Plato's problem. An introduction to mathematical Platonism. Translated from the Italian (Q2846685)

This is a presentation of the main issues in the philosophy of mathematics, presented by focusing on one problem, namely that of the existence and nature of mathematical objects, in other words on Platonism.NEWLINENEWLINE After a short introduction tracing the Greek origins of the philosophy of mathematics, the reader is presented with the situation during the last decades of the 19th century and the first three decades of the 20th century, looking at Frege, Russell, logicism, Hilbert, Brouwer, and Gödel in the space of 45 pages. Intuitionism is presented in this historical section, but makes no later appearance, for space limitations, as we are told on p.\ 90. The remainder of the book is concerned with contemporary issues, all seen as answers to Paul Benacerraf's dilemma, grouped, following a suggestion of Hale and Wright, into conservative and non-conservative responses. The non-conservative responses consist of Harty Field's nominalism, of fictionalism (in both the Field and the Yablo version), Geoffrey Hellman's eliminative modal structuralism, and Penelope Maddy's version of Platonism. The conservative responses consist of the Hale and Wright version of neo-logicism, the Linsky-Zalta ``object theory'', Shapiro's \textit{ante rem} non-eliminative structuralism, and Parsons' version of non-eliminative structuralism. Two chapters are devoted to the Quine-Putnam indispensability argument, of which four versions are presented, with arguments for and against it.NEWLINENEWLINE Although the book is intended as an introduction to the philosophy of mathematics centered on a particular theme, the reader is expected to have a philosophical maturity and familiarity with the essentials of mathematical philosophy that renders it much more useful for a graduate course, or as an excellent comprehensive synthesis of contemporary views in the philosophy of mathematics. It seems less well suited for the undergraduate student, and leaves out much of the traditional material found in introductions to mathematical philosophy, such as an in-depth look at formalism. The emphasis is \textit{contemporary} philosophical throughout, there is little historical orientation that would allow the reader to trace the ideas presented by contemporaries back to earlier times (even though the fictionalist ``as if'' is mentioned on page 130 as part of Yablo's narrative, we are not told anything about Hans Vaihinger's \textit{Philosophie des Als Ob}, in fact his name does not appear anywhere in this book).