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From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s Paperback – September 11, 1997

ISBN-13: 978-0195096323 ISBN-10: 0195096320 Edition: 1st

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Editorial Reviews

Review

"A first-rate contribution to the field."--Mircea Dumitru, Tulane University

"This is an extremely useful collection which makes possible a good course on Intuitionist Logic and Mathematics."--Mary Tiles, University of Hawaii at Manoa

About the Author


Paolo Mancosu is Assistant Professor of Philosophy at the University of California at Berkeley. His main interests are in mathematical logic and philosophy of mathematics. He is the author of Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (OUP, 1996).
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Product Details

  • Paperback: 352 pages
  • Publisher: Oxford University Press; 1 edition (September 11, 1997)
  • Language: English
  • ISBN-10: 0195096320
  • ISBN-13: 978-0195096323
  • Product Dimensions: 9.1 x 0.7 x 6.1 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #818,646 in Books (See Top 100 in Books)

More About the Author

PAOLO MANCOSU is Professor of Philosophy at the University of California at Berkeley. He is the author of numerous articles and books in logic and philosophy of mathematics. He is also the author of Inside the Zhivago Storm. The editorial adventures of Pasternak's masterpiece (Feltrinelli, Milan, 2013). During his career, Mancosu has taught at Stanford, Oxford, and Yale. He has been a fellow of the Humboldt Stiftung, of the Wissenschaftskolleg zu Berlin, of the Institute for Advanced Study in Princeton, and of the Institut d'Études Avancées in Paris. In spring 2014, he was a visiting professor at the Munich Center for Mathematical Philosophy and a LMU-UCB Research in the Humanities fellow.

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23 of 23 people found the following review helpful By A Customer on March 28, 2000
Format: Paperback
This is a reading that contains 25 articles of major European mathematicians of the first half of the twentieth century. With two exceptions, all the papers are translated into English for the first time, most part of them from German, and some of them from Dutch and French. All translations are fluent and, as far as I can tell (I can't check the Dutch originals!), they are sufficiently accurate. The book consists of four sections, which are devoted, respectively, to Brouwer, Weyl, Hilbert and Bernays, and intuitionistic logic. Every section is preceded by a detailed study, in which the selected texts are presented in their historical context. At the end of each introductory study there is a complete bibliography of original sources and secondary literature. The proper understanding of some of these articles presupposes some knowledge of set theory (both Cantor's naive theory and Zermelo's axiomatics)and mathematical analysis (essentially, the concept of continuity). Nonetheless, most of the contributions are accessible to readers with basic notions of first order mathematical logic. Although the level of difficulty of the different articles is somewhat uneven, taken as a whole the book offers a very good text for graduate courses in philosophy of mathematics. It is also of considerable interest for scientists and historians of science.
The leit motiv of the book is the debate between Brouwer and Hilbert - and their respective followers- about the foundations of mathematics in the period between 1920 and 1931, that is, before the impact of Goedel theorems. The debate between formalists and intuitionists touched upon not only logic, but also set theory and the fundamental concepts of analysis, such as that of the continuum.
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4 of 5 people found the following review helpful By Viktor Blasjo on June 8, 2008
Format: Paperback
Hilbert's programme consists in formalising all mathematics and devising a metamathematical "proof theory" to show that one can never deduce a contradiction within this formal system. The programme is meant to establish the certainty of mathematical reasoning, in particular the certainty of questioned methods such as induction, operations with infinite sets (e.g. Dedekind cuts), the principle of the excluded middle applied to infinite sets (e.g. reduction ad absurdum existence proofs), etc. Therefore only safe modes of reasoning may be used at the metamathematical level, namely purely "finitistic" reasoning and a very limited form of induction which allows inferences such as the following: "1. If the + sign occurs at all in a concretely given proof, then in reading the proof one finds a place where it occurs for the first time. 2. If one has a general procedure for eliminating from a proof with a certain concretely given describable property E the first occurrence of the sign Z, without the proof losing the property E in the process, then one can, by repeated application of the procedure, completely remove the sign Z from such a proof, without its loosing the property E." (Bernays, p. 221)

It would be a serious mistake to think that Hilbert settles for consistency and cares not for truth, or that he wishes to reduce mathematics to a game of formulas. On the contrary. "The question has repeatedly been raised whether a proof of consistence suffices as a justification ... This formulation is misleading; it does not take into account the fact that the scientific grounding of the theoretical approach to arithmetic has been achieved for the most part ... and that the proof of consistency is indeed the only desideratum that still needs to be fulfilled. ... If this is successful ...
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From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s
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