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Saturday, January 28, 2012

Craig and actual infinites

William Lane Craig uses Hilbert's hotel in an attempt to illustrate the impossibility of an existing actually infinite multitude, but I find several serious gaps in his arguments which, in my judgment, prevent them from having any force. His approach takes two forms: First, he claims that certain logical contradictions follow from the existence of actual infinites; second, his intuition tells him that an existing infinite multitude is absurd. In response, I want to resolve the alleged contradictions, and argue that intuition is an unreliable guide to the possibility or impossibility of an existing actual infinite.

Metaphysical and logical possibility.

In his "philosophical argument" against the existence of an actual infinite, Craig relies on the notion of "metaphysical" possibility, or as it is sometimes called (as I myself prefer), broad logical possibility, which he laments must follow from the relatively weak force of "intuitions and conceivability arguments" (Blackwell Companion to Natural Theology, p106). However given an accessible characterization of broad logical possibility, we should be careful not to think that just any intuition will do. Take, for instance, my deeply-felt intuition that a traditional God could never exist, which hardly counts as evidence against the possibility of God's existence. Similarly, if Craig has an intuition that Hilbert's Hotel could not exist, then that won't count as evidence either against its possibility---unless of course we have some particular reason to trust his intuition. The role of intuitions ought instead to be rather limited, whose appeals perhaps only serve to make up for our inability to precisely articulate a logical problem. So, for example, consider a person who first encounters the old challenge that an omnipotent God can create a stone so heavy he cannot lift it. Clearly, such a situation is not broadly logically possible. However not everyone can see precisely where the contradiction lies, and so these people must rely instead on their intuition that something is wrong with the picture the statement paints. Notice, though, that this intuition amounts to much more than merely a feeling of distaste for what we might view as a silly or sophistic suggestion. Instead, we genuinely cannot make sense of the statement given our understanding of its individual clauses and terms---that is, we don't know how to fit the statement's disjoint ideas into a consistent mental picture.

In principle, though, when faced with a broadly logically impossible suggestion, given sufficient reflection and ingenuity we ought to be able to find the precise nature of the inconsistencies, and express them as definite logical contradictions. In this way, we can interpret broad logical impossibility (or in Craig's parlance, metaphysical impossibility) as involving a strict logical impossibility which is perhaps hidden in a mass of ideas too tangled for us to penetrate. In other words, to assert that some state of affairs S is not metaphysically possible is to claim either that S involves an apparent logical contradiction, or else that there is some logical contradiction buried in S, but which is hidden from our immediate access. The role of our intuition, then, is to help inform us whether or not we can detect logical contradictions in those ideas where, for whatever reason, we cannot clearly explicate them.

Craig attempts to show an actual infinite is not possible by deciding that Hilbert's Hotel is "ontologically absurd" (p111), and he refers back to this idea of absurdity several times throughout the argument. It's not clear to me whether he intends the "absurd" to materially imply broad logical impossibility, or if he merely takes it as a strong probabilistic indicator; but in either case we have a serious problem. For if he has found no explicit logical contradiction in the actual existence of the hotel, then he must be appealing to an intuitive inability to make sense of it. However this is not the impression I get from his writing on the subject. It seems to me that, putting aside some extraneous confusions, he really does grasp, for the most part, the structure and mechanics of the hypothetical hotel, and can mentally manipulate it (at least apparently) in a logically consistent way. Though he may have a deeply-felt sense that its existence is not possible, this again does not count by itself as evidence against the possibility of the hotel as long as he has a coherent notion of what it means for it to actually exist.

Does the actual existence of the hotel, then, involve a logical contradiction? Craig argues that yes, it does, pointing to two distinct cases.

The first alleged contradiction.

Craig seems to affirm both that

(i) "there are not more things in a multitude M than there are in a multitude M' if there is a one-to-one correspondence of their members;" and

(ii) "there are more things in M than there are in M' if M' is a proper submultitude of M;"

whereas these are, at least according to Craig, inconsistent with the idea that

(iii) "an infinite multitude exists" (p110).

One of these may have to go, but as Craig points out we don't have an argument to keep (iii) over (i) or (ii). On the other hand, he doesn't offer an argument to keep (i) and (ii), either. Instead, he decides that (i) and (ii) are "innocuous," whereas (iii) is not, which he apparently takes as reason enough. However this seems to me entirely too hasty, another appeal to his personal intuition when we have no reason to think his intuition is a good indicator of truth. Further, (i) and (ii) do not seem innocuous at all unless we have already discharged (iii). Now, it is true that in ordinary language our uses of the terms "not more" and "more" are usually equivalent to (i) and (ii), respectively, for the simple reason that in ordinary language we almost invariably consider only finite multitudes. If we extend the meaning of these terms to contexts involving infinite multitudes, then we shall have left our comfort zone, so to speak. So the idea that (i) and (ii) are "innocuous" really depends on determining in advance that (iii) is false, which invites circularity if we wish to use that alleged innocuousness in our reasoning for rejecting (iii).

So if we take (i)-(iii) to involve a logical inconsistency, which is indeed the case as long as we give them appropriate interpretations, then we still haven't demonstrated a logical inconsistency in (iii) until we can first justify both (i) and (ii). Given that the only ready justifications of (i) and (ii) depend on (iii), this alleged contradiction fails to hold up under scrutiny.

The second alleged contradiction.

Craig writes:

In transfinite arithmetic, inverse operations of subtraction and division with infinite quantities are prohibited because they lead to contradictions... But in reality, one cannot stop people from checking out of a hotel if they so desire! In this case, one does wind up with logically impossible situations, such as subtracting identical quantities from identical quantities and finding nonidentical differences (pp111-2).


What is, after all, the result of infinity minus infinity? Well, that really depends on what we mean by "minus," especially as it relates to collections of objects. Craig notes that the mathematical stipulation on subtracting one quantity from another has "no force in the nonmathematical realm" (p112). That is true enough, but only because subtraction has a purely mathematical meaning in any case. Arithmetic, whether finite or transfinite, we apply to quantities, and not directly to the multitudes which those quantities can be said to measure. What we seek, then, in order to have Craig's objection track, is a way to link the arithmetic of quantities to real changes in existing multitudes. In particular, he'll need to affirm something like

(iv) If multitude M has quantity Q and a submultitude M' has quantity Q', then the multitude formed by removing the objects in M' from M has quantity Q minus Q'.

By affirming (iii) and (iv), along with a suitable notion of "quantity," we obtain a contradiction. However, here we seem to face the same sort of situation as we did with the previous alleged contradiction, whereby the most natural way to justify (iv) involves denying (iii) in advance, again threatening circularity. Meanwhile, as Craig himself points out, the set difference operator in mathematics (by which we remove objects of one multitude from another) works quite consistently, and we are free to use it to interpret what it means for guests to leave Hilbert's hotel. He objects only that this move "does not change the fact that in such cases identical quantities minus identical quantities yields nonidentical quantities" (p112). Yet although he is correct that the consistency of set difference operations will not change the inconsistency of transfinite subtraction, he still needs to first establish that indeed transfinite subtraction is required with infinite multitudes---that is, he needs to justify (iv) or something sufficiently similar so as to link transfinite subtraction with guests leaving a hotel. So far, he has not attempted this, and indeed it's hard to see how he could ever succeed apart from an independent argument against (iii).

From the hotel to the general case.

As an additional objection to Craig's appeal to intuition, Graham Oppy's observation that we don't get to infer from the impossibility of Hilbert's hotel the impossibility of an actual infinite in general seems quite powerful (cf. Oppy, Philosophical Perspectives on Infinity, pp51-3). So perhaps an infinitely-roomed hotel exists, but due to physical constraints it is not possible for guests to shuffle about so as to permit infinite arrivals or departures. To the extent that the intuitive absurdities Craig wants to find in Hilbert's hotel depends on such movements, since we have no justification for affirming their physical possibility then we cannot conclude from them that an actually infinite multitude is impossible by itself.

Craig offers two responses to this objection: First, he claims that "Hilbert's Hotel can be configured as we please without regard to mere physical possibilities" (p110). However this is quite obviously false; any reconfiguration of a hotel will still be a physical entity so long as it remains a hotel. In order to avoid minding physical possibilities, then, we must purge from the thought experiement of any significant appeal to physical entities. As it happens, this is precisely what Craig proposes in his second response. In particular, he suggests we generalize Hilbert's hotel by stripping it of all the characteristics which would cause potential mechanical problems, while preserving the same counter-intuitive results. However this does not so much defend against Oppy's objection as it concedes that Oppy has a good point, and that we need to go beyond the one example of Hilbert's hotel before we can infer that an actual infinite cannot exist.

Thus Craig writes: "If a (denumerably) actually infinite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert's Hotel" (p110). So, what does he have in mind, exactly? If the objects in the infinite collection are physical, then he hasn't avoided his need to mind the unknown constraints of physical impossibilities. If on the other hand the objects in the infinite collection are abstract, then we are furnished with an abundance of counter-examples. The only option remaining that I can see is to say that the objects are mental (and non-physical). In that case, though, it's hard to see how he can carry over the concept of fullness of the hotel, and so we lose whatever counter-intuitiveness was associated with the notion of a hotel with no vacancies accommodating new guests. Instead we must imagine mental objects, ideas perhaps, divorced from physical bodies, and, say, blinking into and out of existence. Then we could have an infinite number of ideas existing, followed by the appearance of infinitely many more ideas; then infinitely many ideas suddenly cease to exist, but infinitely many extant ideas nevertheless remain---and so on. In this case, though, any violation of our intuition can easily be chalked up to the bizarre notion of having bodiless ideas pop into and out of existence.

Worse still, Craig seems committed to the position that the multitude of ideas can only be changed by having new ideas blink into existence, and not by having ideas blink out of existence. (This is due to his argument that a series of past events is an existing actual infinite.) This will interfere with his presentation of the second alleged contradiction, which requires us to somehow cut down infinite multitudes to proper submultitudes.

Conclusion.

I have tried to show here that Craig appeals to intuition to decide Hilbert's hotel is not possible, even though we have no reason to trust that intuition. Moreover, we have no means of moving from Hilbert's hotel to the general case without twisting the thought experiment so that it will be almost certain to violate our intuition independent of the presence or absence of infinite multitudes. In addition to these concerns, Craig wants to say that we have reason to think the existence of an actual infinite must involve two distinct logical contradictions, but in each case we find that the only ready justification depends on first showing that an actual infinite does not exist---an invitation to circularity. Given all this, it seems not only that Craig's arguments are insufficient for making his case against an existing actual infinite, but that they require further development before we can use them to throw any force at all to his desired conclusion.

2 comments:

Thomas Larsen said...

I think there are some reasonable arguments against the existence of an actually infinite causal chain, or an actually infinite array of objects that depend on one another; but, as far as I can see, an actually infinite number of disconnected things or objects (independent universes, for example) could possibly exist. Consider, for example, how many universes God could create if God is timeless. One? Two? Three? 10^100? An infinite number? It seems to me that the latter is the most plausible answer.

The reason I'm not willing to use the kalam cosmological argument is that I'm not convinced that the A-Theory of time is correct. (Well, for theological reasons, I think it's more likely to be true than the B-Theory of time, but I'm not entirely convinced.) And I'm not willing to use an argument that I'm not pretty certain of.

K-Dog said...
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