Georg Cantor

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Georg Cantor bigraphy, stories - Mathematician who originated set theory.

Georg Cantor : biography

3 March 1845 – 6 January 1918

Georg Ferdinand Ludwig Philipp Cantor ( ; February 19 (March 3) 1845 – January 6, 1918Grattan-Guinness 2000, p. 351) was a German mathematician, best known as the inventor 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. In fact, 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.

Cantor’s theory of transfinite numbers was originally regarded as so counter-intuitive – even shocking – that it encountered 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. Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God Dauben 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.

The objections to Cantor’s work were occasionally fierce: Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics,Dauben 1979, p. 266. and 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 even 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. It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God.Dauben 2004, pp. 8, 11, 12–13. David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created.".

Life

Youth and studies

Cantor was born in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the oldest of six children, was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm’s brother) was the well-known musician and the soloist in the Russian empire in an imperial orchestra. Cantor’s father 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 winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the University of Zürich. After receiving a substantial inheritance upon his father’s death in 1863, Cantor shifted his studies 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.