Stewart Shapiro's Philosophy of Mathematics *

Abstract

This review essay presents and criticizes the central partof a realistic structuralist account of the subject matterof contemporary mathematics, the view that Shapiroarticulates in Philosophy of Mathematics. It considers thethesis that "a structure is the abstract form of a system",Shapiro's distinction between the "places-as-offices" andthe "places-as-objects" perspectives, his account of thedistinctive features of mathematical structures (formalityand freestandingness), his analysis of what it is forsystems to have the same structure, his (informal) axiom ofcoherence, his attempt to dissolve the question "How can weknow about, or even refer to, places in a structure?", andhis version of ontological relativity.