Physical systems as Topoi

In this talk, I will report on recent work with Chris Isham. The basic idea is that a physical system can be represented by structures within a suitable topos associated with the system. The choice of the topos depends on the type of the system (classical, quantum or even some new kind). Within the topos, there is a state object and a quantity value object, and physical quantitites are suitable morphisms between them. Propositions about physical quantities are represented by subobjects of the state object, which form a Heyting algebra. A tentative set of rules for formulating physical theories within a topos that does not fundamentally depend on the continuum (in the form of the real or complex numbers) is presented.