Luitzen Egbertus Jan Brouwer

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Luitzen Egbertus Jan Brouwer (February 27, 1881 – December 2, 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. The Brouwer fixed point theorem is named in his honor. He proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, 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 Bernays, Ackermann, von Neumman 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^{[citation needed]}. 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 can probably not completely be 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 use 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 [year?], Brouwer made a conscious decision to termporarily 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).

His ideas were initially exposed^{[citation needed]} in Beweis des Jordanschen Satzes für N Dimensionen (1912) ("Proof of Jordan's theorem for N dimensions"). He uncovered some of the main principles, such as triple negation, of intuitionistic logic; which then was taken up by Andrei Kolmogorov and (for a period) by Hermann Weyl, with rather different attitudes. Brouwer spent much time searching for the intuitionistic theory of real numbers, which he called species. This effort would now be considered misplaced: there is no single theory. Intuitionism later became more respectable once Kurt Gödel and later Stephen Kleene had fitted it into mathematical logic; but this was certainly to cut across Brouwer's anti-formal intentions.

He was combative from 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)

[edit] See also

[edit] Bibliography

Primary literature in English translation:

Jean van Heijenoort, 1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk. The original papers are prefaced with valuable commentary.
- 1923. L. E. J. Brouwer: "On the significance of the principle of excluded middle in mathematics, especially in function theory." With two Addenda and corrigenda, 334-45. Brouwer gives brief synopsis of his belief that the law of excluded middle cannot be "applied without reservation even in the mathematics of infinite systems" and gives two examples of failures to illustrate his assertion.
- 1925. A. N. Kolmogorov: "On the principle of excluded middle," p. 414-437. Kolmogorov supports most of Brouwer's results but disputes a few; he discusses the ramifications of intuitionism with respect to "transfinite judgements", e.g. transfinite induction.
- 1927. L. E. J. Brouwer: "On the domains of definition of functions". Brouwer's intuitionistic treatment of the continuum, with an extended commentary.
- 1927. David Hilbert: "The foundations of mathematics," 464-80
- 1927. L. E. J. Brouwer: "Intuitionistic reflections on formalism," 490-92. Brouwer lists four topics on which intuitionism and formalism might "enter into a dialogue." Three of the topics involve the law of excluded middle.
- 1927. Hermann Weyl: "Comments on Hilbert's second lecture on the foundations of mathematics," 480-484. In 1920 Weyl, Hilbert's prize pupil, sided with Brouwer against Hilbert. But in this address Weyl "while defending Brouwer against some of Hilbert's criticisms...attempts to bring out the significance of Hilbert's approach to the problems of the foundations of mathematics."
Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Univ. Press.
- 1928. "Mathematics, science, and language," 1170-85.
- 1928. "The structure of the continuum," 1186-96.
- 1952. "Historical background, principles, and methods of intuitionism," 1197-1207.

Secondary:

Dirk Van Dalen, Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Oxford Univ. Press.
- 1999. Volume 1: The Dawning Revolution.
- 2005. Volume 2: Hope and Disillusion.
Martin Davis, 2000. The Engines of Logic, W. W. Norton, London, ISBN 0-393-32229-7 pbk. Cf Chapter Five: "Hilbert to the Rescue" wherein Davis discusses Brouwer and his relationship with Hilbert and Weyl with brief biographical information of Brouwer. Davis's references include:
- Brouwer, L. E. J., Collected Works, Vol. I", Amsterdam: North-Holland, 1975.
- Brouwer, "Life, Art, and Mysticism," Notre Dame Journal of Formal Logic, vol. 37 (1966), pp. 389-429. Translated by W. P. van Stigt with an introduction by the translator, pp. 381-87. Davis quotes from this work, "a short book... drenched in romantic pessimism" (p. 94).
- W. P. van Stigt, 1990, Brouwer's Intuitionism, Amsterdam: North-Holland, 1990
Stephen Kleene, 1952 with corrections 1971, 10th reprint 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam Netherlands, ISBN 0 7204 2103 9. Cf in particular Chapter III: A Critique of Mathematical Reasoning, §13 "Intuitionism" and §14 "Formalism".