GEORG CANTOR (March 3rd, 1845 – January 6th, 1918)

German mathematician known as the man who tamed infinity.

Main accomplishments:

• Contributed to the explanation of Zeno’s paradoxes.
• Inventor of set theory, defined infinite and well-ordered sets and established the importance of one-to-one correspondence between members of two sets.
• Invented theory of transfinite numbers, proved real numbers are more numerous than natural numbers, and defined cardinal and ordinal numbers and their arithmetic.
• Awarded the Sylvester Medal in 1904 by the Royal Society.

EARLY LIFE

Georg Cantor (1845–1918) was born in St Petersburg, Russia, to Georg Waldemar Cantor, a prosperous merchant, and Maria Bohm, a Russian musician. He was raised Protestant, his father’s religion, and was an outstanding violinist, inheriting musical talent from his mother. As a small child, he was tutored at home, and then attended primary school in St Petersburg.

In 1856, when Cantor was eleven, the family moved to Wiesbaden, Germany. Cantor attended the Gymnasium there until moving to Darmstadt, where he studied at the Realschule, graduating in 1860. During this time, he excelled in mathematics and trigonometry, which he studied at the University of Zurich.

In 1863, one year into Cantor’s studies, his father passed away. Cantor transferred to the University of Berlin where he specialized in math, physics, and philosophy, later studying at the University of Gottingen. He returned to Berlin to complete his dissertation, which was on number theory and received his doctorate in 1867.

CAREER

Cantor’s first job out of university was at a girl’s school in Berlin. He joined the Schellbach Seminar for math teachers and worked on his habilitation, which he presented to Halle when he was appointed in 1869. Here, he solved the problem of the uniqueness of representation of a function as a trigonometric series in 1870, and within the next two years had published papers on trigonometric series.

In 1872, Cantor was promoted to Extraordinary Professor, the same year he published a paper defining irrational numbers in terms of convergent sequences of rational numbers and began a correspondence with Dedekind. The year after, he proved rational numbers are countable in one-one correspondence with natural numbers. He proved algebraic numbers are countable. He proved that real numbers were not countable by the end of the year, publishing it in 1874 in a paper that proved almost all numbers are transcendental.

1874 was also the year Cantor studied the unit square. In the spring, he got engaged to Vally Guttmann, and they went to Switzerland for their honeymoon, which Cantor spent partly visiting German mathematician Richard Dedekind. By 1877, their discussions led Cantor to the discovery that there is a one-to-one correspondence between points on the interval and points in p-dimensional space. This new knowledge affected geometry and ideas about the dimensions of space. He published a paper on it, crystallizing his concepts of one-to-one relationships as denumerable sets, sets of equal power and the concept of dimension.

In 1879, he was promoted to full professor. Within the next five years, Cantor published a set of six papers discussing basic set theory. By 1884, he was suffering from depression, thought to be caused by a combination of stress from math and pressure from critics, and could not prove the continuum hypothesis. In 1886, he bought a house for his growing family of six children, and then was appointed president of the Deutsche Mathematiker-Vereinigung.

In 1893, he quit being president and wrote a paper on even numbers and the sum of prime numbers. His last major papers were published in 1895 and 1897, describing well-ordered sets and ordinal numbers. In 1897, at the first International Congress of Mathematicians in Zurich, Cantor discovered paradoxes in the theory of sets but by 1899, was too mentally ill to solve them.

LATER LIFE

When Cantor suffered mentally, he turned from mathematics to philosophy and studied Shakespearian theory. He published literary pamphlets in the late 1890s, but his mother passed away, soon followed by his brother. When his youngest son died in 1899, his depression increased, but he still studied and lectured on math, attending the International Congress of Mathematics in 1904.

In 1905, he wrote about religious theory and then took some time off from teaching in 1909. He fully retired in 1913, and in June 1917, entered a sanatorium. He died of a heart attack on January 6, 1918.