January 10
Jouko Väänänen,
University of Helsinki
Categoricity arguments and their philosophical uses
Both number theory and set theory have a claim to categoricity, in one form or another, when axiomatized in second order logic. This goes back to Dedekind and Zermelo. It is less well-known that such claims manifest themselves also in first order axiomatizations, however non-categorical such axiomatizations are in the usual setup of mathematical logic (Väänänen, 'An extension of a theorem of Zermelo' BSL, 2019). Parsons and others have written about this e.g. in Parsons, 'The uniqueness of the natural numbers' (Jerusalem Philosophical Quarterly, 1990), and Button and Walsh, 'Philosophy and Model Theory' (Oxford University Press, 2018). We claim that philosophical uses of these arguments do not carry the philosophical weight they are purported to do. To support our claim we analyse the categoricity arguments in detail in the context of both first and second order logic. We expose a common factor of such arguments, internal categoricity, namely categoricity within what the theory in question, be it number theory or set theory, can see. While internal categoricity is a remarkable phenomenon in itself, we argue that it cannot be used to defend the decidability of formal statements in the theory. In conclusion, when categoricity results are used to make certain philosophical claims, even though the categoricity results are by and large correct, they do not support those claims.
Reference: Maddy and Väänänen: Philosophical Uses of Categoricity Arguments, Elements in the Philosophy of Mathematics. Cambridge University Press. (2023).