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
higherorder terms.

