1. Leaving Lake Lemon

I spent much of my time as a student fascinated by mathematics. In high school I felt confined by the conformity of school, and I began taking courses at the local university, Purdue Fort Wayne. Sitting in the offices of the professors there felt like joining a community that had its own language and customs. I vividly remember reading Gauss's work on the arithmetic–geometric mean there and being ecstatic, watching as each trick unfolded before my eyes, at times clapping my hands.

I was at Indiana University when the pandemic began. I would drive out to Lake Lemon, by the train tracks that cut through the hills, to read about schemes, perfectoid spaces, and other otherworldly objects. I pored over Grothendieck's writing on what it means to do good mathematics. I bought every book that might be relevant to my studies and devoured them. I wanted to be a mathematician, maybe not as a profession, but I wanted to learn to think like one: to take an idea apart, find the right vantage point, and see all of its components. Mathematics was its own world, where ideas played and danced off one another like a symphony.

When I finally left academia and stopped taking those trips out to Lake Lemon, I took a step back to work out the value of what I had learned. I wondered what the value of something like the Langlands program was. I was hesitant to conclude that, economically, the program by itself was worth much of anything. However, I had learned a great deal, and mathematics had shaped how I think. That value was real, but it was not something you could put on a balance sheet.

2. What does society get from pure mathematics?

Pure mathematicians' grant proposals rarely dwell on real-world impact. They touch on it briefly, at the start or the end, before diving into the theory. Despite this, mathematicians have largely agreed as a community, and governments have agreed more broadly, that mathematics research and education are worth funding. Former mathematicians such as Jim Simons and Alex Gerko went into finance, and both came back to fund mathematics for its own sake.

Mathematics has long had a productive relationship with industry. The fathers of modern computing, Claude Shannon, Alan Turing, and John von Neumann, all held doctorates in mathematics, and their most influential work moved freely between abstract theory and practical problems. Shannon created information theory, Turing defined what it means to compute and helped break Enigma, and von Neumann described the architecture of the stored-program computer. Claude, the AI model, is named after Shannon.

Since then, much of pure mathematics has moved further from those problems. Its value still reaches society, through the people it trains, through tools that other fields borrow, and through ideas that find uses decades later. But those paths are long and hard to predict, and how quickly mathematics reaches the world has always been limited by how quickly mathematicians can produce it. AI is now changing that, with mathematics becoming a commodity, something that can be produced on demand.

3. But AI can't think for itself

For the past few years, I had heard mathematicians and other theorists say that AI can solve IMO problems but cannot do research, that it cannot conjecture, that it can only repeat what is in its training data, that it cannot think out of distribution, and more. Underneath their objections was what I sensed to be a quiet anthropocentrism, a belief that human intelligence is special.

The first computers were measured against people. According to the mathematician Paul Halmos, when John von Neumann's computer at the Institute for Advanced Study was ready for its first test, someone set it a simple problem about powers of 2. "The machine and Johnny started at the same time," Halmos writes, "and Johnny finished first" [1]. Within a few years, no human could have kept up.

In 1989, two top grandmasters had just lost to chess computers, and Garry Kasparov was asked whether a computer would one day become world champion. "Ridiculous!" he said. "… Never shall I be beaten by a machine! Never will a program be invented which surpasses human intelligence. And when I say intelligence, I also mean intuition and imagination. Can you see a machine writing a novel or poetry?" [2]. Eight years later, he lost to IBM's Deep Blue. Today, language models write poetry. Kasparov's objections rhyme with the ones I had heard from mathematicians.

After Deep Blue, Go became the new example of a game computers could not master. It has far too many possible moves for brute-force search, and strong players describe their moves in terms of intuition and shape. In 2014, Rémi Coulom, the author of one of the strongest Go programs, guessed that a computer would need "maybe ten years" to beat a professional without a handicap [3]. In February 2016, weeks before his match against DeepMind's AlphaGo, Lee Sedol said the program's "level doesn't match mine" [4]. He lost four games to one.

4. On the eve of a revolution

In the past two months, mathematicians have written more about AI than ever before. In a paper based on his ICM lecture, Terence Tao asked what the goals and values of mathematical research actually are, assuming AI will soon be able to do research-level mathematics [5]. Emily Riehl explained why she does mathematical research, and Henry Cohn warned of the "technical debt" of AI-generated mathematics [6, 7]. Timothy Gowers asked what sort of mathematics language models are good at [8]. Jeremy Avigad asked what mathematics is now, and what it should be [9]. Kevin Buzzard described the community's reaction in terms of the five stages of grief [10].

Mathematicians have also organized. The Leiden Declaration on Artificial Intelligence and Mathematics, published in June and endorsed by the International Mathematical Union, asks mathematicians to disclose their use of AI and to keep human authors responsible for their results. More than 4,000 people have signed it [11]. In September, 25 Fields medallists signed a declaration, "A Severe Misalignment of AI in Mathematics," arguing that "the goals of the AI companies and the goals of the mathematical community are severely misaligned." The count has since grown to 28 medallists, joined by more than 8,000 endorsers [12, 13]. Gowers explained why he did not sign it [14].

Even the most optimistic writing on the subject treats AI as an instrument. Christian Szegedy, writing on October 3, compares AI in mathematics to the airplane in the age of the Wright brothers [15]. Mathematicians are explorers of a vast jungle, and the planes are still primitive and expensive; until this year they barely left the ground. They have already improved quickly, and he is clear that they will keep improving. Those who say AI cannot conjecture or build theories, he writes, are "describing last spring's models." But in his picture, AI is always equipment: airplanes, satellites, tools bought at a tool shop, all built to serve human purposes. Humans still set the purpose: we decide why the terrain is being mapped and what we are trying to achieve by mapping it.

5. Not an airplane

I think Szegedy's imagery severely understates where we are heading. AI will not just aid mathematicians. It will choose what to research, with an incomprehensibly deep understanding of why that research is useful and important. In place of the airplane, a better image is an object that keeps accelerating away from us. At first we can follow it. Then it moves faster than we can track, and as it approaches the speed of light, it recedes further and further from our sight. We are not flying it, but rather trying to fly alongside it, and soon we will not be able to see where it is going, or what lies along its path.

Go shows what this looks like in a game with a finite set of rules. Move 37, the one-in-ten-thousand move from AlphaGo's second game against Lee Sedol in 2016, was one that most professionals would not have considered. "When I saw this move," Lee said, "I changed my mind. Surely, AlphaGo is creative" [16]. But the board for mathematics is infinite, and so far it has been built around the limits of what human researchers can comprehend. We will soon face theories that are correct and that we simply cannot understand or grapple with.

Grothendieck thought of himself as a builder, one of the mathematicians "whose spontaneous vocation and joy is to keep building new houses" [17]. A builder constructs new theories around natural ideas. The explorers in Szegedy's essay, by contrast, venture into a world that is already there, looking for what is beautiful and what can be extracted. Rather soon, AI will do both the building and the exploring. As for us, we will feel like children again, in a world full of forces we do not understand.

At the same time, the economic value of mathematics will grow enormously. Here I agree with Szegedy, who expects mathematics to become "the true infrastructure of all scientific and technological progress" [15]. When a new idea or object appears, whether physical or theoretical, AI will let us develop a theory of it almost instantly: what it is, how it should behave, and what follows from it. We will use technologies built on mathematical theories that no human truly understands. Mathematics and the art of creating models will not become less important, but rather become even more woven into the daily lives of everyone.

6. References

  1. Halmos, Paul R. "The Legend of John von Neumann." American Mathematical Monthly 80(4), 1973, pp. 382–394. Link

  2. Kasparov, Garry. Interview with Thierry Paunin, Jeux & Stratégie 55, 1989, pp. 4–5. Quoted in English in Edward Winter, "Kasparov Interviews," Chess Notes. Link

  3. Levinovitz, Alan. "The Mystery of Go, the Ancient Game That Computers Still Can't Win." Wired, May 12, 2014. Link

  4. Yonhap. "Korean Go player confident of beating Google's AI." Korea Herald, February 23, 2016. Link

  5. Tao, Terence. "Mathematics in the age of AI." arXiv:2608.16753, August 17, 2026. Link

  6. Riehl, Emily. "Why I do mathematical research." September 14, 2026. Link

  7. Cohn, Henry. "The technical debt of AI-generated mathematics." September 15, 2026. Link

  8. Gowers, Timothy. "What sort of maths are LLMs good at?" August 12, 2026. Link

  9. Avigad, Jeremy. "What is mathematics now, and what should it be?" August 2026. Link

  10. Buzzard, Kevin. "To grieve, or not to grieve?" Xena, October 1, 2026. Link

  11. "Leiden Declaration on Artificial Intelligence and Mathematics." June 2, 2026. Link

  12. "A Severe Misalignment of AI in Mathematics." Declaration, September 11, 2026. Link

  13. Howlett, Joseph. "25 winners of math's 'Nobel Prize' decry the AI invasion of their discipline." Scientific American, September 14, 2026. Link

  14. Gowers, Timothy. "Why I didn't sign the Fields medallists' letter." September 17, 2026. Link

  15. Szegedy, Christian. "Is Mathematics Over, or Just Graduating?" October 3, 2026. Link

  16. Google DeepMind. "AlphaGo." Link

  17. Grothendieck, Alexander. Récoltes et Semailles, "Promenade à travers une œuvre," §2.5, "Les héritiers et le bâtisseur," 1986. Link