Wittgenstein on Mathematics

by Severin Schroeder

This book offers a detailed account and discussion of Ludwig Wittgenstein's philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege's logicist attempt to provide arithmetic with a foundation and Wittgenstein's criticisms of it, followed by sketches of Wittgenstein's early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein's mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key 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 Wittgenstein's provocative views on Hilbert's Formalism and the quest for consistency proofs and on G�del's incompleteness theorems.

Genres: Philosophy

224 pages, Kindle Edition

Published December 30, 2020

Reviews

[Rafael Almada · Rating 4 out of 5]

An overall good book. Wittgenstein has recently become a rather enigmatic figure to me, and seeing his point of view on mathematics was rather amusing. Not being particularly familiar with the foundations beyond the very basic, I was concerned the book would be hard to approach, but it is surprisingly easy to digest and quite entertaining. If I could give one criticism, the author seemed somewhat defensive of Wittgenstein's points at times, and would dismiss a lot of the contrary arguments. Granted, I'm not familiar enough with the subject matter to say whether that was warranted or not. Wittgenstein's views are seen as controversial amongst philosophers of maths, though that doesn't seem to be the case for conventional mathematicians. I would recommend this book if you are interested in philosophy/foundations of mathematics, even if you are not quite well-versed in the field.