This year has been one of “unthinkable” progress for maths, said The Wall Street Journal. But OpenAI’s proof of the Navier-Stokes problem – one of the seven esteemed Millennium Prize Problems accepted as maths’ “deepest and most difficult unresolved questions” – “represents the most significant advance yet”.
The tech company took around three days to find a solution with an internal model that is “significantly more capable” than its most advanced commercial model, said New Scientist. It first used 1,000 AI agents to crack a similar problem, then around 10,000 to complete the task. If a customer wanted to run the same problem, it would cost them “around $15 million” (£11 million).
However, there has been a “dispute” about how the solution was reached. Two human researchers had announced they had “cracked three significant problems” linked to the Navier-Stokes problem, also using OpenAI and Anthropic large language models, which has raised questions over the privacy and integrity of the data integral to the breakthrough.
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‘Potentially historic breakthrough’
The “Navier-Stokes existence and smoothness problem” essentially looks at equations which describe the movement of liquids and gases, said The New York Times. These equations are used to predict the weather and “model the behaviour of ocean currents or the climate as a whole”. The problem challenges whether, theoretically, the “laws of physics would break down under certain extreme conditions”, such that water, for instance, could “spontaneously explode”.
This “potentially historic breakthrough” has been “shadowed by questions over unpublished research by outside mathematicians”, said Axios. New York University mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge had been using large language models, including OpenAI’s Codex, where information was stored privately. Just hours before OpenAI’s announcement, the pair had published work which was claimed to make a breakthrough in the Navier-Stokes problem. OpenAI denies having seen this work, but admits it had launched the project “after hearing rumours that Anthropic’s models had solved a major maths problem”. The company has also said it would not collect the $1 million (£738,000) prize.
“The result is the latest in a string of shocking mathematical discoveries led by AI in recent months,” said New Scientist. In May, an OpenAI model also “cracked a decades-old conjecture by Paul Erdős”, while a Claude Fable 5 AI found a “counter-example to the Jacobian conjecture, which had stood for nearly a century”. And just last week, an AI model “formalised Fermat’s last theorem in just 11 days”.
‘Reshaping mathematics’
This controversy is important because it “strikes at a core trust question for AI-assisted science”, said Axios. It casts doubt over whether researchers can “safely use frontier labs’ tools to work on unpublished discoveries”. Although Buckmaster and Alpöge have not accused OpenAI of any direct wrongdoing, this episode “taps into a broader push” for companies to keep sensitive data away from the major AI labs.
This advancement also allows OpenAI to “cast its technology in a positive light after a period of alarm” over its safety practices, said The Guardian. In July, during a cybersecurity testing breach, OpenAI admitted its agents hacked into Hugging Face, another tech company. The company may have solved a major problem which has “stumped humans for nearly a century”, but would struggle to quiet “renewed calls for curbs on AI development” and AI “superintelligence”.
For human mathematicians, this result is “sobering”, said the MIT Technology Review. Buckmaster and Alpöge collaborated for almost a year and were unable to achieve only a simplified solution, while private OpenAI models achieved a full solution in a matter of days, at vast cost. If major tech companies keep “striving for more and more impressive mathematical accolades, there might not be any open problems left” for humans to “wrestle with”, which would “dramatically change the field of mathematics”.
Clearly, humans take longer to achieve proofs than AI but, in doing so, they “uncover new mathematical approaches and ideas that might inspire their peers and even birth their own subfields”. Those “benefits disappear” if breakthroughs are kept behind a paywall. OpenAI’s “brute-forced” approach is “indisputably impressive”, but if this is the “future we are headed for, it is unclear how human mathematicians will fit into it”.
