Christian Felix Klein | Biography

Felix Klein is arguably one of the most influential mathematicians of the 19th century. He is best known for building the mathematical community at the University of Göttingen which became a model for research facilities in mathematics worldwide.

Christian Felix Klein was born on November 25, 1849 in Dusseldorf, the son of an official in the local finance department. Klein graduated from Gymnasium (the German equivalent of an academic high school) in Dusseldorf and began studying at the University of Bonn in 1865. At Bonn, he fell under the influence of Julius Plücker, one of the best-known geometers of the century. Plücker had moved the center of his interest to physics, and it had been in physics that Klein originally wanted to work, but Plücker returned to his original interest in geometry and took Klein with him. After Plücker's death in 1867, Klein became responsible for finishing a manuscript of Plücker's, which gave him an early introduction to the scholarly community and, in particular, to Alfred Clebsch, another prominent geometer of the time.

After receiving his doctorate in 1868 Klein spent a year traveling between Göttingen, Berlin, and Paris. Of the three, he enjoyed Göttingen immensely, did not like Berlin, and had to leave Paris ahead of schedule because of the outbreak of the Franco-Prussian War. Some of his travels were spent with the young Norwegian mathematician Marius Sophus Lie, whose ideas on geometry and analysis were much in common with Klein's. Klein's patriotism led him to enlist as a medical orderly during the war, but before the year was over he had returned to Dusseldorf, suffering from typhoid fever. The next year Klein qualified as a lecturer at Göttingen, but the following year he accepted a chair at the University of Erlangen. The complexities of academic promotion within the German university system at the time frequently required moving about from one university to another, merely for the sake of promotion within the original university.

It was the custom for a new professor to deliver an inaugural address at a German university, and in 1872 Klein followed suit at Erlangen. At the time, it was difficult to speak of one geometry, as recent developments had led to a collection of geometries whose relation to one another was unclear. There was the familiar Euclidean geometry, based on the ordinary axioms including the parallel postulate (which stipulated that there was exactly one parallel to a line through a point not on that line). There were at least two non-Euclidean geometries, one denying the existence of any parallels through a point not on a line, the other allowing the existence of an infinite number of parallels. Finally, projective geometry, which had been known since the 17th century, had been given a more quantitative turn in the work of Arthur Cayley, among others.

As outlined by Klein, geometry is the study of the properties of figures preserved under the transformations in a certain group. Which group of transformations one started with determined the geometry in which one was working. For example, if the transformations were limited to rigid motions, then one had Euclidean geometry. If projections were allowed, then one had projective geometry. If an even wider class were included, then one could end up with topology. This view (called the Erlangen program) has infused the spirit, not just of geometry, but of mathematics as a whole ever since.

Also, in 1872 Klein took over editing Mathematische Annalen after the death of Clebsch. Under his editorship this was the leading mathematical journal in the world and it was to remain so until World War II. By 1875, Klein had left Erlangen for the Technische Hochschule in Munich and then in 1880 he went to the University of Leipzig. In 1884 he was invited to take the place of James Joseph Sylvester at Johns Hopkins University in Baltimore, but he declined. He did make several visits to the United States subsequently, where both his personal influence and those of his students were strong. Finally, in 1886, Klein achieved the goal of a chair at Göttingen.

Two factors in particular led Klein to successfully create a mathematical center at Göttingen. One was personal, as he was married to Anne Hegel, a descendant of the German philosopher Georg Wilhelm Friedrich Hegel. Her striking beauty may have been a draw even for those who were not yet convinced of the mathematical attractions of her husband. In the course of their married life the Kleins had one son and three daughters.

The other factor was not so pleasant. One of the subjects on which Klein had been working while at Leipzig were automorphic functions, transformations of the complex plane into itself that satisfied certain conditions. Unfortunately, for Klein the year 1884 turned into a competition with the younger French mathematician Henri Poincare seeking fundamental results. Although Klein's work during this period was of a high quality, he felt that he had not lived up to expectations and suffered a nervous breakdown.

Thereafter, Klein immersed himself in creating a major mathematical center at Göttingen. The mathematical discussions did not stop with the classroom walls, but continued at the Kleins' home or on walks into the woods around Göttingen. One feature of the institute was a room filled with geometrical models to help with visualization. The presence of such a room was a reminder of Klein's antipathy to the abstract style of analysis favored by Karl Weierstrass at Berlin. Klein wanted his mathematics to have intuitive content, which explains why he was anathema to Weierstrass. Klein attracted many of the leading German mathematicians to Göttingen, the most outstanding being David Hilbert. Göttingen's creative atmosphere encouraged the presence of women in the lecture hall and foreign visitors.

At the time of Klein's retirement shortly before the outbreak of World War I, he could take pride in having brought together a mathematical research community the like of which the world had never seen. In 1912, he received the Copley medal of the Royal Society, one of just many honors. His last years were saddened by the death of his son on the battlefield during the war, and he died on January 22, 1925.

Within ten years of his death the Nazi government had undertaken the dismantling of the research community in Göttingen. When the Institute for Advanced Studies was founded at Princeton in the 1930s, it modeled itself after Göttingen. The dream which Klein had brought into reality lived on.