Tim Button (Cambridge): “Mathematical Internal Realism”

Abstract: In “Models and Reality”, Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. The sketch was tantalising, but it was only a sketch. Mathematics was not the focus of any of his later writings on internal realism, and Putnam ultimately abandoned internal realism itself. As such, I have often wondered: What might a developed mathematical internal realism have looked like? In this talk, I try to answer that question.

By combining Putnam’s model-theoretic arguments and Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem–Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, and hence is incomplete.

Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. In so doing, we come to see why “Models are not lost noumenal waifs looking for someone to name them”, but “constructions within our theory itself”, with “names from birth.”

Tim Button is a Senior Lecturer and a Fellow of St John’s College at the University of Cambridge.