The French mathematician Hugo Duminil-Copin, who won the Fields Medal — mathematics’ Nobel — in 2022 for his work on phase transitions in statistical physics, said in a video call of the French Mathematical Society on October 8 that years of his research agenda had been “wiped out” thanks to a tumult of mathematical proofs and results released by OpenAI a day prior.
Sixteen of the AI giant’s claimed results answered questions he had spent years trying to solve. Last month, OpenAI had claimed a proof, certified by the Lean proof-checking software, that Navier-Stokes equations of fluid mechanics are incomplete. Srinivasa Ramanujan did not lack for results but he had to go to Cambridge, less to learn mathematics than to be tutored in the language of proof-writing. Ever since computers began to outpace human calculation, there have been discussions on whether machines understand the way humans do. Imagine an English speaker who knows no Chinese, locked in a room. Through one slit come questions in Chinese characters. A rulebook in English tells him which characters to push out through the other slit, by shape alone, and he follows it without knowing what any symbol means. To someone outside, he seems to understand Chinese. Computers, Searle argued, are mere rapid manipulators of symbols, and their solutions shouldn’t be mistaken for understanding. The latter is more valued than the mere fact that something is true. A deep result in mathematics is not just about examples: it is about finding lasting connections between different areas, connections that are mutually enriching, Dr. “AI is extremely good at solving a well-formulated problem (a problem that has already been formulated by humans through years of work), mostly by pushing well-established strategies with its computational superpower. But there is no evidence that AI has imagination like a human does.”
In one case the model had produced 13 papers at once, standing in for the stages of a programme that takes humans many years. OpenAI released some 700 manuscripts from an unreleased model, claiming results on a weaker form of the Riemann hypothesis, as unsurmountable so far as Mt Kailash is, with implications for prime numbers and their many applications; the nature of the number π; and the speed of multiplying matrices. The answer could potentially settle a $1-million Millennium Problem. Henri Poincaré established the two-dimensional case around 1885 and Luitzen Brouwer proved it comprehensively in 1912. In 1980, the philosopher John Searle offered a thought experiment that, in his view, proved they weren’t.
Duminil-Copin said he cried so loudly that night he woke his daughter, and said his field had been “annihilated”. For example, the ‘hairy ball’ theorem says you cannot comb the hair on a woolly ball flat without leaving a bald patch or a whorl, as on a coconut. “Proof only offers the guarantee of correctness, but a good, elegant exposition of a proof tells us why something is true,” Sabyasachi Mukherjee, mathematician at the Tata Institute of Fundamental Research, Mumbai, whose research spans conformal dynamics and complex analysis, says in an email. Mukherjee says. He adds that he uses AI only for literature surveys, running numerical experiments, and drawing mathematical pictures, and was “uninterested” in purely AI solutions. “To this date, AI has not found such a deep connection,” he says.
