Initial worries for HHA

Let's now turn our focus to HHA. Craig states the argument as follows (ibid.):

1. (A1) An actually infinite number of things cannot exist.

2. (A2) A beginningless series of events in time entails an actually infinite number of things.

3. (A3) Therefore, a beginningless series of events in time cannot exist.

There's supposed to be a close connection between (A3) and premise (2) of the kalam cosmological argument. By establishing (A3), Craig thinks that he is thereby supporting (2).

Before we move on to evaluate Craig's reasons for accepting (A1) and (A2), it's worth noting that the wording of this argument sometimes varies. For example, elsewhere, Craig spells out the argument as follows (Craig (Reference Craig2001b), 221):

1. (A1*) An actual infinite cannot exist.

2. (A2*) A beginningless series of equal past intervals of time is an actual infinite.

3. (A3*) Therefore, a beginningless series of equal past intervals of time cannot exist.

Here there are three noticeable differences from the previous formulation of the argument. First, instead of claiming that an actually infinite number of things cannot exist, (A1*) just says that ‘an actual infinite cannot exist’. Second, whereas Craig talks about ‘events’ in (A2), he instead talks about ‘intervals of time’ in (A2*). And third, (A2) says that a ‘beginningless series … entails an actually infinite number of things’’, but (A2*) says that a ‘beginningless series … is an actual infinite’. Despite these differences, I'll speak loosely about HHA, allowing that to name both of these arguments. I'll consider (A1)–(A3) the first formulation of HHA, and (A1*)–(A3*) the second formulation.Footnote ⁶ Our primary focus will be on the first formulation, but the second formulation will arise later in the discussion.

The justification for (A1) is based on certain thought-experiments which purport to show the absurdity of an actually infinite number of things existing. By ‘actually infinite’ Craig means to be talking about ‘a collection of definite and discrete members whose number is greater than any natural number 0, 1, 2, 3 …’ (Craig (Reference Craig2008a), 116). This is contrasted with a ‘potential infinite’, which is ‘a collection that is increasing toward infinity as a limit but never gets there’ (ibid.).Footnote ⁷ So, according to Craig, if an actually infinite number of things existed, that would have some absurd implications for the kind of world we live in. For example, consider Craig's favourite thought-experiment on this topic: Hilbert's Hotel – a hotel with an actually infinite number of rooms. Now we're to suppose that a new customer comes to the hotel asking for a place to stay when every room is already occupied. In such a hotel, we merely need to shift everybody down a room (so the person in room 1 moves to room 2, and the person in room 2 moves to room 3, etc.). Now room 1 is available for the new customer. This means that, even though every room in the hotel is full, new guests can always be accommodated. Moreover, if an actually infinite number of new customers all request a room, the hotel can accommodate all of them. Each person who already has a room just needs to move to the room that's twice the number of his or her current room. So the person in room 1 moves to room 2, and the person in room 2 moves to room 4, etc. This empties out all of the odd-numbered rooms, and the infinity of new guests can be accommodated. This could be repeated without end. In a world with Hilbert's Hotel and an unlimited supply of new guests (and an unlimited supply of patience!), an infinite number of people can check into the already full hotel every thirty minutes.Footnote ⁸ This, Craig says, is absurd. Moreover, he claims that the thought-experiment can be made even more counterintuitive. Suppose somebody checks out of the hotel. Now we have an empty room, yet the hotel is still housing the same number of guests (because infinity minus one is infinity). In fact, suppose everybody in an odd-numbered room checks out. In this case, an infinite number of people have checked out, yet the same number of people remain in the hotel as before (i.e. infinity). And if we don't like having an infinite number of empty rooms in our hotel, we can simply shift the remaining guests around until all of the rooms are again occupied (ibid., 116–119).

Let's grant Craig (at least part of) what he wants us to take away from this thought-experiment: Hilbert's Hotel really is absurd, and it therefore cannot exist. Let's suppose for the sake of argument that such a place is not a real metaphysical possibility. Given that we've granted the absurdity of Hilbert's Hotel, does it follow that (A1) is true? It certainly follows that it's impossible for there to be a hotel with an actually infinite number of rooms. It might even generalize to the conclusion that it's impossible for there to be any building with an actually infinite number of rooms, or even more generally, that it's impossible for there to be any concrete object with an actually infinite number of equal spatial parts. Moreover, since Craig has elsewhere run the same kind of argument to show that there cannot be an actually infinite number of marbles, baseball cards, books, etc., let's suppose that the argument shows that there can't be an actually infinite number of physical objects. Better yet, let's grant that there cannot be an actually infinite number of concrete objects, physical or not.

Nevertheless, it is certainly not clear that (A1), ‘an actually infinite number of things cannot exist’, follows from the absurdity of Hilbert's Hotel, at least if abstract objects count as ‘things’. Consider numbers, for example. If a number is a thing, and if there are an actually infinite number of them, then (A1) is false. Of course, some philosophers have held that numbers are things, and that there are an actually infinite number of them. Let's call such a view about numbers ‘Platonism’. In response to this worry, Craig reassures us that there are other views which hold that, for example, numbers don't really exist, or that there aren't an actually infinite number of them (ibid., 117). But surely this response misses the thrust of the worry. The obvious point to make is that if Platonism is correct, then it follows that the Hilbert's Hotel thought-experiment doesn't prove (A1). So it's Craig's burden to show that Platonism must be false. Rather than merely point out that we need not be Platonists, Craig needs to give an argument against Platonism. He may think that Hilbert's Hotel will suffice here, but I'm not at all inclined to agree. Should we conclude that just because it would be absurd for an actually infinite number of hotel rooms, marbles, baseball cards, or books to exist, that it would be equally absurd for an actually infinite number of numbers to exist? Or should we think that a suitably reformulated version of the thought-experiment – say, ‘Hilbert's Platonic Heaven‘Footnote ⁹ – would lead us to deny Platonism? If we're relying on intuitions alone here, it's unclear whether the intuitions will carry over. It may well be that our intuitions are better suited to tell us about concrete objects anyway.Footnote ¹⁰

As it turns out, the general conclusion that Craig draws on the basis of the Hilbert's Hotel thought-experiment, (A1), rules out a number of other philosophical theories as well. In addition to Platonism about numbers, one might believe in an infinite number of other abstract objects – e.g. propositions, properties, sets, possible worlds, etc. Alvin Plantinga, for example, believes that possible worlds are abstract objects, and that there are an actually infinite number of them (Plantinga (Reference Plantinga1976), 144).Footnote ¹¹ Likewise, (A1) entails that David Lewis's modal realism view of possible worlds is badly mistaken, since he believed in the existence of an infinite number of concrete worlds.Footnote ¹² Moreover, some people have believed that space is continuous, made up of an infinite number of points, but (A1) seems to rule this out as well. Craig considers this possibility, but he imagines that his opponent must be trying to use this as a clear counterexample to (A1). His response is to point out that the notion that space is continuous is unproven (Craig & Sinclair (Reference Craig, Sinclair, Craig and Moreland2009), 112). Seemingly, Craig thinks that it's up to his opponent to prove it. But again, Craig's premise seems to entail that space is not like this, which is also an unproven claim. True, if one could prove that continuous space is possible, then we'd have a counterexample to (A1). But since Craig is claiming that it's not possible, it's reasonable to expect him to prove it.

We're beginning to see that in order for HHA to go through, it has to settle a number of controversial metaphysical debates. And we're expected to think that these debates can be resolved by the Hilbert's Hotel thought-experiment, or parallel thought-experiments. It seems that there is plenty of room here to doubt that the absurdity of Hilbert's Hotel proves (A1).Footnote ¹³ Perhaps what it actually proves is something quite a bit more modest. Craig will undoubtedly point out that there are independent reasons to reject Platonism about abstract objects, David Lewis's modal realism, and the rest. But notice that there are really two distinct problems here. First, we can see that if any of these views are true, then Craig's premise is false. So Craig seems to have his work cut out for him in showing that all of these various views are false. But secondly, and perhaps more importantly, I want to claim that whatever else can be said against these metaphysical views, we should probably not think that the Hilbert's Hotel thought-experiment disproves all of them. So even if Craig does have good reasons to reject these views, this would not show us that Hilbert's Hotel provides us with a good reason to reject these views, and therefore to affirm (A1). At the very least, we should probably be hesitant to accept (A1) on the basis of Hilbert's Hotel, given that it rules out a number of plausible metaphysical views that have been held by some very intelligent people.

At this point I'm not quite sure what the full import of Hilbert's Hotel actually is. But having registered some of my worries, let's suppose Craig is right to think that the thought-experiment actually shows that (A1) is true. I'm certainly dubious about whether we can really make such a determination on the basis of the intuitions we get about the hotel, but I want to be as charitable to Craig here as I reasonably can be in order to see where the argument goes from there.

When it comes to defending (A2), Craig usually doesn't have as much to say. In addition to claiming that it's ‘obvious’, he writes: ‘If the universe never began to exist, then prior to the present event there have existed an actually infinite number of previous events. Thus, a beginningless series of events in time entails an actually infinite number of things, namely, events’ (Craig (Reference Craig2008a), 120).

Craig doesn't go into great detail explaining exactly what he means by an ‘event’, but it's apparently supposed to be an existing thing. It's notable, however, that in another context Craig seems to acknowledge a chasm between events, on the one hand, and things, on the other:

If being a thing and being an event are different in this way, it's unclear how HHA is supposed to get off the ground in the first place. (A1) tells us that there can't be an actually infinite number of things, and (A2) tells us that a beginningless past would entail an actually infinite number of events (read: events, not things). Craig would need some additional premise to make the argument go through. For example, he would need:

Then, of course, we would like to know what kind of justification there could be for (IE). Hilbert's Hotel wouldn't work here, because it says nothing at all about events.

It's also worth pointing out that some philosophers have denied that events exist. Terence Horgan, for example, argues that ‘it is a mistake to posit events at all’ on the grounds that, ‘despite the initial appearances, there is no real theoretical need to posit events. So, since their elimination yields an important simplification of ontology, we should banish them from existence’ (Horgan (Reference Horgan1978), 28). Similarly, Peter van Inwagen writes: ‘There are, I would say, no events. That is to say, all statements that appear to involve quantification over events can be paraphrased as statements that involve quantification over objects, properties, and times – and the paraphrase leaves nothing out’ (van Inwagen (2011), 220). Craig himself agrees that such a position is plausible, though it's not clear to me whether he is willing to endorse it. In (Craig (Reference Craig2011b), 220), immediately after claiming that events are among ‘the sorts of thing that many metaphysicians plausibly deny exist’, Craig states: ‘These things are real in the sense that they are not illusory, but they are not, properly speaking, existents.’ As it happens, this concession causes problems for HHA, as will soon become apparent.

Elsewhere, as already noted, Craig substitutes ‘intervals of time’ into the argument in place of ‘events’. It's certainly questionable whether an interval of time is a ‘thing’ in the required sense. Things, at least of the non-eternal sort, ordinarily come into and pass out of existence; intervals of time merely pass.Footnote ¹⁴ On the face of it, at least, it seems we're talking about different categories altogether when we use examples about hotel rooms and other physical objects and then apply that reasoning to intervals of time. So I'm inclined to think that someone who accepts the premise that an actually infinite number of things cannot exist has some leeway in which he or she might deny that an event or interval of time is the kind of ‘thing’ to which the argument applies.

Craig has claimed that ‘event’ just means ‘any change’ (Craig & Sinclair (Reference Craig, Sinclair, Craig and Moreland2009), 106). And presumably, a change involves a thing losing or gaining one or more properties. Here again some of the same worries as before begin to crop up. Is a change supposed to be a thing in the way that a physical object is a thing? If a change is just something that happens to a thing, and is not itself a thing, then for all I know there could be a possible world with a finite number of things (i.e. objects) which nevertheless change an infinite number of times. Perhaps there's a world that just consists of a sphere that has been changing colours – say, alternating between being red and being green – from eternity, and will continue to do so forever. How many things exist at that world? Maybe just three: the sphere, redness, and greenness.Footnote ¹⁵ Here's an alternative answer that, I suspect, most of us would find counterintuitive: an infinite number of things exist at that world, since in addition to the sphere and the properties of redness and greenness, you have to count the infinite number of colour changes as existing things. If HHA convinces Craig that there cannot be a world like the one envisioned, then it seems he would endorse this counterintuitive answer to our question. Otherwise, it's unclear why Hilbert's Hotel would allow us to conclude that there can't be an infinite number of changes.Footnote ¹⁶

Let's nevertheless set that worry aside and assume that an event, whatever it is, is a thing which exists – the kind of thing that Hilbert's Hotel shows us to be limited in number. Craig seems to want to say that to deny (2) is to suppose that an actually infinite number of past events (or intervals of time, or changes) compose the past. But given that we have seen that an actually infinite number of things cannot exist, we must deny that there can be an actually infinite number of past events. And to deny that is just to affirm (2).

It's important to pause here and consider exactly how HHA is supposed to work. We've already accepted (for the sake of argument, at least) that an actually infinite number of things cannot exist. Craig thinks that anybody who denies this will be forced to accept an undesirable view. In his words: ‘It is indisputable that if an actually infinite number of things were to exist, then we should find ourselves landed in an Alice-in-Wonderland world populated with oddities like Hilbert's Hotel’ (Craig (Reference Craig2008a), 119–120). So it looks like the argument is supposed to be construed in basically the following way: we know that an actually infinite number of things cannot exist, so we should reject any view that entails that an actually infinite number of things exist. But, as it happens, somebody who believes that the universe is past-eternal is committed to saying that an actually infinite number of things (namely, events) exist. Since this conflicts with what we know to be the case (on the basis of Hilbert's Hotel), we should reject that view.Footnote ¹⁷ One way of formalizing the argument would go like this:

1. (M) It's metaphysically impossible for an actually infinite number of things to exist.

2. (B) The universe began to exist.

   1. (B1) If ∼(B), then ∼(M)

   2. (B2) (M)

   3. (B3) Therefore, (B)

Note that (M) simply paraphrases premise (A1), and (B) is identical to premise (2) of the kalam cosmological argument. Now, when Craig defends HHA in order to prove premise (2), I think what he's after is something like (B1)–(B3). Since the conclusion of this argument is just (B), and since that's precisely the same as (2), this way of formulating the argument at least has the virtue of making it clear how it's supposed to support (2). Notice that on Craig's rendering of HHA, it's not immediately clear what the connection is between (A3) and (2). Finally, keep in mind that Craig's reason for accepting (B1) is that the falsity of (B) entails that an actually infinite number of past events exist.

Problems from the metaphysics of time

It seems to me that, in making this argument, Craig is presupposing a certain view about the metaphysics of time. As such, it would be good here to distinguish briefly two competing theories in the current debate: the A-Theory and the B-Theory. On the A-Theory of time, which is often referred to as the tensed or dynamic theory, there is a real, objective difference between past, present, and future. Generally, according to this view, the present has a privileged ontological status. In fact, some philosophers have thought that the A-Theory is committed to a doctrine called ‘presentism’. On that view, only the present is real. In Craig's own words, presentism is

Craig also seems to believe that A-Theorists are committed to presentism. For example, in one place he apparently conflates the two views, writing: ‘According to the A-Theory, things/events in time are not all equally real: the future does not yet exist and the past no longer exists; only things which are present are real’ (Craig (Reference Craig2008a), 121).

Elsewhere, in a discussion of McTaggart's Paradox, he approvingly notes that other thinkers have concluded that the A-Theory is committed to presentism:

Indeed, Craig believes that by accepting presentism, the proponent of the A-Theory can avoid McTaggart's Paradox, which would otherwise be a good argument against the A-Theory. Thus, regarding certain versions of the A-Theory which reject presentism, such as the so-called ‘growing block theory’ and the ‘moving spotlight theory’, Craig says that it's ‘doubtful … whether these hybrid theories are coherent’ (Craig (Reference Craig and Walls2008b), 597–598). In Craig's view, then, presentism is simply the only viable version of the A-Theory.

The B-Theory, by contrast, is often referred to as the tenseless or static theory of time. On this view, no point in time is ontologically privileged. As Craig puts it: ‘all events in time are equally real’ according to the B-Theory (Craig (Reference Craig2008a), 121). Therefore, the distinction between past, present, and future does not designate anything of ontological significance; past events are just as real as present events. Compared to Napoleon, William the Conqueror is in the past. But compared to Julius Caesar, William the Conqueror is in the future. There is no objective fact about whether something or somebody exists in the past, present, or future, because those characterizations are merely relative to times. B-Theorists usually talk about a different set of temporal relations: earlier than, simultaneous with, and later than.

Now, suppose I deny that the universe had a beginning. Craig would then say that I'm committed to the existence of an actually infinite number of things – namely, the infinite number of past events. If Craig is right, then I either have to give up my belief that the universe never began to exist, or else I have to give up my earlier concession that an actually infinite number of things cannot exist. But suppose I'm a presentist. As Craig himself characterized the view in the quoted passage above, a presentist doesn't believe that past events exist. So a presentist who denies that the universe began to exist is not thereby committed to the existence of an actually infinite number of past events. As a presentist, I would simply respond to Craig by reminding him that, in my view, past events don't exist at all – so there's no way I'm committed to an infinite number of them existing. Thus, the absurdity Craig was aiming for with his argument does not result if we accept presentism. Recall that the absurdity was supposed to result for anyone who believes, contrary to (A1), that the number of things that exist is actually infinite.Footnote ¹⁸

Perhaps, then, the Hilbert's Hotel argument could be geared towards proponents of a B-Theory of time, since that view would maintain that past events really exist. But at the very least, the A-Theorist should be off the hook here. If Craig is correct that the A-Theory entails a commitment to presentism, and if a presentist can easily reject HHA on the grounds that premise (A2) presupposes the falsity of presentism, it follows that the argument is ineffective against a proponent of the A-Theory. One might well conclude that Craig is presupposing the B-Theory in his defence of HHA.

This might not appear too problematic if Craig were to go on to defend the truth of the B-Theory. The problem, however, is that he is a committed A-Theorist and presentist (Craig (Reference Craig2001b), 115–216). This means that, by his own lights, the argument should not be accepted. Furthermore, aside from HHA, Craig uses a second, independent philosophical argument for premise (2) of the kalam cosmological argument, and he is explicitly clear in his writings that this other argument depends upon the truth of the A-Theory. Writing about one of the premises of this argument, he says: ‘As obvious as this premise may seem at first blush, it is, in fact, a matter of great controversy. It presupposes once again an A-Theory of time’ (Craig & Sinclair (Reference Craig, Sinclair, Craig and Moreland2009), 124). Moreover, the kalam cosmological argument as a whole depends upon the truth of the A-Theory, according to Craig:

In short, then, in defending these arguments Craig is presupposing inconsistent metaphysical views. When he defends the first philosophical argument in support of (2), HHA, he seems to presuppose the B-Theory, and when he defends the second philosophical argument in support of (2) he presupposes the A-Theory. And since the kalam argument as a whole requires the A-Theory, Craig's defence of HHA falls flat.

Craig's response

My main contention in the previous section was that Craig's use of Hilbert's Hotel is ineffective against proponents of the A-Theory of time, as long as the only viable version of the A-Theory is presentism. This is because a presentist who claims that the universe is past-eternal is not committed to the existence of an actually infinite number of events. Past events, on that view, don't exist at all.

In a recent defence of the kalam cosmological argument co-authored by Craig and James Sinclair, the authors responded to a related worry. According to them, both Aristotle and Thomas Aquinas accepted (i) the presentist view that past events don't actually exist, and therefore (ii) the notion that ‘the series of past events [is] a potential infinite’ rather than an actual infinite (Craig & Sinclair (Reference Craig, Sinclair, Craig and Moreland2009), 115). Note that these are two separate contentions. I have been arguing that (i), all by itself, is sufficient to cause major problems for HHA.

In responding to Aristotle and Aquinas, it seems to me that Craig and Sinclair unfortunately latch onto (ii) and utterly neglect (i). They write: ‘The question, then, is whether events’ temporal distribution over the past on a presentist ontology precludes our saying that the number of events in a beginningless series of events is actually infinite’ (ibid.). Surely this is a question. But I'm not convinced it's the question. One could agree with Craig and Sinclair that, even on a presentist ontology, if the universe didn't begin to exist, then there have been an infinite number of events. But this does nothing to ease the worry raised by (i), which I take to be the more important problem here. Even though, in such a scenario, the number of events that have occurred is actually infinite, the fact remains that on a presentist ontology none of those events exist.Footnote ¹⁹ Recall that the entire argument was predicated on the notion that an actually infinite number of things cannot exist.

Wes Morriston has made the point that the absurdity of Hilbert's Hotel results from the fact that every member of an infinite set exists at once, and is able to be moved about in relation to one another (Morriston (Reference Morriston2003), 296). As we have seen, the doctrine of presentism presents a problem for Craig if he wants to use the thought-experiment to argue that the number of past events must be finite. But there's another problem lurking here as well. Morriston has pointed out that past events cannot be shuffled around in the way that hotel guests can, which vitiates Craig's attempted use of the thought-experiment (ibid., 296–297). Craig's response to this point is worth considering. He insists that in order to see the absurdity you don't need to be able to manipulate the objects in that fashion. He says:

This response is surely strange, given what Craig has written elsewhere. Consider a separate worry for the Hilbert's Hotel argument that goes something like this: mathematicians know full well that you can't subtract infinities, yet the illustration supposes we ‘subtract’ an infinite number of guests as they check out from the hotel. On this point, Craig writes: ‘In trans-finite arithmetic, inverse operations of subtraction and division are prohibited because they lead to contradictions; but in reality, one cannot stop people from checking out of the hotel if they so desire!’ (Craig (Reference Craig2008a), 120). Yet, notice that this is precisely what Craig is now doing in response to Morriston's objection – namely, imagining the thought-experiment in such a way that the people are prevented from being able to check out of the hotel. If the thought-experiment is described so that the guests are free to switch rooms and check out of the hotel, then it isn't analogous to past events, which can't be moved around in relation to one another. And if the thought-experiment is redescribed, in the way Craig imagines, to make it more analogous to past events, then it comes with a cost: Craig can no longer claim that the hotel guests are free to check out in response to the objection that you can't subtract infinities. Craig thereby sacrifices his response to one objection in his attempt to respond to another objection.

Let's set this problem aside and refocus our attention on the problem that presentism poses for Craig. One way for Craig to respond to this objection, suggested by his response to the previous objection, is to claim that it doesn't matter whether or not the past events exist, as long as they can be numbered. Indeed, he claims that since past events can be counted, we can simply run a Hilbert's Hotel-style thought-experiment using an infinite number of past events rather than an infinite number of hotel rooms. The way to carry out this sort of procedure is, first, to number the events mentally, starting with the present and working our way back. Then, we simply note the same sorts of absurdities that attended the original thought-experiment. In Craig's words, if the past is eternal,

But all of this is supposed to be metaphysically absurd, just like Hilbert's Hotel. And it's supposed to give us reason to reject the view that there are an actually infinite number of past events.

This response won't work for a couple of reasons. First, if we accept this line of argument it looks like we'll have to accept a parallel argument that shows that the universe cannot be future-eternal.Footnote ²⁰ We can mentally number future events and do the same operations to them. So if Craig's argument shows that time must have a beginning, then a parallel argument shows that time must come to an end. But this is unacceptable. Surely the argument doesn't really prove such a thing as that! Craig might want to claim that we can't count events that haven't yet occurred, whereas we can count events that have occurred. This is false. If we construe events as ‘equal intervals of time’, then we can surely mentally number and count future events just as easily as we can past events. Or consider Morriston's proposal that God decrees that two angels take turns praising him for a minute at a time for eternity (Morriston (Reference Morriston2010), 443). I suggest it would not be too difficult to number these events mentally and then mentally shuffle things around in the way Craig is doing for past events.

Craig's preferred response here is to argue that although time will never come to an end, the series of future events is nevertheless only potentially infinite, not actually infinite (Craig (Reference Craig2010b) ). Thus, our Hilbert's Hotel-style reasoning would not apply to the future. This is a mistake. If the angels will be taking turns praising God forever, then the number of future praises must be the same as the number of past praises given the assumption that they have been taking turns praising God for an eternity. Craig insists that the number in the latter case is actually infinite. But if that's right, then he cannot deny that the number in the former case is actually infinite as well, since the past praises and the future praises can be put in a one-to-one correspondence. Craig's insistence that the future praises are not real is irrelevant, since past events aren't real either on the presentist view. Nevertheless, as Craig himself points out, we can still mentally number them and generate (alleged) absurdities. But since nobody thinks this argument seriously shows that time must come to an end, we shouldn't think it shows that time must have had a beginning, either.

The second reason this line of response won't work to Craig's advantage is that it's not clear whether the revised thought-experiment results in a metaphysical absurdity like Hilbert's Hotel. Hilbert's Hotel was absurd because of the counterintuitive implications that resulted from having an actually infinite number of things existing all at once. To claim that the same absurdity can be generated with an actually infinite number of non-existing things like past events makes it seem like Craig's complaint is with the mathematical legitimacy of infinity, not just the idea that the actual infinite is instantiated in reality. It's true that the number of past events is the same as the number of even past events, in Craig's scenario, and this is a straightforward implication of saying that the number of past events is actually infinite. But what is metaphysically absurd about this, if the events don't exist? It still remains that the only conclusion we're able to draw from the absurdities that are generated from mentally numbering and then mentally shuffling past events is this: an infinite number of past events cannot exist. And as we've seen, the presentist is already on board with that, regardless of whether or not she thinks that an infinite number of them have occurred.Footnote ²¹

Because HHA has been variously formulated, Craig might concede that a presentist would be unaffected by the contention that an actually infinite number of things cannot exist. Instead, he might say, we should examine the second formulation of HHA:

1. (A1*) An actual infinite cannot exist.

2. (A2*) A beginningless series of equal past intervals of time is an actual infinite.

3. (A3*) Therefore, a beginningless series of equal past intervals of time cannot exist.

Although the presentist escapes the original argument by avoiding a commitment to an infinite number of things existing, this alternative formulation doesn't explicitly say anything about an infinite number of things existing, but instead talks about the existence of actual infinites themselves.

To assess this response, we need to ask what premises (A1*) and (A2*) are supposed to mean, and whether adequate support has been provided for them. First, we'd want to know what it means to say that ‘an actual infinite cannot exist’. The most natural interpretation of this claim seems to be that an actually infinite number of things cannot exist. Yet we've already seen the problem that accompanies this formulation of the premise, since it allows presentists (Craig included) off the hook. We want to find a reading of this premise which doesn't have this defect. So what can it mean to say that ‘an actual infinite cannot exist’ which does not reduce to ‘an actually infinite number of things cannot exist’? If we speak of things this way, then it apparently must be the case that, if an actual infinite could exist, then it could exist even when the number of existing things is finite.Footnote ²² One way of helping us get a grip on this is to ask what way of formulating the argument will allow Craig to get what he wants – namely, the conclusion that there cannot have been an actually infinite number of events prior to now. Here's a first approximation of the principle he needs:

With that in mind, we can see that (M*) would help Craig's argument go through. But what reason has he given (or could he give) in support of this principle? Recall that Hilbert's Hotel was an illustration in which an actually infinite number of rooms existed all at once. At best, that could justify (M) from earlier, but it leaves open the possibility that (M*) is false. What we'd need is some other illustration which shows that an actually infinite number of things cannot even compose a series in which almost all of the things no longer exist.

One possibility is briefly mentioned by Craig and Sinclair, in their article from The Blackwell Companion to Natural Theology:

While I again agree with Craig that the number of hammer-breaking events in such a scenario is actually infinite,Footnote ²³ it's not clear to me whether such a scenario is really impossible. If the blacksmith has been making a new hammer out of new materials each time his old hammer breaks, then presumably the world would be populated with an actually infinite number of broken hammer pieces. This can be ruled out by the use of a ‘Hilbert's Hammer Collection’ thought-experiment, as I granted earlier in the article. But if the blacksmith has been merely fixing the same hammer every time it breaks, then I'm not yet convinced that the scenario is impossible. Something more must be said to motivate the intuition that the eternal blacksmith scenario is absurd (i.e. metaphysically impossible). While I of course find the scenario hard to fathom, it's no more difficult to imagine than any other past-eternal scenario. And, in any case, it's hard for me to fathom either a world without beginning or a world with an absolute beginning. So I'll leave it to Craig to offer up some other thought-experiment to justify (M*).

Yet another avenue is open to Craig to respond to our hypothetical presentist. Craig might claim that, since the argument would obviously work to show that the past is finite given the B-Theory of time (in which all events are equally real), it's plausible to think that it must work for the A-Theorist as well (ibid., 115). After all, how could one's metaphysical theory of time make a difference to whether past events are finite or infinite in number? It would be strange indeed if one's acceptance of the B-Theory forced her to conclude that the universe had a beginning, whereas her acceptance of the A-Theory would have kept open the possibility of a past-eternal universe. We can formulate this line of argument as follows:

1. (C1) If HHA is sound given the B-Theory, then it's sound given the A-Theory.

2. (C2) HHA is sound given the B-Theory.

3. (C3) Therefore, HHA is sound given the A-Theory.

And here I think there's at least some plausibility to premise (C1). But one person's modus ponens is, of course, another person's modus tollens. I think the following argument is more compelling:

1. (C1) If HHA is sound given the B-Theory, then it's sound given the A-Theory.

2. ∼(C3) HHA is not sound given the A-Theory.

3. ∼(C2) Therefore, HHA is not sound given the B-Theory.

It seems to me that we have at least as much reason to accept ∼(C3) as we have to accept (C2). I've shown, taking words straight from Craig's own writings, that there is a clear inconsistency in what he says about the A-Theory and presentism on the one hand and HHA on the other.

On a similar note, one can see in Craig's more recent writings that he tries dealing with the problem we've raised by noting that, on a ‘growing block’ theory of time, the objection can't even get off the ground, since that view posits that past events do exist (ibid. and Craig (Reference Craig2010b), 456). And he's right about that. The primary objection I've been pressing in this article wouldn't have got any traction if we were evaluating how a growing block theorist could reply to HHA. But I don't see how this helps Craig's case, since he is a presentist, not a growing block theorist. If I agree with Craig that presentism really is the only coherent version of the A-Theory, then I've got to examine how the argument fares against my own position. Craig's argument should not sway the presentist, even if it should sway the growing block theorist. Craig could, I suppose, renounce presentism and start endorsing a view that he's previously said was incoherent. Otherwise, I can't see how the growing block theory is going to offer any comfort to Craig, who remains a presentist.

Lastly, Craig has recently offered one other response that might apply to the presentist's objection. He claims that on a presentist ontology, past events still have some sort of ontological status not shared by future events. He writes:

In context, Craig is here arguing against Morriston's claim that HHA commits Craig to the view that the future must be finite. Craig is trying to show that there is a relevant difference between the past and the future which allows the argument against the infinite past to go through but which does not similarly apply to the future. The strategy is to say that, although neither past nor future events exist, past events are part of the actual world, whereas future events are not. Might this strategy help Craig overcome the problem that HHA is unsound given the truth of presentism? After all, if Craig is right, then the presentist who believes that the universe is past-eternal is positing a world in which, so to speak, an infinite number of things have appeared on the radar – that is, an infinite number of things belong to that world in some sense.

It's not clear to me why this should be a problem – i.e. why I should be motivated to reject the notion that the actual world is something to which an actually infinite number of past events belong, in some sense. The past events aren't real, according to the presentist; they don't exist.Footnote ²⁴ By positing such a world as this, we are not thereby embracing the (allegedly) absurd conclusion that an actually infinite number of things exist, as was the case with Hilbert's Hotel. So I don't know what the problem is supposed to be. Whatever else can be said against this line of response on Craig's behalf, HHA would need to be reformulated if he wants to use this ontology to overcome our objection. Here's a rough sketch of how Craig could reformulate HHA:

1. (D1) There cannot be a world in which an actually infinite number of things have been actualized.

2. (D2) If the actual world is one in which the universe is past-eternal, then there is a world in which an actually infinite number of things have been actualized.

3. (D3) Therefore, the actual world cannot be one in which the universe is past-eternal.

While this argument might be worth investigating, I'm not convinced that Hilbert's Hotel applies to the relevant premise (D1). I'll leave it to Craig to prove otherwise.Footnote ²⁵