AI solves Navier-Stokes raises credit questions
One Setting at the
Center of the Dispute
OpenAI announced on 8 September that it had used an artificial-intelligence model to solve the Navier-Stokes problem, one of the highest-profile open questions in mathematics. [1] The company, based in San Francisco, California, said it verified its proof using a programming language called Lean. The Clay Mathematics Institute, whose scientific headquarters are in Oxford, UK, says it will consider whether a solution is valid only after the results have been published in a peer-reviewed publication and been further vetted by the community. [3] The institute holds a US$1-million award for completing one of seven Millennium Prize Problems it selected at the turn of the century.
One parameter sits at the center of the controversy that followed: a setting inside ChatGPT that lets a user permit or refuse the company’s use of chatbot conversations for training its models. Tristan Buckmaster, a mathematician at New York University in New York City, told Nature that he had three separate accounts on OpenAI’s ChatGPT. [5] On only two of them had he opted out of the setting that grants permission to use conversations in training. Buckmaster says he had used OpenAI tools to work on the problem for a year.
Andreas Thom, a mathematician at the Dresden University of Technology in Germany, describes the same asymmetry in his own case. [6] Until late June, Thom had not opted out of training of the models, meaning there is no way of knowing whether the company’s tools used sessions with him — in addition to his published papers — to help with its work. He had spent roughly a year brainstorming with ChatGPT about the theory of groups, a concept that has pervasive uses across mathematics and physics.
Before OpenAI confirmed rumors that it had solved the Navier-Stokes problem, Buckmaster and his collaborator Levent Alpöge, a mathematician at Harvard University in Cambridge, Massachusetts, had already been working on an aspect of the problem using tools from OpenAI as well as Anthropic. [7] They were told that OpenAI was preparing to announce that it had solved the problem. Buckmaster wrote on social media that the company had jumped on the problem after hearing about the work by him and Alpöge, and that OpenAI’s model could have been learning from their interactions with ChatGPT.
What the Company Says About the Training Data
An OpenAI spokesperson told Nature: “After investigating, we can say with full confidence that no user inputs past July 3rd could have influenced this system in any way.” [2] The company said it started working on the problem on 1 September and that it had not seen “any of their work through any means until they released it publicly.” OpenAI has not stated whether its models drew on Thom’s conversations. A spokesperson said it was up to users to decide whether their conversations help improve models, and emphasized that once users have opted out, OpenAI does not use that data to improve its models.
Thom says he was using a particular strategy to try to construct a type of group called non-sofic — something many mathematicians had long thought impossible. Then in August, OpenAI posted a preprint reporting that it had arrived at the first ever example of such a thing, with a similar strategy to Thom’s. Thom says the credit in this case is unclear. The OpenAI paper correctly references earlier works by him and his collaborators, says Thom.
Mathematicians who work on the Navier-Stokes equations say that a large part of the credit belongs to Buckmaster and Alpöge, and also to Diego Córdoba at the Institute of Mathematical Sciences and Luis Martínez Zoroa at CUNEF University, both in Madrid. [9] In an open letter decrying AI companies’ entry into solving mathematical problems, 25 winners of the Fields Medal — often considered mathematics’ equivalent to the Nobel Prize — write that AI tools muddy the ability to give appropriate credit: “As in all creative professions, this raises severe attribution and plagiarism questions.” [1]
Luke McDonagh, who studies intellectual-property law at the London School of Economics and Political Science, says: “It is quite possible that academic researchers have not fully grasped the consequences of uploading data and knowledge to a personal AI model account.” The widespread use of AI models across most fields of research, and the difficulty of tracing the origin of the information used to train models, could put the very notion of scholarly credit into jeopardy, researchers warn.
How Mathematicians Weigh Credit
Thom draws a direct parallel to human practice. If a human mathematician wanted to break into the field of non-sofic groups, they would probably have conversations with specialists in that field, he says, and learn tricks of the trade that are not represented in the written literature. Typically, they would then credit those conversations in the acknowledgments sections of their papers. “If a human had sat in my office and then had written that paper, I would be angry if he had not given credit to our discussions and explanation,” Thom says, although it is not established whether OpenAI’s model in fact drew on Thom’s conversations. [6]
How to apportion credit in the age of AI will become increasingly complicated, Thom says. His own uncertainty about whether his brainstorming sessions helped train the chatbot stands unresolved. Buckmaster’s year of work, conducted across three accounts with only partial opt-out, leaves a record that cannot be fully reconstructed. Assuming that the OpenAI solution is independently confirmed, it remains to be seen how the maths community will choose to apportion credit — and who deserves the US$1-million award offered by the Clay Mathematics Institute. [3]
The proof, if validated, will pass through peer review and community vetting. The training data, by the company’s own account, cannot be inspected past a fixed date. Thom’s question about whether his conversations were used has no stated answer. Buckmaster’s claim that the company moved after hearing of his work is met by a denial that any input past 3 July could have influenced the system.
Sources
2. OpenAI
4. ChatGPT
6. Dresden University of Technology
8. Anthropic
9. Institute of Mathematical Sciences
10. CUNEF University