A New Dawn For Mathematical Progress

She is a research scientist at DeepMind Institute and an assistant professor at Carnegie Mellon University. Her research is focused on the social and scientific implications of AI. In the past few months, unreleased frontier AI systems have significantly disrupted a classic conception of mathematics.

They successfully established a complete formalization of Andrew Wiles’s proof of Fermat’s Last Theorem, found a solution to the Navier-Stokes problem and — just a few days ago — generated hundreds of manuscripts that each purport to prove an important mathematical problem. Most recently, on October 26, another of the Clay Millennium Prize problems was felled by an AI system. Mathematicians reacted with a volatile mixture of optimism, anxiety and deep disorientation.

In July, for example, Kirwin Hampshire, a Ph.D. student in mathematics, published an essay he titled “The Dark Night of Mathematics” in which he voiced the fear that human mathematicians might soon become obsolete next to AI systems with far greater speed and memory. It feels as though I am living inside of a nightmare.” For Hampshire, the very act of doing mathematics — the human struggle, the human aesthetic judgment, the human feeling and value of finding, understanding and sharing a new proof — could become a quaint artisanal craft, with human mathematicians no longer needed to push the frontier of mathematics. Some mathematicians have realized that they must radically rethink their professional role and identity in light of this new technological reality.

I was recently at an AI conference where a Fields Medal winner gave a talk on the frontier capabilities of AI systems for math problems. They closed their talk by saying: “We need to develop new theories of care and love.” This call was quite telling: a world-class mathematical theorist asking for theories of a very different kind. The mathematician, it seems, might become a philosopher of care and love.

In a short essay, Logan Graves, a mathematical philosopher, went further, suggesting that “the era of human mathematicians is ending, if not ended”; human mathematicians should find a new purpose in life, he wrote, and “the future of the human mathematical community, and eventually all scientific communities, is the practice of priests, then that of monks. Not researchers.” Terence Tao, a prominent Fields Medal winner and mathematics professor, acknowledged the deep complexity of this historical moment. Adopting a cautiously optimistic stance, he forecast that the mathematical “community will emerge stronger and more resilient than before.” Daniel Litt, a professor of mathematics at the University of Toronto, offered a set of core problems that should be solved for building the future of mathematics in an intriguing essay called “The End of Mathematics.” Litt wrote: “I’m broadly optimistic that mathematics will survive and flourish,” but that requires the mathematical community to address complex (not merely formal) questions such as redesigning institutions that incentivize high-quality scientific research and reorienting human capital.

Each of these reactions is entirely understandable; each represents a deeply respectable and admirable attempt to face and process a disruption to established ways of doing, being or becoming — of doing a certain type of mathematics and of being or becoming a particular type of mathematician. Because these reactions respond to different dimensions of a complex concern, it is helpful to distinguish between four central questions that urgently need answers. First: What is mathematical progress, and how is mathematics actually advanced?