An Existential Crisis in Math
· The Atlantic
In late July, one of the world’s most famous mathematicians issued a warning to his field. Math is entering a “turbulent period,” the UCLA professor Terence Tao said in front of a packed lecture hall at the International Congress of Mathematicians. AI is getting so good at math, he said, that it’s causing a “crisis in our mathematical values and practices.”
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Earlier in the same week that Tao gave his talk at math’s most prestigious conference, a researcher at Anthropic had used Claude to disprove an 87-year-old math conjecture. Still, when I asked mathematicians at the conference about their experiences with AI, most hadn’t thought deeply about the topic. Many seemed a little surprised that a journalist was asking them about it at all.
Now mathematicians are paying attention. Last week, OpenAI announced that its bots had cracked the Navier–Stokes problem, one of the seven Millennium Prize problems regarded as among the grandest challenges in the field. Navier–Stokes has stumped mathematicians since the 1930s: “It’s a career-killing problem—you can spend all your career doing it and never get the solution,” David Dritschel, a mathematician at the University of St. Andrews, told me. OpenAI said its swarm of 10,000 AI agents solved the problem in less than four days.
In the math world, the initial shock over OpenAI’s result has morphed into something of an existential reckoning for the discipline. Although OpenAI was able to solve the Navier–Stokes problem only by pulling from nearly a century’s worth of human efforts, academics are realizing just how much power the AI industry now has over math. “We’re seeing how swiftly an entire field can be ravaged by AI,” Max Weinreich, a math professor at CUNY Baruch, told me.
OpenAI’s breakthrough is as notable as how it was achieved. The night before OpenAI announced its solution, the NYU mathematician Tristan Buckmaster released a statement alleging that he had been scooped by OpenAI. With the heavy assistance of AI tools, Buckmaster and his collaborator, Levent Alpöge (who works at Anthropic), had made major progress toward solving the Navier–Stokes problem. Buckmaster implied that OpenAI might have copied their approach by accessing his messages with OpenAI’s agents. “Almost nobody else I know of was working on it,” he wrote.
In an announcement the next day, OpenAI said that it had tasked its AI systems to work on all of the Millennium Problems after hearing rumors that Anthropic had solved two of them. OpenAI claims that it focused on the Navier–Stokes problem only after finding promising partial results, and that it never looked at Buckmaster’s inputs. In any case, OpenAI poured incredible resources into solving this problem—expending millions of dollars in tokens—all because it heard an unfounded rumor of a solution. (OpenAI declined to comment.)
This drama has underlined concerns many mathematicians have about AI developments within the field. AI companies are encouraged to solve problems in a way that generates good headlines for their bots—marshaling resources that academics never could—but those solutions won’t necessarily aid further discoveries or help humans understand mathematics. On Friday, 25 winners of the Fields Medal (sometimes called the Nobel Prize of mathematics) released a statement called “A Severe Misalignment of AI in Mathematics.” “The push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community,” they wrote.
All of this can seem obtuse. Isn’t the point of mathematics to solve problems? But there’s more to a problem than just knowing whether it’s true. Solving a major problem in pure mathematics is often like getting a man on the moon: symbolically meaningful but with little direct value. Even though the Navier–Stokes problem deals with one of the most important models in fluid dynamics—a topic of great interest to airplane designers, climate modelers, and many other scientists—knowing the actual result is basically useless.
However, with both a math problem and the moon landing, the process of reaching that goal can indirectly develop all sorts of useful tools applicable to other problems. As part of the Apollo missions, NASA engineers and contractors had to solve subproblems that ultimately proved useful elsewhere. The need for an onboard computer, for example, helped spark the nascent computer-chip industry. Perhaps more surprising, to ensure that astronauts did not get sick while in space, the Apollo program helped set food-safety standards that have since become standard in the industry.
Mathematicians hope that OpenAI’s Navier–Stokes proof will end up containing ideas that make it easier to study all sorts of other problems, inside and outside of pure mathematics. This has happened before. For instance, efforts to prove Fermat’s Last Theorem produced mathematical tools that now underpin the newest online encryption methods.
It’s still too early to tell what humans might be able to learn from OpenAI’s proof. The paper is 166 pages and “almost incomprehensible,” Dritschel said. Yet over the coming months and years, mathematicians will exert substantial effort to dissect and distill OpenAI’s proof until its central ideas are understood enough to (hopefully) be applied elsewhere.
Humans won’t be able to do that for most AI-generated math results, however. Already, there are AI proofs that no human understands, not because the ideas are impossible to grasp, but because no one has had the time to look through the solution carefully. It’s easy to imagine a future in which mathematicians use AI to prove ever more impressive results ever more quickly—but they don’t understand what they’re proving in much detail. AI systems can speed the rate of problem-solving, but they can’t make it that much easier for human mathematicians to grasp a proof.
So if the field as we know it is to survive, mathematicians need to figure out how to adapt. One idea is to give less credit for solving a problem and more credit for work that communicates the ideas behind a solution in a way that can benefit the broader mathematical community. Yet it’s hard to say exactly how that works in practice. Unlike solving problems, evaluating how well a mathematician has communicated with their peers has no clear metrics.
It will take time for mathematicians to figure out the best response. All the while, AI is continuing to develop incredibly quickly. In less than the time that it takes to complete a Ph.D., systems such as ChatGPT have gone from struggling with basic arithmetic to solving some of the most famous problems in the world. Math’s turbulent period, as Tao deemed it, isn’t ending anytime soon.