Yeah, but is it actually actually infinite?

In response to my recent post about William Lane Craig’s kalām cosmological argument, several readers noted that Craig has replied to an objection like the one I raised, in several places, such as a response to a reader’s question at his Reasonable Faith website, and in his article (co-written with James Sinclair) on the kalām argument in Craig and Moreland’s Blackwell Companion to Natural Theology.  Let’s take a look at what he has to say.

First recall that examples like Craig’s famous Hilbert’s hotel thought experiment involve infinitely large collections of things (e.g. hotel rooms and guests) all of which exist at the same time.  Craig argues that various paradoxical consequences follow from these examples, which shows that the idea of an actually infinite collection of things is metaphysically suspect.   But a universe without beginning would entail an actually infinitely large collection – of days, years, or whatever other unit of time you care to take.  Hence (the argument concludes) the notion of a universe without a beginning is also metaphysically suspect.

My objection noted that Craig is (as I am) an adherent of presentism, i.e. the view that only present things and events exist, and thus that past and future things and events do not exist.  Given presentism, I argued, it is incorrect to say that a universe without beginning entails an actually infinite collection in the relevant sense.  For past things and events (including past days, years, etc.) do not exist.  Hence they do not form an infinite collection of things that exist all at once, as the hotel rooms and guests in the Hilbert’s hotel example do.  The only things and events that exist are present things and events, and they are not infinite in number.  The only day that exists is the current day, and a single day is obviously not an infinite number of days.  Something similar could obviously be said of weeks, months, years, etc.  Hence there is in the case of units of time nothing to parallel the rooms and guests of the Hilbert’s hotel example, so that the parallel fails even if we concede that the example succeeds in showing that an actually infinite collection is impossible.

As far as I can tell, Craig makes three points that might be thought to be relevant to my objection.  First, in both the Reasonable Faith Q & A and in the Blackwell Companion piece, he notes that the fact that past things, events, years, etc. do not exist does not prevent us from being able to count them.  We can correctly say, for example, that it has been fifteen years since 9/11, even though (given presentism) none of the years before the present one exist any longer.  And since we can count them (Craig seems to be saying) there must be a sense in which the years since 9/11 constitute a collection actually having fifteen members in it.  By the same token (so the argument seems to continue, if I understand it correctly) if the universe had no beginning, that would entail that there is an actually infinite collection of years.

Now, I’m not clear how this argument is supposed to constitute a reply to the objection.  No one denies that we can count past years even though they don’t exist anymore.  After all, we can count all sorts of things that don’t exist.  For example, Snow White knew seven dwarfs, and we can go through them, by name, and count them.  (First, Grumpy; second, Sleepy; etc.) But that we can count these dwarfs doesn’t entail that there is an actual collection with seven members in it, because the seven dwarfs, being fictional characters, are themselves not actual.  Similarly, that we can count past years – whether fifteen of them or an infinite number of them – doesn’t entail that they constitute an actual collection, since the past years themselves are also not actual in the relevant sense.

It seems to me that Craig’s argument here might be trading on an ambiguity between two claims:

1. The number of moments that have actually existed is infinite.

2. The number of moments that actually exist is infinite.

A beginningless universe would entail 1, but it would not entail 2, certainly not if presentism is true.  Yet 2, it seems to me, is what Craig needs for a beginningless universe to be relevantly like the Hilbert’s hotel example.  In the Hilbert’s hotel example, it is because we have infinite collections of things all of which exist at once that we get weird results, like wave after wave of infinitely large groups of guests arriving at the hotel and being able to check in even though the hotel is already full.  Precisely because all the past years, days, etc. do not still exist, it is hard to see how they constitute an actual infinite in the same sense of “actual.”  The collection of guests in the Hilbert’s hotel example is “actual” in the sense that the members all do exist; the collection of years in the beginningless universe scenario is “actual” only in the different sense that the members all did exist. 

Here’s another way to look at the problem.  Craig would not deny that it is legitimate for mathematicians to talk about infinite series in various ways, e.g. the infinite series of natural numbers.  The reason this is okay is that numbers (unlike hotel guests, hotel rooms, etc.) are not concrete objects.  Hence when talking about numbers we don’t get the bizarre results we get when considering scenarios in which an infinite collection of concrete objects exist.  But is a collection of units of time (minutes, days, years, etc.) more like a collection of concrete objects like guests and rooms, or is it more like a collection of abstract objects like numbers?  For Craig’s argument to work, it seems that we’d have to say that it is more like the former.  But in fact, this seems false.  It seems instead to be more like the latter.

This is especially plausible if, like Aristotle and Aquinas, we deny that time exists apart from change and the concrete objects that undergo change.  To speak of time apart from change is a bit like speaking of a universal like redness apart from actual red things—it is to engage in abstraction from the concrete conditions under which the thing in question (redness, or time) can actually exist.  Craig may be more inclined to think of units of time as relevantly analogous to concrete objects like hotel guests, etc. because he sympathizes instead with the Newtonian view that time can exist apart from concrete changing objects.

This brings me to a second remark Craig makes that might be thought relevant to my objection.  In the Blackwell Companion article, after developing the point about counting things that no longer exist, he asserts (contrary to what I just claimed) that “all the absurdities attending the existence of an actual infinite” do apply to a beginningless universe, despite past things and events no longer existing (p. 116).  For example, if the number of past events is infinite, he says, then the number of odd-numbered events is no smaller than the number of total events, since the series of odd numbers is of course infinite.  (He also rehearses other points along these lines.)

But the problem with this should be obvious from what has already been said.  Again, Craig does not have a problem with mathematicians talking about infinite series of natural numbers, despite the fact that the series of odd numbers is no smaller than the series of all natural numbers.  The reason is that numbers are abstract rather than concrete objects, whereas examples like Hilbert’s hotel are problematic because hotel rooms and guests are concrete rather than abstract.  A beginningless series of events will be problematic, then, only if it is more like the collection of rooms in Hilbert’s hotel than it is like the series of natural numbers.

But as I have said, the trouble with Craig’s position is that past things (whether events, years, or whatever) do not exist, at least not given presentism.  So they are not relevantly like the concrete objects which all exist together in the Hilbert’s hotel example.  Craig’s reply here thus seems to me to ignore the objection from presentism rather than answering it. 

A third remark from Craig which might be thought relevant to the objection I raised is also to be found on the same page of the Blackwell Companion article, where he cites Aquinas’s example of a blacksmith who has been working for an infinite amount of time and using one hammer after another.  The collection of hammers would constitute an actual infinite.  Now, Craig says that it would constitute an actual infinite even if the hammers did not still exist, and he cites the example only to illustrate his claim that past things need not continue to exist in order to constitute an actual infinite.  But it seems to me that what he should say, in order to try to make of this example a response to the objection from presentism that I have put forward, is this: Suppose such a blacksmith has been working for an infinite number of years, has used a new hammer each year, and has preserved each of these hammers.  Then we would have a collection that is actually infinite even by the presentist’s own lights.  And it would thus be relevantly analogous to the Hilbert’s hotel example.  So wouldn’t this show that the idea of a beginningless universe is paradoxical and metaphysically suspect in just the way Hilbert’s hotel is?

Perhaps this could be developed into a promising reply, but as it stands it too seems to me to fail.  For the most it would show is that there couldn’t be an actually infinite collection of hammers, for the same reason there couldn’t be (if the Hilbert’s hotel argument works) an infinite collection of rooms or guests.  But it doesn’t follow that there couldn’t be an infinitely old universe, precisely because days, years, etc., unlike hammers, don't stick around and thus don't lead to the existence of an “actual” infinite in the relevant sense.  The most the hammer example would show is that even in an infinitely old universe, you couldn't amass an infinite collection of things.  Why not?  Maybe because, though time itself needn't have had a beginning, types of material objects like hammers must have had one.  Nor could Craig easily dismiss this separation of what's true of time from what's true of material things, because, again, like Newton (and unlike Aristotle and Aquinas) he thinks that time can exist apart from the material things that change in time.

So, though I hate to disagree with Craig – I have nothing but respect for him, and have profited much from his work over the years -- it doesn’t seem to me that he has successfully rebutted the objection from presentism.  But maybe there’s another way to do it.  And as I’ve said, I’m not convinced that an infinitely old universe really is possible in principle, and thus I’m agnostic about the kalām argument.