Proofs and Reasons: Toward a Cognitive Science of Real Mathematics

How do mathematicians think? What are the mental and social processes at work when a researcher chooses a theorem, searches for a proof, brainstorms with colleagues at the board, and or finally understands why their proof is true? Our working group will bring research mathematicians together with cognitive scientists, philosophers of mathematical practice, and historians of mathematics to build an integrated, empirical study of research-level mathematics—its cognitive, computational, and conceptual dimensions at once. Our hope for this workshop is that it will serve as the foundation for a new field of inquiry, focused on the reality and heterogeneity of mathematics as it actually exists today.

There is a great deal of work on the necessary prerequisites for such an advance. There is a vast empirical science of reasoning and learning, largely focused on either “ordinary” tasks by untrained human subjects, or in the course of early childhood development. There is also a vast theoretical literature on the formal study of proof in logic, complexity, and type theory. Neither of these has been put into sustained and serious contact with contemporary mathematical practice—despite the vast array of biographies, essays, and informal observations by mathematicians that suggest that this activity is one of the strangest, most interesting and perhaps highest achievements of our species.

The recent arrival of machine theorem-provers and AI-assisted mathematics makes our questions urgent, because we are now building systems that can produce and check proofs faster than we can say what kind of mathematical understanding, if any, they involve. The upcoming Fields Medal award in July 2026 at the International Congress of Mathematics has been described as “the last Fields Medal”, in part because it will almost certainly be the last time someone receives the highest award in mathematics that is not based, in part, on results produced with the aid of advanced AI.

Over three days of discussion-first sessions, our participants will map the shared questions and methods of the area, begin new collaborations and pilot studies, and plan a special issue to anchor the research agenda across communities