|Ideas and Explorations : Brouwer’s Road to Intuitionism
Ideas and Explorations : Brouwer’s Road to Intuitionism / Johannes John Carel Kuiper - [S.l.] : [s.n.], 2004 - Tekst. - Proefschrift Universiteit Utrecht
NBC: 31.02 : filosofie en theorie van de wiskunde
Trefwoorden: constructive second number class, intuitionism, ur-intuition of mathematics, possible cardinalities, logic and mathematics, solipsism, continuum, objectivity, apriori, actual infinite
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.
These five items have far-reaching consequences for Brouwer's mathematical building. To mention the most relevant ones:
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.