UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Internal categoricity and internal realism in the philosophy of mathematics
Tim Button (University of Cambridge)
4:10 pm, Wednesday, April 19th, 2017
Faculty House, Columbia University
Abstract. Many philosophers think that mathematics is about ‘structure’. Many philosophers would also explicate this notion of ‘structure’ via model theory. But the Compactness and Löwenheim–Skolem theorems lead to some famously hard questions for this view. They threaten to leave us unable to talk about any particular ‘structure’.
In this talk, I outline how we might explicate ‘structure’ without appealing to model theory, and indeed without invoking any kind of semantic ascent. The approach involves making use of internal categoricity. I will outline the idea of internal categoricity, state some results, and use these results to make sense of Putnam’s beautiful but cryptic claim: “Models are not lost noumenal waifs looking for someone to name them; they are constructions within our theory itself, and they have names from birth.”