Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...) choice sequences is defective on several counts. (shrink)

Brouwer and Weyl recognized that the intuitive continuum requires a mathematical analysis of a kind that set theory is not able to provide. As an alternative, Brouwer introduced choice sequences. We first describe the features of the intuitive continuum that prompted this development, focusing in particular on the flow of internal time as described in Husserl's phenomenology. Then we look at choice sequences and their logic. Finally, we investigate the differences between Brouwer and Weyl, and argue that Weyl's conception of (...) choice sequences is defective on several counts. (shrink)

The original Brouwerian counter examples were algorithmic in nature; after the introduction of choice sequences, Brouwer devised a version which did not depend on algorithms. This is the origin of the creating subject technique. The method allowed stronger refutations of classical principles. Here it is used to show that negative dense subsets of the continuum are indecomposable.

Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation of (...) mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict. It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century. (shrink)

Volume 1 of this biography of L. E. J. Brouwer was published in 1999.1 The volume under review here covers the period from the early nineteen twenties until Brouwer's death in 1966. It also includes a short epilogue that discusses the disposition of Brouwer's estate after his death, his influence on others, the paths of some of his students and colleagues, and other matters. Van Dalen notes in the Preface that in preparing this volume he consulted some historical studies that (...) appeared after the first volume was published. He also used new material from various archives. The biography contains interesting quotations from unpublished materials in the Brouwer Archive and from correspondence. The bibliographical references to Brouwer's publications, it should be noted, are somewhat different in this volume. This volume, like the first, contains some nice photographs and reproductions. I noted that there were many typographical errors in the earlier book but Volume 2 is relatively free of them.As is the case in Volume 1, the discussion of Brouwer's mathematical and philosophical work is woven into the narrative of Brouwer's life and times. The story in this volume starts with Brouwer's first contacts with Paul Alexandrov and Paul Urysohn in 1923. The interaction began when Urysohn announced that he had found a mistake in Brouwer's definition of dimension in Brouwer's 1913 paper on natural dimension. Was it just a slip of the pen, as Brouwer always maintained, or something more substantial? Urysohn and Alexandrov ultimately came to agree with Brouwer on the matter, and Urysohn was prepared to grant Brouwer priority for the definition of dimension. There were, however, ups and downs along the way. Karl Menger, through his own work in topology and dimension theory, soon got into the picture, and Brouwer and Menger were to …. (shrink)

The situation in constructive mathematics in the nineties is so vastly different from that in the thirties, that it is worthwhile to pause a moment to survey the development in the intermediate years. In doing so, I follow the example of Heyting, who at certain intervals took stock of intuitionistic mathematics, which for a long time was the only variety of constructive mathematics. Heyting entered the foundational debate in 1930 at the occasion of the famous Königsberg meeting.