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/thoughts/infinity/

I have been thinking a lot about the concept of infinity lately, encountering problems related to it when thinking about various — mostly theological — concepts. In this article I will talk about some of these encounters.

Infinite and indefinite

The first such instance was when watching a video by Aarvoll, entitled Metaphysics of the Perennial Philosophy,1 where he brings forward the idea of a problem in the concept of the infinite:

If by finite we mean any kind of limitation whatsoever, then we can't say that the number line is infinite, because even though it is unending, it is only infinite within the dimension that it has to operate with — the natural number — it is still finite insofar as you are not going to have a “pig”, like a pig is not one of the numbers. So you don't have the pig, therefore it is not infinite because there is a limitation in that the numbers don't include pigs.

René Guénon actually wrote a chapter in his book The Metaphysical Principles of the Infinitesimal Calculus entitled Infinite and Indefinite about this problem, where he resolves this “problem with the infinite” by introducing the concept of the indefinite. He describes the infinite as being completely unbounded, that which has no limits, i.e God. Furthermore there can only be one Infinite, for supposedly distinct infinites would limit and therefore inevitably exclude one another. Infinity used in any other sense as that what we just mentioned is then actually indefinite. The Scholastics had the similar concept of infinitum absolutum (the absolute infinite) and infinitum secundum quid (the infinite in a certain respect). René Guénon explains the concept of the Infinite, the finite, and the indefinite and their respective origin as follows:

The finite necessarily presupposes the Infinite — since the latter is that which comprehends and envelops all possibilities — the indefinite on the contrary proceeds from the finite, of which it is in reality only a development and to which it is consequently always reducible, for it is obvious that whatever process one might apply, one cannot derive from the finite either anything more or anything other than that which was already potentially contained therein. To take again the example of the sequence of numbers, we can say that this sequence, with all the indefinitude it implies, is given to us by the law of formation, since it is from this very law that its indefinitude immediately resuts; now this law consists in the following, that given any number, one can form the next by adding a unit. The sequence of numbers is therefore formed by successive additions of the unit to itself, indefinitly repeated, which is basically only the indefinite extension of the process of formation for any arithmetical sum; and here one can see quite clearly how the indefinite is formed starting from the finite.

Paradox of omnipotence

Now that we have resolved this contradiction, which seems to be a semantic one rather than a metaphysical one, Aarvoll goes on stating:

When we are discussing the term [infinite], we are asking “does this apply to the universe?”, and if it does apply, there are all sorts of contradictions that we're going to run into immediately; one of them being, “can the infinite be finite?”... No, clearly that violates the law of identity, but then you are saying “the infinite is limited to the meta-logical laws”. This is really just a variant of the problem of omnipotence; “can God create a boulder so large he can't life it?”. That is saying, is the omnipotent force capable of rendering itself not omnipotent? Can omnipotence negate its identity? or is it restricted by its identity and therefore not fully omnipotent because it can't overcome that identity.

I had a discussion with The Pondering Soul about this problem where he raised a few objections about the way the question is formulated, namely, that the question itself is contradictory — can the infinite be finite? is equal to can A be equal to not A?, a question that by the very way it is formulated is bound to violate the law of identity — and that therefore the answer is bound to be contradictory as well.2 I think however that a mistake is being made here, namely that the definition of the infinite is that which has no delimitation, and the only way for the infinite to be true to its definition is when that which has no limits to not be limited to having no limits, else it would be limited. Again, the whole point of the concept of the infinite is that it has to negate its own identity in order to stay true to its definition and meaning, else it would simply be finite. So indeed, it is not the way the question is formulated that is paradoxical, but rather the very concept of the infinite itself, as Aarvoll also already states himself in the previous quote, clearly that [the infinite] violates the law of identity.

In the book What is Sufism? by Martin Lings, this paradox is laid out in greater detail, albeit it slightly different, where God (the Infinite, the Inward), also wears the name the Outward — the finite world of creation — and the reaction of a saint who encountered this variant of the paradox of omnipotence:

For the mind alone and unaided it is impossible to resolve into Oneness the duality of Creator and creation. The already quoted Moroccan Shaykh, al-'Arabī ad-Darqāwī, tells us in his letters how one day, when he was absorbed in the invocation, a persistent inward voice kept repeating to him the verse of the Qur'an “He is the First and the Last and the Outward and the Inward”. To begin with paid no attention and continued his own repetition of the Name. “But finally” — to quote his words — “since it would not leave me in peace, I answered it: “As to His saying that He is the First and the Last and the Inward, I understand it; but as to His saying that He is the Outward, I do not understand it, for we see nothing outward but created things.” Then the voice said: “If by His words and the Outward anything other than the outward that we see were meant, that would be inward and not outward; but I tell thee and the Outward.” And I realised that there is no being but God nor anything in the worlds of the universe save Him Alone, praise and thanks be to God!”

The Traditional answer to this paradox is, as Martin Lings tells us, outside of the domain of reason and logic:

Every sound mind can see that from the point of view of orthodoxy this commentary constitutes an overwhelming 'proof' of the Oneness of Being because it demonstrates, as with a lightning flash of clarity, that this doctrine can only be denied on pain of the heresy of implying that God is subject to change. But the mind cannot understand how Being can be One any more than it can understand how God can be the Outward as well as the Inward; and in accepting these truths theoretically it brings itself to the extreme limit of its own domain. We are here at the parting of the ways: the exoterist will involuntarily recoil, reminding himself and others that to dabble in theological speculation is strongly discouraged; but the virtual mystic will recognise at once that what lies before him is something other than the domain of dogmatic theology, and far from drawing back he will seek to escape from the apparently firm ground of his purely mental standpoint at the risk of being out of his depth.

The answer to these paradoxes being outside of the domain of reason and logic would be fitting, since how can the Infinite be limited by the meta-logical laws, how can it be encapsulated by our finite minds?

Apophatic theology

This brings us to apophatic theology; any attempt to define or describe God, or the Infinite, is actually an attempt to limit. Trying to limit the Infinite, that which has no limits, with reason and logic or any other way, is clearly impossible and absurd as we we have seen with our example of the paradox of omnipotence. Trying to define God is always trying to encapsulate Him, which is impossible, thus we can only describe god by stating what He is not (though this as well is "pophathic" in a way).3 Once again this shows us that actually the answer to our problem with the Infinite might indeed be outside of the domain of reason. This is a concept for me, someone socialized into atheism and rationalism, that is hard to accept, however that very ratio brought me to this point.

A god that is completely apophatic is something hard (or rather, maybe impossible) to grasp and interact with. Earlier this year I realized that I subconsciously assign a personality to God. Clearly this personality is not God, but a creation of my imagination that makes it easier to interact with Him. Upon doing more research I found out that I am not the first to realize this (of course), Henry Corbin, a 20th century Christian theologian with a strong interest in Islam and other traditions called this projection of mental construct onto God idolâtrie métaphysique (metaphysical idolatry). Rūmī wrote Go, strive towards meaning, O form-worshipper! For meaning is the wing of form's body in the 13th centry, and the Islamic theologians Ibn ʿArabī and and Mullā Ṣadrā also extensively wrote about these "idols of belief", stating things such as:

Most people do not worship God insofar as He is God. They merely worship the objects of their beliefs in accordance with what they have formed for themselves as objects of worship. In reality, their gods are those imaginary idols which they form and carve with the potency of their intellectual or imaginary beliefs.

And:

External idols are also only worshipped because of their worshipper's belief in their divinity. The mental forms are the objects of their worship essentially, and the external forms are their objects of worship accidentally. Thus, the objects of worship of every idol-worshipper are nothing but the forms of his beliefs and the caprices of his soul, as has been alluded to in His saying, Have you seen the one who takes his caprice for his god? (Q 65:23). Just as worshippers of bodily idols worship what their hands have created, so too do those who have partial beliefs concerning God worship what the hands of their intellects have gathered.

Ibn ʿArabī even went as far as stating that there are none but idol-worshippers, that since every belief in God is ultimately a delimitation or an idolized conception of Him:

If God were to take people to account for error, He would take every possessor of a belief to account. Every believer has delimited his Lord with his reason and consideration and has thereby restricted Him. But nothing is worthy of God except non-delimitation... So He delimits, but He does not become delimited. Nevertheless, God pardons everyone.4

Notes


  1. 1Metaphysics of the Perennial Philosophy: 1. Order and Chaos. ↩︎

  2. 2See his blogpost A Response to an Objection: On Omnipotence Paradox. ↩︎

  3. 3The book Unsaying God: Negative Theology in Medieval Islam by Aydogan Kars describes different apopathic theologies within Islam, some being more apophatic than others. The Ismāʿīlis for example employ a double negation, where they first negate an attribute, and afterwards negate the very negation itself, for example God does not exist, and God does not not exist, where the second negation is not a double negation of the attribute but rather a negation of the negation. ↩︎

  4. 4All the previous quotes are taken from the excellent essay Beyond Metaphysical Idolatry: Mullā Ṣadrā on Mental Constructs of God. ↩︎