First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.

Shanker exposes the confusions underlying the currently prevailing interpretations of Wittgenstein. He shows how such approaches and their underlying critical frameworks are incapable of coming to terms with Wittgenstein's arguments in the philosophy of mathematics. This book explains not only Wittgenstein's approach, which was justly heralded as causing a turning point in the philosophy of mathematics, but also the philosophy of mathematics in general.

For Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing 'aspects', or dimensions of meaning, projecting mathematical 'realities'. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein's Later Philosophy of Mathematics (1934–1951) grew from his Early (1912–1921) and Middle (1929–33) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

Covers the state of the art in the philosophy of maths and logic, giving the reader an overview of the major problems, positions, and battle lines. The chapters in this book contain both exposition and criticism as well as substantial development of their own positions. It also includes a bibliography.

The essays in this volume address central themes in Wittgenstein's writings on the philosophy of mind, language, logic, and mathematics.