Ideas and Explorations : Brouwer's Road to Intuitionism

Abstract

This dissertation is about the initial period of Brouwer's role in the foundational debate in mathematics, which took place during the first decades of the twentieth century. His intuitionistic and constructivistic attitude was a reaction to logicism (Russell, Couturat) and to Hilbert's formalism. Brouwer's own dissertation (1907) is a first introduction to his intuitionism, which was the third movement in the foundational debate. This intuitionism reached maturity from 1918 onwards, but one of my aims is to show that there are demonstrable traces of this new development of mathematics as early as 1907 and even before, viz. in his personal notes, which are composed of his numerous ideas in the field of mathematics and philosophy. To mention some important ones: 1. Mathematics is entirely independent of language. Mathematics is created by the individual mind (Brouwer certainly is a solipsist) and the role of language is limited to that of communication a mathematical content to others and is also useful for one's own memory. 2. The ur-intuition of the 'move of time', that is, the experience that two events are not coinciding, is the most fundamental basis of all mathematics. A separate space intuition (Kant) is not needed. 3. A strict constructivism. Only that what is constructed by the individual mind counts as a mathematical object. 4. Logic only describes the structure of the language of mathemqatics. Hence logic comes after mathematics, instead of being its basis. 5. An axiomatic foundation is rejected by Brouwer. Axioms only serve the purpose of describing concisely the properties of a mathematical construction. These five items have far-reaching consequences for Brouwer's mathematical building. To mention the most relevant ones: - The only possible cardinalities for sets are: finite, denumerably infinite, denumerably infinite unfinished and the continuum. - The continuum is not composed of points (Aristotle already said so), but is given to us in its entirety in the ur-intuition. It can be turned into an everywhere dense measurable continuum by constructing a rational scale on it. - Cantor's second number class does not exist as a finished totality for Brouwer, since there is no conceivable closure for the elements of this class. - The continuum problem is a trivial one: Every well-defined subset of the continuum is finite, denumerably infinite, or has the cardinality of the continuum. Finally, in my dissertation the sixth chapter is devoted to Brouwer's view on the application of mathematics to the human evironment and on his outlook on man and on human society in general (chapter 2 of Brouwer's dissertation). His opinion about humanity turns out to be a pessimistic one: All man's effort, when applying mathematics to the surrounding world, is aimed at a domination over his environment and over his fellow men.