Saturday, August 3, 2013

Hedrick on Hilbert's Hotel and the Actual Infinite (Part Two)

(Part One)

This series is about Landon Hedrick’s recent article “Heartbreak at Hilbert’s Hotel”, which analyses and critiques a crucial cog in the argumentative machinery underlying William Lane Craig’s Kalam cosmological argument. I begin with a quick recap of the key ideas from part one.

In the second premise of the Kalam argument, William Lane Craig claims that the universe began to exist. Now, Craig doesn’t just claim that this is contingently true of our universe, he claims it is necessarily true of any temporal universe. In other words, he thinks that there cannot be a temporally extended universe that did not begin to exist. He supports this with two philosophical arguments against the existence of an infinite sequence of events. The one we are examining is called the Hilbert’s Hotel Argument, which is typically stated as follows:

HHA Type I
(1) An actually infinite number of things cannot exist. 
(2) A beginningless series of events in time entails an actually infinite number of things.
(3) Therefore, a beginningless series of events in time cannot exist.

We refer to this version of the argument as “HHA-I”. As you might have gathered from the name, there is another version of the argument (HHA-II) in which the key premises are formulated in a slightly different manner. We’ll be talking about that version of the argument a little bit later on.

Craig defends premise (1) of HHA-I by appealing to a thought experiment involving a hotel with an actually infinite number of guests and rooms (Hilbert’s Hotel). Such a hotel entails a number of (alleged) absurdities and so is thought to prove that an actually infinite number of things cannot exist. But there are many reasons to doubt this. As Hedrick pointed out, at best Hilbert’s Hotel disproves the existence of an actually infinite number of concrete physical objects. The problem for Craig is threefold: (i) there are other kinds of “thing” that may exist in actually infinite quantities; (ii) events may not be “things”; and (iii) even if events are things, they are not things like hotel guests and rooms, so it’s not clear that it would be absurd for an actually infinite number of them to exist.

This is the position we arrived at by the end of part one. In the remainder of this post, we’re going to cover Hedrick’s more original and substantive critique of the HHA. As he sees it, Craig’s preferred metaphysical view of time critically undermines his use of the HHA. To understand this, we’ll have to do three things. First, we’ll need a brief primer on the different theories of time. Second, we’ll need to formulate Hedrick’s argument against Craig. And third, we’ll need to survey and respond to the possible rejoinders from Craig.

1. A Quick Overview of Time
There is a famous quote, attributed I believe to John Wheeler, which states: “Time is what prevents everything from happening at once”. Although this captures an important truth about time (viz. it orders events into a one-after-the-other-sequence), it is not particularly useful or informative when it comes to understanding the nature and ontology of time. That is a topic that has long puzzled philosophers and physicists. There are two general categories of theories — A-Theory and B-Theory — with lots of subdivisions and variations within these general categories.

One of the most striking features of time from our perspective is the fact that it seems to flow forwards. That is to say, we seem to be moving through time: experiencing the present, remembering the past, and anticipating the future. The A-Theory of time takes this “flow” seriously. It holds that time is, essentially and necessarily, a tensed and dynamic thing; that there is a real objective difference between the present, past and future; and that of these three temporal states, the present has some sort of privileged ontological status. On this point, a particularly extreme interpretation of that privileged status can be found in the doctrine of presentism. According to presentism, the only temporal entities that exist are present entities; the past and future do not really exist.

The B-Theory of time is rather different. It holds that the “flow” of time is more illusory than real: a product of our subjective relationship with time, not a property of time itself. Proponents of the B-Theory hold that the past, present and future have are essentially the same, and so the present does not have a privileged ontological status. The analogy here is with physical space. We can think of the universe as a single object (“block”) with multiple dimensions. One of these dimensions represents time. Just as we can take up different locations along the spatial dimensions, without supposing there is a fundamental ontological difference between those dimensions, so too can we take up different “locations” along the temporal dimension. Indeed, past, present and future can be conceived as different “locations” on this block.

William Lane Craig has some well-developed views on time. In the first instance, he is an A-Theorist and predicates the success of the Kalam argument on the success of the A-Theory:

”From start to finish, the kalam cosmological argument is predicated upon the A-Theory of time. On a B-Theory of time, the universe does not in fact come into being or become actual at the Big Bang; it just exists tenselessly as a four-dimensional space-time block that is finitely extended in the earlier than direction. If time is tenseless, then the universe never really comes into being, and, therefore, the quest for a cause of its coming into being is misconceived.”
(Craig and Sinclair, 2009, pp 183-184)

In addition to this, Craig is (or at least appears to be) a presentist. Hedrick provides several illustrations of Craig’s commitment to presentism in his article (pp. 8-9 of the online version). To summarise briefly: Craig seems to conflate the A-Theory with presentism in some of his writings, and, more importantly, he argues that presentism is the only way to avoid certain paradoxes associated with the A-Theory of time. Hence, he rejects other variants of the A-Theory (e.g. the growing block theory and the moving spotlight theory) and endorses presentism.

2. Hedrick’s Argument from Presentism
Craig’s commitment to the A-Theory in its presentist form creates significant problems for his defence of the HHA (both versions). This is the nub of Hedrick’s main objection to the HHA, albeit he expands on this in an elaborate manner. I’ll consider two versions of the objection here: basic and extended. (Hedrick is a little bit too prosaic in his presentation for my liking. Hence why I resort to these different formulations).

It is easy to see how the basic form of the objection might run. The HHA (and the Kalam more generally) assumes that an infinitely extended past sequence of events entails the existence of an actual infinite. But presentism entails that past events do not exist. Consequently, it would seem like one can be a presentist, and believe in a past-eternal universe, without at the same time committing oneself to the existence of an actually infinite number of things. This is doubly disconcerting for Craig since he thinks such a commitment is entailed by the past-eternal view, and because he himself is a presentist.

This gives us Hedrick’s basic argument from presentism. I’ll formulate that argument as follows (numbering continues from part one):

The Basic Argument from Presentism
(11) The HHA assumes that if you deny that the universe began to exist, you accept the existence of an actually infinite number of past events. 
(12) But presentism is a plausible theory of time and presentism holds that past events do not exist. 
(13) Therefore, if one is a presentist, one can deny that the universe began to exist without committing oneself to the existence of an actually infinite number of past events. 
(14) Craig himself is a presentist. 
(15) Therefore, Craig can accept the past-eternality of the universe without committing himself to the existence of an actually infinite number of past events.

Just to be clear, the most important part of this argument is the inference from (11) and (12) to (13). I’ve tacked on (14) and (15) to highlight how the argument from presentism is a particular problem for Craig, but one shouldn’t take away from this that Hedrick’s argument is a purely internal critique of Craig’s reasoning. Rather, his argument will work for anyone willing to defend presentism.

So much for the basic argument. The subsidiary argument, brings a little bit more precision to the presentist view of a past-eternal universe. Such a view would maintain that an actually infinite number of events has occurred, but that they don’t exist. In other words, the current spatio-temporal reality does not consist of an infinite number of events: they’re over and done with, finished, kaput.

That’s a serious problem since, as Wes Morriston points out, the alleged absurdities in Hilbert’s Hotel arise from the fact that all the rooms and all the guests exist simultaneously and can be shuffled about relative to one another. Nothing like this is true on the presentist picture of an eternal universe: past events are not existents that can be manipulated or re-ordered. That suggests the following argument:

The Subsidiary Argument from Presentism
(16) If presentism is true, and if the universe never began to exist, an infinite number of events would have occurred, but would not now exist and could not be moved about or re-ordered relative to one another. 
(17) Hilbert’s Hotel (which is the basis for the HHA) only leads to absurdities because an infinite number of guests and rooms currently exist and can be moved about and re-ordered relative to one another. 
(18) Therefore, the HHA does not work if presentism is true.

I should say by way of clarification that, although Hedrick introduces this argument when discussing presentism, it doesn’t only work for the presentist. Indeed, I can’t think of any temporal theory that would claim that past events could be manipulated and re-ordered in the same way as hotel guests and hotel rooms. So this probably works for everyone. Still, for our purposes, the analysis turns on the intersection between presentism and Morriston’s complaint and so the two arguments will be treated jointly in what follows.

3. Objections and Replies
Now we enter the lengthy final stage of the analysis: we have our initial worries about the HHA-I; we have our two arguments from presentism; we just need to deal with possible objections that Craig might file. I’ll deal with these in a somewhat rapidfire manner. I’ll summarise the objection, and then describe Hedrick’s replies. I was going to formalise and number all of these dialectical thrusts and parries so as to create an argument map, but, unfortunately, the discussion ranges too broadly to make this an easy thing to do.

We’ll start by acknowledging that Craig and Sinclair (2009) do discuss something like the argument from presentism. But their discussion is flawed because it links the argument from presentism to the notion that past events are potentially not actually infinite. This is misleading. The presentist view is not that a potentially infinite number of events have occurred but that an actually infinite number have. It’s just that those events do not exist anymore. Hedrick concludes from this that Craig and Sinclair have sidestepped the argument from presentism.

In response to the claim that past events are not manipulable and re-orderable, Craig suggests that the Hilbert’s Hotel thought experiment can be re-imagined in such a way that the guests and hotel rooms are not manipulable. Just stipulate that the doors are locked and people cannot move from room to room, and then imagine “what it would be like for [the]person in room one to be in room two, and for the person in two - he could be in room four” (Craig, 2009). This, he claims, still generates absurdities.

For my part, I think this re-imagining of the thought experiment involves an element of re-ordering and re-shuffling of the constituent members of the actually infinite set. Furthermore, I think it still plays on intuitions we have about sets of concrete physical objects. But leave those worries to one side.

Hedrick’s complaint is that this re-imagining creates problems for other aspects of Craig’s defence of the HHA. To be as brief as I dare to be, mathematicians argue that arithmetic operations like subtraction are prohibited when dealing with actual infinities because they generate contradictions. Hence, for many mathematicians the HHA is irrelevant since the absurdities in the case of the hotel tend to involve the prohibited operations. Craig responds by arguing that in the real world we can’t simply prohibit subtractions because people can always check out of the hotel. Consequently, he thinks the HHA still stands since it is concerned with the possibility of an actual infinite in the real world. But as Hedrick points out, if we lock the doors to the hotel rooms so that people can’t check out of the hotel, Craig loses this limb of his defence.

To put it more succinctly, Hedrick thinks that the re-imagining of the thought experiment creates a dilemma for Craig:

Hedrick’s Dilemma for Craig: Either Craig allows for the guests can move in and out of the rooms or he prevents them from doing this: 

  • (a) If he allows them to move around, then the Hotel is not analogous to past events and the subsidiary argument holds; or  

  • (b) If he thinks they cannot, then he undermines one of his objections to the illegitimacy of subtracting infinities.

Of course, Craig is slippery guy so I’ll bet he can find a way to slide between the horns of this one.
In the meantime, there’s another objection he has mounted which may avoid the problem altogether. In his article with James Sinclair (2009), Craig suggested that as long as you could number past events (starting with the present and working backward), you could generate Hotel-like absurdities. You could imagine taking away all the odd-numbered events and still being left with an infinite number of events, and so on.

There are two problems with this, one of them pretty straightforward, the other pretty complex. On the straightforward side, if all you are doing is numbering the events, and ruling out manipulation and reshuffling, then you don’t generate the same metaphysical absurdities as you do in the Hotel thought experiment; you simply object to the actual infinite as a mathematical notion. That’s not going to be useful in this debate since you’d then have to refute the entire post-Cantorian edifice of transfinite mathematics.

On the more complex side of things, the numbering thought experiment would allow us to generate a parallel argument against the future eternality of the universe. Starting with the present, you can mentally number an infinite number of future events and perform the same sort of operations that Craig thinks are absurd. Wes Morriston, in fact, has already made this argument with a thought experiment involving two angels who take turns praising God for one minute at a time for eternity. God pre-ordains this from the get-go so if we assume the universe is past eternal, it means that the angels have been praising him an actually infinite number of times and continue to do so in the future.

Craig rejects Morriston’s counterexample by arguing that the sequence of future praises is only potentially infinite, not actually infinite. I’ll just quote Hedrick on this:

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. 
(Hedrick, 2013, p. 13 online version)

I have to say, this is perhaps the major stumbling block for me in Hedrick’s article. I almost bought what he said because of the last sentence, but on reflection I’d like for this to be better explained. I can’t see why, even in this thought experiment, the set of future events isn’t just a set growing without limit and hence actually infinite, but the past is a “completed” set with an actually infinite number of members. Admittedly, I should probably read Morriston’s original paper to get a clearer understanding.

Two final objections to the argument from presentism are possible for Craig. First, note that the HHA (if successful) undermines the possibility of an infinite sequence of past events on the B-theory of time. The only advantage for the B-theorist is that he/she doesn’t need to believe that the universe came into existence. But if the HHA works for the B-theorist, why shouldn’t it work for the A-theorist too? After all, wouldn’t it be odd if one’s metaphysical theory of time made a difference to whether the past could be finite or infinite? The argument here might look like this:

  • (19) If HHA is sound given the B-theory, then it’s sound given the A-theory.
  • (20) HHA is sound given the B-theory.
  • (21) Therefore, HHA is sound given the A-theory.

This is a classic modus ponens inference and as Hedrick highlights: “one man’s modus ponens is another man’s modus tollens.” In other words, one could accept premise (1) and simply argue that since the HHA is not sound given the A-theory (as has been argued in this post), it cannot be sound given the B-theory either. Certainly, there’s no more reason to accept the first version of the argument than this second version. (This rebuttal is fine, insofar as it goes, but I also question whether this is a credible objection in the first place. It seems to me that it conflates one’s theory of time with the reality of time.)

Finally, Craig could argue that on presentism, past events have some sort of ontological status even if they don’t really exist. More precisely, he could argue that past events have been actualised so, even though they don’t exist, they somehow “belong” to the present; they are part of the actual world because they actual world is constituted by what has happened. The problem here is that this forces a reformulation in the HHA. Specifically, the first premise would now have to read:

  • (1*) There cannot be a world in which an actually infinite number of things has been actualised.

But we haven’t been given an argument for that. The Hotel thought experiment doesn’t work and it’s not clear that an infinite number of events “belonging” to the present could generate similar absurdities.

4. Conclusion: Is an infinite sequence of events really absurd?
We have taken a number of twists and turns in this post, but the overall thrust is tolerably clear: if one is a presentist (as Craig is), one doesn’t believe that past events exist. And since HHA-I only concerns itself with things that exist, it doesn’t work against the presentist.

Of course, this analysis hasn’t yet touched upon HHA-II which, as you’ll recall, attempts to argue that an actual infinite simpliciter (i.e. not “an actual infinite number of things”) is impossible. This might be thought to work, even against even the presentist view. Now, unfortunately, Craig’s defence of this version of the HHA relies on the same thought experiments so at least from him there’s no new argument here per se. Still, it’s worth closing by considering whether there is something odd about an actually infinite sequence or number of events, even if they don’t involve “things” that exist.

Aquinas’s famous thought experiment involving the blacksmith, which has been mentioned by Craig in some of his writings, is instructive in this regard:

[Consider the] example of the blacksmith working from eternity who uses one hammer after another as each one breaks… the collection of hammers is an actual infinite. The fact that the broken hammers still exist is incidental to the story; even if they had all been destroyed after being broken, the number of hammers broken by the smith is the same. 
(Craig and Sinclair, 2009, p. 116)

Here, Craig tries to argue that it doesn’t matter whether the hammers exist or not, it is still absurd to imagine a blacksmith working with an infinite number of them. But as Hedrick points out, it’s not clear that any metaphysical absurdities are generated here. Suppose the blacksmith wasn’t working with an infinite number of new hammers here, but, instead, renewing or reforging the hammer using existing material. That way the “destruction” of the old hammer would be built into the thought experiment, and you’d get something more analogous to the notion of finite universe with an infinite sequence of events.

This doesn’t seem absurd. It is hard to fathom, for sure, but hard-to-fathomness isn’t a good criterion in this dialectic since, as Craig himself acknowledges, when it comes to the origins and nature of the universe, all the possibilities are hard to fathom. Where is the logical or metaphysical inconsistency in this example? If we can’t find one, the possibility of an actually infinite sequence of events lives to fight another day.

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