Abstract:

According to structuralism in philosophy of mathematics, arithmetic is about a certain structure. First-order theories are satisfied by (nonstandard) models that do not have this structure. Proponents of structuralism have put forward various accounts of how we succeed to fix one single structure as the intended interpretation of our arithmetical language.

I shall look at a proposal (Volker Halbach & Leon Horsten: Computational Structuralism, Philosophia Mathematica 13, 2005 pp.174-186) that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On this account, the intended models of arithmetic are the notation systems which are computably intertranslatable with our notation system of the Arabic numerals.