Editing Georg Cantor (section)

== Biography ==

===Youth and studies===
[[File:Georg Cantor3.jpg|thumb|left|Cantor, around 1870]]
Georg Cantor, born in 1845 in [[Saint Petersburg]], Russian Empire<!-- DO NOT LINK, see [[MOS:GEOLINK]] -->, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist [[Joseph Böhm]]'s brother) was a well-known musician and soloist in a Russian imperial orchestra.<ref>[http://dic.academic.ru/dic.nsf/enc_music/924/Бём ru: The musical encyclopedia (Музыкальная энциклопедия)].</ref> Cantor's father had been a member of the [[Saint Petersburg Bourse|Saint Petersburg stock exchange]]; when he became ill, the family moved to Germany in 1856, first to [[Wiesbaden]], then to [[Frankfurt]], seeking milder  winters 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 August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the [[Technische Universität Darmstadt]].<ref>{{Cite web|url=http://www-groups.dcs.st-and.ac.uk/history/Biographies/Cantor.html|title=Georg Cantor (1845-1918)|website= www-groups.dcs.st-and.ac.uk|access-date= 2019-09-14}}</ref><ref>{{Cite book|title= Georg Cantor 1845-1918|publisher= Birkhauser|year= 1985|isbn= 978-3764317706}}</ref> In 1862 Cantor entered the [[ETH Zurich|Swiss Federal Polytechnic]] in Zurich. After receiving a substantial inheritance upon his father's death in June 1863,<ref name=":0">{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html|title=Cantor biography|website=www-history.mcs.st-andrews.ac.uk|access-date=2017-10-06}}</ref> Cantor transferred to the [[Humboldt University of Berlin|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.<ref name=":0" /><ref name=":1">{{Cite book|title=Math and mathematicians: the history of math discoveries around the world|last1= Bruno|first1= Leonard C.|year= 1999|publisher= U X L|last2= Baker|first2= Lawrence W.|isbn= 978-0787638139 |location=Detroit, Mich.|page=[https://archive.org/details/mathmathematicia00brun/page/54 54]|oclc= 41497065|url=https://archive.org/details/mathmathematicia00brun/page/54}}</ref>

===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 [[Martin Luther University of Halle-Wittenberg|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.<ref name=":1" /><ref>{{cite web |author1=O'Connor, John J |author2=Robertson, Edmund F. |year= 1998 |url= http://www-history.mcs.st-andrews.ac.uk/Biographies/Cantor.html |title= Georg Ferdinand Ludwig Philipp Cantor |publisher=MacTutor History of Mathematics}}</ref>

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| Harz mountains]], Cantor spent much time in mathematical discussions with [[Richard Dedekind]], whom he had met at Interlaken in Switzerland
two years earlier while on holiday.<ref>{{cite web |last1=O'Connor |first1=JJ |last2=Robertson |first2=E F |title=Georg Ferdinand Ludwig Philipp Cantor |url=https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |website=Maths History |publisher=University of St Andrews |access-date=9 February 2025 |archive-url=https://archive.today/20250209194516/https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |archive-date=9 February 2025 |language=English |date=October 1998 |quote=They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind. |url-status=live}}</ref>

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879.<ref name=":1" /><ref name=":0" /> To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a [[professor|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.<ref name="daub163">[[#Dauben1979|Dauben 1979]], p. 163.</ref> Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,<ref name="daub34">[[#Dauben1979|Dauben 1979]], p. 34.</ref> perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.<ref>[[#Dauben1977|Dauben 1977]], p. 89 ''15n.''</ref> 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.<ref name=":1" /> Kronecker, now seen as one of the founders of the [[Constructivism (mathematics)|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,<ref name=":1" /> so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.<ref>{{cite book |last1=Dauben |first1=Joseph Warren |title=GEORG CANTOR His Mathematics and Philosophy of the Infinite |date=20 September 1990 |publisher=Princeton University Press |isbn=978-0691024479 |url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Cantor/Cantor.html |access-date=9 February 2025 |archive-url=https://archive.today/20250209195322/https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Cantor/Cantor.html |archive-date=9 February 2025 |chapter=Chapter 6}}</ref>

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.<ref>{{cite web |last1=O'Connor |first1=JJ |last2=Robertson |first2=E F |title=Georg Ferdinand Ludwig Philipp Cantor |url=https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |website=Maths History |publisher=University of St Andrews |access-date=9 February 2025 |archive-url=https://archive.today/20250209194516/https://mathshistory.st-andrews.ac.uk/Biographies//Cantor/ |archive-date=9 February 2025 |language=English |date=October 1998 |quote=They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind. |url-status=live}}</ref>

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.<ref>[[#Dauben1979|Dauben 1979]], pp. 2–3; [[#Guinness1971|Grattan-Guinness 1971]], pp. 354–355.</ref> 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''.<ref name="daub138">[[#Dauben1979|Dauben 1979]], p. 138.</ref> He asked Cantor to withdraw the paper from ''Acta'' while it was in proof, writing that it was "...&nbsp;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''."<ref name="daub139">[[#Dauben1979|Dauben 1979]], p. 139.</ref>

Cantor suffered his first known bout of depression in May 1884.<ref name=":0" /><ref name = "daub282"/> 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:

{{Blockquote|... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.<ref>[[#Dauben1979|Dauben 1979]], p. 136; [[#Guinness1971|Grattan-Guinness 1971]], pp. 376–377. Letter dated June 21, 1884.</ref>}}

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.<ref>[[#Dauben1979|Dauben 1979]], pp. 281–283.</ref>

Cantor recovered soon thereafter, and subsequently made further important contributions, including his [[Cantor's diagonal argument|diagonal argument]] and [[Cantor's theorem|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.<ref name=":1" /> 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]],<ref name=":1" /> 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.<ref name=":1" />

===Later years and death===
After Cantor's 1884 hospitalization there is no record that he was in any [[sanatorium]] again until 1899.<ref name="daub282">[[#Dauben1979|Dauben 1979]], p. 282.</ref> Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (Cantor was delivering a lecture on his views on [[Baconian theory]] and [[William Shakespeare]]), and this tragedy drained Cantor of much of his passion for mathematics.<ref name="daub283">[[#Dauben1979|Dauben 1979]], p. 283.</ref> Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by [[Julius König]] at the Third [[International Congress of Mathematicians]]. The paper attempted to prove that the basic tenets of [[transfinite set theory]] were false.  Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.<ref>For a discussion of König's paper see [[#Dauben1979|Dauben 1979]], pp. 248–250. For Cantor's reaction, see [[#Dauben1979|Dauben 1979]], pp. 248, 283.</ref> Although [[Ernst Zermelo]] demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God.<ref name="daub248">[[#Dauben1979|Dauben 1979]], p. 248.</ref> Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years.<ref>[[#Dauben1979|Dauben 1979]], pp. 283–284.</ref> He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory ([[Burali-Forti paradox]], [[Cantor's paradox]], and [[Russell's paradox]]) to a meeting of the [[German Mathematical Society|''Deutsche Mathematiker-Vereinigung'']] in 1903, and attending the International Congress of Mathematicians at [[Heidelberg]] in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the [[University of St. Andrews]] in Scotland. Cantor attended, hoping to meet [[Bertrand Russell]], whose newly published ''[[Principia Mathematica]]'' repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, and lived in poverty and suffered from [[Malnutrition|malnourishment]] during [[World War I]].<ref name="daub284">[[#Dauben1979|Dauben 1979]], p. 284.</ref> The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on 6 January 1918, in the sanatorium where he had spent the last year of his life.<ref name=":0" />