On Sunday afternoon, with the eyes of the rest of the world glued to the World Cup final, an artificial intelligence model solved a problem that has plagued mathematicians since 1939.
By the time Kevin Buzzard woke up in London the next morning, the result was confirmed. By lunchtime that was all his colleagues in the Department of Pure Mathematics at Imperial College London could talk about; At the time of writing, Anthropic employee Levant Alpöge’s post announcing the results has received over 20 million views on X.
“It’s a big day,” Buzzard said. Luck. “Personally, I think it’s a great time to be alive.”
This was the latest in a series of mathematical breakthroughs based on artificial intelligence. AI’s progress (or advance) in unsolved mathematics has accelerated rapidly since mid-2025, when models first solved five of six problems at the International Mathematical Olympiad. Since then, the list of solved problems has grown rapidly: in May, an OpenAI model disproved Erdos’s 80-year-old conjecture about combinatorial geometry, and in June, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics, calling on professionals to set up barriers around transparency, attribution and peer review before AI changes what mathematical knowledge even means.
Mathematicians, assigned to the role of shepherds, have to watch as AI solves these questions one by one, reaching places that the human mind cannot keep up with. Their reaction is a familiar mixture of fear and amazement.
An 87-year-old problem
The problem is called the Jacobian hypothesis, and since 1939 it has been based on the work of the German mathematician Ott-Heinrich Keller. Essentially, the problem is what mathematicians call “maps”, and the conditions under which, given a set of outputs, you can determine the inputs. Since this is mathematics, it was based on another work by a German a century earlier: Carl Gustav Jacobi Jacobi’s determinant of the Jacobian. The main problem for modern practitioners is that they have not yet been able to prove the truth of the Keller hypothesis or find the reason for its falsity.
On Sunday. Alpöge’s result matches the determinant of the Jacobian at every point in space (the determinant remains constant at -2 everywhere), but sends three different starting points to the same destination. This means he failed the test.
This is a “very exciting” result, Buzzard said, and demonstrates the potential for language models to eventually achieve the “supermathematics” that Google Deep Learning scientist Christian Szegedy warned about about five years ago.
But it also leaves much to be desired for mathematicians. The problem with getting modern AI models to solve pure mathematics is that you get the “how” without the “why,” explained Akhil Mathew, a mathematician at the University of Chicago whom Alpöge credits with hinting at the problem. “You can make sure it’s correct,” Matthew said. Luck“but it would be nice to be able to tell the story.”
Alpöge did not respond to Fortune request for comment.
Why do we even have pure mathematics?
Mathematicians have encountered automation before. Most people who study mathematics at the high school level view their work as the kind of calculation that computers conquered decades ago. “Then you get to college, and if you take some advanced math courses, you learn that math is really about reasoning,” Buzzard said.
The calculator multiplies four-digit numbers faster than any human. The mathematician adds a reason: Knowing that 131 times 137 is four million, Buzzard doesn’t need to reach for a calculator—he knows that two odd numbers won’t make an even number. To understand something, he says, you need to “get it into your brain” so well that you can independently regenerate the result from that idea.
Demonstrating your knowledge of formal mathematics is a “proof,” a chain of logical steps, each building on the previous one, that ends with the statement you make. Proofs are both how mathematicians construct and how they are measured. A great piece of evidence can run hundreds of pages and require months of explanation to experts you can trust.
According to Buzzard, AI does not yet have the capabilities to create such evidence. Creating a delicate 150-page proof requires hundreds of steps, and language models have a habit of filling in the gaps with plausible-sounding fillers. because unlike a human colleague, a model does not risk its reputation by being wrong.
However, if and when this bottleneck breaks, it will be Buzzard’s fault. His career project is Lean, a popular computer language in which proofs are verified by a machine rather than exhausted Ph.D. He said that by the time he woke up, the evidence had already been verified in Lean. The moment the proof-writing patterns meet his proof-reading machine, one of man’s last advantages in mathematics disappears.
A matter of “taste”
Matthew was more restrained in his excitement, calling the moment “a very rapid and very worrying change… especially for junior mathematicians.”
Michael Harris, a mathematics professor at Columbia University, wrote in a June essay in the journal Boston Review that the artificial intelligence industry views reasoning or understanding as commercially useless and human mathematicians as “beta intelligence.” But mathematics, he argued, is one of the last examples of unalienated labor, a field that people enter, in the words of Abel Prize laureate Pierre Deligne, because one can make a living by “playing”—what Matthew calls “telling a story.” Even when Deep Blue “solved” chess by defeating Garry Kasparov in 1997, people didn’t stop playing chess; they learned a lesson from it.
But perhaps to society, subsidizing mathematicians who play math sounds boring. Even before AI threatened their jobs, federal funding for mathematics research had dropped by about 72% due to Trump administration cuts to the National Science Foundation. Doctoral enrollment at top research universities fell 15% this fall for the second year in a row; The George Washington University Mathematics Doctoral Program will not accept funded students at all.
Some people think that the death of mathematical professionalization is a good thing, that machines are democratizing the entire enterprise, the “game.” Harry Tan, president of Y Combinator, responded to the news at X by hailing the return of the era of “gentleman scientists”—rich scientists in the vein of Benjamin Franklin funding their own curiosity. But Alpöge is no amateur; he’s the Harvard valedictorian who spent a decade using algorithms to solve exactly these kinds of problems.
And that may be a way to keep people in the math loop, Buzzard said. Beyond calculations and even logical reasoning, “understanding” essentially means knowing what to ask, and what Silicon Valley has come to call “taste.”
“People were trying to get machines to ask questions, and it was terrible,” he said. “All the questions they ask are either boring, obviously true, or obviously false.” The area’s monuments—the Riemann hypothesis or the Keller Jacobian—are named after the people who proposed them, not the people who occupied them, Buzzard noted. “It’s not a coincidence. You’d have to be a brilliant mathematician to come up with the right question.”