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Philosophia Mathematica Advance Access originally published online on May 8, 2009
Philosophia Mathematica 2009 17(3):382-392; doi:10.1093/philmat/nkp008
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© The Author [2009]. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Book Review

CHRISTINE REDECKER. Wittgensteins Philosophie der Mathematik: Eine Neubewertung im Ausgang von der Kritik an Cantors Beweis der Überabzählbarkeit der reellen Zahlen. [Wittgenstein's Philosophy of Mathematics: A Reassessment Starting from the Critique of Cantor's Proof of the Uncountability of the Real Numbers]

Esther Ramharter*

* Institut für Philosophie der Universität Wien, Universitätsstraße 7, 1010 Wien, Austria. esther.ramharter@univie.ac.at

CHRISTINE REDECKER. Wittgensteins Philosophie der Mathematik: Eine Neubewertung im Ausgang von der Kritik an Cantors Beweis der Überabzählbarkeit der reellen Zahlen. [Wittgenstein's Philosophy of Mathematics: A Reassessment Starting from the Critique of Cantor's Proof of the Uncountability of the Real Numbers]. Frankfurt-Hausenstamm: Ontos Verlag, 2006. ISBN 978-3-938793-31-2. Pp. 370

The first 150 words of the full text of this article appear below.


   Introduction and Content
 
Wittgenstein’s notes on mathematics are fragmentary, but nevertheless precise and coherent—this is Christine Redecker’s position with regard to the topic of her book. Starting from Wittgenstein’s critique of Cantor’s diagonal proof she promises to give a reassessment of his philosophy of mathematics. She considers this critique to be neither as groundless as his opponents hold, nor as harmless as some of his defenders present it.

In the latter part of the book Redecker highlights some constructivist, conventionalist, and revisionist elements in Wittgenstein’s philosophy of mathematics.

According to the table of contents the author considers her book as divided into two parts (‘Wittgenstein’s critique of Cantor’s diagonal proof in [RFM II, 1–22]’, and ‘Wittgenstein’s critique in the context of his philosophy of mathematics’), but at least for the purpose of this review it seems more appropriate to split it into three parts: the first dealing in detail with Cantor’s diagonal argument, . . . [Full Text of this Article]


   1. Wittgenstein’s Critique of Cantor’s Diagonal Argument
 
1.1 Does the Diagonal Sequence Define a Real Number?
1.2 Is the Diagonal Number Different from All Elements of the List?
1.3 The Diagonal ‘Proof’
1.4 Countability
1.5 Infinite Sets of Different Cardinality
1.6 Summing up

   2. Wittgenstein and the Real Numbers
 

   3. Wittgenstein and some Isms
 
3.1 Constructivism and Platonism
3.2 Conventionalism
3.3 Revisionism

   4. Objections and Conclusion
 

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