The Future of Mathematics

Jeremy Avigad argues that recent advances in AI are already reshaping mathematical practice: foundation models, having ingested the literature and operating without fatigue, can synthesize respectable publishable results by recombining known techniques, which disrupts traditional workflows where humans craft and publish routine advances. He insists that mathematics’ enduring core - rigorous reasoning, shared language, and aesthetic judgment - remains essential, but notes a real danger that outsourcing problem-solving to AI can decouple theorem-proving from genuine mathematical understanding. He highlights technical limits of current approaches (e.g., reinforcement learning optimizes final success trajectories and may miss long-horizon creativity) and gives historical examples - Riemann, Galois, Poincaré, Grothendieck - whose transformative contributions produced diffuse, delayed benefits that current AI training regimes could not have valued or produced.

The proposed response is practical and cultural: pursue harder problems, cultivate big conceptual projects that prioritize human values, and embrace new tools by actively developing and understanding proof assistants, symbolic automation, and neural methods as legitimate mathematical work. Historical parallels show new technologies repeatedly changed how math is done; engaging with AI will expand opportunity rather than merely erode it. The final message is a pledge of support to students and early-career researchers: institutions must adapt, communities should help preserve the discipline’s health, and mathematicians should lead in shaping how AI augments - not replaces - mathematical judgment and creativity.