MARK VAN ATTEN. Brouwer meets Husserl: On the Phenomenology of Choice Sequences

Abstract

This book summarizes the intense research that the author performed for his Ph.D. thesis (1999), revised and with the addition of an intuitionistic critique of Husserl's concept of number. His starting point consisted of a double conviction: 1) Brouwerian intuitionism is a valid way of doing mathematics but is grounded on a weak philosophy; 2) Husserlian phenomenology (especially as developed in his mature transcendental phase) can provide a suitable philosophical ground for intuitionism. In order to let intuitionism and phenomenology match, he had to solve in general two problems: 1) the question of the reciprocal indifference that the authors had toward each other's theorizing which indeed they knew; 2) Husserl's general attitude of accepting classical mathematics, which contrasts with the critical attitude of the intuitionists. Moreover, in the specific case focused on by this book—concerning choice sequences, that is, sequences that can be completely lawless and that were admitted as mathematical entities by Brouwer—van Atten had to handle a further problem: such sequences do not fit the characteristic of omnitemporality that the late Husserl seemed to consider a necessary attribute of entities if they were to be considered mathematical. In as far as they are lawless we cannot predict at any moment how they will develop in the future, hence they are not determined at any particular time—they will be determined only in the future.