[.ca] From Brouwer To Hilbert: The Debate on the Foundations ... (ISBN 0195096320)

A useful reading with additional commentary:
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.