Proof in Context

Mathematics is unique among the sciences for a variety of reasons. One is that it does not rely on empirical observations, experiments, or similar procedures to justify its conclusions. Instead, its primary means of justification is mathematical proof, a form of justification that relies on reason- ing alone and puts the truth of the proved proposition beyond any rational doubt. The preeminent role of proof in mathematics has been stressed by mathematicians and philosophers ever since Aristotle. And yet, while mathematicians tend to agree on what a proof is when presented with one, it is notoriously hard to pin down what, in essence, a proof actually is. As is well known, de- velopments that started towards the end of the nineteenth century eventually led to investigations into the nature of mathematical proof in the works of Frege, Russell, and others, who developed formal systems to facilitate rigorous mathematical reasoning by making explicit all assumptions on which the validity of a proof depends. Since then, mathematicians and logicians like Hilbert, Gentzen, and others started to investigate formal proofs as objects in their own right and formal proof theory has developed into a thriving field of research. While the study of formal proofs has proven to be a highly successful endeavor, questions regard- ing the relationship between formal proofs and proofs as presented in everyday mathematical contexts, such as textbooks or journal articles, continue to engage philosophers. Specifically, scholars in the philosophy of mathematical practice have recently begun to focus on various issues related to informal proofs and provability. However, despite interesting work on various aspects concerning informal proofs, this research has remained largely fragmented. The goal of the cur- rent project is to provide an overarching theory of the informal concepts of mathematical proof and provability that integrates existing research on these notions and their relationship to their formal counterparts. Specifically, the project aims to offer explications of the concepts of mathe- matical proof and provability based on the premise that the concept of proof is fundamentally context-dependent. Contextual factors are routinely emphasized in discussions of informal proofs. However, despite the prominence of contextualist positions in debates on knowledge and justifi- cation in general epistemology over the past few decades, a systematic account of proof as a con- textual notion has yet to be provided. This project aims to fill this gap by developing a contextualist theory of mathematical proof that draws on insights from the history and philosophy of mathe- matical practice, general epistemology, and the formal study of informal proof and provability. In doing so, it seeks to shed new light on various issues related to one of the most fundamental no- tions in the epistemology of mathematics.