Sunday, July 28, 2013

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

The second premise of William Lane Craig's Kalam Cosmological Argument reads:

  • The universe began to exist.

Typically, Craig presents four arguments in support of this premise: two scientific, two philosophical. Though the scientific arguments raise many interesting points, I think many would agree that the philosophical arguments do the heavy lifting. This is for the simple reason that current physical theories of the origin of the universe are incomplete and hence provide, at best, weak and defeasible support for the premise. The philosophical arguments are much stronger than this, purporting to provide an "in principle" argument for the truth of the premise.

One of the philosophical arguments relies heavily on a thought experiment from the mathematician David Hilbert. The thought experiment describes a hotel with an actually infinite number of rooms and guests. Serious reflection on this thought experiment is alleged (by Craig) to show that an actual infinite cannot exist. This is significant in that if the universe never began to exist, the set of events prior to this moment in time must be actually infinite.

This argument -- which will be formalised more precisely in a moment -- can be called the Hilbert's Hotel Argument (HHA) in honour of its motivating thought experiment. In a recent article entitled "Heartbreak at Hilbert's Hotel" Landon Hedrick has offered an interesting critique of the HHA. Over the next couple of posts, I want to look at what he has to say.

Hedrick's article is interesting for at least three reasons. First, it has a curious, and I think praiseworthy, rhetorical structure. Although Hedrick does not shy away from general and robust criticisms of the HHA, he nevertheless concedes a lot of ground to Craig, repeatedly granting certain premises arguendo, saving most of his argumentative muscle for a subtle internal critique of Craig's argument. The net result is a disarming, yet deceptively comprehensive dismissal of the HHA.

Second, the internal critique that Hedrick offers claims that Craig's commitment to a presentist A-Theory of time undercuts the forcefulness of the HHA. As many will know, Craig has repeatedly emphasised that the Kalam argument "from start to finish" relies on the A-Theory of time. This has led many to rebut the argument on the grounds that the A-Theory is flawed. But if Hedrick is right, these rebuttals are unnecessary, maybe even flawed.

Third, and this is slightly more personal in its interest, Hedrick first entered my consciousness through his blogging several years back. Though never a prolific blogger, he wrote some good pieces on his role in organising the Craig-Carrier debate on the resurrection, and he always struck me as a thoughtful, self-effacing guy. As someone who blogs, while at the same time trying to launch a scholarly career, it's always nice to see someone else making the transition.

Anyway, let's get down to business. In the remainder of this post, I will set out the HHA in more detail, and cover some of Hedrick's initial criticisms of it. I'll leave the argument from presentism for the next day.

1. The Hilbert's Hotel Argument
The HHA comes in two forms. The first of which reads like this (my numbering doesn't follow Hedrick's, he adds letters to code different arguments, but since I won't be going into all those different arguments in my series, I've chosen to ignore them):

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.

Some further argumentation is required to seal the gap between (3) and the second premise of the Kalam argument, but it's pretty easy to see how that might go. You just add: if the universe did not begin to exist, there must be a beginningless series of events prior to the present moment. We'll come back to the link between the HHA and the Kalam a little later on anyway.

As you'll have noticed, HHA-I focuses specifically on the existence of an actually infinite number of things. The second version of the argument, which has been mooted by Craig in some of his work, tries to broaden its scope by dropping the reference to "things":

(1*) An actual infinite cannot exist. 
(2*) A beginningless series of equal past intervals of time is an actual infinite. 
(3*) Therefore, a beginningless series of equal past intervals of time cannot exist.

We won't be dealing with this version of the argument for a while, so keep it filed away in your brain for the time being. For the foreseeable future our main focus will be on HHA-I.

Before proceeding, a quick word about "actual" and "potential" infinities. Those who are familiar with the mathematical history will know that there is an important difference between the two concepts. This can be defined as follows:

Actual Infinite: "Is a collection of definite and discrete members whose number is greater than any natural number." (Craig, 2008). In other words, it is a set with an infinite number of members.
Potential Infinite: "Is a collection that is increasing toward infinity but never actually gets there." (Craig, 2008). In other words, it is a collection which ceaselessly grows in size, but never actually contains an infinite number of members.

The distinction is significant insofar as Craig only objects to the existence of an actual infinite. He's quite comfortable with the existence of potential infinite. Indeed, he thinks the future is a potentially infinite sequence of events. Thus, a classic move for Craig in these debates is to reject an opponent's counterexample by arguing that it appeals to a potential, not an actual infinite.

So why does Craig reject the existence of an actual infinite? The answer lies in his analysis of a set of thought experiments, of which the most famous and prominent is the Hilbert's Hotel thought experiment. In it, we are asked to imagine a hotel with an actually infinite number of guests and an actually infinite number of rooms. Such an entity would entail the existence of an actual infinite number of things. But could it really exist? No; a series of absurdities is thought to follow. Consider:

If one new customer checks to the hotel he/she can be accommodated even though the hotel is full. Simply move every guest in the hotel up one room (i.e. the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on). This way room 1 becomes available.
If an infinite number of guests arrives, they too can be accommodated. Simply move every guest to the room that's twice their current number (i.e. the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on). This empties out all the odd numbered rooms. Indeed, an infinite number of guests can keep checking into the hotel.
Odd things happen when people check out. If one guest checks out, thereby emptying one room, an infinite number remain (infinity minus one is still infinity). If all the guests in odd numbered rooms check out, an infinite number of guests will have left, but they will still leave behind an infinite number of guests. Nevertheless, if every guest in a room number greater than 3 checks out, only three are left.

Mathematicians use these examples to illustrate the fact that certain arithmetic operations are improperly applied to an actual infinite, but Craig uses them to argue that an actual infinite number of things cannot exist in the real world. If they did, absurdities such as those present in Hilbert's Hotel would arise. This suggests, to me at any rate, that Craig defends something like the following argument in order to defend premise (1) of HHA-I:

  • (4) If an actual infinite number of things existed, Hilbert's Hotel would be possible.
  • (5) But Hilbert's Hotel is not possible; it is absurd.
  • (2) Therefore, an actual infinite number of things cannot exist.

(Note: as an aside, one thing that slightly annoys me about Hedrick's article is how he names the different arguments. To my mind, this argument would be more properly called the "Hilbert's Hotel Argument" since it is actually about Hilbert's Hotel, and the argument that he calls the "Hilbert's Hotel Argument" would not since it is really just a general argument against the existence of an actual infinite. I toyed with the idea of re-naming them in this series of posts, but decided against it. It would create too great a chasm between my series and Hedrick's article, which might be confusing for those who wanted to read both.)

In the remainder of this post we'll consider how plausible Craig's defence of the HHA-I is.

2. Does Hilbert's Hotel rule out the existence of an actual infinite?
The first thing we need to do is see whether premise (1) of the HHA-I is properly defended. Craig thinks that the thought experiment does the job but is he right? In answering this question we begin to appreciate Hedrick's concessionary and disarming style.

Hedrick says we can grant that Hilbert's Hotel would be absurd. And since Craig uses very similar thought experiments with other concrete physical objects (e.g. the infinite library), we can grant that this conclusion holds for all concrete physical objects. The problem is that this doesn't rule out the existence of an actual infinite number of things.

Three counterexamples suggest themselves. First, what about the set of natural numbers? According to certain metaphysical views (Platonism), numbers are real objects, and there is an actual infinity of them. Second, what about other abstract objects such as propositions or properties? Surely there are an infinite number of them as well? For example, possible worlds can be understood as sets of propositions, and the notion that there is an actual infinite number of possible has found favour among some religious philosophers (e.g. Plantinga) and some not-so-religious philosophers (e.g. David Lewis). Third, some have even argued that physical space is made up of an actual infinite number of discrete points.

Now, Craig, of course, will reject these counterexamples. He will argue that Platonism is false because numbers aren't things, or there isn't an actual infinite number of them; He will argue that modal realism and cognate views are deeply implausible; or that the notion that space consists of an infinite number of points is unproven. But these counterarguments miss the point. If the alternative metaphysical theses listed above are true, then Hilbert's Hotel doesn't disprove the existence of an actual infinite number of things. The thought experiment only appeals to intuitions we have about concrete physical objects. It does not deal with abstract, non-physical objects, where arguably our intuitions are going to be much less reliable. Sure, the alternative metaphysical theses could be wrong, but to show this one must engage in some pretty abstruse and technical metaphysical arguments. There is no straightforward, intuitively appealing pathway from the thought experiment to premise (1), contrary to what Craig seems to think. To summarise this informally:

  • (6) An actual infinite number of things can (and indeed does) exist: there is an actual infinity of natural numbers; an actual infinity of propositions and properties; and an actual infinity of possible worlds.
  • (7) The metaphysical theories that support these examples are controversial, implausible or unproven.
  • (8) Fair enough, but you have the engage with the arguments for an against these metaphysical theories to support that conclusion; Hilbert's Hotel can't do all the work as it only deals with an actual infinite number of concrete physical objects.

This is a modest dialectic, with a modest conclusion, but it does undercut the force of Craig's use of Hilbert's Hotel.

3. Are past events "things"?
So much for premise (1) of HHA-I. In some ways, premise (2) is even more problematic and, indeed, oft-neglected in debates about the Kalam. The big difficulty is that premise (2) assumes that past events are things, much like rooms in an infinite hotel or books on the shelf of an infinite library. But is this remotely plausible?

For starters, it is worth noting that several leading philosophers have denied that events are things. Terence Horgan, for instance, writing on the topic back in 1978 said:

[I]t is a mistake to posit events at all...despite their 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, 1978, 28)

More recently, Peter van Inwagen has taken a similar view:

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, 2009, 14)

Bizarrely enough, Craig agrees that this is a plausible view. Indeed, he seems to think that whatever it is that events are, they aren't "things" like books and hotel rooms. As he says himself:

"These things [i.e. events] are real in the sense that they are not illusory, but they are not, properly speaking, existents. 
(Craig, 2011, 220)

But if events aren't existents this creates significant problems for his defence of HHA-I. For if events are not objects like rooms and books, then the support that the Hilbert's Hotel thought experiment provides for premise (2) evaporates. The thought experiment only covered concrete physical things. Note, it doesn't get any better if we switch to HHA-II and its talk of "intervals of time" rather than "events". The problem there is that, unlike ordinary things, intervals of time don't come into and pass out of existence. They merely pass.

Leaving all this aside, problems remain when we consider premise (2) in light of Craig's stated views about the nature of events. In his lengthiest, most recent defence of the Kalam argument (Craig and Sinclair, 2009) he says that an event is simply any change. A change, as Hedrick notes, presumably involves something gaining or losing one or more properties. If that's right, a counter-thought experiment suggests itself:

Sphereworld: Imagine a past and future eternal world consisting of one sphere that alternates between green and red perpetually.

This world consists of one object, but an actual infinite number of events (the colour changes). On the face of it, this seems like a possible world. It involves no outright contradictions or apparent absurdities. Craig could perhaps deny its plausibility by arguing that it consists of an actual infinite number of things, much like Hilbert's Hotel. But, as Hedrick notes, this is an extremely counterintuitive interpretation of the thought experiment. At a stretch, you might say the world consists of three things: the sphere, and the two colours (which might be abstract objects). But you wouldn't say it consists of an infinite number. Counting the events as actual things feels bizarre. Furthermore, Sphereworld arguably comes pretty close to being a plausible view of reality: a finite number of things passing through an infinite number of changes. Indeed, Spinoza argued for something like this. He thought reality consisted of a single indivisible substance existing in an infinite number of modes.

To summarise this line of reasoning:

  • (9) One can plausibly argue that events are not "things", or that if they are "things" then they are very different from concrete physical objects like hotel rooms.
  • (10) Sphereworld seems like a non-absurd possible world: it consists of a finite number of things and an infinite number of events.

4. Interim Conclusion
Remember, this is just the warm-up. All Hedrick has done so far is suggest that support for premises (1) and (2) of HHA-I (and premise (2*) of HHA-II) is somewhat lacking. This is because the Hilbert's Hotel thought experiment doesn't apply to abstract objects like numbers or propositions; and because events are probably not "things" in the ordinary sense of the term. But Hedrick is still willing to grant these premises to Craig. That's because he thinks bigger problems are lurking when we combine the defence of the HHA with Craig's views about the ontology of time. We'll get to those problems the next day.

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