Traversal is always understood to be from term to term. But whatever past day is designated, from that (day) to this there are finite days that can be traversed. But the objection proceeds from this, that, positing the extremes, there are infinite terms in between. [ST 1.46.2 ad 6]
Alan in the comments replied to this:
Aquinas didn't come close to refuting the "traversal of the infinite" argument. On the contrary, the fact that traversal requires two termini supports the kalam argument. It's the absence of a beginning terminus given an infinitely old universe that creates the problems, exactly as the kalam arguer contends.
To which I replied:
I'm baffled by the claim that Aquinas doesn't refute the traversal argument. If every traversal requires a beginning and an end, and an infinite past has no beginning, this is a problem only if we already assume that traversal of an infinite past would require traversal of infinite days. But on the infinite past view, every day in the past is finitely distant from the present; it's just that for every finitely distant day there's a day that is more distant. Thus this is true: For every day in the past, traversal of the days from that day to today is traversal of a finite number of days. The fact that there are infinite such days doesn't change this. This is true just as much as it is true that the fact that every integer is a finite distant from 1 is not affected by the fact that there are infinite integers.
That's the background; Alan has a new post up in response to this that I'd like to comment on. But first I want to make some distinctions. With regard to this topic, we often make a distinction between a potential and an actual infinite. It is important that we tread carefully here, because 'actual infinite' does not mean the same as 'actually infinite'. Every actual infinite is actually infinite, but something can be actually infinite without being an actual infinite. If Aristotle's solution to the infinite divisibility problem is right, for instance, the potential divisions of a line segment are actually infinite. This is not the same as to to say that they constitute an actual infinite -- to constitute an actual infinite the line segment would have to be actually divided into infinite parts. Likewise, the set of integers is actually infinite; but it is not an actual infinite. The reason is that in one case -- 'actual infinite' -- the 'actual' means 'not of something potential'; in the other -- 'actually infinite' -- it means 'not merely apparently'. This is significant.
Alan's response to my objection is that it conflates potential and actual infinity. As he says:
(A) is clearly true when we're talking about a potential infinite. We start at the present and run through the time series in reverse, moving farther and farther into the past. Nevertheless, at any point we stop at, we're only a finite remove from the present. But if the distance from past event E to the present is actually finite, then we haven't yet captured the idea of an actually infinite past.
Similarly, the notion of ever larger integers being still a finite remove from 1 is that of a potential infinite, of a magnitude increasing without bound, not of an actual infinite.
(A) is the claim I made in bold above. This response, I think, conflates the actual infinite with the actually infinite. Alan is right that the integers are a potential infinite; it does not follow from this, however, that they are not actually infinite. And that's the key. If anything is actually infinite the integers are; but you cannot start from the premise "The integers are actually infinite" to "Some integer is infinitely distant from some other integer." In fact, the former is necessarily true and the latter is necessarily false: no integer is infinitely distant from any other integer, because every integer is, by its very nature, finitely distant from every other integer. Pick any integer you like, it is a finite distance from every other integer. Nonetheless, the integers are actually infinite, because there are infinitely many such finite distances. Alan is, I think, confusing 'indefinite finite' with 'potential infinite'.
That is the point of Aquinas's response to the traversal argument. In claiming that the past is infinite the advocates of the eternity of the world are not committed to saying that any day is infinitely distant from any other day, only that there are infinitely many days finitely distant from each other. Thus the traversal argument fails because it assumes that the claim of an infinite past means that there is an infinite that is traversed; but this is false: necessarily, there is no infinite to be traversed, only infinitely many possible finite traversals of any size you choose. The case is, if we are considering only the infinity involved, exactly parallel to the case of the integers: within the set of integers, there is no actual infinite to be traversed from integer to integer because no integer is infinitely distant from any other; but the integers are actually infinite.
So, in other words, infinitely many traversals of finite distances is not the same as traversal of an infinite distance. The claim that there is no traversal of an infinite (as opposed to infinite traversals of finites) in the infinite past can't be shown wrong unless some other consideration is added that shows that the days of an infinite past must not only be actually infinite, but must constitute an actual infinite that has to be traversed.
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