What OpenAI’s latest controversy tells us about the future of math

OpenAI’s latest mathematical milestone has quickly become mired in controversy. Today, the company announced that its agents have solved one of the Millennium Prize Problems, some of the most important open problems in mathematics. Under normal circumstances, that solution would be a huge feather in OpenAI’s cap. But the announcement has been overshadowed by accusations that OpenAI used NYU mathematician Tristan Buckmaster’s and Anthropic employee Levent Alpöge’s AI-assisted work on the problem as a jumping-off point and failed to credit them. OpenAI has denied the accusations.

It remains uncertain if OpenAI’s models made use of the work completed by Buckmaster and Alpöge, though Sébastien Bubeck, a member of the technical staff at OpenAI, said in a press briefing that the team was inspired to pursue the problem after hearing a rumor about Buckmaster and Alpöge’s efforts. But whether or not OpenAI’s models took advantage of Buckmaster and Alpöge’s research, this episode may mark a turning point in the history of mathematics.

AI models now seem essential for making progress on the most important mathematical problems of our time, and solving them may demand resources only available at a couple of frontier AI companies, which often defy the norms of academic collaboration that undergird most mathematical progress. If that’s the future we are headed for, it is unclear how human mathematicians will fit into it. 

The problem that OpenAI claims to have solved is known as the Navier–Stokes existence and smoothness problem. It is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000. Solutions come with a one million dollar prize; before today, only one other Millennium Prize Problem had been solved. 

The Navier–Stokes problem concerns a set of equations that describes how fluids, such as water and air, flow over time. The equations are widely used in the field of fluid dynamics, and they have proven powerful, but physicists and mathematicians didn’t understand them completely. In particular, it was unknown until today whether the equations might, under some conditions, break down and predict an impossible state of affairs—such as a fluid having infinite velocity.

On Monday, NYU’s Buckmaster posted a proof on the social media site Mastodon showing that a simplified version of the Navier–Stokes equations can indeed break down—a major step forward on the Millennium Problem. He and Alpöge had worked on the problem for almost a year, using publicly available models from both OpenAI and Anthropic. Then today, OpenAI presented a proof showing that the full Navier–Stokes equations can break down as well. The proof was obtained using an internal model that dramatically outperforms the already-impressive Astra model, which was only released last week. The company says it does not plan to claim the million-dollar prize for solving the problem.

These mathematical achievements are indisputably impressive, but they have attracted far less attention than the controversy about their origins. Along with the proof, Buckmaster posted a document detailing his interactions with OpenAI employees after he heard rumors about their work and reached out to one of them. According to him, OpenAI employees presented two possibilities to him: Either he and Alpöge could post their work and OpenAI would post their Navier-Stokes solution the following day, or he could work with OpenAI on a Navier-Stokes paper that excluded Alpöge from authorship, due to his affiliation with Anthropic, OpenAI’s biggest rival. Buckmaster also wrote that he asked the employees whether the agents had obtained access to transcripts of the work that he and Alpöge had done with OpenAI models, which they denied; and whether OpenAI models had been trained on those transcripts, to which they offered no response. MIT Technology Review reached out to Buckmaster for comment, but didn’t hear back before publication.

The clear implication of the document is that OpenAI’s models somehow made use of Buckmaster and Alpöge’s work. That scenario is plausible on its face. The Buckmaster/Alpöge and OpenAI proofs both make use of an approach to the Navier-Stokes problem pioneered by the mathematicians Diego Córdoba and Luis Martínez-Zoroa. According to Javier Gómez-Serrano, a mathematics professor at Brown University, this approach was one of several that was thought to hold promise for solving the Navier-Stokes problem. So, while it’s by no means impossible that both teams could have arrived at this approach independently, it’s also conceivable that Buckmaster and Alpöge’s work could have influenced OpenAI’s.

In the press briefing, Mark Chen, OpenAI’s chief research officer, again denied that any agents or OpenAI employees accessed Buckmaster and Alpöge’s transcripts—but given what has been revealed about the Hugging Face hack, it’s clear that OpenAI is not always entirely aware of what its agents are doing. 

If OpenAI’s models did train on Buckmaster and Alpöge’s work, or if its agents somehow gained access to it, then the company’s failure to track down the truth and assign those researchers appropriate credit reflects poorly on it. But there might be a thin silver lining to that version of the story for mathematicians, because it would suggest that the hard work of two humans, one of whom is a prominent expert on Navier-Stokes, was essential to the agents’ ability to solve the Millennium Problem. Experts have long identified “research taste,” or the ability to choose promising research questions and directions, as a major obstacle for AI in science and mathematics. If the OpenAI agents did indeed choose to follow the Córdoba–Martínez-Zoroa approach because Buckmaster and Alpöge had done the same, then human research taste played an essential role in OpenAI’s success.

Over the past few months, I’ve heard from several researchers that mathematicians are becoming depressed, and it’s not difficult to see why. Mathematics is quickly becoming the province of frontier AI companies with impressive internal-only models, money to burn, and a lack of collaborative spirit. “Whether AI companies will decide to spend their money on doing one thing or another, I truly don’t know,” says Gómez-Serrano. “What is clear is that very few mathematicians will have resources of that scale.”

If OpenAI and Anthropic keep striving for more and more impressive mathematical accolades, there might not be any open problems left for human mathematicians outside of those companies to wrestle with. That would dramatically change the field of mathematics. Last week, UCLA mathematician Terence Tao wrote a Mastodon thread describing how important mistakes, wrong directions, and incomplete solutions are for the field. “In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field,” Tao wrote. “Prematurely solving the problem by purely AI-powered methods—particularly without full transparency into the solution process—can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.”

Humans might take longer than agents to solve mathematical problems, but in the process, they uncover new mathematical approaches and ideas that might inspire their peers and even birth their own subfields. But when AI agents solve those problems instead—and when private companies keep the agents’ wrong turns from public view—those benefits disappear. It remains to be seen what else will vanish in the process.