Aquinas and Potential Infinities

One of the only areas where two of my absolute favorite Medieval saints, who were both towering philosophers, disagreed sharply and even violently, was on the question of whether the Aristotelian suggestion of Eternal Duration (meaning here that the world has an infinite past history) was coherent. Bonaventure, on the one hand, offered at least six arguments to demonstrate the absurdity of thinking there actually was an infinte past to the world, and further suggests that even the most amateur of philosophers should be able to see this as something self-evident. Aquinas, along with all Medieval peripatetics, defended the idea that it was logically possible that the universe had always existed. He did, however, depart with Averroes and even Christian philosophers where they agreed with Aristotle on this point; instead he argued from the authority of the Catholic Church that creation ‘ex nihilo’ was still logically possible. He said that this was a matter which neither science nor philosophy would ever resolve, however, and suggested an epistemic stalemate between those who thought the universe had always existed, and those who thought it began to exist some finite time ago.

As luck would have it, it seems he was wrong. Today most people recognize that science can more or less license the claim that the universe came into existence as we know it in a cataclysmic episode called the ‘Big Bang’ some 14 billion years ago, give or take. Moreover, philosophers like William Lane Craig have argued for a form of Cosmological argument in which one of the premises explicitly states “the universe began to exist”. Leaving aside for the moment the very good arguments from science in support of this premise, Craig also turns to philosophical arguments to support this premise. He turns in particular to the philosophy of math; providing demonstrations such as Hilbert’s Hotel, Craig attempts to make the case that actual infinites cannot exist in the world – they cannot instantiate.

Hilbert’s Hotel is a thought experiment in which we are asked to imagine a hotel with an infinite number of rooms, all of which are full. Suppose a person came in wishing to have a room, the clerk might respond that the hotel is full. However, the clerk could also, if the client insisted, make room for them by simply moving person in room 1 over to room 2, and person in room 2 over to room 3, and so on, so that every person in the hotel simply moves to the next room down. This way room has been made, but now there are still only just as many people, it would seem, in the hotel as were there before. Moreover, imagine that an infinite number of people wanted entry – the clerk could have simply moved everyone from their room # (call it room #n) to room #2n such that person in room 1 goes to room 2, person in room 2 goes to room 4, person in room 3 goes to 6, and so on, so that now all the odd numbered rooms are free. The clerk has managed to accommodate an infinite number of people. Problems keep getting worse however; for instance, imagine that an infinite number of people want to check out all at once – are there now an infinite number of people left, a finite number, or zero? All three possibilities are open, such that every logically possible answer to the question is possibly right.

These kinds of difficulties have always troubled able philosophers, and the solution which Craig proposes is for us to realize that infinity is just an idea in the mind, and not something which can instantiate in reality. Therefore, one can only speak of ‘potential infinities’ rather than ‘actual infinities’. Potential infinity is the limit towards which something endlessly reaches, whereas actual infinity is some infinite which has been actualized. For instance, imagine a person or a computer made to count from 1,2,3.. up to positive infinity; most people would agree that there is no way to traverse that gulf, even if the person or computer counting were given an infinite amount of time in which to count upwards – but not everyone is agreed.

Now, when considering the universe, this argument against actual infinites implies that there cannot be an actually infinite number of events in the past. Imagine that there were, that would imply that from the infinite past time has been counting down to the present moment – but before it could get to the present moment, it would have to have gotten to the moment previous, and before that one it would have to have gotten to the previous one to it – and so on literally ad infinitum. Similarly we might imagine counting upwards from negative infinity to get to {…-2,-1,0…}. Since the series of past events are actual, this would qualify as an actual infinite. Moreover, Craig maintains that the universe might plausibly last forever (as is required for an orthodox Christian view of heaven in the eschaton – as everlasting). This kind of infinite, however, is potential, since infinity here acts as the limit towards which our world endlessly approaches. This distinction seems clear to me.

Now, the curious part is that when Bonaventure offers similar arguments for the finitude of the universe’ past, Aquinas responded by saying that the series of past events could be infinite since this is a potential infinite. This had always confused me, until I realized through reading some of Craig’s article in the Blackwell Companion for Natural Theology that this was precisely because Aquinas seems to have subscribed to a kind of ‘presentism’ according to which only that which is ‘now real’ qualifies in ontological rigor as that which cannot be infinite, since it alone is ‘actual’. This is a very interesting thought. I used to wonder how Aquinas could have made such an obvious mistake as to think that the series of past events was potentially infinite, and now that I understand the grammar of his argument I feel it is worth some reflection (I knew, after all, that he could never have blundered so obviously). I still fundamentally agree with Bonaventure, over against Aquinas on this question, but it is insightful to reflect on the state of these arguments as they were judged by the Medievals, and by Aquinas in particular.

Finally, Craig suggested that this suggestion from Aquinas seems to require Aquinas’ commitment to the A theory of time. I’m not sure I agree with that. I do think he required some kind of ontological or semantic constraint of presentism, but I’m not yet sure what exactly to make of that. Almost all the rationalists of the early modern period, such as Spinoza, agreed that some actual infinities are implausible, while others, they argue, are logically required by their definitions, such as God. Perhaps I’ll blog in the short future on the topic of what the Rationalists made of the notion of infinities. In any case, concerning Aquinas, if he did not make a more subtle mistake of accepting the A theory of time implicitly, then I would like to explore this distinction between something’s being an ‘actual and concrete infinite’ and something’s being merely a ‘potential infinite – in such a way that it may be actual metaphysically. Perhaps there can be no distinction here except some kind of relevant semantic distinction – if this could be worked out, one might semantically marry B theory and Aquinas’ suggestion that the series of past events constitute possibly a potential infinite and not an actual infinite.

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About tylerjourneaux

I am an aspiring Catholic theologian and philosopher, and I have a keen interest in apologetics. I am creating this blog both in order to practice and improve my writing and memory retention as I publish my thoughts, and in order to give evidence of my ability to understand and communicate thoughts on topics pertinent to Theology, Philosophy, philosophical theology, Catholic (Christian) Apologetics, philosophy of religion and textual criticism.
This entry was posted in Infinities, Metaphysics, Philosophy of Time. Bookmark the permalink.

One Response to Aquinas and Potential Infinities

  1. Pingback: Spinoza on Infinity | Third Millennial Templar

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