In September, an unreleased OpenAI model solved a maths problem that had been open for close to 90 years. It needed 88 hours.The Navier-Stokes problem asks whether the equations that describe how fluids move always behave sensibly, or whether they can break down. It is one of seven Millennium Prize Problems set out in 2000, each carrying a million-dollar reward, and only one of them had been solved before.That was by Grigori Perelman, who famously refused the money. OpenAI says it won’t be claiming its prize either.Then, on October 6, the company went much bigger. It put more than 700 manuscripts on GitHub in a single night, grouped into 372 families of results that cover almost every branch of the subject, from number theory and topology to probability and theoretical computer science. The model behind them had attempted around 4,000 problems, and OpenAI says the average result took the equivalent of about three hours of ChatGPT Pro’s thinking time. One of those results proves the quasi-Riemann hypothesis, a close relative of the most famous open problem in mathematics. Rutgers mathematician Alex Kontorovich wrote on X that a human who managed it would be in line for an “instant Fields Medal”.The pace is the part that is hard to take in. In May, OpenAI announced that one of its models had disproved a decades-old conjecture by Paul Erdős. In August, it announced ten long-standing problems at once. Two months later, it was hundreds.You would expect the field to be celebrating.Instead, more than two dozen Fields medallists, holders of the discipline’s highest honour, have signed a statement calling the goals of AI companies and mathematicians severely misaligned, and over 8,000 researchers have backed it since. PhD students are openly asking what their degrees will be worth by the time they graduate. It is tempting to read all of this as a profession sulking about being outdone by a machine, and plenty of people online have read it exactly that way. Very few mathematicians, though, are arguing that the AI got the maths wrong.The problem lies in what comes with those answers, or rather what doesn’t. They arrive without anyone who understands them, they skip the slow, painful work that actually trains mathematicians, and they come from labs that are now racing the very people expected to check them.
The proofs arrived without anyone to explain them
To most people, a proof simply confirms that something is true. To mathematicians, it is the explanation of why it is true, and that explanation is the part other researchers build on, borrow from and teach. It is also why a new result usually comes with an author attached, someone who gives talks on it, takes questions from sceptical colleagues and defends every step in front of referees before the field accepts it.None of the October manuscripts has one.Researchers working through them describe odd notation, arguments that are hard to follow and reference lists that look thin for results of this size. By OpenAI’s own account, only one of the 700-odd papers was edited by a human, and just 10 include the model’s chain of thought, the record of how it reached its conclusions. The Institute for Advanced Study in Princeton summed up the shift in a statement, warning that AI can now produce mathematical arguments that the person who prompted it can neither understand, verify nor take responsibility for. Terence Tao, among the most respected mathematicians alive, put it more pointedly, criticising prompters who chase a famous problem, declare it solved and then cannot answer basic questions about the result.The usual safety net is Lean, a programming language that checks proofs line by line, much like a compiler checks code. By OpenAI’s own figures, only about 42% of the release has been run through it so far, which leaves most of it resting on prose alone.Even the formalised portion isn’t a clean bill of health. Researchers at Cambridge and King’s College London compared the written Navier-Stokes proof with its Lean version and found at least two places where the two don’t say the same thing. That doesn’t sink the result, but it does show that a model translating its own argument into a checker can’t simply be taken at its word. Lean can also be gamed, with models sometimes finding loopholes in the checker instead of solving the actual problem, a behaviour researchers call reward hacking.Within days of the release, three papers had been withdrawn over a sign error, and more than a dozen others had been revised.The work itself is strong in places. Researchers who know these areas describe several results as top-journal material, and a handful as the kind that would make a career.The issue is scale, and who carries the cost of it. A lab can publish 700 manuscripts in a night, while a specialist can take weeks to read just one of them properly, and there are only so many specialists to go around. Peer review, the process that normally catches mistakes before they spread, has been left to happen after the fact.Mathematics has been in a similar spot once before, on a far smaller scale. Srinivasa Ramanujan’s notebooks held thousands of results, most of them without proofs, and the number theorist Bruce Berndt spent more than two decades working through them one by one. But those results came from a single mind, reached the world over years, and nobody was promising another batch the following month.
Mathematics has always moved forward by getting stuck
Ask mathematicians what they got out of a problem they never solved, and many will tell you it was worth it anyway. Hugo Duminil-Copin, a 2022 Fields medallist, wrote this year that his failed attempts at one well-known conjecture in probability produced dozens of ideas he later used elsewhere, leading to discoveries he would not otherwise have made.In the same essay, he described AI proofs as ones that “nuke the mathematical landscape”.The subject has always moved forward this way. Imaginary numbers, the square roots of negative numbers, were dismissed as useless by the 16th-century mathematicians who first stumbled on them. Today engineers use them to design the circuits in your phone, and quantum physics cannot be written down without them. Very often, the tools built on the way to an answer end up outlasting the question itself.That middle stretch is exactly what AI skips. Researchers reading the release say the model mostly pushed known methods further and combined them in clever ways, filling gaps in what the field already knew rather than opening up anything new. Stephen Wolfram, speaking at a public event on the day of the release, made the point that producing new theorems is easy, and producing ones that anybody cares about is not.William Thurston, one of the great geometers of the 20th century, saw a small version of this coming in 1994. In an essay on how mathematics makes progress, he described how his own rapid run of results in an area called foliations put others off working on it. Once they moved on, much of what they had understood faded with them. Mathematical knowledge, he argued, lives mostly in the heads of the people working on it, with the papers playing a supporting role.Thurston emptied one corner of the field. The model has done the same to dozens at once.The people with the most to lose are the ones still learning. For generations, PhD advisers have handed students an open problem that might take a year or two to crack, knowing the answer mattered less than the instincts the student built along the way. Those are precisely the problems AI is now clearing in bulk. Some students have already watched their dissertation topics get solved for them, and some researchers have seen grant proposals built around now-answered questions lose their purpose overnight. A field that stops producing people who understand its answers will eventually struggle to tell which ones matter.Someone still has to decide which results are worth building on, which ones lead somewhere new, and which were never worth the compute. That kind of judgement comes from years of being stuck on hard problems, and so far there is no shortcut to it.
Open problems are now fair game for AI labs
There is a quieter cost too, and it is about trust. Mathematicians routinely share half-finished ideas with colleagues long before they are ready to publish, on the understanding that nobody will run off with them. That arrangement held because so few people understand any given problem well enough to get there first.A lab with enough compute no longer needs to.OpenAI says it only went after Navier-Stokes after spotting chatter online that someone else was close. That someone turned out to be NYU’s Tristan Buckmaster, working with Levent Alpöge, an Anthropic researcher collaborating in a personal capacity. The company threw roughly 10,000 AI agents and millions of dollars in compute at the problem, and announced its solution a day after the pair published related findings. What happened in between is still disputed. Buckmaster has said publicly that he used OpenAI’s Codex while working on the problem, and that he never got a clear answer on whether those sessions fed into the model’s training. OpenAI says no specific user data was accessed, but admits it can’t rule out that de-identified data from his usage helped improve its models. Sébastien Bubeck, the OpenAI researcher Buckmaster named, disputes parts of his account and says the two proofs differ significantly.Andreas Thom, a Dresden mathematician whose work underpinned one of OpenAI’s August results, raised a similar concern and made the sharper point: removing a name from data does not remove the idea in it.With nothing settled, the uncertainty alone is changing behaviour. Some researchers have stopped listing open conjectures at the end of their papers, and people who maintain public lists of unsolved problems are wondering if they are simply handing out targets. The companies involved also sit in an uncomfortable double role, selling research tools to the same mathematicians they are competing with for results.A field that has always worked in the open is becoming more guarded.OpenAI is hardly the only lab chasing these results. In early 2024, Google DeepMind made headlines with a system that could solve olympiad-level geometry problems, and by last year its models were scoring gold at the International Mathematical Olympiad. Anthropic’s Claude was used to disprove the decades-old Jacobian conjecture in July, and Meta has published papers mathematicians wrote with its Muse Spark model.Part of the appeal has little to do with mathematics itself. Maths demands long chains of careful reasoning, and many in the industry see cracking it as a sign of more capable, more general AI. A famous solved problem has become the cleanest way for a lab to show that its model is the smartest, which goes some way to explaining why the field’s own needs have so often come second.After months of criticism, OpenAI has adjusted parts of its approach. It worked with an independent advisory group of nine leading mathematicians hosted by the Institute for Advanced Study, published the papers with a public revision history, disclosed how many problems the model attempted and promised to fund workshops to help the field digest the results. What it refused was the group’s very first request: stop testing advanced problems on proprietary models nobody else can use. OpenAI argues that this is how it builds better tools for scientists. Its critics worry about a two-tier system, in which a few labs race ahead on unreleased models while everyone else is left checking their work.
Maths is only the first field this will happen to
Mathematics was always going to feel this first, because it is one of the few fields where a machine can check its own work. The rest of science is not far behind.A study of 41.3 million papers, published in Nature in January, found that scientists who use AI publish more and get cited more, but also that AI narrowed the range of topics being studied by almost 5% and cut engagement with one another’s work by 22%. In a Google survey of more than 600 scientists, nearly half said AI pushed them towards safer, incremental projects, the opposite of the risky work that tends to produce big breakthroughs. Theoretical physicists, whose work sits closest to pure maths, say AI became their default tool for calculation in a matter of months.So the problem with AI doing mathematics comes down to what happens after the answer. October 6 delivered a mountain of them and handed everything else back to mathematicians: the checking, the explaining, the credit and the training of whoever comes next. Rumours of the next batch were circulating before most of the first had even been read.When Perelman refused his million, he argued that Richard Hamilton, whose work his proof was built on, deserved as much credit as he did.Mathematics has always treated the people behind an answer as part of the answer itself. Whoever does the maths next, human or machine, will have to work that way too.