AI Breakthrough in Mathematics
OpenAI has made a significant announcement today regarding the resolution of a long-standing mathematical problem: a 200-year-old equation that describes the natural behaviour of fluids, such as water and air. This declaration showcases the remarkable potential of artificial intelligence to accelerate advancements in the field of mathematics. However, the news has been overshadowed by allegations from mathematician Tristan Buckmaster, who claims that OpenAI hastily reached a solution after becoming aware of his and his colleague Levent Alpöge’s research. Buckmaster further asserts that OpenAI attempted to manipulate the attribution of credit for this achievement.
The Navier-Stokes Equation
The resolution in question pertains to the Navier-Stokes equation, which is one of the unresolved problems identified by the Clay Millennium Prizes, each offering a reward of one million dollars. Sebastien Bubeck, a mathematician and AI researcher at OpenAI, revealed during a press conference that the company began training a new AI model with advanced mathematical capabilities on 28th August. After hearing rumours that Anthropic was making progress towards solving the Navier-Stokes equation, Bubeck explained that OpenAI decided to allocate more resources to tackle the problem. Over 1,000 agents were engaged in this endeavour for more than 50 hours, ultimately leading to the discovery of a solution.
Bubeck noted, “I thought there must be an error somewhere. By Sunday morning, we had the definitive solution, formalised in Lean and everything.” Lean is a programming language that facilitates the formalisation of mathematical proofs. OpenAI indicated that addressing this problem necessitated significantly more computational power than had been previously applied to mathematical challenges, amounting to “millions of dollars”, according to Mark Chen, OpenAI’s research director.
Controversy Arises
On the following Monday, Alpöge and Buckmaster, a mathematician from New York University (NYU), released papers claiming to have made key progress in an area relevant to the Navier-Stokes problem. They noted that they had utilised various AI models, including Claude and Codex, to complete their work. Buckmaster also issued a statement highlighting that he had recently learned that OpenAI was aware of his and Alpöge’s research and had begun allocating substantial resources to solve the issue. He stated that he inquired with OpenAI officials whether the company had accessed Codex records from both researchers. According to him, they claimed that the model “did not consult user data”, but Buckmaster reports that they did not respond to inquiries about the training process.
He further recounted that OpenAI presented several “proposals”, including one in which Buckmaster could publish an article announcing that the Navier-Stokes problem had been solved by an internal OpenAI model, but without mentioning Alpöge’s name. In a social media post, Bubeck denied the suggestion that the company had proposed omitting Alpöge’s name. OpenAI’s CEO, Sam Altman, also intervened in the controversy, defending the team’s work.
Clarifying the Situation
Buckmaster, Alpöge, and Anthropic did not immediately respond to requests for comment. During the press conference, Bubeck and other OpenAI executives denied any inspection of Codex prompts from the two mathematicians to inform their own work. “We, whether as researchers or agents, did not see any of their work until it became public last night,” he stated. In a blog post detailing their work, the company remarked, “While unlikely, we cannot rule out that anonymised data derived from their use of our products may have contributed to improving our models.”
Bubeck emphasised, “I want to make it very clear that we recognise the significance of the work by Levent Alpöge and Tristan Buckmaster on the unsforced Euler equations, and we can only congratulate them on this monumental achievement. To clarify, we did not use their prompts or their proofs to instruct our models or guide our agents.” The unsforced Euler equations describe the flow of an ideal fluid without friction when no external forces act upon it. Ven Chandrasekaran, a mathematician at the company, also highlighted that their solution was of a significantly different nature compared to that generated by OpenAI’s model.
Future Implications
The debate over credit for resolving one of the most significant mathematical problems could persist for some time. As AI takes on an increasing role in the search for proofs, such disputes may become more frequent. The implications of this situation extend beyond the immediate controversy, raising questions about intellectual property, collaboration, and the evolving relationship between human mathematicians and artificial intelligence.