**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 1897 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 1904. 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.

He obtained his master's degree in 1904 and in the same year married Lize de Holl who was eleven years older that Brouwer and had a daughter from a previous marriage. 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. He published his own ideas on this topic in 1905 in his treatise *Leven, Kunst, en Mystiek* Ⓣ. In this work he [1]:-

... considers as one of the important moving principles in human activity the transition from goal to means, which after some repetitions may result in activities opposed to the original goal.

Brouwer's doctoral dissertation, published in 1907, made a major contribution to the ongoing debate between Russell and Poincaré on the logical foundations of mathematics. His doctoral thesis [13]:-

... revealed the twin interests in mathematics that dominated his entire career; his fundamental concern with critically assessing the foundations of mathematics, which led to his creation of intuitionism, and his deep interest in geometry, which led to his seminal work in topology...

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 was fiercely independent and did not follow in anybody's footsteps, but he apparently took his teacher's advice...

Brouwer continued to develop the ideas of his thesis in *The Unreliability of the Logical Principles* published in 1908.

The research which Brouwer now undertook was in two areas. He continued his study of the logical foundations of mathematics and he also put a very large effort into studying various problems which he attacked because they appeared on Hilbert's list of problems proposed at the Paris International Congress of Mathematicians in 1900. In particular Brouwer attacked Hilbert's fifth problem concerning the theory of continuous groups. He addressed the International Congress of Mathematicians in Rome in 1908 on the topological foundations of Lie groups. However, after studying Schönflies' report on set theory, he wrote to Hilbert:-

I discovered all of a sudden that the Schoenfliesian investigations concerning topology of the plane, on which I had relied in the fullest way, could not be taken as correct in all parts, so that my group-theoretic results also became doubt.

In 1909 he was appointed as an privatdocent at the University of Amsterdam. He gave his inaugural lecture on 12 October 1909 on 'The nature of geometry' in which he outlined his research programme. A couple of months later he made an important visit to Paris, around Christmas 1909, and there met Poincaré, Hadamard and Borel. 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 1912 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 1951. Hilbert wrote a warm letter of recommendation which helped Brouwer to gain his chair in 1912. 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.

Although he had helped Brouwer to obtain his chair in Amsterdam, in 1919 Hilbert tried to tempt him away with an offer of a chair in Göttingen. 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 1919 to 1923, 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. Before the next week's lesson, his assistant came to me to say that Brouwer did not want questions put to him in class. 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. Controversy surrounded Cantor's general set theory because of the set-theoretic paradoxes or contradictions. 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 1909 and 1913. He discovered characterisations of topological mappings of the Cartesian plane and a number of fixed point theorems. His first fixed point theorem, which showed that an orientation preserving continuous one-one mapping of the sphere to itself always fixes at least one point, came out of his researches on Hilbert's fifth problem. 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 1913.

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 1907 attacked the logical foundations of mathematics and marks the beginning of the Intuitionist School. His views had more in common with those of Poincaré and if one asks which side of the debate between Russell and Poincaré he came down on then it would have with the latter.

In his 1908 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. In 1918 he published a set theory developed without using the Principle of the Excluded Middle *Founding Set Theory Independently of the Principle of the Excluded Middle. Part One, General Set Theory*. His 1920 lecture *Does Every Real Number Have a Decimal Expansion?* was published in the following year. The answer to the question of the title which Brouwer gives is "no". Also in 1920 he published *Intuitionistic Set Theory*, then in 1927 he developed a theory of functions *On the Domains of Definition of Functions* without the use of the Principle of the Excluded Middle.

His constructive theories were not easy to set up since the notion of a set could not be taken as a basic concept but had to be built up using more basic notions which, in Brouwer's case, were choice sequences. 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.

The later part of Brouwer's career contains some controversial episodes. He had been appointed to the editorial board of *Mathematische Annalen* in 1914 but in 1928 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 1943). He tried to remove Brouwer from the board in a way which was not compatible with the way the board was set up. Brouwer vigorously opposed the move and he was strongly supported by other board members such as Einstein and Carathéodory. 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 1935 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 1941. He was also active setting up a new journal and he became a founding editor of *Compositio Mathematica* which began publication in 1934.

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 1943 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.

After retiring in 1951, Brouwer lectured in South Africa in 1952, and the United States and Canada in 1953. His wife died in 1959 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 1962, despite being well into his 80s, he was offered a post in Montana. He died in 1966 in Blaricum as the result of a traffic accident.

Kneebone writes in [3] about Brouwer's contributions to the philosophy of mathematics:-

Brouwer is most famous ... for his contribution to the philosophy of mathematics and his attempt to build up mathematics anew on an Intuitionist foundation, in order to meet his own searching criticism of hitherto unquestioned assumptions. 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.

Kneebone also writes in [3] about the influence that Brouwer's views on the foundations of mathematics had on his fellow mathematicians:-

Brouwer's projected reconstruction of the whole edifice of mathematics remained a dream, but his ideal of constructivism is now woven into our whole fabric of mathematical thought, and it has inspired, as it still continues to inspire, a wide variety of inquiries in the constructivist spirit which have led to major advances in mathematical knowledge.

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. Other honours included election to the Royal Society of London, the Berlin Academy of Sciences, and the Göttingen Academy of Sciences. He was awarded honorary doctorates the University of Oslo in 1929, and the University of Cambridge in 1954. He was made Knight in the Order of the Dutch Lion in 1932.

**Article by:** *J J O'Connor* and *E F Robertson*

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