After AI Cracks a Hard Math Problem, Do We Still Need Mathematicians?

Something big happened in mathematics while I was writing this.

Late on the night of 7 September, two mathematicians put three papers online, saying that with heavy help from AI they had cracked a set of fluid dynamics problems that had been open for decades. The next day OpenAI announced that an unreleased internal model had solved the Navier-Stokes equations, a problem carrying a million-dollar prize.

A colleague of mine who works on fluid dynamics in astrophysics sent me the link with a single question attached. What is this actually good for?

He makes his living off those equations and he could not think of a use. If he cannot see the point, nobody outside the field is going to. So that is what I want to talk about.

Seven mountains

In 2000 the Clay Mathematics Institute picked seven problems and put a million dollars on each of them, the Millennium Prize Problems. Seven problems is one way to describe them. Seven mountains is closer. Twenty-six years on, exactly one has been climbed, the Poincaré conjecture, and Grigori Perelman, the Russian mathematician who did it, turned the money down. Navier-Stokes is one of the six still standing. These mountains are famous not only for their height. They are famous because a century of people hiking towards them has worn an entire network of paths into their slopes.

The equations sound remote. What they govern is all around you. They are named after two nineteenth-century scientists, Claude-Louis Navier and George Gabriel Stokes, who wrote down how a fluid moves. Weather forecasting, aircraft wings, blood in an artery, the accretion disk around a black hole that my colleague studies. All of it runs on those equations.

The embarrassment is that we have used them for more than a century without ever proving that they always work. Nobody knows whether a fluid starting out perfectly smooth can have its velocity run off to infinity in finite time. Mathematicians call this blowup. Put another way, every engineer alive is operating the same machine, the machine has never once failed, and nobody has checked whether it can explode on its own.

That is the question. Can it?

Eighty-eight hours, ten thousand agents

Take OpenAI first. By their account, they began on 1 September and set roughly ten thousand AI agents running at once, the kind earlier columns have described, the ones that take an instruction and get on with it unsupervised. Eighty-eight hours later they stopped, with a 166-page paper and a formal version in Lean, a language that lets a computer verify a proof line by line.

One thing most of the coverage skipped. The original prize notice lists several different answers that count as a solution. OpenAI answered the version that permits an external force to be applied. What fluid specialists actually care about is the version with no force at all, because that is the one that corresponds to the world. Same mountain, gentler face.

The money is not going anywhere soon either. Clay requires publication in a journal, then a two-year wait, then general acceptance by the mathematical community. OpenAI has said it does not intend to claim the prize. Reaching a summit and opening a route are not the same thing.

The other story that week tells you a great deal more.

The two who posted the night before were Tristan Buckmaster, a professor at NYU's Courant Institute, and Levent Alpöge, a mathematician working at Anthropic. They proved that three simpler fluid equations do blow up, again with a force applied, and they released the machine-checkable files alongside, so that anyone can download them and verify the result for themselves.

By Buckmaster's own account, the two of them worked for most of a year making only slow progress, and broke through in mid-August with help from several AI models. Their proofs lean heavily on AI as well. But the approach they were following had been laid down years earlier by two other mathematicians, Diego Córdoba and Luis Martínez-Zoroa. What consumed the year was working out which of those routes to follow and crossing the rest off.

Choosing the direction is the hard part, and Alpöge showed that more clearly in July. Using Claude Fable 5, which Anthropic had released weeks before, he found a counterexample to the Jacobian conjecture, overturning a claim that had stood since 1939 and defeated everybody for eighty-seven years. The counterexample is short enough to fit inside a single social media post.

Eighty-seven years, and the answer runs to a few lines. What defeated generations was never the writing down. It was that nobody had looked in that direction. And the direction came from another mathematician, who suggested he go and have a look.

AI will walk without ever tiring. Where the route goes still has to occur to a person.

My own working life is a budget version of the same arrangement. Before bed I hand the model whatever I have not managed to think through that day, let it run all night, look at what came out in the morning and decide whether it is worth chasing. Some ideas it kills overnight and saves me weeks. What it still cannot tell me is which question was worth handing over.

Why mathematics went first

That all of this is happening in mathematics is not an accident. Mathematics is a cheap trail to set out on. One person, one piece of chalk, one blackboard, and you are moving. No experiment to wait for, no telescope time to apply for. More to the point, every step tells you immediately whether you have gone wrong, and immediate feedback is precisely what these systems need. My own field is slower. What stops us is usually not that we cannot think of a theory. It is that we do not have the data.

So mathematics has moved fast. Last July, AI reached gold-medal standard at the International Mathematical Olympiad. This year, old conjectures have been falling one after another, and now it is Navier-Stokes. But mathematics is only out in front. These stories will arrive more and more often, and my colleague's question is one that every field is going to end up asking itself.

The point is the hike

So let me answer him honestly. In the near term this proof is of no use whatsoever to the black hole accretion disks on his desk.

Then why does anybody climb? The mathematician Terence Tao has put it about as plainly as it can be put. Solving these problems, he says, "is only a proxy goal for the primary goal of developing mathematical understanding and insight." He has an image for it that I like very much. These problems, he has said, are like distant locations that you would hike to. The journey is what lets you lay down trail markers and draw the maps that other people build on. AI tools are like taking a helicopter to drop you off at the site, and you miss all the benefits of the journey itself.

What is it that you miss? The paths that go nowhere. Which gully cannot be crossed, which slope looks like a shortcut and is not. None of that ever reaches a paper. The photograph from the summit goes around the world. The dead ends do not. But a dead end is never walked for nothing. The routes one generation wears out rot down into the soil the next generation stands on, and where the next flower comes up depends on how deep that layer is.

The row this week is a footnote to the same point. Buckmaster has said publicly that after he wrote to OpenAI on 3 September, the company pressed him for calls, and that on those calls he was asked to publish alone and drop Alpöge from the authorship; he refused. He also asked whether OpenAI's model had been trained on, or had access to, the Codex sessions where the two of them had kept every draft. He was told the model did not look up user data. On training, he says, he got no answer. OpenAI says no specific user data was accessed, while adding that it cannot rule out that de-identified data from their use of its products helped improve its models.

The only thing I have left to teach

For anybody who supervises students, the loss is not abstract.

Most researchers are carrying a few years on the books that produced nothing. A problem that never worked out, no paper at the end of it, and at the time it felt like waste. Then for the next decade and a half, the thing you rely on when judging whether a direction will go anywhere is the feel you picked up in the dark during those years.

Earlier columns have talked about scholarly taste. What a supervisor still has to give a student is no longer technique. AI writes better code than I do and writes it faster. The only thing I have left worth handing over is a map I walked out myself, with the dead ends already marked.

Give a student a machine on their first day that answers everything, and they will answer beautifully. They may never learn which way to go.

So that is what I told my colleague. In the near term, no use at all. But the first purpose of science was never to reach summits. What moves a field is rarely the answer to any one of them. It is the eye a whole generation grinds sharp on the trail.

None of which makes AI a bad thing. I use it every day and I enjoy it. What gives me pause is not that it is too strong. It is something else. Once every mountain we have already named has been taken by helicopter, who decides which one to climb next.

Perhaps one day AI will think up the questions as well. Until that day, the person set down on the summit is not the one who comes back with a map.

About the author: Graduate of Chong Hwa Independent High School, Kuala Lumpur. Earned his PhD in Astrophysics from Harvard University in 2017, then held a NASA Hubble Fellowship at the Institute for Advanced Study (IAS) in Princeton. Former joint professor of Astronomy and Computer Science at the Australian National University, and currently Associate Professor of Astronomy at The Ohio State University, where his work focuses on using AI to accelerate discovery in astronomy and the fundamental sciences. Now on sabbatical in Malaysia, watching the local scene as an outsider, and welcomes letters from readers. Personal website: https://www.ysting.space