A mathematician is a machine for turning coffee into theorems. This was a line often invoked by the Hungarian mathematician Paul Erdős, who was fuelled by coffee and amphetamines, slept little, travelled near-constantly, never married or had children, and published almost fifteen hundred papers in his lifetime, in collaboration with hundreds of other mathematicians. But Erdős was, in most ways, not machinelike. He called lecturing in mathematics “preaching,” woke friends at odd hours by asking, “Is your brain open?,” wore only silk underclothing on account of his sensitive skin, and claimed not to have learned how to butter toast until he was twenty-one.

He gave away most of his earnings to charitable causes and to young mathematicians in need of funds for their education, and he offered bounties (sometimes ten dollars, sometimes a thousand) for solutions to mathematical problems he’d proposed. Mathematics was his life. Mathematicians were his family. It has been said that, even above his remarkable contributions to the mathematical literature, he helped the discipline most profoundly by turning people who might otherwise have become hedge-fund managers or F.B.I. agents or lost souls into mathematicians.

When he died, in 1996, at the age of eighty-three, he left behind about a thousand problems. Solving an Erdős problem is a great honor, as is having a low “Erdős number,” a term that describes how many degrees of separation one is from having published work with him.

In another era of mathematics, which is to say, in April, I visited Tao at his office in the Mathematical Sciences Building at U.C.L.A. The building was connected through a mazelike series of corridors and stairs to Boelter Hall, where, in 1969, the first message was sent on , the precursor to the internet. One of the hallways had a telephone booth. Tao’s office was spacious and mostly unadorned, with beige carpeting and a beige sofa, six blackboards mounted onto two long walls, and stacks and shelves with hundreds of books bearing titles such as “Advances in Analysis: The Legacy of Elias M. Stein,” “Number Theory Volume I: Tools and Diophantine Equations,” and “The Arithmetic of Elliptic Curves.”

Near the light switch, there was a drawing of a toothy monster in a black suit and tie, next to a message in code (“LWKIRLCS,” part of it read) written on a sheet of paper. His oldest child had made it for him years ago. “I don’t think I remember how to decode it,” Tao said. None of his many prizes and honors were on display, but there was a wall hanging—made by the Mathematics Genealogy Project, of North Dakota State University—recording generations upon generations of who had studied under whom.

Much of the math we learn as kids is mechanical, which may be why mathematics is sometimes misperceived as an unimaginative field. A more accurate understanding might come from recalling that Charles Dodgson, better known as Lewis Carroll, was a professional mathematician. In some ways, the writing of “Alice’s Adventures in Wonderland” resembles mathematical research. Mechanical calculation, in this analogy, is the basics of composition: spelling, subject-verb agreement.

But, in order to explore logic and whimsy and wonder and paradox, Carroll invented characters like the White Rabbit, the Cheshire Cat, and the Queen of Hearts, and situations like the Mad Tea Party and the Caucus Race. These characters and scenes are akin to the objects and techniques mathematicians invent as they explore mathematical ideas. When Andrew Wiles proved Fermat’s Last Theorem, more than three hundred and fifty years after its initial postulation, mathematicians were thrilled, but not because of the answer, which wasn’t surprising. (No, there are no positive whole numbers such that if > 2.)

What was wonderful about Wiles’s proof was the mathematical techniques he’d developed along the way. That was what advanced mathematics.

The mathematician Geordie Williamson, of the University of Sydney, in Australia, described one A.I.-assisted solution to me as “like this beautiful thing for the whole math community” and another as “like the weed killer Roundup being sprayed on a garden.”Both results were published by Levent Alpöge, a mathematician at Anthropic.

The beautiful thing concerned an eighty-seven-year-old problem, the Jacobian conjecture. We can think of the problem as having to do with the possibility of a one-to-one mapping from one shape to another. On July 19th, the day of the Spain-Argentina soccer game, Alpöge posted on X, “hello there the jacobian conjecture is false thanx to my close friend akhil”—a human—“for asking about it and my other close friend fable”—an A.I. model—“for working during the world cup final.”

Claude Fable 5 had produced a counterexample to the conjecture, much like how an ostrich is a counterexample to the conjecture that all birds can fly. Williamson said, “Within thirty-six hours, mathematicians had kind of understood where the counterexample comes from and why it’s natural.”A handful of mathematicians, including Tao, explored the proof, came to understand it—a process that I kept hearing mathematicians refer to as proof digestion—and reconstructed the algebraic geometry that underlay it.

On July 21st, Tao posted the work on his blog, including the transcript of a conversation he’d had with ChatGPT as he looked into the details. By July 22nd, Semon Rezchikov, of Barnard College, had created a visualization of the counterexample using only high-school-level mathematics. The proof had opened up thinking, not decapitated it.

“I thought the Fields medallists’ letter was valuable,” John Urschel, a mathematician at M.I.T., told me. “Though I understand that maybe it doesn’t translate well outside of the mathematical community. I saw one person be, like, ‘We’re getting answers to cancer research too quickly, we must slow down for the sake of the field!’ ” Urschel is good friends with Alpöge, whom he remembers saying three or four years ago that A.I. was going to end math as they knew it. “I thought he had lost all sense,” Urschel said.