Tristan Buckmaster, a professor of mathematics at New York University, on Tuesday published three proofs along with preliminary findings regarding one of the major open problems in theoretical mathematics. This discovery, made in collaboration with anthropomathematician Levent Alpöge using both Codex and Claude AI models, is significant in its own right, but it also comes with an unusual controversy surrounding OpenAI’s attempts to solve the same problem.
“There is another part to this story,” Buckmaster wrote in a statement announcing his evidence.
As Buckmaster and Alpöge were finalizing their results, they learned that “information about our progress was being passed to OpenAI.” When I contacted OpenAI about this, I was told that OpenAI had already achieved a complete proof of the central problem. But when asked additional questions about when OpenAI started researching this problem and how much human input was involved, the answer became even more ambiguous.
“It became clear that the entire team was working on this problem, and it turned out that an insane amount of compute was being used… It was ultimately agreed that (the first prompt) had been sent within the past few days after information about our work had reached OpenAI,” Buckmaster said.
If true, it would suggest that the OpenAI team was convinced that Buckmaster and Alpöge’s approach was correct and decided to use their material advantage in computing resources to arrive at a formal proof first.
Sebastian Bubeck, head of mathematical research at OpenAI, says these claims are “false and inflammatory.”
“To be clear, I participated in the discussion in accordance with academic norms and am disappointed that it turned out the way it did,” he wrote in a post after Buckmaster’s statement. “Anyone who knows me knows that academic standards are of the utmost importance to me,” Bubeck said, adding that he would issue a more complete statement in the near future.
The controversy centers on the presence and smoothness of Navier Stokes, one of the seven Millennium Prize issues. This is a series of major unsolved math problems, each with a $1 million reward from the Clay Mathematics Institute for the first person or group to provide a solution. Although the Navier-Stokes equations are widely used in fluid mechanics, they are poorly understood in theoretical terms. A solution would be a major step forward in our collective understanding of mathematical physics.
Although this problem is widely pursued among mathematicians, the specific tactics used by Buckmaster and his collaborators are less common. As a result, Buckmaster wondered if OpenAI was taking the same approach at the same time.
“The smooth-force routes to the Clay problem in Pfefferman’s problem statement, options c and d, are the routes that Luis and Diego carved out and that Levent and I quietly chose to attack,” Buckmaster wrote. “As far as I know, very few people were working on it,” he continued. “It’s not a direction you can give a problem statement to a model and arrive at in a few days.”
Although Mr. Alpöge is employed by Anthropic, he was not conducting this research on behalf of the company. As a result, they used a variety of models in their work, relying primarily on OpenAI’s Codex. Still, Alpöge’s partnership with a rival lab appears to have hurt OpenAI, with Buckmaster claiming that Bubeck asked him to remove Alpöge’s credit as part of a compromise.
When Buckmaster asked him to make the controversy public, Bubeck reportedly responded, “Why ruin your career?” Buckmaster said that when Buckmaster pushed back, Bubeck followed up by saying, “You don’t have to be nice to me if you don’t want me to be nice to you.”
Buckmaster also expressed concern that because he used Codex extensively in assembling the project, information from his work may have influenced OpenAI’s own efforts to solve the problem. OpenAI reserves the right to train models on Codex interactions, but you may opt out. If the OpenAI team had used a model trained on Buckmaster’s own Codex interactions, it is conceivable that they might have regurgitated his work when faced with a similar problem.
OpenAI did not respond to a request for comment on this possibility.
Either way, this issue could reignite the ongoing debate about the role of AI in mathematics research and the specific incentives for OpenAI. Buckmaster seems to believe that the best solution is to make as much information about research available to the public as possible.
“I have not seen any evidence of OpenAI,” Buckmaster wrote. “I don’t know what their model did or how they did it. I don’t know if our data was used. I’m not accusing anyone of anything. I’m saying what I was told and when I was told what was suggested to me. I’m saying that because the alternative is to have them say things in a series of announcements that I know to be false.”
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