Brouwer's intuitionism; A re-appraisal of Brouwer's contribution to the study of the foundations of mathematics

Abstract

Brouwer's contribution to the study of foundations of mathematics is generally cepted, yet the greater part of his work has remained inaccessible because of language and for lack of a bibliography.In this study of the fundamental concepts of Brouwer's intuitionism and philosophy use has been made of all Brouwer1's published papers. Chapter I: Some relevant biographical details are given, as well as a survey of Brouwer's foundational works. A largely forgotten work, Leven. Kunst en Mystiek. has been included as a useful source of information on Brouwer's character, his general views, and his mystical tendencies.A bibliography of all Brouwer's work has been compiled, and is included. Chapter II: An analysis is made of Brouwer's philosophy, which determined his intuitionism as early as 1905.The central place in this philosophy is taken by Brouwer's theory of intuition and mathematics: intuition as the human mind actingindependently of all data of experience, and mathematics as nothing else but this intuitive mental activity. It is shown that this philosophy is the foundation of Brouwer's intuitionism; all his intuitionist theories and practices ultimately stem from his conception of mathematics. Chapter IIII: in particular, Brouwer's criticism of classical mathematics, logicism, formalism, and Poincare's neo-intuitionism is based on his absolute distinction between mathematics and language, i.e. expression of this mental activity in sounds or symbols. Neither language norlogic, the post-factum analysis of this language can contribute anything to mathematics; any device, such as the Principle of the Excluded Middle which claims to produce mathematical results from a purely verbal structure, is suspect. Chapter IV: Brouwer's conception of mathematics places greater emphasis on the active human role; this leads to an entirely new concept of the infinite sequence, of sets, and of the continuum,A survey is given of these fundamental notions of Brouwer's analysis, especially in as far they diverge from classical mathematics. Chapter V summarizes the main conclusions drawn in this work.