Cantor’s religious beliefs and his transfinite numbers

Page last updated 27 Feb 2022

How much of Georg Cantor’s invention of transfinite numbers arose on account of his religious beliefs? A recent study of the relationship between the reasoning process of an individual and their religiosity suggested that the results “accord closely with the hypothesis that religious dogmatism correlates with a cognitive-behavioral tendency to forgo logical problem solving when an intuitive answer is available.”  (Footnote: Richard E Daws & Adam Hampshire, “The negative relationship between reasoning and religiosity is underpinned by a bias for intuitive responses specifically when intuition and logic are in conflict”, Frontiers in Psychology 8, 2017.)

Georg Cantor

Was this the case with Georg Cantor? A vast amount of literature has been written about Cantor and his religious beliefs, but rather than add further unnecessary verbiage, we can let Cantor’s own words speak for him. Cantor was in frequent written communication with high ranking religious leaders regarding connections between his notions of the infinite and how that could be considered to be in harmony with the notion of a god that has capabilities that are infinite. The result is that there is a sizable stack of Cantor’s writings about god, numbers and sets, of which only a small selection is included below:

Georg Cantor professes his conviction that his notion of the transfinite, and the reality of its actual existence, follow directly from god: (Footnote: Letter from Cantor to Cardinal Franzelin Halle January 22, 1886, as in Georg Cantor, Briefe, H. Meschowski, and W. Nilson, eds., Berlin: Springer, 1991, p 255, my translation.)

Cantor sets out in some detail his justification of his notion of the transfinite by appealing to aspects of a god that he believes in: (Footnote: Letter from Cantor to Jeiler (1888), pp 414-417 in Tapp, Christian, and Georg Cantor. Kardinalität und Kardinäle: wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit, Vol. 53, Franz Steiner Verlag, 2005, translation by Joanna Van Der Veen & Leon Horsten in “Cantorian infinity and philosophical concepts of god”, European journal for philosophy of religion 5.3 (2013): pp 117-138.)

Cantor states that his ideas on set theory arise from metaphysical beliefs: (Footnote: Pages 309-310 in Tapp, Christian, and Georg Cantor. Kardinalität und Kardinäle: wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit, Vol. 53, Franz Steiner Verlag, 2005, translation by Joanna Van Der Veen & Leon Horsten in “Cantorian infinity and philosophical concepts of god”, European journal for philosophy of religion 5.3 (2013): pp 117-138.)

In a letter of 1883 Cantor claims that he is only a vessel by which eternal truths are communicated: (Footnote: Letter from Cantor to Gösta Mittag-Leffler Halle, Dec. 23, 1883, as in Georg Cantor, Briefe, H. Meschowski, and W. Nilson, eds., Berlin: Springer, 1991, p 160, my translation.)

Cantor suggests that his mathematics will lead to a greater understanding of god: (Footnote: From a letter from Georg Cantor to A. Eulenberg, Feb. 28, 1886, as in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Georg Cantor, ed by Ernst Zermelo, Springer-Verlag, 2013, translation by Gabriele Chaitkin in On the Theory of the Transfinite.)

Cantor also claims that the transfinite occurs in some way in the real world: (Footnote: From Mitteilungen zur Lehre vom Transfiniten (Communications on the theory of the transfinite), section V, Zeitschrift für Philosophie und philosophische Kritik 91, pp 81–125; 92 (1887), as in pp 405–406, Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts, ed Zermelo, Springer-Verlag, republished, 2013, my translation.)

Cantor claims that the transfinite is god’s invention; (Footnote: Letter from Cantor to Jeiler (13 October 1895), p 427 in Tapp, Christian, and Georg Cantor. Kardinalität und Kardinäle: wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit, Vol. 53, Franz Steiner Verlag, 2005, as in p 218 of “Idealist and realist elements in Cantor’s approach to set theory” by Ignasi Jané, Philosophia Mathematica 18.2 (2010): pp 193-226.)

In case it might be thought that perhaps Cantor was simply a man of his time, no more or no less religious than the average person of that era, the mathematician Gerhard Kowalewski, who had quite regular contact with Cantor in the 1890’s, wrote in his autobiography that: (Footnote: Gerhard Kowalewski (1876 - 1950), Bestand und Wandel, Oldenbourg Wissenschaftsverlag, Munich 1950, p. 107 and p. 200 respectively, my translations.)

“Cantor was a deeply religious person. What he experienced while building his set theory moved his innermost soul.”

and:

“These cardinal numbers, the Cantorian Alephs, were something sacred for Cantor, in a sense the steps that lead up to the throne of infinity, to the throne of God. According to his convictions, all conceivable cardinal numbers were populated by these Alephs.”

And in an article of 1887, Cantor quotes an entire chapter from St. Augustine, in Latin, here is a translated excerpt: (Footnote: St. Augustine, De Ovitate Dei in Latin, Book XII, Chapter 19, quoted in Cantor’s Mitteilungen zur Lehre vom Transfiniten 1, II (Communications on the theory of the transfinite), Zeitschrift fur Philosophie und philosophische Kritik. English translation by Henry Bettenson, City of God, Penguin, Harmondsworth, 1972, pp. 496-497.)

For further reading, you can read some of Cantor’s religious/mathematical correspondence in the articles referred to in the above. In the article On the Theory of the Transfinite, where Cantor, who initiates a dialogue with a church Cardinal, discusses at great length the question of whether the notion of the absolute infinite is compatible with the church teachings, and suggests that some religious writers are incorrect in claiming that the notion of an absolute infinite is incompatible with church teachings. Some interesting points include:

You can read online an English translation of one of Cantor’s major works, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets), which lays out his philosophy on different sizes of infinity and transfinite numbers, and the page The Origins of Transfinite Numbers gives the details of the process by which Cantor arrived at his notion of transfinite numbers. For an overview of current set theories, see the overview that starts at Overview of set theory: Part 1: Different types of set theories.

Are transfinite numbers necessary in maths?

Some mathematicians are so enamored with transfinite numbers that attempt to justify disregarding the religious origins in various ways - one such way is their belief that transfinite ordinal numbers are necessary to obtain certain results. One such result is that a Goodstein sequence always terminates by reaching zero; Reuben Goodstein wrote a proof to this effect that was based on transfinite numbers. Since that time, many mathematicians appear to believe that it is impossible to prove that result without using transfinite numbers, but in fact there is a straightforward proof that Goodstein sequences always terminate and which does not use transfinite numbers at all, see A Proof of Goodstein’s Theorem without Transfinite numbers.