Biography[ edit ] 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 simple 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.

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Biographies index L E J Brouwer is usually known by this form of his name with full initials, but he was known to his friends as Bertus, an abbreviation of the second of his three forenames.

He attended high school in Hoorn, a town on the Zuiderzee north of Amsterdam. His performance there was outstanding and he completed his studies by the age of fourteen. He had not studied Greek or Latin at high school but both were required for entry into university, so Brouwer spent the next two years studying these topics. During this time his family moved to Haarlem, just west of Amsterdam, and it was in the Gymnasium there in that he sat the entrance examinations for the University of Amsterdam.

Korteweg was the professor of mathematics at the University of Amsterdam when Brouwer began his studies, and he quickly realised that in Brouwer he had an outstanding student. While still an undergraduate Brouwer proved original results on continuous motions in four dimensional space and Korteweg encouraged him to present them for publication. This he did, and it became his first paper published by the Royal Academy of Science in Amsterdam in Other topics which interested Brouwer were topology and the foundations of mathematics.

He learnt something of these topics from lectures at the university but he also read many works on the topics on his own. After the marriage, which would produce no children, the couple moved to Blaricum, near Amsterdam. Three years later Lize qualified as a pharmacist and Brouwer helped her in many ways from doing bookkeeping to serving in the chemists shop. However, Brouwer did not gain the affection of his step-daughter and relations between them was strained.

From an early stage Brouwer was interested in the philosophy of mathematics, but he was also fascinated by mysticism and other philosophical questions relating to human society. In this work he [ 1 ] His doctoral thesis [ 13 ] He quickly discovered that his ideas on the foundations of mathematics would not be readily accepted [ 13 ]:- Brouwer quickly found that his philosophical ideas sparked controversy.

Korteweg , his thesis advisor, had not been pleased with the more philosophical aspects of the thesis, and had even demanded that several parts of the original draft be cut from the final presentation. Korteweg urged Brouwer to concentrate on more "respectable" mathematics, so that the young man might enhance his mathematical reputation and thus secure an academic career. Brouwer continued to develop the ideas of his thesis in The Unreliability of the Logical Principles published in The research which Brouwer now undertook was in two areas.

He addressed the International Congress of Mathematicians in Rome in on the topological foundations of Lie groups. In he was appointed as a privatdocent at the University of Amsterdam. Prompted by discussions in Paris, he began working on the problem of the invariance of dimension. Brouwer was elected to the Royal Academy of Sciences in and, in the same year, was appointed extraordinary professor of set theory, function theory and axiomatics at the University of Amsterdam; he would hold the post until he retired in Hilbert wrote a warm letter of recommendation which helped Brouwer to gain his chair in Despite the substantial contributions he had made to topology by this time, Brouwer chose to give his inaugural professorial lecture on intuitionism and formalism.

In the following year Korteweg resigned his chair so that Brouwer could be appointed as ordinary professor. He was also offered the chair at Berlin in the same year. These must have been tempting offers, but despite their attractions Brouwer turned them down. Perhaps the exceptional way he was treated by Amsterdam, mentioned in the following quote by Van der Waerden , helped him make these decisions. Van der Waerden , who studied at Amsterdam from to , wrote about Brouwer as a lecturer see for example [ 14 ] :- Brouwer came [to the university] to give his courses but lived in Laren.

He came only once a week. In general that would have not been permitted - he should have lived in Amsterdam - but for him an exception was made. I once interrupted him during a lecture to ask a question. He just did not want them, he was always looking at the blackboard, never towards the students. Even though his most important research contributions were in topology, Brouwer never gave courses on topology, but always on -- and only on -- the foundations of 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.

He was a very strange person, crazy in love with his philosophy. As is mentioned in this quotation, Brouwer was a major contributor to the theory of topology and he is considered by many to be its founder. The status of the subject when he began his research is well described in [ 13 ]:- When Brouwer was beginning his career as a mathematician, set-theoretic topology was in a primitive state.

Point set theory was widely applied in analysis and somewhat less widely applied in geometry, but it did not have the character of a unified theory. There were some perceived benchmarks. For example; the generally held view that dimension was invariant under one-to-one continuous mappings He did almost all his work in topology early in his career between and He discovered characterisations of topological mappings of the Cartesian plane and a number of fixed point theorems.

Originally proved for a 2-dimensional sphere, Brouwer later generalised the result to spheres in n dimensions. Another result of exceptional importance was proving the invariance of dimension. As well as proving theorems of major importance in topology, Brouwer also developed methods which have become standard tools in the subject.

In particular he used simplicial approximation, which approximated continuous mappings by piecewise linear ones. He also introduced the idea of the degree of a mapping, generalised the Jordan curve theorem to n-dimensional space, and defined topological spaces in Van der Waerden , in the above quote, said that Brouwer would not lecture on his own topological results since they did not fit with mathematical intuitionism.

In fact Brouwer is best known to many mathematicians as the founder of the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws. His doctrine differed substantially from the formalism of Hilbert and the logicism of Russell. His doctoral thesis in attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School.

In his paper The Unreliability of the Logical Principles Brouwer rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false. Part One, General Set Theory. The answer to the question of the title which Brouwer gives is "no". Also in he published Intuitionistic Set Theory, then in he developed a theory of functions On the Domains of Definition of Functions without the use of the Principle of the Excluded Middle.

Loosely speaking, that the elements of a set had property p, meant to Brouwer that he had a construction which allowed him to decide after a finite number of steps whether each element of the set had property p. Such ideas are fundamental to theoretical computer science today. He had been appointed to the editorial board of Mathematische Annalen in but in Hilbert decided that Brouwer was becoming too powerful, particularly since Hilbert felt that he himself did not have long to live in fact he lived until He tried to remove Brouwer from the board in a way which was not compatible with the way the board was set up.

In the end Hilbert managed to get his own way but it was a devastating episode for Brouwer who was left mentally broken; see [ 26 ] for details. In Brouwer entered local politics when he was elected as Neutral Party candidate for the municipal council of Blaricum.

He continued to serve on the council until He was also active setting up a new journal and he became a founding editor of Compositio Mathematica which began publication in Further controversy arose due to his actions in World War II.

Brouwer was active in helping the Dutch resistance, and in particular he supported Jewish students during this difficult period. However, in the Germans insisted that the students sign a declaration of loyalty to Germany and Brouwer encouraged his students to do so. He afterwards said that he did so in order that his students might have a chance to complete their studies and to work for the Dutch resistance against the Germans.

However, after Amsterdam was liberated, Brouwer was suspended from his post for a few months because of his actions. Again he was deeply hurt and considered emigration. His wife died in at the age of 89 and Brouwer, who himself was 78, was offered a one year post in the University of British Columbia in Vancouver; he declined.

In , despite being well into his 80s, he was offered a post in Montana. He died in in Blaricum as the result of a traffic accident. Brouwer was somewhat like Nietzsche in his ability to step outside the established cultural tradition in order to subject its most hallowed presuppositions to cool and objective scrutiny; and his questioning of principles of thought led him to a Nietzschean revolution in the domain of logic.

He in fact rejected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle, handed down with very little change into modern times, and very recently extended and generalised out of all recognition with the aid of mathematical symbolism. Despite failing to convert mathematicians to his way of thinking, Brouwer received many honours for his outstanding contributions.

We mentioned his election to the Royal Dutch Academy of Sciences above. He was awarded honorary doctorates the University of Oslo in , and the University of Cambridge in He was made Knight in the Order of the Dutch Lion in

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## L. E. J. Brouwer

Korteweg — was professor of mathematics, mechanics and astronomy at the University of Amsterdam from —; the last five years as extraordinarius, so as to make place for Brouwer. Around , Korteweg was the most important and influential mathematician in the Netherlands, and he contributed much to the internationalisation of Dutch mathematics. A mathematical physicist trained by van der Waals, he had a particular interest in mechanics and thermodynamics. Among his many results, the best-known is probably the Korteweg-de Vries equation, describing the behaviour of waves in a shallow channel. Korteweg also had a strong historical interest, and was chief editor of vols. Part of the correspondence with his PhD student Brouwer concerned the unification of physical theories see van Stigt, , pp.

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## Luitzen Egbertus Jan Brouwer

Sein Vater war, wie auch einige Verwandte, Lehrer. Unter den studentischen Bekanntschaften Brouwers sticht der Dichter Carel Adema van Scheltema — hervor, mit dem Brouwer eine lebenslange Freundschaft verband. Brouwer selbst schrieb Gedichte und unterhielt stets literarische Interessen. Nach seiner Graduierung nahm er aufmerksam die seit kurzem propagierte Philosophie von G. Die Schrift Zur Analysis Situs bezog sich ganz auf die Entwicklungen der damaligen mengentheoretischen Topologie. Daneben entwickelte er die Methode der simplizialen Approximation.

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Biographies index L E J Brouwer is usually known by this form of his name with full initials, but he was known to his friends as Bertus, an abbreviation of the second of his three forenames. He attended high school in Hoorn, a town on the Zuiderzee north of Amsterdam. His performance there was outstanding and he completed his studies by the age of fourteen. He had not studied Greek or Latin at high school but both were required for entry into university, so Brouwer spent the next two years studying these topics. During this time his family moved to Haarlem, just west of Amsterdam, and it was in the Gymnasium there in that he sat the entrance examinations for the University of Amsterdam.