Chapter 4 - Logic, mathematics, and conceptual structuralism

Summary

Logic is integral to mathematics and, to the extent that is the case, a philosophy of logic should be integral to a philosophy of mathematics. The aim of the mathematician working in the mainstream is to establish truths about mathematical concepts by means of proofs as the principal instrument. There are other dimensions of mathematical practice that reward metamathematical study motivated by the philosophy of conceptual structuralism. One is the open-ended nature of certain principles such as that of induction for the integers and comprehension for sets. This accords with the fact that in the development of mathematics what concepts are recognized to be definite evolve with time. Thus one cannot fix in advance all applications of these open-ended schematic principles by restriction to those instances definable in one or another formal language, as is currently done in the study of formal systems.