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.

Par des suites de choix, nous comprenons des suites qui ne sont pas déterminées complètement par une loi arithmétique. Elles sont des objets caractéristiques de l’intuitionnisme de Brouwer. Nous prétendons qu’à partir de 1927, l’utilisation par Brouwer de suites de choix particulières n’est pas reconnu comme tel. Nous prétendons que l’utilisation de ces suites dans la méthode du sujet créatif, après la seconde guerre mondiale, n’a pas à être mis en relation avec l’utilisation de celles-ci dans les années vingt et (...) qu’elles sont mal interprétées. Nous montrons où se trouvent ces suites de choix dans l’œuvre de Brouwer et comment elles doivent être traitées. (shrink)

Par des suites de choix, nous comprenons des suites qui ne sont pas déterminées complètement par une loi arithmétique. Elles sont des objets caractéristiques de l’intuitionnisme de Brouwer. Nous prétendons qu’à partir de 1927, l’utilisation par Brouwer de suites de choix particulières n’est pas reconnu comme tel. Nous prétendons que l’utilisation de ces suites dans la méthode du sujet créatif, après la seconde guerre mondiale, n’a pas à être mis en relation avec l’utilisation de celles-ci dans les années vingt et (...) qu’elles sont mal interprétées. Nous montrons où se trouvent ces suites de choix dans l’œuvre de Brouwer et comment elles doivent être traitées. (shrink)

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But (...) other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. (...) But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. (shrink)