From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920's (Hardcover)

~ Paolo Mancosu (Editor) "Luitzen Egbertus Jan Brouwer (1881-1966) is a central figure in the history of contemporary mathematics and philosophy..." (more)
Key Phrases: intuitive number theory, absurdity predicates, intuitionist set theory, Mathematische Annalen, Primordial Intuition, Hermann Weyl (more...)

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 --This text refers to the Paperback edition.

Product Description

This book offers the first comprehensive introduction to the debate on the foundations of mathematics in the 1920s. Most contemporary work in this field originates from the groundbreaking contributions of Hilbert, Brouwer, Bernays, and Weyl. This volume makes available the first English translations of 25 central articles on the debate by these major figures and many others. Each section of the book contains an introduction which provides the necessary historical and technical context for understanding the articles.

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Product Details

• Hardcover: 352 pages
• Publisher: Oxford University Press, USA; illustrated edition edition (September 11, 1997)
• Language: English
• ISBN-10: 0195096312
• ISBN-13: 978-0195096316
• Product Dimensions: 9.3 x 5.9 x 0.8 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon.com Sales Rank: #2,299,682 in Books (See Bestsellers in Books)

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Most Helpful Customer Reviews

18 of 18 people found the following review helpful:

4.0 out of 5 stars A useful reading with additional commentary, March 28, 2000

By A Customer

This review is from: From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s (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. The idea of infinity has always been at the center of the disputes. Should we accept in mathematics the existence of infinite entities, such as set of points or cardinal and ordinal transfinite numbers? Hilbert and the formalists answered that we can, provided that our theories are logically consistent, that is, imply no contradictions. Brouwer and the intuitionists, on the contrary, thought that consistency is a necessary although not a sufficient condition for mathematical existence. They demanded an effective method of construction for every mathematical entity, and this stringent condition lead them to reject some fundamental portions of set theory and classical mathematics.

The outcome of the debate was rather inconclusive. Goedel's theorems about the incompleteness of formal arithmetic and the unprovability of consistency for formal theories -such as set theory- put severe limitations on Hilbert's program. On the other hand, intuitionists were unable to reconstruct large fragments of elementary and higher mathematics, and where they succeeded, the results were very complicated and extremely awkward. Intuitionistic logic and Hilbert metamathematical program are still alive, yet we now know that neither of them can be accomplished in the way they were conceived in the 1920s.

1 of 1 people found the following review helpful:

5.0 out of 5 stars Very useful, June 8, 2008

By	Viktor Blasjo - See all my reviews (REAL NAME)

This review is from: From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s (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 ... we can then rely on the results of the application of the basic postulates of arithmetic just as if we were in the position to verify these intuitively. For by recognizing the consistency of the application of these postulates, it is established at the same time that an intuitive proposition that is interpretable in the finitistic sense, which follows from them, can never contradict an intuitively recognizable fact. In the case of a finitistic proposition, however, the determination of its irrefutability is equivalent to the determination of its truth." (Bernays, p. 259)

Hilbert's programme looked promising in some sandbox examples (pp. 208-210) but was ultimately a great failure. In 1927, after many years of hard work by Hilbert and his clan, Hilbert has only one result: the principle of the excluded middle for integers (p. 229). All questions regarding the reals and more complicated sets are wide open, and would remain so. In fact, even this one feeble result falls apart since it turned out that the axioms for the integers were incomplete. (The latter by Gödel's results, whose impact on Hilbert's programme is addressed here only once, and then extremely briefly, by Bernays on p. 263.)

According to Brouwer, mathematics is a language-less activity. Mathematical language and the principles of logic are merely imperfect ways of representing this intuitive activity. Thus "the aim of the Formalist School ... is based on a false belief in the magical character of language [and] imprudent trust in classical logic" (pp. 48-49). Grounding mathematics in formalism and logic is "like considering the human body to be an application of anatomy" (p. 9). Indeed, Brouwer points to a vicious circle in Hilbert's programme. "The (contentual) justification of formalist mathematics by means of o proof of its consistency contains a circulus vitiosus since such justification is based on the (contentual) correctness of the assertion that correctness of a proposition follows from its noncontradictority, that is, is based on the (contentual) correctness of the Principle of the Excluded Middle" (p. 41), as is confirmed in the second Bernays quotation above.

Brouwer's theory of the continuum is his main positive contribution. A flavour of this theory is conveyed by its most conspicuous theorem: all functions are uniformly continuous (p. 39). The simplest classical counterexample is a function with y=0 if x<0 and y=1 otherwise. There are several good arguments for why the intuitionistic theorem is in better accord with intuition. First, the classical counterexample relies on "magical language" since, constructively, the function cannot be evaluated; for example, there are numbers that are "neither equal to 0 nor distinct from 0" (p. 51). Second, the theorem implies that the continuum cannot be split, in accordance with intuition (whereas it is split in the classical counterexample). "Indeed, Democritus argues with good reason that if I can break a stick, then it was from the outset not a whole. Strictest atomism is the inescapable conclusion of this." (Weyl, p. 135; see p. 124).

Weyl calls himself "in the middle of the war of the factions" (p. 141). He praises Brouwer's theory but maintains that Hilbert's programme can be pursued all the same since one may "fuse mathematics with physics and assume that the mathematical concepts of number, function, etc. (or Hilbert's symbols), generally partake in theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc." (p. 140).