In 1869 Cantor qualified for a teaching position at the University of Halle, soon becoming associate professor and, in 1879, full professor. He carried on his work there until his death.

In 1869 Cantor qualified for a teaching position at the University of Halle, soon becoming associate professor and, in 1879, full professor. He carried on his work there until his death.

Contributions to the Founding of the Theory of Transfinite Numbers

(One of the greatest mathematical classics of all time, th...)

One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc., as well as in the entire field of modern logic.

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.

Background

Cantor was born on March 3, 1845, in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. The oldest of six children, he was regarded as an outstanding violinist.

Cantor's father, Georg Waldemar Cantor, had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg. Cantor's mother, Marie Bohm, was from a family of musicians.

Education

Cantor attended the Gymnasium in Wiesbaden, and later the Grossherzoglich-Hessische Realschule in Darmstadt. It was there that he first became interested in mathematics. In 1860, he graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862 he began his university studies in Swiss Federal Polytechnic, resuming them in Berlin in 1863, after the sudden death of his father. At that time Karl Weierstrass, famed as a teacher and as a researcher, was attracting many talented students to the University of Berlin. His lectures gave analysis a firm and precise foundation, and later many of his pupils proudly proclaimed themselves members of the “Berlin school” and built on the ideas of their teacher.

Cantor spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. He was a good student, and he received his doctorate degree in 1867. He also submitted his dissertation on number theory at the University of Berlin in 1867.

Cantor’s unique contribution to mathematics was that his special way of asking questions opened up vast new areas of inquiry, in which the problems were solved partly by him and partly by successors. In Berlin, Cantor was a member (and from 1864 to 1865 president) of the Mathematical Society, which sought to bring mathematicians together and to further their scientific work. In his later years, he actively worked for an international union of mathematicians, and there can have been few other scholars who did as much as he to generate and promote the exchange of ideas among scientists.

He conceived a plan to establish an Association of German Mathematicians and succeeded in overcoming the resistance to it. In 1890 the association was founded, and Cantor served as its first president until 1893. He also pressed for international congresses of mathematicians and was responsible for bringing about the first ever held, in Zürich in 1897. Thus Cantor was no hermit living within his own narrow science. When, later, he did sever ties with many of his early friends - as with H. A. Schwarz in the 1880’s - the reasons lay in the nature of his work rather than in his character.

In 1869 Cantor qualified for a teaching position at the University of Halle, soon becoming associate professor and, in 1879, full professor. He carried on his work there until his death. In those days a professor at Halle was so poorly paid that without other income he would have been in financial straits. It was Cantor’s hope to obtain a better-endowed, more prestigious professorship in Berlin, but in Berlin the almost omnipotent Kronecker blocked the way. Completely disagreeing with Cantor’s views on “transfinite numbers,” he thwarted Cantor’s every attempt to improve his standing through an appointment to the capital. Recognition from abroad came early, however. Cantor’s friend Mittag-Leffler accepted his writings for publication in his then new Acta mathematica. He became an honorary member of the London Mathematical Society (1901) and of other scientific societies, receiving honorary doctor’s degrees from Christiania (1902) and St. Andrews (1911).

The closing decades of Cantor’s life were spent in the shadow of mental illness. Since 1884 he had suffered sporadically from deep depression and was often in a sanatorium. He died in 1918 in Halle University’s psychiatric clinic.

Like his father, Cantor was a Protestant; his mother was a Catholic. The link with Catholicism may have made it easier for him to seek, later on, support for his philosophical ideas among Catholic thinkers.

Membership

Mathematical Society
,
Germany

1864 - 1865

Association of German Mathematicians
,
Germany

1890 - 1893

Connections

Cantor's marriage in 1874 to Vally Guttmann was born of deep affection, and the sunny personality of the artistically inclined “Frau Vally” was a happy counter to the serious, often melancholy, temperament of the great scholar. They had six children, and an inheritance from his father enabled Cantor to build his family a house.

Father:

Georg Waldemar Cantor

Mother:

Maria Anna Böhm

Spouse:

Vally Guttmann

Friend:

Karl Hermann Amandus Schwarz

References

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for his larger attempts to meld spirituality and science.

The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more.

1989

Georg Cantor: His Mathematics and Philosophy of the Infinite
One of the greatest revolutions in mathematics occurred when Georg Cantor promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.