Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. The most celebrated result was his proof of the topological invariance of
dimension. Among his further results, the
Brouwer fixed point theorem is also well known. Brouwer also proved the
simplicial approximation theorem in the foundations of
algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of
simplicial complexes, of the treatment of general continuous mappings.
Brouwer in effect founded the mathematical philosophy of
intuitionism as an opponent to the then-prevailing
formalism of
David Hilbert and his collaborators
Paul Bernays, Wilhelm Ackermann, John von Neumann and others (cf. Kleene (1952), p. 46-59). As a variety of
constructive mathematics, intuitionism is essentially a philosophy of the
foundations of mathematics. It is sometimes and rather simplistically characterized by saying that its adherents refuse to use the
law of excluded middle in mathematical reasoning.
Brouwer was member of the
Significs group, containing others with a generally
neo-Kantian philosophy . It formed part of the early history of
semiotics -- the study of symbols -- around
Victoria, Lady Welby in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group.
In 1905, at the age of 26, Brouwer expressed his philosophy of life in a short tract
Life, Art and Mysticism described by Davis as "drenched in romantic pessimism" (Davis (2002), p. 94). Then Brouwer "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions"; indeed his thesis advisor refused to accept his Chapter II " 'as it stands, ... all interwoven with some kind of pessimism and mystical attitude to life which is not mathematics, nor has anything to do with the foundations of mathematics' " (Davis, p. 94 quoting van Stigt, p. 41). Nevertheless, in 1908:
:"... Brouwer, in a paper entitled "The untrustworthiness of the principles of logic", challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384--322 B.C.) have an absolute validity, independent of the subject matter to which they are applied" (Kleene (1952), p. 46).
"After completing his dissertation (1907 - see Van Dalen), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess" (Davis (2000), p. 95); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem. Hilbert -- the formalist with whom the intuitionist Brouwer would ultimately spend years in conflict -- admired the young man and helped him receive a regular academic appointment (1912) at the University of Amsterdam (Davis, p. 96). It was then that "Brouwer felt free to return to his revolutionary project which he was now calling
intuitionism " (ibid).
He was combative for a young man. He was involved in a very public and eventually demeaning controversy in the later 1920s with Hilbert over editorial policy at
Mathematische Annalen, at that time a leading
learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student
Arend Heyting.
About his last years, Davis (2002) remarks:
:"...he felt more and more isolated, and spent his last years under the spell of 'totally unfounded financial worries and a paranoid fear of bankruptcy, persecution and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house." (Davis, p. 100 quoting van Stigt. p. 110.)
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