# Bartel Leendert van der Waerden

### Quick Info

Born
2 February 1903
Amsterdam, Netherlands
Died
12 January 1996
Zürich, Switzerland

Summary
Van der Waerden was a Dutch mathematician best known for his "Modern Algebra" text-book. He also worked on topology and on the history of mathematics.

### Biography

Bartel van der Waerden's parents were Theodorus van der Waerden and Dorothea Adriana Endt. Theo's father, Bartel's paternal grandfather, was Hendricus Johannes van der Waerden who owned a large blacksmith business. Theo was born in Eindhoven on 21 August 1876 and studied civil engineering at the Delft Technical University. Then, after teaching mathematics and mechanics in Leeuwarden and Dordrecht, he moved to Amsterdam in 1902 where again he taught mathematics and mechanics. At university he had become interested in politics and played a role in politics throughout his life as a left wing Socialist. He married Dorothea on 28 August 1901. Her parents were Coenraad Endt and Maria Anna Kleij who were Dutch Protestants. Bartel was the eldest of their three children, the other two boys being Coenraad (born 29 December 1904) and Benno (born 2 October 1909). In 1911 Theo was awarded the degree of Doctor of Technical Sciences, and was elected as a member of the SDAP (Sociaal-Democratische Arbeiderspartij) to the Provincial government of North Holland in 1910.

As a child, van der Waerden was not allowed to read his father's mathematics books but was told to play outside. This made him fascinated to discover mathematics for himself. After elementary school, van der Waerden entered the Hogere Burger School of Amsterdam in 1914. As a school pupil at the Hogere Burger School, van der Waerden showed remarkable promise and he developed for himself the laws of trigonometry. He studied mathematics at the University of Amsterdam, beginning his course in 1919 at the age of sixteen. He learnt topology from Gerrit Mannoury who was a friend of his father (Mannoury was a Communist and Theo, although not a Communist himself, had many friends in that party). He also learnt invariant theory from Roland Weitzenböck. Dirk van Dalen [3] paints a picture of van der Waerden as an undergraduate:-
The study of mathematics was for him the proverbial 'piece of cake'. Reminiscing about his studies he said: "I heard Brouwer's lectures, together with Max Euwe and Lucas Smidt. The three of us listened to the lectures, which were very difficult." ... Van der Waerden meticulously took notes in class, and usually that was enough to master all the material. Brouwer's class was an exception. Van der Waerden recalled that at night he actually had to think over the material for half an hour and then he had in the end understood it.

Van der Waerden was an extremely bright student, and he was well aware of this fact. He made his presence in class known through bright and sometimes irreverent remarks. Being quick and sharp (much more so than most of his professors) he could make life miserable for the poor teachers in front of the blackboard. During the, rather mediocre, lectures of Van der Waals Jr he could suddenly, with his characteristic stutter, call out: "Professor, what kind of nonsense are you writing down now?" He did not pull such tricks during Brouwer's lectures, but he was one of the few who dared to ask questions.
Brouwer however, did not like to be questioned and his assistant spoke to van der Waerden asking him to ask no further questions during lectures. After taking his first degree in Amsterdam he went to Göttingen for seven months to study under Emmy Noether. Brouwer wrote to Hellmuth Kneser at Göttingen on 21 October 1924 before van der Waerden went there (see [3]):-
In some days my student (or actually Weitzenböck's) will come to Göttingen for the winter semester. His name is Van der Waerden, he is very clever and has already published (namely in Invariant Theory).
At Göttingen, van der Waerden learnt much topology from Hellmuth Kneser. He said [7]:-
... from the beginning I was in contact with him, and from him I really learned topology. Kneser and I used to have lunch together; after having eaten he went home, but on occasion we first took a brief walk. We strolled through the woods of Göttingen, and he taught me many things.
Dirk van Dalen writes [3]:-
Once in Göttingen under Emmy's wings, Van der Waerden became a leading algebraist. Emmy was very pleased with the young Dutchman, "That Van der Waerden would give us much pleasure was correctly foreseen by you. The paper he submitted in August to the Annalen is most excellent (Zeros of polynomial ideals) ... ", she wrote to Brouwer on 14 November 1925.
Van der Waerden returned to the Netherlands in 1925 where he both wrote his doctoral dissertation, supervised by Hendrik de Vries, and undertook military duty at the marine base in Den Helder. Dirk van Dalen writes [3]:-
In mathematics Van der Waerden was easily recognised as an outstanding scholar, but in the 'real world' he apparently did not make such a strong impression. When Van der Waerden spent his period of military service at the naval base in Den Helder, a town at the northern tip of North-Holland, his Ph.D. advisor [Hendrik de Vries] visited him one day. He said that the commander was not impressed by the young man, "he is a nice guy but not very bright."
His doctoral thesis De algebraiese grondslagen der meetkunde van het aantal was submitted to the University of Amsterdam and he defended it in the grand hall of the University on 24 March 1926. He had been awarded a Rockefeller fellowship for a year and, following the semester in Göttingen with Emmy Noether, he went to Hamburg to study for a semester with Hecke, Artin and Schreier. There he attended Artin's algebra course and took notes with the aim of writing a joint book with him. However, when later Artin saw the part of the text van der Waerden was writing, he suggested that he write the whole book without any chapters being contributed by Artin. This eventually became van der Waerden's famous text Moderne Algebra . The year 1927 was a busy one for van der Waerden. He was offered a position at the University of Rostock but was appointed to a lectureship at Groningen in the same year. He returned to Göttingen as a visiting professor in 1929 and in July of that year he met Camilla Rellich, sister of the Franz Rellich who was completing his doctoral thesis under Richard Courant. Van der Waerden married Camilla in September 1929 and the two returned to Groningen. There he continued working on Moderne Algebra which contained much material from Emmy Noether's lectures as well as those of Artin. Volume I was published in 1930 while volume II, which contains much of van der Waerden's own work, was published in the following year.

In 1931 he was appointed professor of mathematics at the University of Leipzig where he became a colleague of Werner Heisenberg. His interaction with Heisenberg and other theoretical physicists led to van der Waerden publishing Die gruppentheoretische Methode in der Quantenmechanik in 1932. He then began to publish a series of articles in Mathematische Annalen on algebraic geometry. In these articles, van der Waerden defined precisely the notions of dimension of an algebraic variety, a concept intuitively defined before. His work in algebraic geometry uses the ideal theory in polynomial rings created by Artin, Hilbert and Emmy Noether. His work also makes considerable use of the algebraic theory of fields. However, he later changed his approach as is evident in his book Einführung in die Algebraische Geometrie (1939). Dan Pedoe writes in a review:-
About ten years ago, van der Waerden, already eminent as an algebraist, began, in a series of papers in the Mathematische Annalen, to create rigorous foundations for algebraic geometry. The implication-that there was some-thing unsound in the magnificent structure of Italian geometry-was vigorously contested by Severi. Fortunately, van der Waerden continued his researches, but with the implicit sub-title, "An algebraist looks at algebraic geometry". With increasing knowledge of the powerful methods of the Italian school, he has gladly modified his own methods. Ideal-theory, the weapon of attack in his first papers, he has found almost completely unnecessary ... As a result of the experience gained in writing these papers, and in giving various courses of lectures, Professor van der Waerden has produced a work which must sooner or later find a place on every geometer's bookshelves.
In 1934 van der Waerden joined the main editorial board of Mathematische Annalen. This was a difficult time to take on such a role since he came under pressure from the Nazis not to publish papers by Jewish authors. This pressure made him think about resigning, but when he was informed that if he did so it was likely that one of the Nazis Wilhelm Blaschke or Ludwig Bieberbach would replace him, he decided to continue. Both before and after the start of World War II, van der Waerden, as a foreigner, had problems from the Nazis. Although working in Germany he refused to give up his Dutch citizenship and his life was made difficult. He wrote on 16 May 1940 (see [1]):-
In itself I have nothing against German citizenship, however, at this moment, since Germany has occupied my homeland, I would not gladly give up my previous neutrality and throw myself in a certain measure publicly on the German side.
This letter was written one day after the German invasion of the Netherlands was complete. At this time his parents were still in the Netherlands. They had moved from Amsterdam to Laren near the end of the 1920s after their children had left home. Theo had built a fine home there but shortly after the German invasion, on 12 June 1940, he died of cancer. Dorothea continued to live there with her daughter but she was so distressed by the German occupation that she committed suicide on 14 November 1942, drowning herself in a lake near her home. On 4 December 1943 the van der Waerden home in Leipzig was bombed and Bartel and Camilla van der Waerden, together with their three children Helga, Ilse and Hans, left for Dresden where Camilla's brother Franz Rellich was professor of mathematics. Here the situation was equally bad so they accepted an invitation from one of van der Waerden's students to live with her in Bischofswerda, a small town near Dresden. They remained there for nearly a year before returning to Leipzig. The city was under continual air attack and in 1945, unable to take the strain any longer, they moved to Austria to live in the country at Tauplitz, near Graz, with Camilla's mother. In July 1945 American soldiers arrived in Tauplitz and told everyone to return to their country of origin. The van der Waerdens returned to the Netherlands and lived in the house Theo van der Waerden had built in Laren.

Van der Waerden now had no job and hardly any money to buy food for his family. He was offered a post at Utrecht University, arranged by Hans Freudenthal, but because he had worked throughout the war in Germany, the government refuse to allow him to take up the position. Freudenthal then managed to obtain a position for van der Waerden working for Shell in Amsterdam on applied mathematics. In 1947 he visited the United States, going to Johns Hopkins University where he was offered a permanent post. He refused the offer and returned in 1948 to a chair of mathematics at the University of Amsterdam where he remained until 1951. In 1950 Karl Fueter died and van der Waerden was appointed to fill the vacant chair in Zürich in 1951. His impact on the department in Zürich was very great. As well as an almost unbelievable range of mathematical research interests, van der Waerden stimulated research in Zürich by supervising over 40 doctoral students during his years there. In fact van der Waerden was to remain in Zürich for the rest of his life.

Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science. We have already mentioned Van der Waerden's most famous book, Moderne Algebra published in 1930-1931. In Galois theory he showed the asymptotic result that almost all integral algebraic equations have the full symmetric group as Galois group. He produced results in invariant theory, linear groups, Lie groups and generalised some of Emmy Noether's results on rings. In group theory he studied the Burnside groups $B(3, r)$ with $r$ generators and exponent 3. These are solutions of the Burnside problem. These groups were shown to be finite by Burnside. In 1933 van der Waerden found the exact order and structure of the groups $B(3, r)$. He showed that the order of $B(3, r)$ is $3^{N(r)}$ where the exponent
$N(r) = r + r(r - 1)/2 + r(r - 1)(r - 2)/6.$
Among his many historical books are Ontwakende wetenschap (1950) translated into English as Science Awakening (1954), Science Awakening II: The Birth of Astronomy (1974), Geometry and Algebra in Ancient Civilizations (1983), and A History of Algebra (1985). Dirk Struik, reviewing the first of these, writes:-
This is the first book which bases a full discussion of Greek mathematics on a solid discussion of pre-Greek mathematics. Carefully using the best sources available at present, the author acquaints the reader not only with the work of Neugebauer and Heath, but also with that of the philological critics who centered around the "Quellen und Studien." ... This book contains a wealth of material, critically arranged, and reads exceedingly well. It has an original approach and contains much novel material.
As to A History of Algebra, Jeremy Gray writes in a review:-
It is almost unfailingly clear. The arguments presented are summarized with a deftness that isolates and illuminates the main points, and as a result they are frequently exciting. Since nearly 200 pages of it are given over to modern developments which are only now receiving the attention of historians, this book should earn itself a place as an invaluable guide. Its second virtue is the zeal with which the author has attended to the current literature. Almost every section gives readers an indication of where they can go for a further discussion. As a result, many pieces of information are here presented in book form that might otherwise have languished in the scholarly journals. Since one must be cynical of the mathematicians' awareness of those journals, the breadth and generosity of van der Waerden's scholarship will do everyone a favour.
The history of mathematics was not a topic he just turned to late in life. He explained [7]:-
When I was a student, Hendrik de Vries gave a course on the history of mathematics. After that I read Euclid and some of Archimedes. Thus, my interest began very early. At Göttingen - the first time I was there - I attended the lectures of Neugebauer, who gave a course on Greek mathematics.
Van der Waerden's important paper Die Arithmetik der Pythagoreer appeared in 1947 followed by Die Astronomie der Pythagoreer in 1951.

In 1973 van der Waerden retired from his chair in Zürich. He continued to undertake research in the history of mathematics publishing around 60 papers after he retired. The papers which appeared in the years 1986-88 include: Francesco Severi and the foundations of algebraic geometry (1986), On Greek and Hindu trigonometry (1987), The heliocentric system in Greek, Persian and Hindu astronomy (1987), The astronomical system of the Persian tables (1988), On the Romaka-Siddhanta (1988), Reconstruction of a Greek table of chords (1988), and The motion of Venus in Greek, Egyptian and Indian texts (1988). Although several of his publications appeared after 1988, all were taken from lectures he had given earlier.

### References (show)

1. S L Segal, Mathematicians under the Nazis (Princeton University Press, Princeton, NJ, 2003).
2. A Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators (Springer, New York, 2008).
3. D van Dalen, Mystic, geometer, and intuitionist : The life of L E J Brouwer (The Clarendon Press, Oxford University Press, Oxford, 1999).
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