L. E. J. Brouwer

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L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer.jpeg
Born
Luitzen Egbertus Jan Brouwer

(1881-02-27)27 February 1881
Died2 December 1966(1966-12-02) (aged 85)
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
Proving hairy ball theorem
Intuitionism
Awards Foreign Member of the Royal Society [1]
Scientific career
Fields Mathematics
Institutions University of Amsterdam
Doctoral advisor Diederik Korteweg [2]
Doctoral students Arend Heyting [2]
Influences Immanuel Kant [3]
Arthur Schopenhauer
Influenced Hermann Weyl
Michael Dummett
Ludwig Wittgenstein
Brouwer (right) at the International Mathematical Congress, Zurich 1932 Bohr Brouwer Zurich1932.tif
Brouwer (right) at the International Mathematical Congress, Zurich 1932

Luitzen Egbertus Jan Brouwer ( /ˈbr.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 is known as the founder of modern topology, [6] particularly for establishing his fixed-point theorem and the topological invariance of dimension. [7]

Contents

Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics in which math is argued to be a cognitive construct rather than a type of 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.

Biography

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. [8] He was an Invited Speaker of the ICM in 1908 at Rome [9] and in 1912 at Cambridge, UK. [10]

Brouwer founded intuitionism, a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays, Wilhelm Ackermann, and John von Neumann (cf. Kleene (1952), p. 46–59). A variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics. [11] 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, 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. [12] [13] [14] 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. 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." [15]

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

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See also

Related Research Articles

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a convex compact subset of Euclidean space to itself.

David Hilbert German mathematician

David Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.

In logic, the law of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the so called three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or De Morgan's laws.

In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Mathematical logic, also called formal logic, is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, philosophy, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.

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In the mathematical philosophy, the pre-intuitionists were a small but influential group who informally shared similar philosophies on the nature of mathematics. The term itself was used by L. E. J. Brouwer, who in his 1951 lectures at Cambridge described the differences between intuitionism and its predecessors:

Of a totally different orientation [from the "Old Formalist School" of Dedekind, Cantor, Peano, Zermelo, and Couturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue. These thinkers seem to have maintained a modified observational standpoint for the introduction of natural numbers, for the principle of complete induction [...] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof. For the continuum, however, they seem not to have sought an origin strictly extraneous to language and logic.

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In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. The debate concerned fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. Much of the controversy took place while both were involved with the prestigious Mathematische Annalen journal, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board.

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In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if X is a normal connected locally connected topological space, then the following two properties are equivalent:

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References

  1. Kreisel, G.; Newman, M. H. A. (1969). "Luitzen Egbertus Jan Brouwer 1881–1966". Biographical Memoirs of Fellows of the Royal Society . 15: 39–68. doi: 10.1098/rsbm.1969.0002 .
  2. 1 2 3 L. E. J. Brouwer at the Mathematics Genealogy Project
  3. van Atten, Mark, "Luitzen Egbertus Jan Brouwer", The Stanford Encyclopedia of Philosophy (Spring 2012 Edition).
  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". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy .
  6. Larios, Pablo. "The Room is Sound, The Objects Abstractions: The Art of Catherine Christer Hennix". frieze. Retrieved 26 October 2020.
  7. Luitzen Egbertus Jan Brouwer entry in Stanford Encyclopedia of Philosophy
  8. "Luitzen E.J. Brouwer (1881 - 1966)". Royal Netherlands Academy of Arts and Sciences. Retrieved 21 July 2015.
  9. Brouwer, L. E. J. "Die mögliche Mächtigkeiten." Atti IV Congr. Intern. Mat. Roma 3 (1908): 569–571.
  10. Brouwer, L. E. J. (1912). Sur la notion de «Classe» de transformations d'une multiplicité. Proc. 5th Intern. Math. Congr. Cambridge, 2, 9–10.
  11. L. E. J. Brouwer (trans. by Arnold Dresden) (1913). "Intuitionism and Formalism". Bull. Amer. Math. Soc. 20 (2): 81–96. doi: 10.1090/s0002-9904-1913-02440-6 . MR   1559427.
  12. "...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.
  13. 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)
  14. “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)
  15. "Interview with B L van der Waerden, reprinted in AMS March 1997" (PDF). American Mathematical Society. Retrieved 13 November 2015.
  16. Kreisel, G. (1977). "Review: L. E. J. Brouwer collected works, Volume I, Philosophy and foundations of mathematics ed. by A. Heyting" (PDF). Bull. Amer. Math. Soc. 83: 86–93. doi: 10.1090/S0002-9904-1977-14185-2 .

Further reading