Georg Cantor

Georg Cantor
Cantor, c. 1910
Born
Georg Ferdinand Ludwig Philipp Cantor

(1845-03-03)3 March 1845
Saint Petersburg, Russian Empire
Died6 January 1918(1918-01-06) (aged 72)
Halle, Province of Saxony, German Empire
NationalityGerman-Russian
Alma mater
Known forSet theory
Spouse
Vally Guttmann
​
(m. 1874)​
AwardsSylvester Medal (1904)
Scientific career
FieldsMathematics
InstitutionsUniversity of Halle
ThesisDe aequationibus secundi gradus indeterminatis (1867)
Doctoral advisor
Signature

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918[1]) 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 he was well aware of.[2]

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 Poincaré[3] 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,[4] believed the theory had been communicated to him by God.[5] 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[6] – on one occasion equating the theory of transfinite numbers with pantheism[7] – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).[8]

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".[9] 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".[10] 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,[11] though some have explained these episodes as probable manifestations of a bipolar disorder.[12]

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.[13] David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."[14][15]

  1. ^ Grattan-Guinness 2000, p. 351.
  2. ^ The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources.
  3. ^ Dauben 2004, p. 1.
  4. ^ Dauben, Joseph Warren (1979). Georg Cantor His Mathematics and Philosophy of the Infinite. princeton university press. pp. introduction. ISBN 9780691024479.
  5. ^ Dauben 2004, pp. 8, 11, 12–13.
  6. ^ Dauben 1977, p. 86; Dauben 1979, pp. 120, 143.
  7. ^ Cite error: The named reference daub77102 was invoked but never defined (see the help page).
  8. ^ Dauben 1979, chpt. 6.
  9. ^ Dauben 2004, p. 1; Dauben 1977, p. 89 15n.
  10. ^ Rodych 2007.
  11. ^ 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.
  12. ^ 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".
  13. ^ Cite error: The named reference daub248 was invoked but never defined (see the help page).
  14. ^ Hilbert (1926, p. 170): "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.")
  15. ^ Cite error: The named reference encomium was invoked but never defined (see the help page).