Over the last century, the philosophy of mathematics has been primarily focused on the quest to establish logical foundations and to justify axioms. But there are also philosophical issues that have to do with how living, breathing human agents practice mathematics. This is the elusive middle ground in which this project develops, between the perfect abstract realm of mathematical theories and the messy human world inhabited by mathematicians. From our contemporary perspective, the correctness of a mathematical argument is understood in terms of formalizability, and our justification in mathematical beliefs is cashed out in terms of formal proofs. Ordinary proofs, however, present gaps and include diagrams; they are dissimilar to formal proofs. What, then, is the relationship between formal and ordinary proofs? What are the epistemic norms governing mathematical practice? Can proofs include diagrams? Can computers help us prove new results, and if so, how? These questions are pressing because technology is rapidly changing the way in which we practice mathematics and the epistemic norms governing it. This project’s primary goal is to provide a multifaceted theory of mathematical knowledge and justification that makes sense of how mathematical knowledge is produced and shared among human agents.

The project has interdisciplinary roots: philosophy and mathematics. It is also relevant to mathematical education, cognitive sciences, and sociology. Moreover, the societal impact is substantial because it will contribute to changing the folk image of mathematics: unveiling the human side of mathematics makes it more accessible and more attractive.

1. Develop a fallibilist theory of mathematical justification that satisfies social constraints.

2. Explore the interdependence between knowledge-that and knowledge-how in mathematics by focusing on how mathematicians use notations and diagrams.

3. Investigate the different roles that computers play in mathematical research, focusing on interactive proof assistants and generative AI applied to mathematics.

Humanizing Mathematical Knowledge: Fallibility, Technology, Know-How is a three-year (2025-2028) FIS Consolidator Grant project, funded by the Ministero dell’Università e della Ricerca. The project is located at the University School for Advanced Studies IUSS Pavia under the direction of Silvia De Toffoli. The total amount of the grant is 1,660,388€.

We're hosting a workshop on Annalisa Coliva's forthcoming monograph Social and Applied Hinge Epistemology on June 10. See our Workshops page for updates about this event.