Proof Validity in Historical Perspective
Organização:  CFCUL & CIUHCT
Ciências ULisboa
23 / 06 / 2022

Several studies have revealed the limits of an understanding of the practice of proving theorems in mathematics only in terms of the notion of logical validity. What these studies highlight is that, when looking at their proofs, mathematicians sometimes adopt a richer and more general notion of validity that is not limited to logical analysis. For instance, several studies on the notion of purity of methods in mathematics have shown how, according to some mathematicians, a legitimate proof of a theorem is only that which appeals to notions that are not extraneous to the theorem in question. And this even when other proof approaches to the same theorem, equally valid from a logical point of view, are available. Another example comes from the study of the role of diagrams in ancient Greek mathematics. In that context, diagrams did not have a merely heuristic function and mathematicians relied on the manuscripts’ diagrams as part of the arguments. The goal of the present talk is to contribute to these second order discussions on proofs by providing an investigation of the cross relations that exist between the practice of proving theorems and the social dimension in which this practice takes place. More precisely, using a case study from the history of mathematics, I will show how a philosophical view may constrain the conceptual tools used to prove theorems and affect the standards of proof validity that are adopted by a specific community of mathematicians. I will argue that such an analysis highlights how the concept of proof validity cannot be captured solely in terms of purely intra-mathematical considerations. Moreover, it reveals the key role that an historically informed study of mathematical practice may have in deepening our understanding of notions, like that of valid proof, that are central to philosophy of mathematics.

| |