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The autonomy of mathematical knowledge. Hilbert’s program revisited. (English) Zbl 1192.00012
Cambridge: Cambridge University Press (ISBN 978-0-521-51437-8/hbk; 978-0-511-63660-8/ebook). xiii, 213 p. (2009).
The book under review is devoted to Hilbert’s program. The author claims that its true philosophical significance is that it makes the autonomy of mathematics evident. He argues that Hilbert’s central insight was that mathematical techniques and practices do not need grounding in any philosophical principles.
The book grew out of the author’s doctoral dissertation. It consists of 6 chapters. In Chapters 1 and 6 the significance of Hilbert’s anti-foundational stance is shown. Hilbert’s program is presented there in the context of historical events. Its influence on contemporary philosophy of mathematics is indicated as well. Chapters 2–5 are devoted to more technical issues. In Chapter 2 it is explained that Hilbert’s program was primarily an effort to demonstrate that mathematics could answer questions about how its own methods work. In Chapter 3 Hilbert’s attempts to solve this problem are discussed. The meaning of Gödel’s incompleteness theorems as well as Herbrand’s contribution to Hilbert’s program (in particular the partial progress Herbrand made to Hilbert’s goal of mathematical autonomy) are considered. In Chapter 4 the discussion is rephrased in terms close to S. Feferman’s notion of intensionality. It is examined to what extent Gödel’s and Herbrand’s techniques of arithmetization are intensionally correct. In Chapter 5 the point of view from previous chapters is applied to a problem whether the fact that a version of Gödel’s second incompleteness theorem for Robinson’s weak arithmetic Q can be understood as showing that Q cannot prove its own consistency. The author argues that weak arithmetic theories are the natural place to turn to investigate how mathematics speaks for itself.
The book brings an interesting interpretation of Hilbert’s program and a vision of the early history of modern logic.

00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century