1 Review of Stewart Shapiro , Philosophy of Mathematics : Structure and Ontology

This book is an important contribution to the philosophy of mathematics. It aims to clarify and answer questions about realism in connection with mathematics, in particular whether there exist mathematical objects (ontological realism) and whether all meaningful mathematical statements have objective and determinate truth-values (truth-value realism). The author develops a novel structuralist account of mathematics that answers both questions affirmatively. By regarding mathematics as ‘the science of structure’ (p. 5), he attempts to render both forms of realism naturalistically respectable. The resulting philosophy of mathematics is extremely interesting and deserves the attention of anyone with a serious interest in the field. The book is not an easy read, however. The prose is often quite dense, and although this is in part due to the difficulty of subject and the sheer number of ideas discussed, part of the blame must also be attributed to the fact that the discussion occasionally is poorly organized and that it can be hard to tell exactly what the author’s view is. The introductory chapter is particularly dense and would serve better as a summary of the book than an introduction to it. Despite these imperfections, however, a careful study of the book will be enormously rewarding to anyone with some prior exposure to the field. In what follows, I will go through the book, focusing on what I take to be its main themes.