L. E. J. Brouwer

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L. E. J. Brouwer
Indecomposability
Indecomposable continuum
Invariance of domain
Spread
Proving hairy ball theorem
Intuitionism
RelativesHendrik Albertus Brouwer (brother)[3]
AwardsForeign Member of the Royal Society[1]
Scientific career
FieldsMathematics
InstitutionsUniversity of Amsterdam
ThesisOver de grondslagen der wiskunde (1907)
Doctoral advisorDiederik Korteweg[2]
Doctoral studentsArend Heyting[2]
Brouwer (right) at the International Mathematical Congress, Zurich 1932

Luitzen Egbertus Jan "Bertus" Brouwer

measure theory and complex analysis.[2][4][5] Regarded as one of the greatest mathematicians of the 20th century, he is known as one of the founders of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension.[6][7][8]

Brouwer also became a major figure in the

objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl
. In addition to his mathematical work, Brouwer also published the short philosophical tract Life, Art, and Mysticism (1905).

Biography

Brouwer was born to

Dutch Protestant parents.[9] Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem, the topological invariance of degree, and the topological invariance of dimension
. Among mathematicians generally, the best known is the first one, usually referred to now as the Brouwer fixed point theorem. It is a corollary to the second, concerning the topological invariance of degree, which is the best known among algebraic topologists. The third theorem 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.[10] He was an Invited Speaker of the ICM in 1908 at Rome[11] and in 1912 at Cambridge, UK.[12] He was elected to the American Philosophical Society in 1943.[13]

Brouwer founded

constructive mathematics, intuitionism is a philosophy of the foundations of mathematics.[14] It is sometimes (simplistically) characterized by saying that its adherents do not admit the law of excluded middle
as a general axiom in mathematical reasoning, although it may be proven as a theorem in some special cases.

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 cannot 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, which has been described by the mathematician Martin 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.[15][16][17] 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, 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 as a young man. According to Mark van Atten, this pugnacity reflected his combination of independence, brilliance, high moral standards and extreme sensitivity to issues of justice.

Bergen-Belsen, as his assistant in 1948.[19] However, Brouwer did encourage his students to sign a loyalty oath to the Nazis in 1943.[20]

In later years, 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."[21]

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

Notes

  1. ^ The Dutch pronunciation of the phrase Luitzen Egbertus Jan Brouwer is [ˈlœytsə(n) ɛɣˈbɛrtə ˈʃɑm ˈbrʌuər]. The words in isolation are pronounced [ˈlœytsə(n)], [ɛɣˈbɛrtəs], [ˈjɑn] and [ˈbrʌuər]. The surname Brouwer is pronounced /ˈbraʊ.ər/ in English.

References

  1. hdl:10077/30385
    .
  2. ^ a b c L. E. J. Brouwer at the Mathematics Genealogy Project
  3. ISSN 0012-2017
    .
  4. ^ O'Connor, John J.; Robertson, Edmund F., "L. E. J. Brouwer", MacTutor History of Mathematics Archive, University of St Andrews
  5. ^ a b Atten, Mark van. "Luitzen Egbertus Jan Brouwer". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
  6. ISBN 9781134672455
    . p. 53.
  7. ^ Van Atten, Mark (2016), "Brouwer, L.E.J.", Routledge Encyclopedia of Philosophy
  8. ^ Luitzen Egbertus Jan Brouwer entry in Stanford Encyclopedia of Philosophy
  9. ISBN 9781447146162
    .
  10. ^ "Luitzen E.J. Brouwer (1881 - 1966)". Royal Netherlands Academy of Arts and Sciences. Retrieved 21 July 2015.
  11. ^ Brouwer, L. E. J. "Die mögliche Mächtigkeiten." Atti IV Congr. Intern. Mat. Roma 3 (1908): 569–571.
  12. ^ Brouwer, L. E. J. (1912). Sur la notion de «Classe» de transformations d'une multiplicité. Proc. 5th Intern. Math. Congr. Cambridge, 2, 9–10.
  13. ^ "APS Member History". search.amphilsoc.org. Retrieved 2023-04-12.
  14. MR 1559427
    .
  15. ^ "...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.
  16. ^ 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)
  17. ^ “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)
  18. Abraham A. Fraenkel, ‘Hitler’s Math,’ Tablet
    8 February 2008
  19. Times Literary Supplement
    19 November 2021 p.6.
  20. ^ "Dostoevsky's beliefs". TLS. Retrieved 2023-11-21.
  21. ^ "Interview with B L van der Waerden, reprinted in AMS March 1997" (PDF). American Mathematical Society. Retrieved 13 November 2015.
  22. doi:10.1090/S0002-9904-1977-14185-2
    .

Further reading

External links

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