On Sunday afternoon, while the rest of the world’s eyes were glued to the World Cup final, an AI model resolved a problem that had tortured mathematicians since 1939.
By the time Kevin Buzzard woke up in London the next morning, the result had been verified. By lunch, it was all his peers at the Imperial College London’s pure mathematics department could talk about; at the time of writing, Anthropic employee Levant Alpöge’s post announcing the result has drawn more than 20 million views on X.
“It is a big day,” Buzzard told Fortune. “I think it’s a great time to be alive, personally.”
It was the latest in a series of AI-driven mathematical breakthroughs. AI’s progress (or assault) on unsolved mathematics has compounded rapidly since mid-2025, when models first solved five of six problems at the International Mathematical Olympiad. From there, the list of fallen problems grew fast: OpenAI’s model disproved an 80-year-old Erdős conjecture on combinatorial geometry in May, and in June, 16 researchers from 15 universities published the Leiden Declaration on Artificial Intelligence and Mathematics, urging the profession to set guardrails around transparency, attribution, and peer review before AI reshapes what mathematical knowledge even means.
Mathematicians, relegated to the role of shepherd, are left to watch AI close these questions one-by-one, reaching places where the human mind can’t follow. Their reaction is a now familiar mixture of dread and amazement.
An 87-year-old problem
The problem is called the Jacobian conjecture and since 1939, it has rested on the work of German mathematician Ott-Heinrich Keller. Fundamentally, the problem is about what mathematicians call “maps,” and the conditions under which, given a set of outputs, you can determine the input. This being math, it was based on yet another German’s work a century earlier: Carl Gustav Jacob Jacobi’s Jacobian determinant. The main issue for modern practitioners is that they were unable to prove Keller’s conjecture true, or find a reason that it was false—until now.
On Sunday. Alpöge’s result met the Jacobian determinant at every point in space—the determinant holds steady at −2 everywhere—yet sends three different starting points to the same destination. Meaning, it didn’t pass the test.
It’s a “very exciting” result, Buzzard said, one that demonstrates the potential of language models to eventually reach the “supermathematician” Google Deep Learning scientist Christian Szegedy warned about half a decade ago.
But it also leaves mathematicians wanting. The trouble with getting current AI models to solve pure math is that you get the “how” without the “why,” explained Akhil Mathew, the University of Chicago mathematician who Alpöge credits with suggesting the problem to him. “One can check out that it’s correct,” Mathew told Fortune, “but it would be nice to be able to tell a story.”
Alpöge didn’t respond to Fortune’s request for comment.
Why we even have pure mathematics
Mathematicians have faced down automation before. Most people with high-school level math see the job as calculation, which computers conquered decades ago. “Then you go to college, and if you do some advanced math classes, you learn that actually math is all about reasoning,” Buzzard said.
A calculator multiplies four-digit numbers faster than any human. What a mathematician adds is the reason: told that 131 times 137 is four million, Buzzard doesn’t need to reach for a calculator—he knows two odd numbers can’t make an even one. To understand something, he said, is to “make it fit in your brain,” so well that you could regenerate the result yourself from the idea.
The demonstration of your knowledge,in formal mathematics, is a “proof,” a chain of logical steps, each following from the last, that ends at the claim you’re making. Proofs are both how mathematicians build and how they’re measured. A great proof can run hundreds of pages and take months of explaining to experts to trust.
So far, AI doesn’t yet have the capabilities to create such a proof, Buzzard said. Creating a delicate 150-page proof requires hundreds of steps, and language models have a habit of bridging gaps with plausible-sounding filler. because unlike a human colleague, the model risks no reputation by being wrong.
If and when that bottleneck cracks, it’ll be Buzzard’s own doing, though. His career project is Lean, a popular computer language in which proofs are checked by machine rather than by exhausted PhDs. He said the proof was already checked in Lean by the time he woke up. The moment proof-writing models meet his proof-checking machine, one of human’s last advantages in mathematics disappears.
The question of “taste”
Mathew was more tempered in his excitement, calling this moment “a very rapid and very unsettling change… especially for junior mathematicians.”
Michael Harris, a professor of mathematics at Columbia, wrote in a June essay in Boston Review that the AI industry treats reasoning, or understanding, as commercially worthless, and human mathematicians as a “beta version of intelligence.” Yet mathematics, he argued, is one of the last examples of unalienated labor, a field that people enter, in the words of Abel Prize winner Pierre Deligne, because one can earn a living “by playing” — what Mathew calls “telling a story.” Even when Deep Blue “solved” chess by beating Garry Kasparov in 1997, people didn’t stop playing chess; they learned from it.
But perhaps it sounds uninspiring for the public to subsidize mathematicians to play. Even before AI threatened their work, federal funding for mathematics research has fallen roughly 72% under the Trump administration’s cuts to the National Science Foundation. PhD admissions at top research universities are down 15% this fall, the second consecutive year of contraction; George Washington University’s math doctorate will admit no funded students at all.
Some think that the death of mathematical professionalization is good, that the machines democratize the whole enterprise, the “playing.” Garry Tan, president of Y Combinator, reacted to the news on X by hailing the return of the age of the “gentleman scientist”— rich savants à la Benjamin Franklin, funding their own curiosity. But Alpöge is no hobbyist; he is a Harvard valedictorian who has spent a decade using algorithms to calculate exactly this kind of problem.
And that might be the trick to keeping humans in the mathematical loop, Buzzard said. Beyond calculating, beyond even logical reasoning, “understanding,” at the bottom, is knowing what to ask, what Silicon Valley has taken to calling “taste.”
“People have tried to get machines to ask questions, and they’re abysmal,” he said. “All the questions they ask are either boring or obviously true or obviously false.” The monuments of the field—the Riemann hypothesis or Keller’s Jacobian—are named for the people who posed them, not the people who settled them, Buzzard pointed out. “It’s not a coincidence. You have to be a brilliant mathematician to come up with the right question.”
