Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918Grattan-Guinness 2000, p. 351.) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri PoincaréDauben 2004, p. 1. and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God.Dauben 2004, pp. 8, 11, 12–13. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of GodDauben 1977, p. 86; Dauben 1979, pp. 120, 143. – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher was in favor of it, and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).
The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth".Dauben 2004, p. 1; Dauben 1977, p. 89 15n. Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries,Dauben 1979, p. 280: "... the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression. though some have explained these episodes as probable manifestations of a bipolar disorder.Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression".
The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended Cantor's work from its critics by declaring, "No one shall expel us from the paradise that Cantor has created.": "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.")
Biography
Youth and studies
Georg Cantor was born on March 3, 1845, in Saint Petersburg, Russian Empire, and was brought up in the city until the age of eleven. The eldest of six children, he was regarded as an outstanding violinist. His father, Georg Waldemar Cantor, was Danish, but fled with his family to Russia during the Napoleonic Wars; his mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg.Purkert and Ilgauds 1985, p. 15. Cantor's family was Lutheran, though his mother was raised Catholic.Georg Cantor at the Encyclopedia Britannica, cited 13 August 2025. He described his paternal grandparents as having belonged to the Portuguese Jewish community of Copenhagen.Cantor gave this account of his father's origin in a letter to Alexander Baumgartner in 1889 (Tapp, Christian (2005) Kardinalität und Kardinäle, Springer-Verlag, Berlin, p. 236), and again to Paul Tannery in 1896 (Tannery, Paul (1934) Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, p. 306.). His maternal grandfather Franz Böhm (1788–1846; the violinist Joseph Böhm's brother) was a well-known musician and a soloist in a Russian imperial orchestra.ru: The musical encyclopedia (Музыкальная энциклопедия).
Cantor's father worked as a broker in the Saint Petersburg stock exchange until an illness in 1856, which forced the family to seek out a more temperate climate. They moved to Germany, first to Wiesbaden, then to Frankfurt. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867.
Teacher and researcher Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on holiday in Gersau in Switzerland, and with whom a lively mathematical correspondence had developed.
Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.Dauben 1979, p. 163. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,Dauben 1979, p. 34. perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.Dauben 1977, p. 89 15n. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.
In 1881, Cantor's Halle colleague Eduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.
In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.Dauben 1979, p. 138. He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica."Dauben 1979, p. 139.
Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of Elizabethan literature, thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.Dauben 1979, pp. 281–283.
Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
In 1889, Cantor was instrumental in founding the German Mathematical Society, and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which took place in Zürich, Switzerland, in 1897.
Later years and death