OpenAI claims Navier-Stokes proof, 25 Fields Medal winners push back

The Navier-Stokes equations were developed by 19th-century scientists to describe the flow of viscous fluids, and engineers use them to model airflow for airplane design. But mathematicians became enamored with the equations themselves, not their applications. As Jared Speck of Vanderbilt University puts it, the community's main interest was certainly not engineering — it was the mathematical richness, the puzzle aspect. The community had for years been developing what Speck calls a deep and beautiful theory around the equations, and were on the verge of cracking the problem before OpenAI's proof found that yes, the equations did imply a sci-fi fluid explosion.

That order of events is what unsettles mathematicians. When humans write proofs, they organize their arguments based on previous research, which they cite, and spend months or years convincing colleagues of each step. LLMs do not reliably cite their work and do not explain themselves clearly to humans. The result, says Speck, is that basically nobody in his community really understands what is going on — first the result was given to them by the computer, and now people are trying to understand it. The declaration that 25 Fields Medal winners signed is not against AI itself, says Juspreet Singh Sandhu, who has signed it and already uses AI in his work; it states that mathematical culture values understanding, and that the speed through which such systems tear through proofs could mean we will have solved without understanding.

The stakes may extend beyond this one proof. Lorenzo Gavassino of the University of Cambridge warns that when a calculation turns out harder than expected — to the point of being discouraging — that is when the greatest progress is possible, as with the 16th-century invention of the imaginary number i, which eventually underpinned complex analysis and quantum mechanics. The mathematical community has long relied on a teacher-apprentice structure, with senior mathematicians handing juniors problems they could have solved faster themselves, as an investment in the next generation. If AI scoops that lowest-hanging fruit, it could narrow that training path. OpenAI's new advisory group is a first response, but whether it changes how such results are produced and credited is likely to depend on what the peer review of the 166-page proof actually finds.