The Future of Mathematics

Our present situation would be much more depressing if mathematics were a matter of ticking off problems imposed on us by aliens, an endless sequence of exercises and exams. The good news is that when we do mathematics, we get to choose the problems, grade the solutions, and favor the ones we like best. We decide what’s interesting to us, what questions to pursue, and why. Our destiny is in our hands.

The things we admire most in mathematics often seem to come out of nowhere. In 1853, the young Bernhard Riemann submitted three potential topics to his advisor, Carl Friedrich Gauss, to choose from for his Habilitationsvortrag, a lecture he was required to give to secure a teaching position at the University of Göttingen. Gauss reportedly chose the topic for which Riemann was least prepared, to see what he would make of it. The resulting lecture, “On the Hypotheses Which Lie at the Foundations of Geometry,” was published posthumously in 1868, and it revolutionized the field. The lecture distinguished a space’s metric properties from its topological properties and introduced the general notion of a manifold, though the latter did not receive a fully rigorous treatment until the twentieth century. The focus on intrinsic properties of a space—in Riemannian geometry, those determined by the metric and independent of embedding in a larger space—was key to Einstein’s theory of general relativity decades later. Riemannian geometry has had applications that Riemann himself could never have imagined, from robotics and medical imaging to statistical analysis.

William Ewald’s excellent sourcebook, From Kant to Hilbert, provides a lovely introduction to the paper and quotes Felix Klein’s description of the work:

“The publication of [Riemann’s lecture] occurred just at the time when I was beginning to occupy myself independently with mathematical problems. So I still have vivid memories of the extraordinary impact Riemann’s train of thought made on the young mathematicians of the day. Much seemed to us dark and difficult to understand, and yet of unfathomable depth, where the modern mathematician, who has already absorbed all these things into his mode of thought from the outset, only admires the clarity and fecundity of the exposition.”

What seemed dark and mysterious to the young Klein is now part of the canon, something that foundation models have absorbed and internalized through their training, just as young mathematicians do. I am grateful to Ballard once again for suggesting this example and pointing out that no reinforcement-learning setup could have evaluated Riemann’s decisions: the benefits are diffuse and hard to track, and the time horizon is much too long.

The same can be said for countless mathematical developments that have opened up new vistas, such as Galois’ focus on groups of permutations in the study of solvability of algebraic equations, Poincaré’s qualitative studies of dynamical systems, or Grothendieck’s far-reaching conceptual innovations. Will AI eventually be able to make advances like these? That’s not even the right question to ask. Telling us that some AI agent is spinning out theorems that are highly interesting to it and other AI agents does nothing for us. We should care about AI only insofar as the results are interesting and important to us, and, at the end of the day, it’s up to us to decide what that means. The values we assign to mathematical developments are embedded in our history and culture.

Imagine planning a trip to go backpacking in the Alaskan wilderness for exercise and recreation. You might be happy to let AI help book your flight, but not to let AI take the hike for you and send you pictures. In mathematics, what is at stake isn’t our recreation but agency over our reasoning and deliberation. Whether or not AI can think, it can’t think for us. We have our own lives to live; mathematics is our story to tell, and it’s up to us to decide how to tell it. With AI, there’s even more to explore, and no shortage of avenues for discovery.