I will begin the talk with a brief sketch of Aristotle’s original philosophy of mathematics. This, I will argue, is based on two postulates. The first is the embodiment postulate, which states that mathematical objects do exist, though not in a separate Platonic world, but embodied in the material world. The second is that infinity is always potential and never actual. I will then consider the extent to which this Aristotelian approach holds for contemporary mathematics. I will assume that most contemporary mathematicians accept ZFC. This rules out Aristotle’s second postulate since ZFC’s axiom of infinity implies the existence of an actual infinity. However, I will claim that the embodiment postulate can still be defended for contemporary mathematics. At first sight this seems a curious claim since Cantor’s theory of transfinite alephs can be developed within ZFC, and surely transfinite alephs are not embodied in the material world. I will discuss this difficulty at length, and try to overcome it using ideas from Fictionalist and If ..then-ist philosophies of mathematics.

**Date**: 6 March 2017, 16:30 (Monday, 8th week, Hilary 2017)**Venue**: Radcliffe Humanities

Woodstock Road OX2 6GGSee location on maps.ox**Details**: Ryle Room, First Floor**Speaker**: Donald Gillies (King’s College London)**Organising department**: Faculty of Philosophy**Organisers**: James Studd (University of Oxford), Daniel Isaacson (University of Oxford), Volker Halbach (University of Oxford)**Part of**: Philosophy of Mathematics Seminar**Topics**: Philosophy, Mathematics**Booking required?**: Not required**Audience**: Members of the University only- Editor: Andy Davies