L. E. J. Brouwer

L. E. J. Brouwer
Born Luitzen Egbertus Jan Brouwer
27 February 1881
Overschie
Died 2 December 1966 (aged 85)
Blaricum
Nationality Dutch
Fields Mathematics
Institutions University of Amsterdam
Alma mater University of Amsterdam
Doctoral advisor Diederik Korteweg[1]
Doctoral students Maurits Belinfante
Johanna Geldof
Bernardus Haalmeijer
Arend Heyting
Frans Loonstra
Barend Loor
Wilfrid Wilson[1]
Known for Brouwer-Hilbert controversy
Phragmen–Brouwer theorem
Brouwer fixed-point theorem
Proving Hairy ball theorem
Influences Immanuel Kant[2]
Arthur Schopenhauer
Influenced Hermann Weyl
Notable awards Foreign Member of the Royal Society[3]

Luitzen Egbertus Jan Brouwer ForMemRS[3] (Dutch: [ˈlœy̯tsə(n) ɛɣˈbɛrtəs jɑn ˈbrʌu̯ər]; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.[1][4][5] He was the founder of the mathematical philosophy of intuitionism.

Biography

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.[6] 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. 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 24, 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). Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions.[7][8][9] Brouwer then "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.)

Bibliography

Primary literature in English translation

Secondary

See also

References

  1. ↑ 1.0 1.1 1.2 L. E. J. Brouwer at the Mathematics Genealogy Project
  2. ↑ van Atten, Mark, "Luitzen Egbertus Jan Brouwer", The Stanford Encyclopedia of Philosophy (Spring 2012 Edition).
  3. ↑ 3.0 3.1 Kreisel, G.; Newman, M. H. A. (1969). "Luitzen Egbertus Jan Brouwer 1881–1966". Biographical Memoirs of Fellows of the Royal Society 15: 39. doi:10.1098/rsbm.1969.0002.
  4. ↑ O'Connor, John J.; Robertson, Edmund F., "L. E. J. Brouwer", MacTutor History of Mathematics archive, University of St Andrews.
  5. ↑ Luitzen Egbertus Jan Brouwer entry by Mark van Atten in the Stanford Encyclopedia of Philosophy
  6. ↑ L. E. J. Brouwer (trans. by Arnold Dresden) (1913). "Intuitionism and Formalism". Bull. Amer. Math. Soc. 20 (2): 81–96. doi:10.1090/s0002-9904-1913-02440-6. MR 1559427.
  7. ↑ "...Brouwer and Schopenhauer are in many respects two of a kind." Teun Koetsier, Mathematics and the Divine, Chapter 30, "Arthur Schopenhauer and L.E.J. Brouwer: A Comparison," p. 584.
  8. ↑ Brouwer wrote that "the original interpretation of the continuum of Kant and Schopenhauer as pure a priori intuition can in essence be upheld." (Quoted in Vladimir Tasić's Mathematics and the roots of postmodernist thought, § 4.1, p. 36)
  9. ↑ “Brouwer’s debt to Schopenhauer is fully manifest. For both, Will is prior to Intellect." [see T. Koetsier. “Arthur Schopenhauer and L.E.J. Brouwer, a comparison,” Combined Proceedings for the Sixth and Seventh Midwest History of Mathematics Conferences, pages 272–290. Department of Mathematics, University of Wisconsin-La Crosse, La Crosse, 1998.]. (Mark van Atten and Robert Tragesser, “Mysticism and mathematics: Brouwer, Gödel, and the common core thesis,” Published in W. Deppert and M. Rahnfeld (eds.), Klarheit in Religionsdingen, Leipzig: Leipziger Universitätsverlag 2003, pp.145–160)

External links