We had a sweeping discussion touching upon many issues in the philosophy of mathematics, including the nature of mathematical truth, mathematical abstraction, the nature of mathematical existence, the meaning and role of proof in mathematics, the completeness theorem, the incompleteness phenomenon, infinity, and a discussion about the motivations that one might have for studying mathematics.
In this insightful and remarkable work, Professor Novaes defends and explores at length the philosophical thesis that mathematical proof and deduction generally has a fundamentally dialogical nature, proceeding in a back-and-forth dialogue between two semi-adversarial but collaborative actors, the Prover and the Skeptic, who together aim to find mathematical insight. This view of proof-as-dialogue, she argues, carries explanatory power for the philosophy of mathematical practice, explaining diverse aspects of proof-writing, refereeing, and more, including the multifaceted roles of proof, including proof as verification, proof as certification, proof for communication and persuasion, proof as explanation, and proof as a driver of mathematical innovation.
In extensive, refined scholarly work, Novaes explores the historical and intellectual roots of the dialogical perspective on deduction, tracing the idea from ancient times through medieval philosophy and into the present day, including case studies of current mathematical developments, such as Mochizuki’s claimed proof of the abc conjecture, as well as recent psychological experiments on the role of group reasoning in resolving certain well-known disappointing failures of rationality, such as in the Wason card experiment. Truly fascinating.
On the basis of her work, I have nominated Novaes for the Lakatos Award (given annually “for an outstanding contribution to the philosophy of science, widely interpreted, in the form of a book published in English during the current year or the previous five years”). Lakatos himself, of course, was a friend of dialogical mathematics—his famous Proofs and Refutations proceeds after all in a dialogue between mathematicians of different philosophical outlooks. Novaes engages with Lakatos’s work explicitly, pointing out the obvious parallels, but also highlighting important differences between her Prover/Skeptic dialogical account and the kind of proof dialogues appearing in Proofs and Refutations. In light of this connection with Lakatos, I would find it especially fitting for Novaes to win the Lakatos Award.