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 adhered to an intuitionist philosophy of mathematics. This is a variety of constructive 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 in effect founded mathematical intuitionism, as an opponent of the prevailing trend towards formalism.

He was member of the Significs group, containing others with a generally neo-Kantian philosophy. It formed part of the early history of semiotic study, 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.

His ideas were initially exposed 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 David Hilbert, over editorial policy at Mathematische Annalen, at that time a leading learned journal. Politically Brouwer was pro-German. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.

[edit] Bibliography

Primary literature in English translation:

Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
- 1923. "On the significance of the principle of excluded middle in mathematics, especially in function theory." With two Addenda and corrigenda, 334-45.
- 1927. "The foundations of mathematics," 464-80
- 1927. "Intuitionistic reflections on formalism," 490-92.
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:

Van Dalen, Dirk. 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.