This will be a talk in the Logik Café at the University of Vienna Department of Philosophy. Slides here.
Abstract: It is commonplace to assume that if our discourse involving natural numbers and arithmetic is coherent at all, then it determines a unique structure (up to isomorphism). Recently, however, some mathematicians have challenged this assumption (especially the “New York School” of set theorists), arguing that the independence phenomenon in mathematics suggests that our talk about natural numbers is, in fact, indeterminate. In particular, an assumption of this view is that only statements that can be given first-order formulation have determinate content. In this paper we argue that such a view runs the risk of being self-undermining. We will suggest that modifying the view to include a schematic commitment to *feasibly checkable* fragments of set theory and mathematics as interpreted in these fragments yields a coherent conception of mathematics. We will, however, suggest that the view comes at the price of significant metamathematical reinterpretation, with several open philosophical and technical questions. This is joint work with Daniel Waxman.