數學家觀察 AI 對數學未來的衝擊

I t’s 4:26 a.m., and I’m walking around the house while my wife sleeps upstairs. I’m feeling disoriented and discombobulated, just as I have for months, since May, when AI came for math.

The leaves are ablaze here in Ithaca, fiery reds and oranges, and I saw them swirling around in my backyard yesterday, following the streamlines of an eddy described by the Navier-Stokes equations. You’ve heard of Navier-Stokes, right? By now you must have—in early September, OpenAI claimed to have solved the Navier-Stokes Millennium Prize problem. An internal AI model unleashed about 10,000 agents that ran for 88 hours straight and gnawed on the old problem until it yielded.

I am a math professor, and my field is in crisis. Nobody in math is ever going to forget the year 2026. Unless this is really the end of the subject, and then nobody will remember anything about math anyway. Well, the machines will remember. But for them, this is something like day one.

Read: OpenAI just carpet-bombed mathematics

I’ve been trying not to talk like this. It’s really discouraging for the graduate students and undergraduates, who have been dreaming of a life as a mathematician, or maybe a math professor—a life of solving puzzles, asking questions, working with friends who love math too, thinking about our sublime, transcendently beautiful subject that is so misunderstood. Meanwhile, the wider public is confused. Don’t mathematicians _want_ their problems to be solved?

Yes and no.

For “applied” mathematicians like me—ones who look outward and take inspiration from nature and the real world, and who hope to use math to help improve science, medicine, business, engineering, and everyday life—AI may prove to be a boon, the latest tool that lets us see further and deeper, in the tradition of the telescope and the microscope. So yes, absolutely, some of us are excited to use it.

But for many “pure” mathematicians—those who look inward at math’s most theoretical aspects, at its inner beauty and structure, using human creativity, imagination, and logic—AI feels more ambiguous because machines can discover the truths we’re seeking, at the cost of robbing us of the struggle and joy of discovering them ourselves.

Pure math is the study of ideal patterns and structures: numbers and shapes, waves and vibrations, order and chaos, symmetry and infinity. There are no perfect circles in nature, only in our minds. Except—and this is eerie—the truths of mathematics _also_ seem to exist outside of human minds. They inhabit a mysterious Platonic realm, casting shadows on our reality.

Our species has been playing this game for at least 4,000 years. It’s as much an expression of the human spirit as music and literature. If AI continues to bulldoze pure math, it’s going to crush the hopes of the young people who were dreaming of joining this tradition.

E veryone I know is reacting in their own way. Let me tell you three stories, about what we’re gaining and what we’re losing in the aftermath of the detonation of OpenAI’s biggest bomb yet on pure math—its October 6 report that close to 400 landmark open problems had been solved by its latest model. These were puzzles that had served as beacons in the darkness, calling and challenging and inspiring us for decades. On average, the model took just three hours’ worth of computation to dispose of each of them.

Earlier this month, I was heading for my weekly lunch with my pal Alex Townsend. He’s much younger than I am—he could be my son—and he thinks a lot about the future. With fortuitous timing, we recently finished writing a book about computers and math, and what happens when humanity makes a pact with the machine to start outsourcing our intelligence to something alien and voracious.

As we walked to lunch, Alex seemed agitated. Eyes open wider than normal. Gesticulating. Shouting. He said he had collected something like 200 important unsolved problems in his subfield, numerical linear algebra. Rather than let OpenAI or Anthropic get to them first, he decided to curate the problems himself, in the hope that all of his friends and colleagues could solve them on their own (possibly with the assistance of AI) before the big companies stomped on them.

This is how Alex is reacting to the crisis: embracing the technology and making it ours. Using it to extend the frontiers of math, pure and applied, and solve problems that really matter—problems in medicine and energy and climate, hard problems that AI will help us with. That’s the message of hope he wants to give to grad students. He says we need to clear all of the existing open problems—pull out all the Excaliburs—to pave the way to a new kind of mathematics, in which the most important human contributions will be asking good questions, developing theories, interpreting machine-generated proofs, finding new applications, deciding which problems matter, and enjoying one another’s company as we explore all of this together.

So how did his colleagues feel about his plan? Many were exhilarated, but some were furious. They saw him as a collaborator with everything bad about what’s happening now. They worry that he’s helping to destroy the pipeline for training young researchers, and that he’s encouraging the premature solution of problems that should have been reserved for students.

Alex believes that resisting AI is futile. He thinks we need to adapt, radically and rapidly. He told me he was releasing even more problems—600 long-standing problems across all of applied math—for people to solve, presumably with the help of AI, so that genuinely new work can begin.

J ust the day before my lunch with Alex, the whole math department—faculty, staff, and students—at Cornell was called to a town-hall meeting. The meeting was meant to provide solace and support, and to address issues such as how to assess future job candidates, and what to expect of a Ph.D. student now that an AI can solve their dissertation problem and even write their thesis for them.

At one point, one of my pure-math colleagues stood up. Justin Moore, known for his work in logic and set theory—foundational subjects for the rest of mathematics—told us about a certain problem that he has long loved and has returned to often over the past 19 years. He said that he had bought a bottle of XO cognac in Luminy, outside Marseille, back in 2010. He planned to open the bottle when the problem was solved, and to invite others who worked on it to share in the celebration.

Read: Something weird is happening in math

The problem had to do with something I couldn’t understand in the least—something about non-amenable groups and Thompson’s F. But it didn’t matter that it was impenetrable to many of us. He had been saving that bottle of cognac for a problem that had just been killed by a machine. It was just another trophy in OpenAI’s bag, one of hundreds, presented like severed rhino heads.

Someone asked him what he thought of the AI proof. Justin said he didn’t know. He hadn’t read the paper yet. He said he’d been “too busy.”

That astonished me. Too busy to look at a problem he had been dreaming about for 19 years? “I am genuinely busy and probably wouldn’t be able to get to reading a human’s proof immediately either,” he told me afterward, by email. “It does not help that AI does not write mathematics particularly well.”

He still planned to open the bottle, he said, “once a group of peers I trust all agree that the proof is correct, and once we collectively understand the proof.” But he added, “I have a feeling that the cognac tasting will feel more like a wake than a celebration.”

T he object of our desire, as mathematicians, is a mixture of knowing and understanding. But pure and applied mathematicians don’t always see eye to eye on what AI portends for mathematics, because our communities attach different weights to the worth of knowing versus understanding.

Roughly speaking, applied mathematicians strive to discover both what’s true and, when possible, why it’s true. We use rigorous math when we can, but we aren’t embarrassed to lean on physical intuition, computers, experiments, data, analogies—anything to get to the answer and make sense of it.

Pure math, on the other hand, is all about the why. Without the reasoning behind it, mere knowledge of the right answer—to a pure mathematician—feels shallow, frustrating, and incomplete. This fixation on understanding can be hard for the public to sympathize with. If an AI model correctly predicts that you’re at heightened risk of developing breast cancer in the next five years, should you care that it can’t explain why, and neither can your radiologist? When lives are at stake, understanding becomes less urgent than just knowing the answer.

Pure mathematicians agree, but this tension is not part of their work. As Curtis McMullen, a professor at Harvard and a Fields Medalist, said to me in a recent email after he heard me make a similar observation on NPR’s _On Point_: “Mathematics is not radiology. It is more like art or philosophy, and in fact, human understanding is what it is all about.”

Applied mathematicians want to understand math too, because it’s mysterious, and alluring, elusive, inchoate. We’re not even sure whether math is out there in the universe—because what does _out there_ even mean in terms of abstract concepts that live in our imagination? But we chase math anyway, and sometimes it materializes, or reveals itself to us, and then, a century or two later, it miraculously gives us GPS, and lasers for eye surgery, and remote TV transmissions, and ultrasounds for pregnant women, and all of the songs and photographs in your pocket, compressed with fantastically ingenious mathematical algorithms that people have thought up.

But that defense of math is too easy, and anyway, that’s pure math that morphed into applied math. What about pure math that stays pure, the part of math that’s more like art or music or poetry, not yet of any real use to anyone, except that it makes life worth living for the people who love it? Isn’t that useful too?

At the town hall, a student named Aria Beaupré asked something like that. _What’s the point?_ she wondered out loud. This is a bright young graduate student. _Someone that age shouldn’t be asking that question,_ I thought. That’s for old people like me, looking back, not for young people like her, looking ahead. We were all reeling, too numb to make sense of what had just happened.

Since then, I think I’ve groped my way toward an answer. A couple of weeks ago, during a conversation with Rachel Abrams on _The Daily_, the _New York Times_ podcast, I wandered off the topic of math and AI and onto the meaning of life. Before that conversation, I would have said that the point of it all is learning what’s true, even more so than understanding why.

Now I think the point, even for math professors, may be human connection. Among the hundreds of emails I received after the episode came out, one from an undergraduate moved me deeply. The student described many failures and tribulations, and a strict but inspiring teacher. “The very human things that you expressed, and that I’ve poured into this email, are not something that AI can achieve or experience,” the student wrote, and later added, “We give these ideas and concepts meanings, and we use math to connect with one another, and bring meaning to each other’s lives and experiences. AI can’t even begin to replicate this impact, and it never will.”

Read: Math can’t go on like this

But is human connection really a sufficient justification for why anyone would devote themselves to math? I’m not sure it would satisfy pure mathematicians like Justin. The deeper question is whether mathematics is fundamentally a human enterprise, like art or sports, or whether it’s more like science, the study of objective truths that exist independently of us.

If mathematical truths exist outside of human minds, awaiting discovery by any intelligence powerful enough to pursue them, then AI’s achievements are surely something to rejoice about. Why should it matter who or what makes the discoveries? But if mathematics is a creative _human_ endeavor—a way of exercising our ingenuity and sharing those experiences with other people—then something achingly precious may be lost when the machines take over and leave us behind.

“Even though my heart wants math to be like an Olympic sport going forward, I know that it’s a science,” Justin wrote recently. He finds himself torn between these competing conceptions of the subject we both love. “AI is separating those two things in an uncomfortable way and making us look in the mirror at ourselves.”

Perhaps the hardest part is that both conceptions of math can be true. Mathematical truths may exist outside of us, but the search for them has always taken us deep inside ourselves, to find the best that our minds can muster. Until now, discovery and the joy of discovering went hand in hand. Suddenly, AI is ripping them apart.

That’s why I’ve been losing sleep for months, and why Justin still hasn’t opened his bottle. Neither of us knows what comes next. Will there ever be a time for cognac again?