Editing Georg Cantor (section)

{{Short description|German mathematician (1845–1918)}}
{{Featured article}}
{{Use dmy dates|date=August 2016}}
{{Infobox scientist
| name              = Georg Cantor
| image             = Georg Cantor (Porträt).jpg
| caption           = Cantor, {{circa|1910}}
| birth_name        = Georg Ferdinand Ludwig Philipp Cantor
| birth_date        = {{Birth date|1845|3|3|df=y}}
| birth_place       = [[Saint Petersburg]], [[Russian Empire]]
<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->
| death_date        = {{Death date and age|1918|1|6|1845|3|3|df=y}}
| death_place       = [[Halle (Saale)|Halle]], Province of Saxony<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->, German Empire<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->
| nationality       = [[Germans|German]]-[[Russians|Russian]]
| alma_mater        = {{plainlist|
* [[ETH Zurich|Swiss Federal Polytechnic]]
* [[Humboldt University of Berlin|University of Berlin]]
* [[University of Göttingen]]
}}
| thesis_title      = De aequationibus secundi gradus indeterminatis
| thesis_url        = https://web.archive.org/web/20150704015954/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN237853094&DMDID=DMDLOG_0008&L=1
| thesis_year       = 1867
| doctoral_advisor  = {{plainlist|
* [[Ernst Kummer]]
* [[Karl Weierstrass]]}}
| doctoral_students = 
| known_for         = [[Set theory]]
| spouse            = {{marriage|Vally Guttmann|1874}}
| field             = [[Mathematics]]
| work_institutions = [[University of Halle]]
| prizes            = [[Sylvester Medal]] (1904)
| signature         = Georg Cantor Signature.png
}}

'''Georg Ferdinand Ludwig Philipp Cantor''' ({{IPAc-en|ˈ|k|æ|n|t|ɔr}} {{respell|KAN|tor}}; {{IPA|de|ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯|lang}}; {{OldStyleDate|3 March|1845|19 February}}&nbsp;– 6 January 1918<ref>[[#Guinness2000|Grattan-Guinness 2000]], p. 351.</ref>) was a <!--please do not add a nationality--> mathematician who played a pivotal role in the creation of [[set theory]], which has become a [[foundations of mathematics|fundamental theory]] in mathematics. Cantor established the importance of [[one-to-one correspondence]] between the members of two sets, defined [[infinite set|infinite]] and [[well-order|well-ordered sets]], and proved that the [[real number]]s are more numerous than the [[natural number]]s. Cantor's method of proof of this theorem implies the existence of an [[infinity]] of infinities. He defined the [[cardinal number|cardinal]] and [[ordinal number|ordinal]] numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.<ref>The biographical material in this article is mostly drawn from [[#Dauben1979|Dauben 1979]]. [[#Guinness1971|Grattan-Guinness 1971]], and [[#Purkert|Purkert and Ilgauds 1985]] are useful additional sources.</ref>

Originally, Cantor's theory of [[transfinite number]]s was regarded as counter-intuitive&nbsp;– even shocking. This caused it to encounter resistance from mathematical contemporaries such as [[Leopold Kronecker]] and [[Henri Poincaré]]<ref>[[#Dauben2004|Dauben 2004]], p. 1.</ref> and later from [[Hermann Weyl]] and [[L.&nbsp;E.&nbsp;J. Brouwer]], while [[Ludwig Wittgenstein]] raised [[philosophical objections to Cantor's theory|philosophical objections]]; see [[Controversy over Cantor's theory]]. Cantor, a devout [[Lutheran Christian]],<ref>{{cite book|last1=Dauben|first1=Joseph Warren|title=Georg Cantor His Mathematics and Philosophy of the Infinite|url=https://archive.org/details/georgcantorhisma0000daub|url-access=registration|date=1979|publisher=[[Princeton University Press]]|isbn=9780691024479|pages=introduction}}</ref> believed the theory had been communicated to him by God.<ref name = "xdpfir">[[#Dauben2004|Dauben 2004]], pp.&nbsp;8, 11, 12–13.</ref> Some [[Christian theology|Christian theologians]] (particularly [[Neo-Scholasticism|neo-Scholastics]]) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God<ref name = "nuozkv">[[#Dauben1977|Dauben 1977]], p. 86; [[#Dauben1979|Dauben 1979]], pp.&nbsp;120, 143.</ref>&nbsp;– on one occasion equating the theory of transfinite numbers with [[pantheism]]<ref name = "daub77102"/>&nbsp;– a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher {{ill|Konstantin Gutberlet|de|Konstantin Gutberlet|it}} was in favor of it and Cardinal [[Johann Baptist Franzelin]] accepted it as a valid theory (after Cantor made some important clarifications).{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}}

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".<ref>[[#Dauben2004|Dauben 2004]], p. 1; [[#Dauben1977|Dauben 1977]], p. 89 ''15n''.</ref> 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".{{sfn|Rodych|2007}} 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,<ref name="daub280">[[#Dauben1979|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.</ref> though some have explained these episodes as probable manifestations of a [[bipolar disorder]].<ref name="bipolar">[[#Dauben2004|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".</ref>

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.<ref name = "daub248"/> [[David Hilbert]] defended it from its critics by declaring, "No one shall expel us from the [[Cantor's paradise|paradise that Cantor has created]]."<ref>{{harvtxt|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.")</ref><ref name="encomium"/>