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Oran Magal, Wilfried Sieg. Hilbert's Programs and Beyond. Oxford: Oxford University Press, 2013. ISBN 978-0-19-537222-9 (hbk); 978-0-19-970715-7 (e-book). Pp. xii + 439, Philosophia Mathematica, Volume 22, Issue 3, October 2014, Pages 417–423, https://doi.org/10.1093/philmat/nku019
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Extract
This book is a collection of previously published papers by Wilfried Sieg (with minor corrections marked in footnotes), together with a newly written Introduction (in two parts), and brief, informative ‘guides’ at the start of each of the sections into which the book is divided. The unifying theme of the book is the foundational work of Hilbert and Bernays, and, to a lesser extent, that of closely related figures such as von Neumann, Gentzen, and Gödel, with certain chapters discussing proof theory and its philosophical significance more generally.
The book opens with two introductions; the first offers, as its title indicates, a ‘perspective on Hilbert's Programs’; the plural form, which also figures in the title of the book, indicates Sieg's view that Hilbert's work on the foundations of mathematics is spread across different time periods, in which different goals are set and different means used. The second introductory essay is a summarized timeline of the various papers and lecture notes of which these ‘programs’ are composed. Especially worth noting here is Sieg's view regarding the main tasks for the philosophy of mathematics in the aftermath of the foundational and proof-theoretic work examined in the book: first, to investigate the rôle of abstract structures in mathematical practice and in mathematical understanding. Second, to analyze the function of accessibility notions in both technical and philosophical work on the foundations of mathematics.
