Philosophy of Mathematics seminar

Neil Barton, Konstanz

Algebraic Levels and Incomplete Structures

There seems to be a feeling in contemporary philosophy of mathematics that model theory does not have much to tell us about mathematical structuralism. The idea is roughly the following: Contemporary model theory makes heavy use of set theory, and set theory itself is a mathematical theory, and so the use of model theory in elucidating the notion of structure is doomed to be viciously circular. Alongside this we have a discussion of whether it is possible to have incomplete structures - should every structure be determined up to isomorphism (or some close variant thereof) or can we have structures with indeterminate properties (e.g. no fixed cardinality)? In this talk, I'm going to argue that model theory can be brought to bear on exactly this question. In particular I'll argue that the use of strongly minimal sets, algebraic closure, and developments of the Zilber trichotomy conjecture support the claim that there is a kind of structure that is indeterminate in cardinality but which can be thought of as composed of `building blocks' built over a base cardinality. I'll then provide some tentative suggestions as to how we can think of these structures as cashed out in higher-order terms.