数学家直面AI可能超越人类的现实

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— Timothy B. Lee

At a July 23 press conference in Philadelphia, the Canadian mathematician Jacob Tsimerman announced that he was joining the safety team at OpenAI. The timing was jarring: Tsimerman had just received a Fields Medal, perhaps math’s most prestigious prize.

“Because I have some publicity on me now,” he told me the next day, “I’m trying to direct people into AI safety as much as I can.”

Rapid AI progress hasn’t just made Tsimerman worried about AI safety; it’s also made him pessimistic about the future of mathematics as a profession.

Jacob Tsimerman. (Photo courtesy of the Simons Foundation. CC BY 4.0)

“I feel quite confident that very shortly AI will become robustly superhuman at what professional mathematicians currently do,” he told me. “I mostly want people to grapple with that reality.”

I had traveled to Philadelphia to attend the International Congress of Mathematicians (ICM), the world’s most prestigious math conference, because I wanted to find out how mathematicians felt about the rapid pace of AI progress in their field.

Three years ago, leading AI models struggled with arithmetic. Last year they reached near-parity with the world’s top high schoolers in math competitions.

Now AI systems are autonomously solving open problems that stumped human mathematicians for decades:

• In May, an internal OpenAI model disproved the Erdős unit distance conjecture, which Princeton mathematician Noga Alon described as “arguably the best known problem” in the mathematical subfield of discrete geometry.
• In July, a mathematician working at Anthropic tweeted that Claude Fable had found a counterexample to the Jacobian conjecture in higher dimensions.
• On Saturday, OpenAI announced that an internal version of Astra, its next major model family, had “solved ten major open problems” — including several “of broad interest across mathematics as a whole.”

Developments like these have led some to claim that mathematics is close to being “solved” by AI systems.

How do mathematicians feel about this? I spoke with over 20 mathematicians in Philadelphia, ranging from prominent professors such as Tsimerman to incoming graduate students.

To my surprise, many were optimistic about the impact of AI on their own work, at least in the near future. A fair number said that AI systems had been helpful in their own research — albeit in limited ways — and seemed to expect that AI systems would continue to complement human talent rather than replace it.

And even those who thought AI systems might eventually get better than humans at all mathematical tasks bristled at the notion that math would then be “solved.” They argued that mathematics has a diverse array of goals and values, only some of which are about solving open problems. While AI can change which values humans should pursue, they argued, it does not change why humans might want to do math in the first place.

The traditional response to automation

Yu Deng, John Pardon, Jacob Tsimerman, and Hong Wang sit onstage after receiving their Fields Medals in Philadelphia on July 23. (Photo by Erin Blewett/AFP via Getty Images)

That July 23 press conference featured mathematicians who had just won a Fields Medal or another prestigious math award at the ICM. A high school reporter asked each panelist what they would tell students anxious that AI systems might narrow their future place in mathematics.

Tsimerman said he wanted young people to keep “learning and improving themselves because you don’t know how the world will turn out.” He encouraged students to “engage with AI because it’s going to be a big part of our world going forward.”

At the same time, he thought students were right to pay attention to how AI is disrupting the math profession. “I don’t think it’ll exist the way it exists right now,” he said.

Not everyone agreed. Yu Deng, a University of Chicago professor who also just won a Fields Medal, described himself as “on the more optimistic side.” He predicted that “AI is going to be helping mathematicians instead of replacing them.”

“What we may expect in the future is that mathematicians will come up with new theories, new ideas, new frameworks and the AI is going to do some of the technical details,” Deng said. “The AI will get stronger, but then we’ll redefine what are technical details. I believe that the way we study math will change, but the joy we get from studying math will not change.”

I spoke to many mathematicians whose views were close to Deng’s; he was effectively describing how mathematicians have historically dealt with automation. As computers have made certain types of calculations easy — like multiplication or algebraic manipulations — humans have been able to find new problems computers can’t solve.

The mathematician Jordan Ellenberg encapsulated this viewpoint in his 2014 book How Not to Be Wrong. He wrote that unless machines completely surpass humans’ mental powers and end civilization, math will probably be fine.

After all, math has already been computer aided for decades. Many calculations that once would have counted as “research” are now considered no more creative or praiseworthy than adding a series of ten-digit numbers; once your laptop can do it, it’s not mathematics anymore.

But this hasn’t put mathematicians out of work. We’ve managed to stay just ahead of the ever increasing sphere of computer dominance, like action heroes outracing a fireball. And if machine intelligences of the future can take over from us much of the work we know as research now? We’ll reclassify that research as “computation.”

Today’s AI is far more capable than computers in 2014. Still, this viewpoint seems to be functionally how a lot of mathematicians think about current AI systems in their own research.

The most common use case I heard about was mathematicians using AI to learn about techniques from unfamiliar areas of the mathematical literature.

The Brandeis grad student Vasiliy Neckrasov said that previously, if he wanted to use tools from an unfamiliar area of math, he’d have to read through “a giant textbook for 500 pages.” Going in, he wouldn’t know if the textbook applied to his specific research, so it might be a waste. Today, AI can quickly point him to the right resources — and he feels “more focused, more motivated” reading them “because I really needed to learn exactly these” results.

Jeremy Avigad, a professor at Carnegie Mellon, told me that a lot of colleagues use systems this way. He said that “people feel less threatened” by AI systems that serve as powerful search engines than AI systems directly proving mathematical results.

Some mathematicians told me they’d used AI tools to directly solve problems — but only as part of a larger project. Alonso Castillo-Ramirez said that ChatGPT had been able to construct an example of a cellular automaton that had special properties relevant to his research. He was impressed. “Otherwise, even with a computer program, it would have been very difficult to find” the example. But ChatGPT’s example was only one part of a larger research project.

Neckrasov uses AI more aggressively than anyone else I talked to. He pays $200 per month to use Codex for a variety of mathematical tasks like searching the literature, filling gaps in proofs, and reviewing drafts of his papers. But he still uses it as a tool.