Felix Hausdorff Biography

Felix Hausdorff laid the foundations of set theoretic topology, which has evolved into an elaborate discipline that interacts with nearly every other field of mathematics. He precisely developed such basic notions as limits, continuous maps, connectedness, and compactness, which have become fundamental in building many kinds of mathematical structures. One of Hausdorff's revolutionary ideas, spaces of non-integer dimension,plays an important role in various topics, including geometric measure theory, the theory of dynamical systems, and in the description of the popularized notion of fractals. He was also a philosopher and author.

Hausdorff was born on November 8, 1868, in Breslau, Germany, which is now Wrocaw, Poland. His mother was Johanna Tietz Hausdorff; his father, Louis Hausdorff, was a dry goods merchant. The family moved to Leipzig, Germany, in 1871. The young Hausdorff eventually attended Leipzig University, where he studied astronomy and mathematics, earning his Ph.D. in 1891. His early research concentrated in the areas of optics and astronomy. After graduation, Hausdorff volunteered to serve in the German infantry. He achieved the rank of vice-sergeant before removing himself from consideration for further promotion in 1894. Hausdorff was Jewish, and no acknowledged Jews had been commissioned as officers in the German military for nearly 15 years. In 1896, following his father's death, Hausdorff succeeded him as a partner in the publishing firm Hausdorff and Company, which produced the leading trade magazine for spinning, weaving, and dyeing. That same year, he was accepted as a lecturer at Leipzig University.

The Literary Hausdorff

Hausdorff had a lively interest in the fine arts and in philosophy. An accomplished pianist, he occasionally composed songs. Like many others of his generation, Hausdorff was deeply influenced by the philosophy of Friedrich Nietzsche, though he maintained a critical distance to certain parts of Nietzsche's work.

The first of Hausdorff's four full-length literary works was published in 1897. He wrote under the pseudonym Dr. Paul Mongréso that he could express himself freely without jeopardizing his university position. The first book, Sant' Ilario: Thoughts from the Landscape of Zarathustra, was primarily a collection of aphorisms relating to Nietzsche's influential volume Thus Spake Zarathustra. It was published by the same company that had published Nietzsche's works and was even produced with a similar book cover.

Chaos in Cosmic Choice, the second book written under the name Mongré, dealt with relationships between space and time and was intended as a radical continuation of Immanuel Kant's criticism of traditional metaphysics. Hausdorff presented the same concepts in "The Space Problem," his 1903 inaugural lecture after being appointed as an associate professor at Leipzig University. Mongré's third major literary work was Ecstasies, a volume of sonnets and poems published in 1900. He also wrote The Doctor's Honor, a satirical play that was successfully produced in Hamburg and Berlin.

The Mathematical Hausdorff

In 1897, Hausdorff began publishing papers on topics in mathematics, including non-Euclidean geometry, complex numbers, and probability. He became interested in Georg Cantor's work on set theory, and during the summer semester of 1901 he taught what may have been the first course on set theory to be presented in Germany. Also, about this time, David Hilbert was publishing work applying set theory to geometry; this work may have been the inspiration for Hausdorff's greatest mathematical accomplishment.

In 1910, Hausdorff accepted a position as associate professor at the University of Bonn. Although he had written one or two articles on set theory each year for two decades, he published nothing from 1910 until 1914; apparently this was a period of intense work on the creation of point set topology. Hausdorff moved to Greifswald in 1913 to become a professor at the university there, and the following year, he published his monumental Grundzüge der Mengenlehre ("Basic Features of Set Theory")

The Grundzüge was a comprehensive text dealing with set theory,point set topology (now more commonly called set theoretic topology), and real analysis.Although the book was written for students at the advanced undergraduate level, Hausdorff noted in the preface that the volume also offered new ideas and methods to his professional colleagues. By organizing point set theory with just the right choice of axioms, he so thoroughly revised the related existing work that his book became the foundation on which modern topology has been developed.

Topology generalizes concepts such as continuity and limits to sets other than real and complex numbers. A topological space is free of all imposed structure not relevant to the continuity of functions defined on it. While Hausdorff's definitions and axioms were so general that an unlimited variety of geometric interpretations was possible, he developed the Euclidean plane as a special case by adding appropriate postulates. As Carl B. Boyer wrote in A History of Mathematics, "Topology has emerged in the twentieth century as a subject that unifies almost the whole of mathematics, somewhat as philosophy seeks to coordinate all knowledge."

Because of its generality, topology gives rise to apparent paradoxes that violate intuition,two of which Hausdorff addressed in the Grundzüge. One involves the transfinite numbers developed by Cantor, in which there are different magnitudes of infinity, and an infinite, proper subset may have the same number of elements as its superset. The other, now called the Hausdorff Paradox, shows that the surface of a sphere can be decomposed into three equal, nonintersecting sets so that the original sphere may be represented as the union of any two of them.

In 1919, Hausdorff introduced another revolutionary concept. He generalized the notion of dimension (e.g., a two-dimensional triangle or a three-dimensional cube) to include the possibility of objects with fractional dimensions. This has proven highly fruitful in various areas of mathematics, besides being popularized in the form of computer-generated fractal images.

Hausdorff returned to the University of Bonn in 1921, where he worked as a professor for the rest of his career. He was respected as the most capable mathematician in Bonn and as a professor whose lectures were well reasoned and clearly delivered. He taught until 1935, when he reached the mandatory retirement age of 67. Hausdorff continued to publish mathematical papers until 1938.

The Personal Hausdorff

In 1899, Hausdorff married Charlotte Goldschmidt; his only child, a daughter named Lenore (usually called Nora), was born the following year. Although Charlotte came from a Jewish family, she had been baptized a Protestant Christian in 1896, and Lenore was similarly baptized. The family moved to Bonn in 1921, and the street on which they lived would be renamed Hausdorffstrasse in 1949.

The anti-Semitism that had blocked Hausdorff's promotion in the infantry and threatened to prevent his promotion at Leipzig University continued to plague him throughout his lifetime. For instance, a young professor whose appointment Hausdorff had supported in 1926 became openly anti-Semitic in 1933, repudiating any former contact with Jews and refusing to join the rest of the faculty in attending seminars given by Jewish mathematicians. Some of Hausdorff's Jewish friends left Germany to escape the persecution; others whose emigration was thwarted committed suicide.

Suicide was a topic often addressed by Nietzsche; consequently, it had been a subject for reflection by Hausdorff. Zarathustra advocated "voluntary death" as a consummation of life for the noble man. "Death and Return," an 1899 essay by Mongré, broached the subject in the form of a letter to a fictitious, depressed friend. In it, the author advised that "this final remedy really helps, that it does not [merely] plunge one into a futile expense for morphine or revolver cartridges." Apparently, Hausdorff viewed suicide as an effective impediment to the Nazis' strategy of destroying their victims' human dignity: in death, the individual had no future, but his past was indestructible.

The infamous November pogrom of 1938, in which 20,000 Jews were arrested and government-sanctioned attacks resulted in 25 million marks' worth of damage to hundreds of Jewish homes, shops, and synagogues, occurred the day after Hausdorff's seventieth birthday. Charlotte Hausdorff and her sister, Edith Pappenheim (who had come to live with them a few months earlier), tried to bolster Hausdorff's spirits. He continued to work on his mathematics, but he put his writings into storage rather than publishing them.

In mid-January of 1942, the Hausdorffs were ordered to report to an internment camp located at a former monastery; this would probably be followed by deportation to a concentration camp. After organizing their affairs and leaving property disposal and cremation instructions with trusted friends, Hausdorff, his wife, and her sister committed suicide on January 26, 1942, by taking an overdose of sedatives.

On January 25, 1980, a memorial plaque honoring Hausdorff was placed at the entrance of the Mathematical Institute at the University of Bonn. In 1992, an exhibition of photographs and personal, literary, and mathematical documents was held at the University of Bonn, commemorating the fiftieth anniversary of Hausdorff's death.