History of Science and Technology Colloquium

I will argue that representation (and correspondences between representations) should replace ontology as the central explainer in the philosophy of mathematics. The argument has two parts. The core of the argument shows how distinctive qualities of competing representations help explain the ability of their users to grasp (and prove) mathematical relationships. I refer to several case studies (Descartes vs Euclid, but especially: ways of representing knots). The closing argument contrasts ways in which traditional ontology-oriented inquiries are unproductive in explaining the intellectual power of mathematics.