Luitzen Egbertus Jan Brouwer ForMemRS[1] (/ˈbraʊ.ər/; 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.[2][4][5] He was the founder of the mathematical philosophy of intuitionism. Contents 1 Biography 2 Bibliography 2.1 In English translation 3 See also 4 References 5 Further reading 6 External links Biography[edit]
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
"... 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."[13] 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[edit] In English translation[edit] Jean van Heijenoort, 1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk. The original papers are prefaced with valuable commentary. 1923. L. E. J. Brouwer: "On the significance of the principle of excluded middle in mathematics, especially in function theory." With two Addenda and corrigenda, 334-45. Brouwer gives brief synopsis of his belief that the law of excluded middle cannot be "applied without reservation even in the mathematics of infinite systems" and gives two examples of failures to illustrate his assertion. 1925. A. N. Kolmogorov: "On the principle of excluded middle", pp. 414–437. Kolmogorov supports most of Brouwer's results but disputes a few; he discusses the ramifications of intuitionism with respect to "transfinite judgements", e.g. transfinite induction. 1927. L. E. J. Brouwer: "On the domains of definition of functions". Brouwer's intuitionistic treatment of the continuum, with an extended commentary. 1927. David Hilbert: "The foundations of mathematics," 464-80 1927. L. E. J. Brouwer: "Intuitionistic reflections on formalism," 490-92. Brouwer lists four topics on which intuitionism and formalism might "enter into a dialogue." Three of the topics involve the law of excluded middle. 1927. Hermann Weyl: "Comments on Hilbert's second lecture on the foundations of mathematics," 480-484. In 1920 Weyl, Hilbert's prize pupil, sided with Brouwer against Hilbert. But in this address Weyl "while defending Brouwer against some of Hilbert's criticisms...attempts to bring out the significance of Hilbert's approach to the problems of the foundations of mathematics." 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. Brouwer, L. E. J., Collected Works, Vol. I, Amsterdam: North-Holland, 1975.[14] Brouwer, L. E. J., Collected Works, Vol. II, Amsterdam: North-Holland, 1976. Brouwer, L. E. J., "Life, Art, and Mysticism," Notre Dame Journal of Formal Logic, vol. 37 (1996), pp. 389–429. Translated by W. P. van Stigt with an introduction by the translator, pp. 381–87. Davis quotes from this work, "a short book... drenched in romantic pessimism" (p. 94). W. P. van Stigt, 1990, Brouwer's Intuitionism, Amsterdam: North-Holland, 1990 See also[edit] Gerrit Mannoury George F C Griss Bar induction References[edit] ^ a b 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.
^ a b c
Further reading[edit] Dirk van Dalen, 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.
2013. L. E. J. Brouwer: Topologist, Intuitionist, Philosopher. How
Martin Davis, 2000. The Engines of Logic, W. W. Norton, London,
ISBN 0-393-32229-7 pbk. Cf. Chapter Five: "Hilbert to the Rescue"
wherein Davis discusses Brouwer and his relationship with Hilbert and
Weyl with brief biographical information of Brouwer. Davis's
references include:
Stephen Kleene, 1952 with corrections 1971, 10th reprint 1991,
Introduction to Metamathematics, North-Holland Publishing Company,
Amsterdam Netherlands, ISBN 0-7204-2103-9. Cf. in particular
Chapter III: A Critique of Mathematical Reasoning, §13 "Intuitionism"
and §14 "Formalism".
Koetsier, Teun, Editor,
External links[edit] Wikiquote has quotations related to: L. E. J. Brouwer Works by or about
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