Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer (
February 27,
1881 â€"
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.
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.
*
*
Stanford Encyclopedia of Philosophy entry
