L. E. J. Brouwer

L. E. J. Brouwer
Born Luitzen Egbertus Jan Brouwer
(1881-02-27)27 February 1881
Overschie
Died 2 December 1966(1966-12-02) (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
Michael Dummett
Ludwig Wittgenstein
Notable awards Foreign Member of the Royal Society[3]
Brouwer (right) at the International Mathematical Congress, Zurich 1932

Luitzen Egbertus Jan Brouwer ForMemRS[3] (/ˈbraʊər/; 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, 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 in the emerging field of topology. The main results were his fixed point theorem, the topological invariance of degree, and the topological invariance of dimension. The most popular of the three among mathematicians is the first one called the Brouwer Fixed Point Theorem. It is a simple corollary to the second, about the topological invariance of degree, and this one is the most popular among algebraic topologists. The third is perhaps the hardest.

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. In 1912, at age 31, he was elected a member of the Royal Netherlands Academy of Arts and Sciences.[6]

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.[7] 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 a 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.[8][9][10] 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.

Dutch mathematician and historian of mathematics, Bartel Leendert van der Waerden attended lectures given by Brouwer in later years, and commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on—and only on—the foundations of his intuitionism. It seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, and he judged everything he had done before, his greatest output, false according to his philosophy."[11]

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

In English translation

See also

References

  1. 1 2 3 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. 1 2 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. ↑ Atten, Mark van. "Luitzen Egbertus Jan Brouwer". Stanford Encyclopedia of Philosophy.
  6. ↑ "Luitzen E.J. Brouwer (1881 - 1966)". Royal Netherlands Academy of Arts and Sciences. Retrieved 21 July 2015.
  7. ↑ L. E. J. Brouwer (trans. by Arnold Dresden) (1913). "Intuitionism and Formalism". Bull. Amer. Math. Soc. 20 (2): 81–96. MR 1559427. doi:10.1090/s0002-9904-1913-02440-6.
  8. ↑ "...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.
  9. ↑ 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)
  10. ↑ “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)
  11. ↑ "Interview with B L van der Waerden, reprinted in AMS March 1997" (PDF). American Mathematical Society. Retrieved 13 November 2015.

Further reading

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