public category. His interlocutor stated: ‘as a researcher […], I was constructed like an AI.’ Axel Cypel, for his part, points out that AI differs from human thought in how it learns: a human does not need to see a million cats to recognize a cat.
Recent Mathematical Results
An editorial note, dated October 2026, sheds light on the question in light of current events. On October 26, 2026, OpenAI announced a refutation of the exponential time hypothesis, a conjecture in theoretical computer science. Anthropic for its part published a result attributed to a Claude search model. Both labs provided formalizations in Lean, a proof assistant, i.e., software that verifies a proof step by step. OpenAI also organized an external review of its result with outside researchers.
A refutation of this type relies on constructing an algorithm and demonstrating its properties. If confirmed, it constitutes an original theoretical result. Google Research for its part described mathematical structures discovered with AlphaEvolve, integrated into proofs that enabled the establishment of new results.
A Proof Does Not Tell Everything About Its Origin
Axel Cypel emphasizes that the model learned by an AI is not a copy of its training data, but an abstract synthesis. Its use can produce an interpolation, i.e., a point that was not in the training set. The question remains whether this constitutes creation. The author believes there is, a priori, no reason to exclude that a strong AI, capable of producing a thought process, could one day exist. He considers it doubtful, however, that machine learning could produce one.
A valid proof establishes a result, but it does not, by itself, reveal the nature of the process that produced it. This is why the author does not propose to settle this debate. Instead, he invites us to use AI to produce new answers and to ‘combine this strength with our own to find, why not, new theories.’
