Editing Georg Cantor (section)

==Philosophy, religion, literature and Cantor's mathematics==
The concept of the existence of an [[actual infinity]] was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the [[orthodoxy]] of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.<ref name="daub295">[[#Dauben1979|Dauben 1979]], p. 295.</ref> He directly addressed this intersection between these disciplines in the introduction to his ''Grundlagen einer allgemeinen Mannigfaltigkeitslehre'', where he stressed the connection between his view of the infinite and the philosophical one.<ref>[[#Dauben1979|Dauben 1979]], p. 120.</ref> To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications&nbsp;– he identified the [[absolute infinite]] with God,<ref>[[#Hallett|Hallett 1986]], p. 13. Compare to the writings of [[Thomas Aquinas]].</ref> and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.<ref name = "xdpfir"/> He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science.<ref>{{cite journal |last=Hedman |first=Bruce |date=1993 |title=Cantor's Concept of Infinity: Implications of Infinity for Contingence |url=https://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html |journal=Perspectives on Science and Christian Faith |volume=45 |issue=1 |pages=8–16 |access-date=5 March 2020}}</ref> [[Joseph Dauben]] has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.<ref>{{cite book |last=Dauben |first=Joseph Warren |date=1979  |title=Georg Cantor: His Mathematics and Philosophy of the Infinite |url=https://www.jstor.org/stable/j.ctv10crfh1 |location= |publisher=Princeton University Press |doi=10.2307/j.ctv10crfh1 |jstor=j.ctv10crfh1 |isbn=9780691024479|s2cid=241372960 }}</ref><ref>{{cite journal |last=Dauben |first=Joseph Warren |date=1978 |title=Georg Cantor: The Personal Matrix of His Mathematics |url=https://www.jstor.org/stable/231091 |journal=Isis |volume=69 |issue=4 |pages=548 |doi=10.1086/352113 |jstor=231091 |pmid=387662 |s2cid=26155985 |access-date=5 March 2020 |quote=The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.}}</ref>

Debate among mathematicians grew out of opposing views in the [[philosophy of mathematics]] regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.<ref name="daub225">[[#Dauben1979|Dauben 1979]], p. 225</ref> Mathematicians from three major schools of thought ([[Constructivism (mathematics)|constructivism]] and its two offshoots, [[intuitionism]] and [[finitism]]) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that [[nonconstructive proof]]s such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that [[constructive proof]]s are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.<ref name="daub266">[[#Dauben1979|Dauben 1979]], p. 266.</ref> Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.<ref>{{Cite journal |last=Snapper |first=Ernst |year=1979|url=<!-- http://math.boisestate.edu/~tconklin/MATH547/Main/Exhibits/Three%20Crises%20in%20Math%20A.pdf -->http://www2.gsu.edu/~matgtc/three%20crises%20in%20mathematics.pdf|title=The Three Crises in Mathematics: Logicism, Intuitionism and Formalism|journal=Mathematics Magazine|volume=524|issue=4|pages=207–216|access-date=2 April 2013|archive-url=https://web.archive.org/web/20120815055019/http://www2.gsu.edu/~matgtc/three%20crises%20in%20mathematics.pdf|archive-date=15 August 2012|url-status=dead|doi=10.1080/0025570X.1979.11976784}}</ref> Mathematicians such as [[L.&nbsp;E.&nbsp;J. Brouwer]] and especially [[Henri Poincaré]] adopted an [[intuitionism|intuitionist]] stance against Cantor's work.  Finally, [[Ludwig Wittgenstein|Wittgenstein]]'s attacks were finitist: he believed that Cantor's diagonal argument conflated the [[intension]] of a set of cardinal or real numbers with its [[Extension (semantics)|extension]], thus conflating the concept of rules for generating a set with an actual set.{{sfn|Rodych|2007}}

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.<ref name = "nuozkv"/> In particular, [[neo-scholasticism|neo-Thomist]] thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".<ref>{{Cite journal |last=Davenport|year=1997 |first=Anne A. |title=The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century|journal=Isis|volume=88|issue=2|pages=263–295|jstor=236574|doi=10.1086/383692|s2cid=154486558 }}</ref> Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:<ref name = "daub7785"/> "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.".<ref>{{Harvnb|Cantor|1932|p=404}}. Translation in [[#Dauben1977|Dauben 1977]], p. 95.</ref> Prominent neo-scholastic German philosopher Konstantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}}

Cantor also believed that his theory of transfinite numbers ran counter to both [[materialism]] and [[determinism]]&nbsp;– and was shocked when he realized that he was the only faculty member at Halle who did ''not'' hold to deterministic philosophical beliefs.<ref name="daub296">[[#Dauben1979|Dauben 1979]], p. 296.</ref>

It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 ''Grundlagen'', he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz.<ref>{{Cite journal|last=Newstead|first=Anne|date=2009|title=Cantor on Infinity in Nature, Number, and the Divine Mind|journal=American Catholic Philosophical Quarterly|volume=83|issue=4|pages=533–553|doi=10.5840/acpq200983444|url=https://philarchive.org/rec/NEWQOI}}</ref> In making these claims, Cantor may have been influenced by [[Friedrich Adolf Trendelenburg|F. A. Trendelenburg]], whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's ''Ethica''. Trendelenburg was also the examiner of Cantor's ''[[Habilitation#Germany|Habilitationsschrift]]''.<ref>{{Cite journal|last=Newstead|first=Anne|date=2009|title=Cantor on Infinity in Nature, Number, and the Divine Mind|journal=American Catholic Philosophical Quarterly|volume=84 |issue=3 |pages=535}}</ref><ref>{{Cite journal|last=Ferreiros|first=Jose|date=2004|title=The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions|journal=Science in Context|volume=17 |issue=1–2 |pages=49–83|doi=10.1017/S0269889704000055|pmid=15359485|s2cid=19040786|url=http://philsci-archive.pitt.edu/17321/1/Ferreir%C3%B3s%202004.pdf |archive-url=https://web.archive.org/web/20200921124322/http://philsci-archive.pitt.edu/17321/1/Ferreir%C3%B3s%202004.pdf |archive-date=2020-09-21 |url-status=live}}</ref>

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as [[Tilman Pesch]] and [[Joseph Hontheim]],<ref>[[#Dauben1979|Dauben 1979]], p. 144.</ref> as well as theologians such as Cardinal [[Johann Baptist Franzelin]], who once replied by equating the theory of transfinite numbers with [[pantheism]].<ref name="daub77102">[[#Dauben1977|Dauben 1977]], p. 102.</ref> Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's.{{sfn|Dauben|1979|loc=chpt. 6|ref=Dauben1979}} Cantor even sent one letter directly to [[Pope Leo XIII]] himself, and addressed several pamphlets to him.<ref name="daub7785">[[#Dauben1977|Dauben 1977]], p. 85.</ref>

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this [[Metaphysics|metaphysical]] system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom."<ref>[[#Dauben1977|Dauben 1977]], pp. 91–93.</ref> These ideas parallel those of [[Edmund Husserl]], whom Cantor had met in Halle.<ref>On Cantor, Husserl, and [[Gottlob Frege]], see Hill and Rosado Haddock (2000).</ref>

Meanwhile, Cantor himself was fiercely opposed to [[infinitesimal]]s, describing them as both an "abomination" and "the [[cholera]] [[bacillus]] of mathematics".<ref name="Infinitesimal" />

Cantor's 1883 paper reveals that he was well aware of the [[Controversy over Cantor's theory|opposition]] his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."<ref name="daub96">"[[#Dauben1979|Dauben 1979]], p. 96.</ref>

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of [[contradiction]] and defined in terms of previously accepted concepts. He also cites Aristotle, [[René Descartes]], [[George Berkeley]], [[Gottfried Leibniz]], and [[Bernard Bolzano]] on infinity. Instead, he always strongly rejected [[Immanuel Kant]]'s philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical [[Philistinism|Philistine]] who knew so little mathematics."<ref>Russell, Bertrand ''The Autobiography of Bertrand Russell'', George Allen and Unwin Ltd., 1971 (London), vol. 1, p. 217.</ref>