Absolute space and time are said to emanate from Aristotle. The Church acted as custodian of these concepts from early on up to recent times. I am thinking about another issue, namely that of mathematical sets, i.e. the simultaneous reference to all numbers in a set, particularly those with an infinite number of elements. The view that such a reference is impossible also emanates from Aristotle’s writings.

The Church might be thought of as being charged with authority on infinite issues and my question is now whether the medieval (and newer) Church can be said to have undertaken a similar custodian role of this concept?

It doesn’t come to mind as easy as absolute space in astronomy.

EDIT: I understand now that the term absolute space and time is strongly connected to Newtonian physics vs. theory of relativity. I was thinking more loosely in medieval theological terms where the quality “absolute” was almost monopolized by divine concepts. Let me rephrase both the background and the question. i believe my introduction has tainted peoples’ view of the question (which does not contain much of a personal view, with the possible exception of the word “similar”).

Let me also give a hint of my thoughts about the question. I was wondering, inter alia, about the dualism of nominalism vs. universalism. I have been told in another post that these issues are heavily obfuscated, and I speculated that perhaps a view of the infinite might be coupled to this controversy. It has lead to accusations of heresy in the case of William of Ockham.

EDITED BACKGROUND The medieval (and later) church had strong opinions about space and time – and motion – some of which is clear from its stance on the Solar system and from Thomas de Aquinas’ writings. It also clear that some of these concepts came from Aristotle.


Is there any view expressed by the medieval (or later) church regarding so called actual infinity?


Nobody has mentioned any action or expressed opinions from the church, so I lean towards the position that it hasn’t occurred.

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    Aristotle was a precursor of the relational theory of space, see Absolute and Relational Theories of Space and Motion, absolute space and time "emanate" from Newton's natural philosophy. They have little relation to the Church. The denial of actual infinity does go back to Aristotle, and was commonly accepted until Cantor, not specifically due to the Church either, see How does actual infinity (of numbers or space) work?
    – Conifold
    Aug 13 '19 at 0:53
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    It might improve the reception of this question (which I do like and have upvoted and voted to reopen) if it were shortened and clarified - for example, as "What official positions (or at least very strong opinions loudly expressed by people in authority), if any, has the (Catholic?) church taken regarding 'actual' vs. 'potential' infinities?" Granted, this rephrasing probably misses some nuances of what you're asking - taking an official position is probably different than "undertaking a custodial role" - but I think those nuances are currently unclear. Aug 13 '19 at 11:28
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    A relevant source: Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite by Joseph W. Dauben. Aug 13 '19 at 11:32
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    Aristotelianism (in its Thomist form) was and is the official philosophy of the Church. There was an episode with Jesuits banning infinitesimals from their curriculum in 1632, it had more to do with opposition to atomism than to actual infinity, see Math SE. A good historical survey of debates over actual infinity from middle ages to Cantor is Mancosu, Measuring the size of infinite collections.
    – Conifold
    Aug 14 '19 at 4:32
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    What does "absolute space and time" have anything to do with set theory? If your question is about set theory, why even bother mentioning "absolute space and time." Does it matter whether the church was custodian of "absolute space and time" or not? Do you understand when two ideas are related, or do you only jump between different red-herrings? Mar 4 '20 at 3:28

Wikipedia Aug 13 '19 on Cantor says:

Cantor identified the Absolute Infinite with God, 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.

His elder and quondam coworker later fiercest critic Kronecker responded to Cantor's cogitations with :

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk

Usually translated as

The good Lord made the ??? all the rest is the work if man.

Whether the ??? should be integers or natural numbers or something else (un-real!) is not clear.

Cantor's choice of the adjective "real" for ℝ was likely an act of defiance against Kronecker.

Likewise how literally Kronecker spoke is hard to figure out.

We can't easily verify on whose side God happens to be!

We can however confirm these more mundane facts:

Cantor's father, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans.  His mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:

Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ...

("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.

The Wikipedia entry has many more interesting facts eg. Cantor wrote to the Pope etc.

Wrt religion we may I think reasonably conclude that 3 diverse faiths/theologies collided within Cantor to produce some interesting effects!

Even Wittgenstein had this to say:

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".

Added later

After comments of @Conifold and @NoahSchweber

The "real number" matter...

...is rather a mess.

Yes Descartes used the word real

In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones.

Which means Descartes real is not the modern post-Cantorean real, the cartesian real being closer to what we moderns call algebraic number.

More than the math what matters is

The human context

of these discussions/differences.

Descartes was using real as against imaginary; where "imaginary" was a cute-word like the "strangeness" and "charm" that physicists give their particles.

Cantor OTOH, was operating in the context of the objections (to non-constructivism) of the majority of mathematicians of his time saying things like :

This is not mathematics; this is theology.

IOW (I hear) Descartes "real" as a repartee to "imaginary" which is a bit of a joke.

Cantor's "real" OTOH is a belligerent response in an atmosphere which in today's fashion would he called "toxic".

Coming back to...

The math

Descartes real is denumerable; Cantor's famously not.

Given that by most accounts Cantor's major contribution to math – or theology if you disapprove! – is his account of the cardinality of transfinite sets I think the distinction is very significant.

Of course whether Cantor deliberately chose that word to cock a snook at his detractors is debatable; I think it's a reasonable conjecture.

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    "Cantor's choice of the adjective "real" for ℝ was likely an act of defiance against Kronecker." I believe the term "real number" predates Cantor substantially - IIRC, it was introduced by Descartes (to distinguish from general complex numbers). Aug 13 '19 at 11:17
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    Re: the Kronecker quote's translation, I've only ever seen "ganzen Zahlen" translated as "integers" - is there a source backing up the claimed ambiguity here? Aug 13 '19 at 11:23
  • @NoahSchweber Wikipedia on pre-intuitionists Another one: Natural numbers were created by God And then goes on to say let's not split hairs about integers vs natural numbers. IOW just run google with »"God made the natural numbers" Kronecker« Aug 14 '19 at 2:50
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    @NoahSchweber "Real number" was indeed introduced by Descartes in La Geometrie (1637), it appears in English since 1668. The literal translation of ganzen Zahlen is "whole numbers", which traditionally refers to positive integers. The origin of the "quote" is Weber's hearsay, and there are conflicting opinions expressed in what Kronecker published, see Did Kronecker attribute immutable origin to the integers? on hsm.
    – Conifold
    Aug 14 '19 at 23:19
  • @NoahSchweber added "appendix" on "real nos" Aug 15 '19 at 5:50

The only specific cases I can remember reading about are Scotus' answer to the question of language referring to God, where he proposed that "infinity" is to language about God what Aquinas had thought "simplicity" to be (this is a macerated citation from the SEP, the Scotus entry iirc), and then Aquinas discoursing somewhere in the SUMMA about whether divine infinity requires infinitarianism instead of trinitarianism about the divine persons. Scotus has a positive presence of infinity in mind; but Aquinas uses a negative concept of infinity to judge God infinite even if existent as a finite number of persons (I believe; it's been years and the SUMMA's a maze).

Aquinas' position would probably be closer to an "official" position, I suppose (and Scotus was accused of heretical vibes at times).

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