Remarlzable Mathematicians from Euler to Von Neumann

Remarlzable Mathematicians From Euler to von Neumann loan James Mathematical Institute, Oxford THE MATHEMATICAL ASSOCIATION OF AMERICA ·: .,. CAMBRIDGE ,:: UNIVERSITY PRESS THE PENNSYLVANIA STATE l.JN!VERS!TY COMMONWEALTH CAMPUS LIBRARIES From Cantor to Hilbert DAVID HILBERT (1862-1943) We now return to Germany. An inspiring teacher as well as a great re searcher, Hilbert was arguably the leading figure in the mathematical world in the years following the death of Poincare. Many of the mathematicians who came to Gottingen, as students, as faculty members or as visitors, fell under his spell. Through the reminiscences of some of them we can obtain a faint idea of Hilbert's exceptional charisma. He revolutionized several branches of mathematics and opened up new fields of research, while his stimulating writings have exerted a powerful influence throughout the mathematical world. The most important phase of Hilbert's career lies in the first quarter of the twentieth century. David Hilbert was born on January 23, 1862 in Wehlau, near Konigsberg, then the principal city of East Prussia. He was descended from a Protestant middle-class family which had settled near Freiberg in Saxony in the seventeenth century. On his father's side his more immediate forbears were professional men who practised in the capital and coronation city of the Prussian kings, which had developed a special cultural tradition. His David Hilbert (1862- 1943) 247 father Otto was a judgei his mother, Maria Theresa (nee Erdtmann), was interested in mathematics, and Hilbert is said to have inherited his gifts from her side of the family. The future mathematician was their only son, but he had a younger sister who died in childbirth at the age of twenty-eight. When Otto Hilbert became a more senior judge the family moved into Konigsberg itself, where his son was enrolled at the Royal Friedrichskolleg at the age of eight. This was a traditional grammar school where the curriculum emphasized the classicsi no science was taught, apart from a little mathematics. It was not the most suitable school in Konigsberg for someone like David Hilbert, and those who would later become close friends of his attended the more progressive Wilhelms-Gymnasium. He only began to display his true abilities when he completed his school career with a year at the Gymnasium, followed by three years as a student at the University of Konigsberg, apart from the customary two semesters elsewhere which he spent in Heidelberg. Another Konigsberg student at that time was the precocious Hermann Minkowski, already mentioned in the proB.le of Henry Smith. Minkowski was two years younger than Hilbert but being more advanced academically was in a position to act as his mentor. They formed a lasting friendship. An other early influence was Adolf Hurwitz from Gottingen, three years senior to Hilbert and already an associate professor at the university. In Hilbert's student years the leading mathematician was the able and versatile Heinrich Weber, Dedekind's collaborator on the memoir 'Algebraic functions of a single variable'. In 1883 Weber left and was replaced by Lindemann, celebrated for having proved the transcendence of n:. Lindemann was hardly in the first rank of mathematicians, but he had a wide range of interests, one of which was the theory of invariants. This was the subject on which Hilbert wrote his dissertation. Hilbert passed the doctoral examination at the end of 1884 and embarked upon the customary tour of other centres. He went first to visit Klein in Leipzig, thus beginning a long-lasting friendship, and then went on to Paris, where he met Hermite, Jordan, and Poincare amongst others. On the way back to Konigsberg he stopped in Berlin and saw Kronecker, who had just succeeded Kummer at the university. By 1886 Hilbert had qualified as a privatdozent at Konigsberg, and remained in this position for the next six years. In 1892 he was appointed associate professor to replace Hurwitz who had moved to Zurich. In the same year he married Kathe Jerosch, the independent-minded daughter of a local merchanti their only child Franz was born the following year. He suffered from mental illnessi from the age From Cantor to Hilbert of twenty-one he was often a patient in a psychiatric hospital, and never lived a normal life. Hilbert's primary research interest during this period was in invariant theory, and in 1893 he created a sensation with the paper he contributed to the Chicago Mathematics Congress describing the outcome of his work. First Cayley and Sylvester in England, and then Clebsch and Gordan in Germany, had been laboriously constructing tables of invariants. Their investigations led them to conjecture that the set of invariants always had a finite basis, but the algorithmic methods they used had only very limited success in proving this. By a tour de force of abstract algebra, but using ideas which went back to Dedekind, Hilbert proved the conjecture in a few pages. When he heard of this Gordan exclaimed, 'That is not mathematics. That is theology.' Much later Courant described Hilbert's method of dealing with a problem as follows: 'He was a most concrete, intuitive mathematician who invented, and very consciously used, a principle: namely, if you want to solve a problem first strip the problem of everything that is not essential. Simplify it, specialize it as much as you can without sacrificing its core. Thus it becomes simple, as simple as it can be made, without losing any of its punch, and then you solve it. The generalization is a triviality which you do not need to pay too much attention to. This principle of Hilbert's proved extremely useful for him and also for others who learned it from him; unfortunately it has been forgotten.' However in 1893 von Lindemann moved from Konigsberg to Munich and Hilbert, who was in his first year as associate professor, was chosen to succeed him, against strong competition. Although only just turned thirty, he was now a full professor at a first-rate university. After some wrangling Minkowski, who had moved to Bonn the previous year, returned to take the post Hilbert had just vacated, although it was not long before he followed Hurwitz to Zurich. During the ten years he was at Konigsberg, Hilbert lectured on a wide range of subjects, from number theory, function theory, and projective and differential geometry to Galois theory, hydrodynamics, and differential equations. Moreover, with the exception of a one-hour course on determinants, he never lectured on the same subject twice. He spoke slowly and without frills. When giving a course it was his practice to devote the first part of each lecture to a summary of the previous one, as would be done at school. Although Hilbert was deeply attached to his hometown he had am bitions to move to a place where there were better mathematics students David Hilbert (1862- 1943) 249 than at Konigsberg. The Hilberts used to sit each morning reading the newspaper at breakfast just for news about the state of health of profes sors of mathematics all over Europe. It was a very healthy period but in 1895 the opportunity he had been waiting for came up. When a certain mathematician died this led to a cascade of professorial appointments at German universities. At the end of this process Klein had secured Hilbert for the Georgia Augusta, the university with which he was to be so closely identified and where he remained for the rest of his professional career. At the same time Hilbert succeeded Klein as principal editor of the Annalen. According to Courant, 'He took this very seriously and the violence with which he rejected papers was completely without sympathy. As long as he and Caratheodory and such people were editors it was a very high ranking, maybe the highest ranking, mathematical journal in the world, and it really meant something to have a paper printed there.' In these years Hilbert was in his prime while Klein, although immensely powerful, had long since ceased research activity. Klein's emphasis on the importance of unifying pure and applied mathematics was in contrast to Hilbert's view in which a secure axiomatic foundation was essential and intuitive arguments were not worth much. They also differed in other ways. Unlike Hilbert, Klein believed in maintaining the traditional distant relationship between professors and students. According to Courant: In the Gottingen society, if you read old chronicles, a Gottingen professor was a demi-god and very rank conscious - the professor, and particularly the wife of the professor. Hilbert came to Gottingen and it was very, very upsetting. Some of the older professors' wives met and said 'Have you heard about this new mathematician who has come? He is upsetting the whole situation here. I learned that the other night he was seen in some restaurant, playing billiards in the backroom with privatdozents.' It was considered completely impossible for a full professor to lower himself to be personally friendly with younger people. But Hilbert broke this tradition completely, and this was an enormous step towards creating scientific life; young students came to his house and had tea or dinner with him. Frau Hilbert gave big, lavish dinner-parties for assistants, students, etc. Hilbert went with his students and also everybody else who wanted to come, for hour-long hikes in the woods during which mathematics, politics and economics were discussed. From Cantor to Hilbert Courant also gave a picture of Hilbert at work: He spent his whole time gardening and in between gardening and little chores, he went to a long blackboard, maybe twenty feet long, covered so that also in the rain he could walk up and down, doing his mathematics in between digging some flower beds.

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