L. E. J. Brouwer | |
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| Born | Luitzen Egbertus Jan Brouwer (1881-02-27)27 February 1881 Overschie, Netherlands |
| Died | 2 December 1966(1966-12-02) (aged 85) Blaricum, Netherlands |
| Nationality | Dutch |
| Alma mater | University of Amsterdam |
| Known for | Brouwer–Hilbert controversy Brouwer fixed-point theorem Brouwer–Heyting–Kolmogorov interpretation Jordan-Brouwer separation theorem Kleene–Brouwer order Phragmen–Brouwer theorem Tietze-Urysohn-Brouwer extension theorem Simplicial approximation theorem Bar induction Degree of a continuous mapping Indecomposability Indecomposable continuum Invariance of domain Spread Provinghairy ball theorem Intuitionism |
| Relatives | Hendrik Albertus Brouwer (brother)[3] |
| Awards | Foreign Member of the Royal Society[1] |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of Amsterdam |
| Thesis | Over de grondslagen der wiskunde (1907) |
| Doctoral advisor | Diederik Korteweg[2] |
| Doctoral students | Arend Heyting[2] |
Luitzen Egbertus Jan "Bertus"Brouwer[a] (27 February 1881 – 2 December 1966) was a Dutchmathematician andphilosopher who worked intopology,set theory,measure theory andcomplex 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 hisfixed-point theorem and thetopological invariance of dimension.[6][7][8]
Brouwer also became a major figure in thephilosophy ofintuitionism, aconstructivist school of mathematics which argues that math is acognitiveconstruct rather than a type ofobjective truth. This position led to theBrouwer–Hilbert controversy, in which Brouwer sparred with hisformalist colleagueDavid Hilbert. Brouwer's ideas were subsequently taken up by his studentArend Heyting and Hilbert's former studentHermann Weyl. In addition to his mathematical work, Brouwer also published the short philosophical tractLife, Art, and Mysticism (1905).
Brouwer was born toDutch Protestant parents.[9] Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were hisfixed point theorem, the topological invariance of degree, and thetopological 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 thesimplicial approximation theorem in the foundations ofalgebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision ofsimplicial complexes, of the treatment of general continuous mappings. In 1912, at age 31, he was elected a member of theRoyal Netherlands Academy of Arts and Sciences.[10] He was an Invited Speaker of theICM in 1908 at Rome[11] and in 1912 at Cambridge, UK.[12] He was elected to theAmerican Philosophical Society in 1943.[13]
Brouwer foundedintuitionism, a philosophy of mathematics that challenged the then-prevailingformalism ofDavid Hilbert and his collaborators, who includedPaul Bernays,Wilhelm Ackermann, andJohn von Neumann (cf. Kleene (1952), p. 46–59). A variety ofconstructive mathematics, intuitionism is a philosophy of thefoundations of mathematics.[14] It is sometimes (simplistically) characterized by saying that its adherents do not admit thelaw 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 theSignifics Group. It formed part of the early history ofsemiotics—the study of symbols—aroundVictoria, 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 tractLife, Art and Mysticism, which has been described by the mathematicianMartin 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:
"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 particularthe 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 callingintuitionism " (ibid).
Both Brouwer and Hilbert worked as editors at the prestigious mathematical journal,Mathematische Annalen. Hilbert was the chief editor of the journal alongsideBlumenthal,Carathéodory, andEinstein, and Brouwer was an associate editor. The relationship between Brouwer and Hilbert began to deteriorate as their intellectual feud began to become a personal one, culminating in the eventual dismissal of Brouwer by Hilbert. Carathéordory sought out Einstein's advice on this matter and Einstein chose to remain neutral in the feud, leading to Brouwer's dismissal.[18]Abraham Fraenkel argues that the reason for this dismissal was Brouwer's turn towards pushing ideas ofGermanic Aryan supremacy;[19] however, there does not exist any evidence of Brouwer's involvement with the GermanNazi party, nor theNational Socialist Movement in the Netherlands.[20] Brouwer was not known to hold onto any nationalistic views by his contemporaries, and his views were akin to theGermanic idealists andromanticists of the time.[21]
In later years Brouwer became relatively isolated; the development of intuitionism at its source was taken up by his studentArend Heyting. Dutch mathematician and historian of mathematicsBartel 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."[22]
About his last years, Davis (2002) remarks:
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