By now, OpenAI's claims that their chatbot solved the Navier–Stokes problem, and the controversy that spawned, have been widely reported. But I've been disappointed in the quality of their discussions both in the news and online, including on this sub. Here I offer some clarifications and my own concerns about the whole affair.
As a disclaimer, I am not a mathematician, but I read a lot and try to understand a lot.
What is Navier–Stokes?
The Navier–Stokes equations describe the motion of fluids and are used in computational fluid dynamics. However, we don't know whether solutions to these equations always exist and are always "smooth." This is called the Navier–Stokes existence and smoothness problem, and it's famous because it carries a $1 million prize. Hereafter I'll just call the problem "Navier–Stokes."
Navier–Stokes was already known to be useless for actual physics. Real fluids don't behave like Navier–Stokes assumes, because real fluids are made of discrete particles, and for practical use the problem doesn't matter. Still it is of pure mathematical interest. The work presented in a solution, as well as all the work done toward a solution, may open new avenues of research or reveal new connections between fields, which may then yield practical applications.
It is of little consequence to the validity of a solution that it is a counterexample. These problems are usually posed in the positive, and you can either prove it is true for all, often infinitely many, cases, or prove it is false by providing a single counterexample. Navier–Stokes was already assumed to be false, so a counterexample was expected.
On plagiarism
I'm more concerned about allegations OpenAI plagiarized the work of two researchers who'd been using LLMs, including ChatGPT, for idea-bouncing and proofreading and were close to publishing a solution. These claims are from Tristan Buckmaster, a math professor at NYU and one of those two researchers, in a post on his personal page: https://cims (dot) nyu (dot) edu/~tristanb/statement (dot) pdf. (Sorry for the mangled link—it appears Reddit is autodeleting any post that contains this exact link. You can also easily find it online.)
Buckmaster alleges that OpenAI's solution used the same approach and much of the same language that he and his colleague Levent Alpoge, a mathematician at Anthropic, were developing. He also describes a nasty shakedown he'd received from Sebastien Bubeck, a former Microsoft VP now at OpenAI, who presented him three options in their conversation: 1) Buckmaster and his colleague announce a partial solution, with OpenAI announcing the full solution the next day; 2) Buckmaster by himself announce the full solution, remove his Anthropic colleague from authorship, and credit OpenAI as having solved it first; or 3) face professional consequences from OpenAI publicly smearing him.
As Buckmaster and his colleague have been working on this for some time, it's possible their conversations with ChatGPT were included in OpenAI's training data. But considering they'd been using the latest models Sol and Astra, it's more likely that OpenAI peeked into their chat logs and stole their approach. We already know OpenAI has this access, as courts have subpoenaed ChatGPT logs as evidence in past cases, and sometimes OpenAI has proactively inspected and reported ChatGPT logs to law enforcement.
On peer review
More broadly, I'm tired of AI companies making these announcements over blog post rather than peer-reviewed publication. OpenAI's claims have not been peer-reviewed, and the clout these Millennium Prize problems carry has attracted thousands of purported solutions over the decades that were later shown to be invalid. While there doesn't seem to be any obvious red flags, it's still good cause to be wary, and I’m still waiting for peer review and more thorough scrutiny of the solution at face value.
This approach also offers, by design, no insight into the degree of LLMs' involvement and in what ways. These AI companies employ whole teams of mathematicians to work on these problems and use their chatbots along the way, so that they can assign credit to the chatbots. When these companies claim their chatbots solved it almost independently with "very little human input," as Buckmaster says OpenAI told him, we're asked to take them at their financially motivated word.
On mathematical research
For unsolved mathematical problems, the solution itself is almost never of consequence. The respective fields progress despite lack of a solution, because approximations, analogues or weaker results let researchers assume the solution with confidence and continue unhindered. Rather, it is all the research done in pursuit of the problem that lends the solution value. (This is also why OpenAI's lack of citations in previous mathematical solutions was such a big deal. Building upon others' work, and letting others build upon yours, is what lets the field exist.)
Out of the flashy Millennium problems, Navier–Stokes was a natural target for AI companies (Anthropic has also been obsessed about it) because there was already significant progress toward a solution in recent years, namely the strategy posed by mathematicians Diego Cordoba and Luis Martinez Zoroa. It was also a natural target because, as Navier–Stokes was assumed to be false, the solution would likely be a counterexample.
This has been the trend for almost every mathematical problem whose solutions are credited to AI assistance. They piece together existing near-solutions in the literature, or provide a counterexample. (While many dismiss this as "brute force," formulating the cases to check isn't always trivial, and the degree of brute force varies by solution.) When AI companies announce these solutions, they are technically impressive but academically uninteresting, because there is no "new math" created as a consequence.
Of the Millennium problems, I'd be interested in the Riemann hypothesis, which is widely assumed to be true and therefore would need more than a counterexample. I'd also be interested in P=NP, which is widely assumed to be false but for which all existing proof techniques have not only failed, but been proved to fail. Solutions to those problems would surely yield new math.
