Cantor, Georg: Mathematician who was born on January 3, 1845 in St. Petersburg, †and died on January 6, 1918 in Halle. Cantor was quite probably the most important mathematician of the 19th century. He single-handedly developed ►set theory and discovered basic facts about infinity, but ended up in an asylum.

Georg Cantor was born on March 3, 1845, in St. Petersburg. His father, Georg Woldemar Cantor, was the owner of a successful commercial establishment. Forced by a lung ailment to retire from business, Georg moved the family to Frankfurt on the Main in 1856.

Cantor first attended private schools in Frankfurt and Wiesbaden. He enrolled in a Darmstadt junior high school at the age of fourteen, then transferred a year later to the local Advanced Trade School of the Hesse Grand Duchy to prepare for a college program in engineering. While at the school, however, he decided to major in mathematics. After receiving his father's permission, he completed high school in the summer of 1862 with high honors and enrolled as a mathematics major at the Federal Institute of Technology in Zurich.

When Cantor's father died in the summer of 1863, the family relocated to Berlin, where Cantor transferred to Friedrich Wilhelm's University. The institution was one of the major centers of mathematical research at the time, and Cantor had the opportunity to study with professors such as Karl Weierstrass, Friedrich Arndt, Ernst Eduard Kummer, and Leopold Kronecker. He specialized in number theory, completing a brilliant dissertation in that field in 1867 titled De aequationibus secundi gradus indeterminatis.

After his habilitation (the German postdoctoral lecturer qualification) and a brief stint as a high school teacher, Cantor accepted the first academic position offered to him: a lectureship at a provincial second-tier university in Halle. Cantor regarded this position as a transitory stage in his career; his ultimate goal was to obtain a professorship at a famous university such as the University of Berlin. In the meantime, he kept himself occupied with research in analysis.

But Cantor's dream of becoming a full professor in Berlin or at some other top-tier university was never fulfilled. He remained in Halle until the end of his life. Paradoxically, the reasons for this lie precisely in his spectacular mathematical research and publications. Almost single-handedly he provided a foundation for set theory, a proof of the countable infinity of rational numbers, and a proof of the non-countability of the continuum of real numbers. These publications, however, drew hostility from his former professor in Berlin, the influential Berlin mathematician Leopold Kronecker. Kronecker was a fanatical believer in ►finitism who regarded set theory as a mathematical aberration and despised everything remotely related to the concept of infinity. He therefore attempted to undermine his former student's work in every way possible and used his influence to prevent Cantor from getting an appointment in Berlin.

In the summer of 1874, Cantor married Vally Guttmann, a friend of his youth from Berlin. They settled down in the province as comfortably as possible, given his modest lecturer's salary. In 1877, Cantor discovered a proof that lines and planes have the same number of points and that, in general, the ►dimension number has no impact on the degree of infinity. Accordingly, spaces with any number of dimensions have just the same number of points as a two-dimensional line.

In 1880, Cantor moved a step further and developed the theory of transfinite numbers, which are infinite by definition. In doing so, he came across the question of the ►continuum hypothesis. He used all his talents and skills to find a proof for this hypothesis but never found one. Over the ensuing years, Cantor would think time and again that he had found either the proof or, alternatively, a refutation, but the question of the hypothesis's truth remained stubbornly inaccessible to any mathematical solution. What Cantor did not know, but would be demonstrated by the American mathematician Paul Cohen in 1963, is that the continuum hypothesis can be neither proven nor refuted within set theory. Cantor's frustrations over his continuing failures concerning the continuum hypothesis, as well as the persistent attacks by Kronecker and his followers, eventually led to a breakdown and mental illness in early summer 1884.

Cantor was hospitalized in the psychiatric unit at the Halle University Clinic for several months. By the time he was allowed to leave the clinic, he was a profoundly changed man. He now believed he had gone too far in his exploration of infinity. Convinced that only by completely withdrawing from mathematics could he regain his mental health, he devoted himself to non-mathematical topics. Among other things, he began to develop the thesis that the British philosopher Francis Bacon was the real author of Shakespeare's plays.

But Cantor could not resist the temptation of infinity for long. Over the subsequent years, he again and again attempted to find his long-sought proof of the continuum hypothesis, each attempt ending in a nervous breakdown and hospitalization. In fact, Cantor spent almost a third of the remainder of his life in the psychiatric unit of his university's clinic. The university administration made every effort to assist him, giving him paid leave during his phases of illness while maintaining his status as university professor.

In the name of restricting the almighty power of individual scholars such as his arch-enemy, Cantor called for the founding of an association of German mathematicians. This effort was successful, and the German Mathematical Association was founded in 1890 with Cantor himself as chairperson. During the subsequent years he enjoyed watching his set theory be gradually accepted and developed further by a new generation of mathematicians such as Hilbert, Minkowski, Zermelo, Poincaré, and others.

This success, however, did little to improve Cantor's mental health. At a mathematicians' conference in 1904, a Hungarian scholar happened to present a refutation -- erroneous, as it eventually turned out -- of the continuum hypothesis. Rather than welcoming the acceptance of his ideas among younger scholars, Cantor felt personally attacked. His illness had long since taken on symptoms of paranoia. In 1911, during a period of relative mental stability, he accepted an invitation to be a guest lecturer in Britain. Instead of lecturing on mathematics, however, he presented his theses on Bacon and Shakespeare, to the astonishment of his audience. In 1917, he was hospitalized for the last time at the Psychiatric Clinic in Halle, where he died on January 6, 1918, in a state of mental derangement.


© Johann Christian Lotter   ■  Infinity  ■  Links  ■  Forum