Wittgenstein on Mathematics

This book offers a detailed account and discussion of Ludwig Wittgensteins philosophy of mathematics. In Part I, the stage is set with a brief presentation of Freges logicist attempt to provide arithmetic with a foundation and Wittgensteins criticisms of it, followed by sketches of Wittgensteins early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgensteins mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity - calculation - rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgensteins provocative views on Hilberts Formalism and the quest for consistency proofs and on Gِdels incompleteness theorems.