Cantor’s religious beliefs and his transfinite numbers
How much of Cantor’s invention of transfinite numbers arose on account of his religious beliefs? A vast amount of literature has been written about this, and I do not intend to add further unnecessary verbiage. Suffice it to point out that 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. Cantor wrote a plethora of material regarding god, numbers and sets, of which only a small selection is included here; Cantor’s own words speak for him.
Cantor professes his conviction that his notion of the transfinite, and the reality of its actual existence, follow from 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. )
One proof [of the reality of the transfinite] proceeds from the concept of God and infers from the greatest perfection of God’s essence the possibility of the creation of a transfinite order, from his supreme goodness to the necessity of the creation of the transfinite.
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. )
God knows infinitely many things in reality and categorically outside Himself (possible objects, objects that occur throughout the time series in the future), that admittedly do not always occur together on the side of the things, but that do have a simultaneity in their being known in God’s Intellect. If only one would have convinced oneself always of this secure and unshakable proposition in its full content (I mean, not just in general, but also in special and concrete cases), then one would have recognized in it without any trouble the truth of the transfinite, and millennia of disputes and errors would have been avoided.
If we now apply this proposition to a special class of objects of God’s intellect, then we arrive at the elements (elementary propositions) of the theory of infinite numbers and order types.
Every single finite cardinal number (1 or 2 or 3, etc) is contained in God’s Intellect both as an exemplary idea, and as a unitary form of knowledge of countless compound objects, to which the cardinal number applies: all finite cardinal numbers are therefore distinctly and simultaneously present in God’s mind (Augustine 2003: book XII, chapter 19: ‘Against those, who say that, what is infinite, can also not be comprehended by God’s knowledge’). (Footnote: Actually Cantor’s reference should be to chapter 18, not chapter 19. )
They build in their totality a manifold, unified, and from the other contents of God’s Intellect separated thing in itself, that itself forms an object of God’s Knowledge. But since the knowledge of a thing presupposes a unitary form, by which this thing exists and is known, there must in God’s intellect be a determinate cardinal number available, which relates itself to the collection or totality of all finite cardinals in the same way as the number 7 relates itself to the notes in an octave.
For this, which can be shown to be the smallest transfinite cardinal number, I have chosen the sign ω.
On the other hand the finite cardinal numbers 1, 2, 3, … form in their natural order a well-ordered collection […]; the general form, under which this well-ordered collection of all finite cardinal numbers is necessarily conceived by God’s Intellect (on reflection on them belonging to the ordering that I have just described), I call the ordinal number of this well-ordered collection, or its order type, and I signify it with ω; here, too, it can easily be shown that ω is the smallest transfinite ordinal number.
When we abstract in ω from the ordering of its elements (which are then just units), then we will naturally obtain the cardinal number that we have denoted above by ω.
This is to explain my notation; the bar over the ω should remind of the abstraction from the ordering of the elements in the cardinal number ω; one can say, that the cardinal number ω originates from the ordinal number ω, when we abstract in the latter from the ordering of the units that are contained in it.
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. )
The general theory of manifolds … belongs entirely to metaphysics. You can easily convince yourself from this, when you examine the categories of cardinal number and ordinal type, these fundamental concepts of set theory, with respect to the degree of their generality and besides will remark, that in them thought is completely pure, so that there is not even the least scope for the imagination to play a role. This is not altered in the least by the images, to which I, like all metaphysicians, from time to time help myself, and also the fact that the works of my pen are published in mathematical journals, does not modify the metaphysical content and character of them.
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. )
I am far from claiming my discoveries are due to personal merit, because I am only an instrument of a higher power that will continue to work long after me, just as it revealed itself thousands of years ago to Euclid and Archimedes.
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, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Ernst Zermelo, ed. Berlin: Verlag von Julius Springer Cantor, Georg, and Ernst Zermelo. Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts. Springer-Verlag, 2013, translation by Gabriele Chaitkin as in On the Theory of the Transfinite. .)
The Transfinite with its abundance of formations and forms, points with necessity to an Absolute, to the “truly Infinite”, to whose Magnitude nothing can be added or subtracted and which therefore is to be seen quantitatively as absolute Maximum. The latter exceeds, so to speak, the human power of comprehension and eludes particularly mathematical determination; whereas the Transfinite not only fills the vast field of the possible in God’s knowledge, but also offers a rich, constantly increasing field of ideal inquiry and attains reality and existence, I am convinced, also in the world of the created, up to a certain degree and in different relations, to bring the Magnificence of the Creator, following His absolute free decree, to greater expression than could have occurred through a merely “finite world”. This will, however, have to wait a long time for general recognition, especially among the theologians, as valuable as this knowledge would prove to be as a resource for the promotion of their domain (religion).
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. )
The transfinite with its abundance of shapes and forms necessarily points to an absolute, to the “truly infinite”, the size of which cannot be added or decreased and which is therefore to be considered in quantitative terms as the absolute maximum. The latter to some extent exceeds human comprehension and above all eludes a mathematical determination; whereas the transfinite not only fills the wide area of what is possible in God’s knowledge, but it also offers a rich, ever-increasing field of ideal research and in my opinion, in the created world it also occurs to a certain extent in various relationships to reality and existence, in order to express the glory of the Creator according to his absolutely free will, more strongly than it could have happened through a mere “finite world”.
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. )
The transfinite is capable of multiple formations, specifications and individuations. In particular, there are transfinite cardinal numbers and transfinite ordinal numbers, which possess a mathematical regularity as definite and as humanly researchable as the finite numbers and forms. All these particular modes of the transfinite exist from eternity as ideas in the divine intellect.
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:
In his first reply Cardinal Franzelin suggests that Cantor may be suggesting Pantheism, which would be unacceptable to the church.
Cantor protests against this stating that:
“… no system is further removed from my essential beliefs than pantheism, apart from materialism, with which I have absolutely nothing in common.”
Cardinal Franzelin replies again, this time rebuking Cantor for suggesting that the transfinite is necessarily created (as in the quotation above of 22 Jan 1886) since that would imply that god has no choice in the creation of infinities.
Cantor’s response is that what he actually meant was not that god was bound by any necessity, but that is necessary for humans to recognize that god created not only the infinite but also the transfinite.
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