The Meaning of Mathematics: On the Meaning of Mathematical Statements in a Structuralist Context

Abstract

The aim of this thesis is to provide an account of the meaning of second-order formulae in a structuralist context. We first give the necessary background in notions of formality in logic and argue for the use of second-order logic as language of mathematics, as proposed by Shapiro. We then formalize Giovannini’s and Schiemer’s theory of structural definitions. Afterwards, we argue that the meaning of a second-order formula in a structuralist context is the isomorphism class of structures that satisfy its propositional function. Finally, we show how this theory of meaning works with respect to the structure of the natural numbers.