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2013 | Buch

Child and Student Kapitel lesen Erstes Kapitel lesen

L.E.J. Brouwer – Topologist, Intuitionist, Philosopher

How Mathematics Is Rooted in Life

verfasst von: Dirk van Dalen

Verlag: Springer London

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Dirk van Dalen’s biography studies the fascinating life of the famous Dutch mathematician and philosopher Luitzen Egbertus Jan Brouwer. Brouwer belonged to a special class of genius; complex and often controversial and gifted with a deep intuition, he had an unparalleled access to the secrets and intricacies of mathematics. Most mathematicians remember L.E.J. Brouwer from his scientific breakthroughs in the young subject of topology and for the famous Brouwer fixed point theorem. Brouwer’s main interest, however, was in the foundation of mathematics which led him to introduce, and then consolidate, constructive methods under the name ‘intuitionism’. This made him one of the main protagonists in the ‘foundation crisis’ of mathematics. As a confirmed internationalist, he also got entangled in the interbellum struggle for the ending of the boycott of German and Austrian scientists. This time during the twentieth century was turbulent; nationalist resentment and friction between formalism and intuitionism led to the Mathematische Annalen conflict ('The war of the frogs and the mice'). It was here that Brouwer played a pivotal role. The present biography is an updated revision of the earlier two volume biography in one single book. It appeals to mathematicians and anybody interested in the history of mathematics in the first half of the twentieth century.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Child and Student
Abstract
L.E.J. (Bertus) Brouwer was born in 1881 in Overschie, now a part of Rotterdam. His father, a headmaster, moved to the town Medemblik in North Holland and subsequently to Haarlem. Bertus was mostly taught at home (his mother was a teacher too), and entered high school at Hoorn at the age of 9. After moving to Haarlem, he exchanged after two years the high school for the gymnasium (the Latin school). In 1897 he enrolled in mathematics, physics and astronomy at the Amsterdam University. The most prominent of his teachers was Korteweg, who had provided the mathematics for the theories of Van der Waals. At the student corporation Brouwer met Adama van Scheltema, later the leading socialist poet, who became his life long friend. Already as a gymnasium student Brouwer moved in artistic circles, a habit he kept up in later life.
Dirk van Dalen
Chapter 2. Mathematics and Mysticism
Abstract
Brouwer showed during his study his potential in mathematics by proving a significant decomposition theorem for four-dimensional rotations. During his student years Brouwer suffered from nervous breakdowns and minor physical problems. The cause was his military service, which brought him in a company with little patience for a young clever boy. The two basic exams at the university were passed cum laude. After the last one (1904) Korteweg accepted him as a Ph.D. student. 1904 brought more changes: he married the daughter, Lize, of the widow of a family doctor, bought a cottage, called the hut, in the countryside not far from Amsterdam. The architect was another friend from his student years, Ru Mauve (son of the famous painter). The hut allowed him to work in peace on his ambitious dissertation. In 1905 Brouwer lectured in Delft on “Life, Art, and Mysticism”, a provocative view of a true mystic on life, society, language, …, which contained some elements of his philosophical views that came to underlie his intuitionistic mathematics.
Dirk van Dalen
Chapter 3. The Dissertation
Abstract
Brouwer’s Thesis is a somewhat ambiguous book, it contains a purely mathematical part, dealing with ‘Hilbert 5’, i.e. the elimination of differentiability conditions in the theory of Lie groups, and a number of geometrical investigations. But the larger part was the presentation of a personal approach to the foundations of mathematics together with well-argued criticism of contemporary schools. The chapter makes extensive use of archive material, that allows us to follow how Brouwer’s ideas evolved. It contains the fundamental material on Brouwer’s ur-intuition, the genesis of the natural numbers and the continuum. Furthermore Brouwer’s views and first steps in intuitionistic logic are discussed. The dissertation and the archive material shows that Brouwer’s philosophical principles went beyond just mathematics.
Dirk van Dalen
Chapter 4. Cantor–Schoenflies Topology
Abstract
When Brouwer continued his investigations into Hilbert 5, he discovered that his main topology source, Schoenflies Bericht, was far from correct. He set himself to straighten out the defective parts; the best known fall out of this research was his work on indecomposable continua, with the spectacular example of three domains with one common boundary. The chapter also contains the story of Brouwer’s research on fixed points on the sphere and his translation theorem (on fixed point free continuous maps of the plane onto itself). He simultaneously produced a number of papers on vector field on surfaces. The best known result was the hairy ball theorem: a continuous vector field on a sphere must be zero or infinite at at least one point.
Dirk van Dalen
Chapter 5. The New Topology
Abstract
The next step in Brouwer’s topological research was the study of continuous maps on manifolds. The program opened with a bang: in a brief note Brouwer proved the invariance of dimension under homeorphisms. This publication led to an unpleasant altercation with Lebesgue, who claimed to have already found a proof. In fact he had deduced the invariance from the paving principle, but failed to prove the paving principle. In the end Brouwer’s priority and superior insight was fully vindicated. In subsequent papers Brouwer enriched the arsenal of basic notions of topology with simplicial approximation and the mapping degree. The contacts with Baire, Hadamard, Blumenthal, and Hilbert, are described. Brouwer’s name became lastingly attached to his fixed point theorem. Brouwer also proved the invariance of domain theorem, which he subsequently used to salvage Klein’s continuity method for proving uniformisation. This brought him into a conflict with Paul Koebe, who was the uncrowned king of uniformisation and complex function theory. This first topological period closed with a significant feat: Brouwer defined, following Poincaré’s first approach, the general notion of dimension, and proved its ‘correctness’, i.e. showed that ℝ n is n-dimensional.
Dirk van Dalen
Chapter 6. Making a Career
Abstract
After his successes in topology, it was about time for Brouwer to find a position. There was no doubt that here was an exceptionally bright mathematician, but that was not quite enough for the board of the Amsterdam university. Korteweg started a campaign to get him a position as a lecturer or an extraordinary professor. As a first step he managed to get Brouwer accepted as a member of the Netherlands Royal Academy. From there it was not so difficult to get him a university position. In 1909 he was accepted as a private teacher, in 1912 as an extraordinary professor, 1913 full professor (Korteweg was so generous to exchange his chair with Brouwer’s extraordinary one). Recognition was now coming his way—the Mathematische Annalen invited him to join the board of editors (an honor indeed!). In 1913 Brouwer offered his assistance to Schoenflies in the preparation of a new edition of his Bericht; this brought him into fruitless discussions with Schoenflies, who could not master the intricacies of modern topology.
Dirk van Dalen
Chapter 7. The War Years
Abstract
Cut off from his international contacts, in particular Göttingen, Brouwer returned to his foundational research. He lectured on The theory of point sets, a course on constructive set theory. In the 1916/17 course he introduced choice sequences and the continuity principle. Now that he saw how to exploit the apparent weakness of his intuitionistic mathematics, he decided to build his new intuitionism in a systematic way. There were some signs during the war years that something was brewing, but the first papers appeared after the war. Part of his efforts were directed at a project called Significs, a study of language and meaning following Lady Welby and Frederik van Eeden, the author and first psychiatrist in Holland. A considerable effort was made to create a group of people with a common interest in the subject, including an International Academy, etc. In the later years of the war Brouwer proposed, together with colleagues, to found a section of the Dutch Airforce (which was hardly existing at the time) for the study and application of air reconnaissance (photogrammetry). Finally, from now on Brouwer was spending more time on university/faculty/Academy matters.
Dirk van Dalen
Chapter 8. Mathematics After the War
Abstract
The post war years brought Brouwer back to topology and to intuitionism. Mostly ‘unfinished business’; mathematicians were catching up with Brouwer’s innovations, hence an exchange of ideas and problems. As the topology editor for the Mathematische Annalen Brouwer also got more papers to handle (e.g. Nielsen and Kerékjártó).
Most of Brouwer’s efforts, however, went into his intuitionism; from 1918 on he published substantial papers to put the subject on a firm footing. The first paper in the series introduced choice sequences and a constructive, but not finitistic, notion of set, now known as spread; furthermore the continuity principle—which was immediately applied to prove that the set of all number theoretic functions is not denumerable.
Brouwer got offers from Göttingen and Berlin, he remained however in Amsterdam on favorable conditions. One of those was that he could offer a position to Hermann Weyl, who in turn used the offer to improve his conditions in Zürich. The first international conference Brouwer attended after the war was the one in Nauheim, where he gave his first talk on intuitionistic mathematics, ‘Does every real number have a decimal expansion?’.
Dirk van Dalen
Chapter 9. Politics and Mathematics
Abstract
The main theme of this chapter is the boycott of Germany after World War I. The newly founded Conseil internationale de recherches excluded scientists from the Ax’s countries and barred them from conferences, and even discouraged contacts between their subjects and German/Austrian scientists. Brouwer took the internationalist position and tried to obstruct the policies of the Conseil wherever possible. This brought him into numerous conflicts, including one with the Utrecht professor A. Denjoy.
Dirk van Dalen
Chapter 10. The Breakthrough
Abstract
The meetings of various scholars in the so-called Signific Circle continued on a fairly informal level. It was mostly a small group of followers of Van Eeden, that met every fortnight in the pharmacy of Brouwer’s wife, in the colony of Van Eeden, or in Brouwer’s cottage or garden.
Brouwer’s time and attention were mostly devoted to getting his intuitionism going. This was a success indeed; Brouwer managed to establish the ‘shocking’ facts that surprised, and partly alienated, the mathematical community. On the basis of his continuity theorem and an suitable form of transfinite induction the continuity theorem; every real function is (locally uniformly) continuous. This established beyond any doubt that intuitionism was not a poor subsystem of classical mathematics, but a system with its own strong principles and results.
Fraenkel was one the first to appreciate Brouwer’s enterprise, he gave intuitionism ample space in his books on set theory.
Brouwer’s Ph.D. student Heyting had started to join Brouwer in his project. He started to give presentations of intuitionism that could be grasped by the general mathematician, which did change the position of intuitionism as a mathematical doctrine.
Dirk van Dalen
Chapter 11. The Fathers of Dimension
Abstract
In this chapter we find a complete return to topology, to be precise, to dimension theory. In 1923 Brouwer attended the Marburg meeting of the German Math Society, at that same meeting there was a young Russian topologist, Urysohn, who had given a definition of dimension unaware of the fact that Brouwer had already done so. When his attention was called to this fact he checked Brouwer’s 1913 paper and found a mistake. He informed Brouwer and the latter immediately set out to check his old paper. Indeed there was a erroneous detail in the definition. The two corresponded about the matter and eventually agreed that Brouwer’s mistake was a slip of the pen (born out by solid evidence). Almost at the same time Menger had studied the topic and given his own definition of dimension. In 1924 Urysohn and Alexandrov visited Germany, went on to see Brouwer and then went on to France. Urysohn died when swimming in rough weather at the coast of Brittany. Alexandrov and Brouwer were inconsolable, they decided to edit Urysohn’s scientific estate.
Dirk van Dalen
Chapter 12. Progress, Recognition, and Frictions
Abstract
Brouwer was extremely critical of Hilbert’s formalism, which was completely opposed to the intuitionistic philosophy of mathematics. Brouwer trod softly in his publications in order to avoid conflict. However, when Hermann Weyl joined Brouwer’s intuitionism, his ‘New crisis-paper’ the tone changed. Hilbert told Brouwer and Weyl in unmistakable terms how wrong they were. The Grundlagenstreit was born. The conflict dragged on for years.
From time to time the scientific discussion was replaced by external causes. One such was the planned publication of the Riemann memorial volume by the Mathematische Annalen. A conflict arose when the participation of French mathematicians could in this project of their arch enemy was questioned. One section of the editorial board objected to the French participation, which was advocated by Hilbert. The conflict left bad feelings.
Inspired by the contributions of Alexandrov, Menger, and Urysohn, Brouwer collected in 1925 a small group of topologists in Amsterdam, including Vietoris and Wilson. The brief concentration of topologists was later called the Dutch topological school.
Dirk van Dalen
Chapter 13. From Berlin to Vienna
Abstract
The two main topics here are Brouwer’s Berlin lectures and the Vienna lectures. The Berlin lectures were held in 1927 on invitation of the math faculty. Reports tell us that Brouwer made furor in Berlin, students form all over the country flocked to Berlin to hear the mysterious great man. Students called themselves putschists, in reaction to a reaction of Hilbert to Weyl’s battle cry ‘And Brouwer, that is the revolution’. The future for intuitionism looked rosy for a while.
A year later Brouwer lectured in Vienna, this was essentially the first public exposition of his philosophy since his dissertation. Wittgenstein attended the lecture, which influenced his return to philosophy.
Dirk van Dalen
Chapter 14. The Three Battles
Abstract
Brouwer had to defend himself on three fronts: the Formalism-Intuitionism conflict, the War of the frogs and the mice, The Menger dimension conflict. The War of the frogs and the mice is extensively discussed, since it brought in fact the end of the foundational conflict. The Menger conflict deals with the priority for the dimension definition.
Dirk van Dalen
Chapter 15. The Thirties
Abstract
After the War of the Frogs and the Mice Brouwer more or less retired from the scene. Freudenthal, Hopf’s student, was appointed as his assistant in Amsterdam. Brouwer fought an investment scandal involving a health spa in Budapest, and founded a new mathematics journal, Compositio Mathematica. Heyting introduced his formal system for intuitionistic logic and arithmetic. The foundational atmosphere clearly was improving. The rise of the nazi regime is discussed, including an attempt of the new authorities to lure Brouwer to Göttingen; which predictably failed. At the end of the thirties Brouwer briefly returned to topology with a proof of the triangulation property for differentiable manifolds, only to find out that an American mathematician, Cairns, had already solved that case.
Dirk van Dalen
Chapter 16. War and Occupation
Abstract
The war brought hardship to all of Holland. The universities had their fair share of the consequences of the occupation and the new regime. Jews were dismissed, the adventures of Freudenthal illustrate the fate of a man who was fortunate enough to survive the occupation. Brouwer, as a professor had to play a role in policy making of the university; unlike the world wise, who practiced a chameleon tactics, Brouwer did not hesitate to speak out on touchy topics. This, in the end, did not make friends. The art of survival in wartime under a hostile regime is illustrated by the fortunes of the Brouwer family and their friends. The famous hut was destroyed by a fire in which a large part of his archive was lost.
Dirk van Dalen
Chapter 17. Postwar Events
Abstract
After the liberation the process of normalization set in, including the purification of all institutions. Here Brouwer was punished for his independent position in wartime; he was in fact suspended for a couple of months, which he considered a gross insult. Moreover, his colleagues used the occasion to take over the power in the faculty. Although Brouwer fiercely opposed the new policies of the faculty and the department, his influence was limited. He was even sidetracked in the running of his own journal, Compositio; he used to speak of “the theft of my journal”. The days of his domination were over. A most welcome balm for his wounded feelings was a standing invitation to lecture on intuitionism in Cambridge. He also returned to his research. This time he published a number of papers on extensions of his previous intuitionistic œvre, and a few substantial expositions of his philosophical and foundational views. A planned monograph based on his Cambridge lectures appeared only posthumously.
Dirk van Dalen
Chapter 18. The Restless Emeritus
Abstract
After his retirement, Heyting became the resident intuitionist. Brouwer himself resumed his habit of traveling. He visited and lectured abroad, e.g. in Finland, the US, and South Africa. In particular the grand tour of Canada and the US gave him much pleasure; he met many old friends who had emigrated, including Einstein, Gutkind, Gödel. At home he was involved in the management of his wife’s pharmacy, a complicated affair with relocating the shop, reconstructing the building, etc. His wife died in 1959 and he died when crossing the road in front of his house, hit by a couple of cars.
Dirk van Dalen
Backmatter
Metadaten
Titel
L.E.J. Brouwer – Topologist, Intuitionist, Philosopher
verfasst von
Dirk van Dalen
Copyright-Jahr
2013
Verlag
Springer London
Electronic ISBN
978-1-4471-4616-2
Print ISBN
978-1-4471-4615-5
DOI
https://doi.org/10.1007/978-1-4471-4616-2

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