Editing L. E. J. Brouwer (section)

==Biography==
Brouwer was born to [[Dutch Protestant]] parents.<ref>{{Cite book | url=https://books.google.com/books?id=gFbqtEMohBYC&dq=%22lej+brouwer%22+%22reformed+church%22&pg=PA16 | isbn=9781447146162 | title=L.E.J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics is Rooted in Life | date=4 December 2012 | publisher=Springer }}</ref> Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his [[Brouwer fixed-point theorem|fixed point theorem]], the topological invariance of degree, and the [[Invariance of domain|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 complex]]es, 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]].<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00004406 |title=Luitzen E.J. Brouwer (1881 - 1966) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=21 July 2015}}</ref> He was an Invited Speaker of the [[International Congress of Mathematicians|ICM]] in 1908 at Rome<ref>Brouwer, L. E. J. [https://babel.hathitrust.org/cgi/pt?id=iau.31858027749682;view=1up;seq=593 "Die mögliche Mächtigkeiten."] Atti IV Congr. Intern. Mat. Roma 3 (1908): 569–571.</ref> and in 1912 at Cambridge, UK.<ref>Brouwer, L. E. J. (1912). [https://web.archive.org/web/20171112131913/https://pdfs.semanticscholar.org/b90a/89587ec4436cb0a7d03124fea2430b7e8bdc.pdf Sur la notion de «Classe» de transformations d'une multiplicité]. Proc. 5th Intern. Math. Congr. Cambridge, 2, 9–10.</ref> He was elected to the [[American Philosophical Society]] in 1943.<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Luitzen+E.J.+Brouwer&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2023-04-12 |website=search.amphilsoc.org}}</ref>

Brouwer founded [[intuitionism]], a philosophy of mathematics that challenged the then-prevailing [[formalism (mathematics)|formalism]] of [[David Hilbert]] and his collaborators, who included [[Paul Bernays]], [[Wilhelm Ackermann]], and [[John von Neumann]] (cf. Kleene (1952), p.&nbsp;46–59). A variety of [[Constructivism (mathematics)|constructive mathematics]], intuitionism is a philosophy of the [[foundations of mathematics]].<ref>{{cite journal|author=L. E. J. Brouwer (trans. by Arnold Dresden)|title=Intuitionism and Formalism|journal=Bull. Amer. Math. Soc.|year=1913|volume=20|issue=2|pages=81–96|mr=1559427|doi=10.1090/s0002-9904-1913-02440-6|doi-access=free}}</ref> 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 (mathematician)|Martin Davis]] as "drenched in romantic pessimism" (Davis (2002), p.&nbsp;94). [[Arthur Schopenhauer]] had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions.<ref>"...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.</ref><ref>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)</ref><ref>“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)</ref> 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.&nbsp;94 quoting van Stigt, p.&nbsp;41). Nevertheless, in 1908:
: "... Brouwer, in a paper titled '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.&nbsp;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.&nbsp;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.<ref name ="sep"/> He was involved in a very public and eventually demeaning controversy with Hilbert in the late 1920s over editorial policy at ''[[Mathematische Annalen]]'', at the time a leading journal. According to [[Abraham Fraenkel]],  Brouwer espoused [[Aryan race|Germanic Aryanness]] and Hilbert removed him from the editorial board of  [[Mathematische Annalen]] after Brouwer objected to contributions from [[Eastern European Jewry|Ostjuden]].<ref>[[Abraham A. Fraenkel]], [https://www.tabletmag.com/sections/arts-letters/articles/hitlers-math ‘Hitler’s Math,’ ] [[Tablet (magazine)|Tablet]]  8 February 2008</ref>

In later years Brouwer 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."<ref>{{cite web|url=https://www.ams.org/notices/199703/interview.pdf|title=Interview with B L van der Waerden, reprinted in AMS March 1997 |publisher=American Mathematical Society |access-date=13 November 2015}}</ref>

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.)