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A Mystic and Mathematical Revolutionary
L.E.J. Brouwer's Role in Topology and Intuitionism
To the uninformed observer it seems that the claim of the Low Countries to a place in the Olympus of mathematics is based on a few heroes of the past: Christian Huygens, Simon Stevin, Thomas Jan Stieltjes. In partial explanation, it may be remarked that mathematics, which at the beginning of the nineteenth century was still supposed to be part and parcel of a civilised man's education, has evolved at such an unnerving pace that recent mathematics is a hermetically closed book, even to those who claim to understand dna, quantum physics and white dwarfs. And so one of the most remarkable mathematicians of our century reposes in quiet oblivion. I am referring here to Luitzen Egbertus Jan Brouwer, a man who single-handedly organised revolutions in mathematics.
Brouwer was born in the small town of Overschie, which is now a part of Rotterdam, in 1881. His father was a schoolmaster (later headmaster), and according to the custom of the teaching profession he climbed the ladder of his profession by moving from job to job and from town to town. After some years the Brouwer family settled for good in Haarlem. There the boy Bertus finished his high school and gymnasium education. He was a prodigious student - always number one in his class, now and then skipping classes. At the age of 9 he had already entered high school - a feat which earned him a place in the Guinness Book of Records.
Having completed his gymnasium education in 1897 Brouwer enrolled in the faculty of Mathematics and Physics of the University of Amsterdam. Contrary to what one would expect of such a bright boy, he did not rush through the academic training. The most plausible explanation of his slow pace was his poor health. During the years of study he had many physical problems, and on top of that he was prone to nervous attacks. Correspondence between Brouwer and his friend C.S. Adama van Scheltema records Brouwer's ups and downs. It is a touching document that shows unexpected glimpses of the life and worries of a brilliant but high-strung boy.
In December 1900 Brouwer passed his first degree examination (the ‘candidaats examen’), and in 1904 he obtained his doctoral degree (comparable to the M.Sc.). The second part of his study was interrupted by military service and a number of illnesses, including nervous breakdowns. In spite of
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his slow progress, his brilliance was well recognised. Already before his final examination Brouwer had published a couple of research papers on rotations in a four-dimensional space, which greatly impressed his future Ph.D. adviser D.J. Korteweg.
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Against Aristotle
Between the final examination and the doctor's title Brouwer managed to finish a number of papers on force fields in non-Euclidean spaces. But his most striking achievement was the publication of a mystical treatise with the title Life, Art and Mysticism (Leven, Kunst, mystiek, 1905). The book was the result of a series of lectures that Brouwer gave at the Institute of Technology at Delft in 1904. Although Art, Life and Mysticism is an isolated item among Brouwer's many other publications, its significance can hardly be overrated. Indeed, Brouwer himself was a genuine mystic, who strove for the ultimate introspection - the ‘turning into oneself’. Everything else was of secondary importance, and a number of social, economic and scientific aspects of the world were denounced as sinful. For example, the domination of nature (an old Dutch custom) and of one's fellow creatures. A number of themes that reappear in Brouwer's later ground-breaking work occur here for the first time: the unreliability of language, the jump from goal to means, Brouwer's ideas on science and social organisation. The idea is that an individual observes that certain strings of sensations occur in a (more or less) specific order. So in order to realise the final sensation of a
L.E.J. Brouwer (1881-1966).
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series (the end) it might be a good idea to realise an intermediate sensation (the means) in the hope and expectation that it will more or less automatically lead to the end.
In spite of his other activities, Brouwer finished his dissertation within 3 years. The dissertation contains a wealth of ideas and results. One part deals with a nineteenth-century theory of geometric transformations: the theory of Lie groups. Here Brouwer gave a partial solution to problem no. 5 of the famous list of problems which the master mathematician Hilbert had presented to the world in 1900 at the international conference in Paris as a sort of homework for the twentieth century.
The two remaining chapters dealt with fundamental problems of mathematics, to wit the relation of mathematics with the world (i.e. its applications) and with basic notions. Brouwer, as a true mystic, claimed that mathematics is a constructive mental activity of the individual. Numbers, for example, are made by us, they do not exist out there, independent of us. Mathematics, he used to say, is rather an action than a theory. Postulating properties is therefore no good, one has to construct things and prove properties. This led him to criticise set theory and axiomatic mathematics, in particular the so-called formalist school. Shortly after the dissertation Brouwer published one of the most shocking attacks on traditional science: he denied the validity of Aristotle's principle of the excluded third - that something is true or false. Even in mathematics, he said, one cannot make a decision between a statement and its negation. E.g., in his famous Brouwerian counterexamples he used an unsolved problem, such as ‘in the decimal expansion of ¼ there is a sequence of ten digits “9”’. Let us call this statement a. Now, a person who claims that a or not-a is true, has according to Brouwer, the obligation to really produce this sequence 9999999999, or he has to show that such a sequence is impossible. And as long as this has not been accomplished, one may not assert ‘a or not-a’.
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Topology
Immediately after this epochal assault on traditional science, Brouwer moved on to a particular kind of geometry: the geometry in which one studies how properties behave under continuous deformation. Think of a rubber sheet that may be stretched, twisted, folded,...., but not torn or cut, for example the behaviour of a picture on a child's balloon when one pinches and twists it. In this area, called topology, he forced a number of the fundamental breakthroughs that mathematics had been waiting for. The best known result is Brouwer's fixed point theorem. Let us make a simplified example: take a circular disk and map it continuously into itself, then at least one point remains in its place. To put it more picturesquely, consider a rubber circular sheet, stretch it, twist it fold it, crumple it - as long as you don't cut or tear it. Now put it in this form in its old place, then at least one point of the sheet will be in its old place. Or in another form: take a cup of coffee, stir it arbitrarily but slowly so that no drops fly around. When the coffee is at rest again, at least one coffee particle will be at its old place. Brouwer proved his ‘fixed point theorem’ (1911) quite generally for arbitrary dimensions. The fixed point theorem (and its later generalisations) became very
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important in many areas, for example economics or game theory.
Another spectacular result was the hairy-ball theorem: if you try to comb a hairy ball, there will always be a singularity (i.e. a crown). Brouwer's new methods allowed him to prove a famous open problem which goes back to the father of set theory, George Cantor (1878): dimension is invariant under topological transformations. That is to say, if one takes a n-dimensional object, say a basic cube, and deforms it continuously and reversibly then the dimension remains the same.
His work in topology brought Brouwer almost instant recognition; the unknown Dutchman became overnight the leading expert in topology. The striking feature of Brouwer's work was his perfect geometric intuition. The new discipline of topology so heavily taxed the imagination that even great men stumbled into its treacherous pitfalls. One particular geometric exploit brought Brouwer's name to everybody's lips: by means of a clever approximation construction he had split a circular disk into three domains with one common boundary curve. His predecessors had never thought that the notion of boundary could be so pathological!
Brouwer's genius was soon recognised by his colleagues; it took a bit longer before the authorities followed, but in 1912 he was made a professor in Amsterdam. Foreign recognition came in 1915 when he was asked to join the editorial board of the Mathematische Annalen, the most prestigious mathematics journal of the period. In 1919 Brouwer was offered a chair at the Universities of Göttingen and of Berlin. Göttingen in the reign of Felix Klein and David Hilbert was considered to be the Mecca of mathematics, so Brouwer definitely had reached the top of the profession! Nonetheless he remained in Amsterdam, where he obtained some concessions from the board of the University. It was from Amsterdam that he launched the intuitionist revolution in mathematics. During the war years he had returned to his first love: the foundations of mathematics and philosophy.
As a side activity, Brouwer took part in philosophical activities of a mainly socio-linguistic nature, part of the enterprise known under the name Signific Circle. Significs was concerned with ‘meaning’ in its widest sense, including social and psychological aspects; it was introduced in the Netherlands by the author-psychiatrist Frederik van Eeden. Eventually Brouwer's teacher and colleague Gerrit Mannoury took it upon himself to further the study of significs.
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Grundlagenstreit
Where significs was considered a harmless occupation for philosophers, intuitionism created havoc in mathematical circles. Intuitionism had a negative and a positive aspect. The negative aspect became best known: Brouwer criticised the role of language and of logic. He held that no linguistic or logical technique could replace the need for actual effective construction or proof. His rejection of the principle of the excluded third and of the proof by reductio ad absurdum significantly reduced the body of mathematical results. On the other hand there was the positive aspect: the recognition of indeterministic procedures, which culminated in choice sequences, i.e. sequences of objects chosen more or less arbitrarily. This introduction of sub- | |
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jectivity into mathematics was the source of many heated discussions.
The negative aspect in particular concerned the so-called formalist school under David Hilbert. Hilbert, when confronted with the paradoxes at the turn of the century, had devised a method to safeguard mathematics: formalisation (the younger brother of axiomatisation). The method asked for a rigorous formalisation of mathematics and its principles, so that this formalisation could itself be handled mathematically. In this way he expected to prove that in the formal system one could never prove a statement and its negation; this is called the consistency of the system.
Brouwer denounced the method as just dealing with the language of mathematics and not with mathematics itself. Moreover, one had to trust mathematics first before a non-contradiction proof could be accepted. So what would be achieved, after all? Hermann Weyl, Hilbert's favourite student, joined Brouwer in his revolution, much to the chagrin of Hilbert. The 1920s saw a violent struggle between Brouwer and Hilbert; curiously enough Brouwer - an emotional and spontaneous person - fought the battle with an almost superhuman scholarly objectivity, whereas Hilbert - the authoritarian Prussian professor - strewed invectives around in his paper.
The case of intuitionism looked hopeful for some time. Brouwer addressed large audiences, the best known occasion being the Vienna Lectures in March 1928. The Viennese, including Ludwig Wittgenstein, flocked to the lecture hall and wondered at Brouwer's curious philosophy. It is a well-established fact that Brouwer's lectures induced Wittgenstein to return to philosophy. Wittgenstein's language-oriented philosophy got a vigorous impetus from the doctrines of Brouwer on language and mathematics.
Eventually this so-called ‘Grundlagenstreit’ was settled by brute force. Hilbert kicked Brouwer out of the editorial board of the Mathematische Annalen, an unprecedented insult. Even the struggle that followed, aptly called ‘The War of the Frogs and the Mice’ by Einstein, could not change Brouwer's fate. In the end loyalty to the revered master, Hilbert, prevailed over justice and common sense. In a sense the intuitionist-formalist conflict was complicated by extra-mathematical influences. Brouwer had since the Great War fought the injustice of the boycott of German scientists by the new international organisations. His strong sense of justice, combined with his general regard for the underdog, made him a champion of the German case. This led to clashes with Hilbert, who advocated a quiet return to recognition by the former Entente powers. One particular clash occurred on the occasion of the International Mathematics Conference in 1928 in Bologna, where the Germans were admitted to the conference as observers. Brouwer thought this an insult which no German mathematician should swallow. So he joined in with a German boycott of the conference. Hilbert, on the other hand, urged his colleagues to accept the invitation, and to go to Bologna. In the end Hilbert won, but he did not forgive Brouwer's opposition.
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Blaricum
After the affair of the Mathematische Annalen, Brouwer felt so discouraged and insulted that he withdrew from mathematics for more than to years. Only after the Second World War did Brouwer return to the pages of mathematics journals.
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Even though Brouwer had suffered badly under the onslaught of Hilbert's opposition, the mathematical world had not forgotten him; the University of Oslo granted him an honorary doctorate on the occasion of the centenary of the death of the famous Norwegian mathematician Abel.
Because of his treatment after the war, Brouwer tended to feel that he had been victimised by jealous colleagues. He carried out his duties and travelled abroad where he felt better appreciated, remaining aloof from his Dutch colleagues. The appreciation of his foreign colleagues was demonstrated by his appointment as a foreign member of the Royal Society in London in 1948.
From 1946 he taught for several terms a course on intuitionism at the University of Cambridge. That same University granted him an honorary doctorate in 1953.
In the summer of 1953 Brouwer gave a number of talks at the Canadian Summer Conference of Mathematics, and in the following autumn he made a lecture tour through the United States. On his tour he met a number of old friends and colleagues from Germany who had emigrated to the States in the thirties; in Princeton he visited Kurt Gödel, who had attended his Vienna Lecture in 1928, and Albert Einstein, with whom Brouwer had personal relations in the twenties.
Throughout his professional life he lived in the small town of Blaricum, not far from Amsterdam. The atmosphere of artists and social reformers suited him. In his own home he lived a simple clean life, sticking to vegetarian meals and practising health exercises, open air baths, etc. His wife, who ran a pharmacy in Amsterdam, was a specialist in herb lore; she shared Brouwer's food habits, and administered potions to her husband when the occasion called for it. His house (with its surrounding huts) was a favourite meeting place for his friends and colleagues. It was here that in December 1966 he was knocked down by a car when crossing the lane in front of his house to take a St Nicholas present to friends across the street. He died instantly.
Brouwer's heritage is nowadays well-recognised. In a number of areas in mathematics and computer science, his methods and ideas are accepted as exactly right. Topology has undergone a metamorphosis after Brouwer left the area, strong and efficient algebraic tools have replaced the ingenious geometric arguments and constructions of Brouwer. Nonetheless, his influence is generally acknowledged.
dirk van dalen
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Further reading
| brouwer, l.e.j., Collected Works I, II. Amsterdam, 1975 / 1976. |
| dalen, d. van, ‘The War of the Frogs and the Mice’, Mathematical Intelligencer, 1990, pp. 17-31. |
| dalen, d. van, L.E.J. Brouwer. A Biography. Oxford, forthcoming. |
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