Georg Cantor (nonfiction)

Georg Cantor (1894).

Georg Ferdinand Ludwig Philipp Cantor (March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician.

He invented 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.

On July 28, 1899, Cantor asked Richard Dedekind whether the set of all cardinal numbers is itself a set, because, if it is, it would have a cardinal number larger than any other cardinal.

On August 31, 1899, Cantor wrotes to Dedekind, remarking that his "diagonal process" could be used to show that the power set of a set has more elements than the set itself.

Cantor criticized the non-Archimedean extensions of the real numbers proposed by Otto Stolz and Paul du Bois-Reymond, calling their work an "abomination". Cantor published a "proof-sketch" of the inconsistency of infinitesimals. The errors in Cantor's proof are analyzed by Ehrlich (2006).

Cantor's work is of great philosophical interest, a fact of which he was well aware.

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 Poincaré (nonfiction), and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections.

Cantor, a devout Lutheran, believed that set theory had been communicated to him by God.

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 -- 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: Henri Poincaré (nonfiction) referred to his ideas as a "grave disease" infecting the discipline of mathematics, and Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."

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 (nonfiction) curriculum.

Writing decades after Cantor's death, Wittgenstein lamented that mathematics (nonfiction) 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, though some have explained these episodes as probable manifestations of a bipolar disorder.

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.

Mathematician David Hilbert defended the award from its critics by declaring:

"No one shall expel us from the Paradise that Cantor has created."

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  August 31, 1899: Georg Cantor writes to Dedekind, remarking that his "diagonal process" could be used to show that the power set of a set has more elements than the set itself.

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  Cantor said to be "flattered but slightly embarrassed" that the Cantor set has become associated with crime-fighter The Sigil.

Fiction cross-reference

• Anarchimedes
• Georg Cantor
• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematician
• Mathematics
• Niles Cartouchian
• The Sigil

Nonfiction cross-reference

• Abraham Fraenkel (nonfiction)
• Cantor's diagonal argument (nonfiction)
• David Hilbert (nonfiction)
• Paul du Bois-Reymond (nonfiction)
• Hermann Weyl (nonfiction)
• L. E. J. Brouwer (nonfiction)
• Leopold Kronecker (nonfiction)
• Ludwig Wittgenstein (nonfiction)
• Mathematician (nonfiction)
• Mathematics (nonfiction)
• Set theory (nonfiction)

External links

• Georg Cantor @ Wikipedia