Infinity

I discussed this topic previously, but without coming to a definite conclusion. Here I will give what I think is the correct explanation.

In his book Infinity, Causation, and Paradox, Alexander Pruss argues for what he calls “causal finitism,” or the principle that nothing can be affected by infinitely many causes:

In this volume, I will present a number of paradoxes of infinity, some old like Thomson’s Lamp and some new, and offer a unified metaphysical response to all of them by means of the hypothesis of causal finitism, which roughly says that nothing can be affected by infinitely many causes. In particular, Thomson’s Lamp story is ruled out since the final state of the lamp would be affected by infinitely many switch togglings. And in addition to arguing for the hypothesis as the best unified resolution to the paradoxes I shall offer some direct arguments against infinite regresses.

Thomson’s Lamp, if the reader is not familiar with it, is the question of what happens to a lamp if you switch it on and off an infinite number of times in a finite interval, doubling your velocity after each switch. At the end of the interval, is it on or off?

I think Pruss’s account is roughly speaking correct. I say “roughly speaking” because I would be hesitant to claim that nothing can be “affected” by infinitely many causes. Rather I would say that nothing is one effect simultaneously of infinitely many causes, and this is true for the same reason that there cannot be an infinite causal regress. That is, an infinite causal regress removes the notion of cause by removing the possibility of explanation, which is an intrinsic part of the idea of a cause. Similarly, it is impossible to explain anything using an infinite number of causes, because that infinity as such cannot be comprehended, and thus cannot be used to understand the thing which is the supposed effect. And since the infinity cannot explain the thing, neither can it be the cause of the thing.

What does this imply about the sorts of questions that were raised in my previous discussion, as for example about an infinite past or an infinite future, or a spatially infinite universe?

I presented an argument there, without necessarily claiming it to be correct, that such things are impossible precisely because they seem to imply an infinite causal regress. If there an infinite number of stars in the universe, for example, there seems to be an infinite regress of material causes: the universe seems to be composed of this local portion plus the rest, with the rest composed in a similar way, ad infinitum.

Unfortunately, there is an error in this argument against a spatially infinite world, and in similar arguments against a temporally infinite world, whether past or future. This can be seen in my response to Bertrand Russell when I discuss the material causes of water. Even if it is possible to break every portion of water down into smaller portions, it does not follow that this is an infinite sequence of material causes, or that it helps to explain water. In a similar way, even if the universe can be broken down into an infinite number of pieces in the above way, it does not follow that the universe has an infinite number of material causes: rather, this breakdown fails to explain, and fails to give causes at all.

St. Thomas gives a different argument against an infinite multitude, roughly speaking that it would lack a formal cause:

This, however, is impossible; since every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental.

By this argument, it would be impossible for there to be “an infinite number of stars” because the collection would lack “a species of multitude.” Unfortunately there is a problem with this argument as well, namely that it presupposes that the number is inherently fixed before it is considered by human beings. In reality, counting depends on someone who counts and a method they use for counting; to talk about the “number of stars” is a choice to break down the world in that particular way. There are other ways to think of it, as for example when we use the word “universe”, we count everything at once as a unit.

According to my account here, are some sorts of infinity actually impossible? Yes, namely those which demand an infinite sequence of explanation, or which demand an infinite number of things in order to explain something. Thus for example consider this story from Pruss about shuffling an infinite deck of cards:

Suppose I have an infinitely deep deck of cards, numbered with the positive integers. Can I shuffle it?

Given an infinite past, here is a procedure: n days ago, I perfectly fairly shuffle the top n cards in the deck.

This procedure is impossible because it makes the current state of the deck the direct effect of what I did n days ago, for all n. And the effect is a paradox: it is mathematically impossible for the integers to be randomly shuffled, because any series of integers will be biased towards lower numbers. Note that the existence of an infinite past is not the problem so much as assuming that one could have carried out such a procedure during an infinite past; in reality, if there was an infinite past, its contents are equally “infinite,” that is, they do not have such a definable, definite, “finite” relationship with the present.

3 thoughts on “Infinity

  1. I’m very pleased to see a new post, and hope that you’re doing well.

    I have a few general questions about material causality, as related to this problem of infinities. When speaking about ordinary objects and their material causes, enumerating their parts helps us to understand the nature of the object–that my glasses are made of two glass lenses, several bits of metal and a few pieces of plastic genuinely sheds light on understanding the object (at least in a rudimentary way). This is a simple case largely because glasses are a functional object, and knowing what it is made of is related to how it achieves its function. Though I could split my glasses into smaller and smaller parts, and conceivably imagine an infinite list of parts, this enumeration is not aiding in understanding my glasses.

    However, the case of the “universe” gets trickier for me to understand, because the universe is not “functional” in the same way my glasses are. It is natural for us to say that the universe is composed of stars and dust and dark matter, and this seems to give us some understanding–but having conceptually divided the universe into these parts, we seem to be stuck saying that there might be an infinite amount of these things. But infinite causality does not contribute to understanding, which makes it seem like considering the universe as stars (if we understand stars to be infinite in number) are -not- in fact helpful for understanding the universe. Because the universe is not functional in the same way as artifacts, it’s harder for me to understand which ways of conceptualizing are better for understanding.

    Does saying “the universe is composed of stars” without actually giving a number of stars escape this problem? Does just referring to the universe as a single wave function help? Do we have to avoid concepts that even -might- end up being described as infinite if we want to further our understanding?

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  2. If you made an exhaustive, or even potentially exhaustive, list of sorts of stuff and said the universe was made out of those things (as with stars, dust, and dark matter), that seems to be a sort of material causality. But not in the sense that you took a physical object and divided it up into parts: “stars” is not some particular part of the universe. Rather it is more of a conceptual division, and these are conceptual parts. But no infinity is involved here. The problem with infinity arises if you attempt to take individual quantitative parts and attempt to exhaustively enumerate them. So the general statement about the universe does help us understand something, but “the universe has a star here, some empty space there, a cloud of dust there, and so on….” does not. No one would even attempt the latter as an explanation, whereas the former would be common.

    As with your example of a wave function, there could be other approaches to the question of the material cause of the universe. A wave function in particular might be more of a formal thing, but certainly Aristotelian prime matter would be one way of thinking about it which would not result in an infinite list.

    Not sure if that helps or not — I’m not completely sure what the exact issue is.

    I’m doing ok. I expect to be posting again at least from time to time, although perhaps not as often as in the past.

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