Talk: Matteo Zicchetti (Konstanz)

Title:

Internalism, Intolerance, and the Determinacy of Arithmetic

Abstract:

It is a standard view in the Philosophy of Mathematics that arithmetic is about a unique subject matter, the natural numbers, and that arithmetical truth is (to some degree) determinate. The uniqueness of the arithmetical subject matter is usually understood in terms of categoricity, i.e., that arithmetic has one intended structure (up to isomorphism). Determinacy of truth is understood as the claim that all arithmetical statements are either true or false (in a sense, all arithmetical questions have, in principle, a definite answer). Unfortunately, standard ways employed to prove and argue for the uniqueness and determinacy of arithmetic have turned out to be philosophically problematic. From these issues, a 'challenge' of explaining the uniqueness and determinacy of arithmetic arises.
Mathematical Internalism promises to overcome the major issues resulting from the standard approach to uniqueness and determinacy. In particular, Internalism aims to explain the determinacy of arithmetical truth by arguing for the intolerance of arithmetic: That all arithmetical structures must agree (to some extent) on what is true in them. Recently, the internalist explanation of determinacy has been criticised. If successful, this critique puts internalism in danger of not being able to meet the challenge of explaining arithmetical determinacy. This talk aims to discuss the internalist argument for the determinacy of arithmetic, analyse its critique and propose a new argument for determinacy, which should be able to grant the determinacy of arithmetic and be immune to the recent critiques.