Tuesday, March 17, 2009

What a Kalamity!

Andrew's post over at Prosblogion reminded me of this:



I think it's around 3:20 that you get something like this:
Philosophically, the idea of an infinite past seems absurd. Just think about it. If the universe never had a beginning then the series of past events in the universe is infinite. But, mathematicians recognize that the existence of an actually existing number of things leads to self-contradictions. For example, "What is infinity minus infinity?

I can't tell you how many times I've heard some variant on this in class or in coffee shops from kids yammering on and on about actual and potential infinities. Having taken three semesters of Calculus (during which time I completed Calc I and II!), I think I'm almost qualified to weigh in on this one.

Craig likes to argue that there must have been only a finite number of moments prior to this one on the grounds that there cannot be an actual infinity of anything. To bring out the absurdity of an actual infinite collection, he asks us to consider the following example:
Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? ... But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds!

Maybe this is too quick, but isn't an answer to Craig's question that the mathematicians and the cleaning lady are both right in their own way? If the guest in room #1 is, say, Wes and he checks out at 9:00 a.m., then when the cleaning lady says at 10:00 a.m. "There is one less person in the hotel now than there was earlier this morning, his name was Wes" I think she speaks truthfully. No mathematician would deny this. If the mathematician says that "There are not fewer members in the set of guests in HH at 8:59 a.m. than there are in the set of guests at 10:00 a.m." I think the mathematician speaks truthfully. Here the mathematician is working from the idea that one set has fewer members than another only if you can't put the members of the first set into 1-1 correspondence with the members of the second. No cleaning lady would deny this.

We can do the same thing with numbers. If you have the set of primes and then kick out the smallest prime, there's a sense in which there is one less member in the first set than the second (i.e., 2 is contained in the first set only). There is a sense in which there is not one less member in the first set than the second because if you wanted to have a dance you could put the members of the sets into 1-1 correspondence.

Now, if I understand Craig's view, it is that everything I've said about the sets of numbers is correct but you can't have "in reality" collections of things that are infinite because that would lead to absurdities. But, to the extent that I've defused the absurdities in the mathematical case I think I've done so in the previous case with Hilbert's Hotel. The appearance of contradiction is really due to an ambiguity exposed. Close the hotel down.

8 comments:

exapologist said...

Nicely stated. His arguments against the possibility of infinite traversals doesn't work, either. (Warning: pet peeve-induced rant coming...)


I believe you already know this, but people like Paul Draper (Senor's notes here) and Wes Morriston have published replies to both sorts of a priori arguments -- about a decade ago. Does Craig bring up these replies to his readers (in his popular apologetics books) or his listeners (in his debates)? Not so much.

To be fair, Craig has responded to Morriston's criticisms of his a posteriori arguments for a finite past, his causal premise, and his arguments for a personal cause. But I just can't help but question his integrity by ingnoring, and even failing to mention or cite, the criticisms of his a priori arguments in his books. It's hard not to resist concluding that he wants to keep his audiences in the dark, knowing that few if any will look up and read these replies.

He's a professional philosopher. How can he live with himself when he does this sort of thing?

Geoff said...

Maybe you can put it this way. The sentence "there's one less person in the hotel today than there was yesterday" is ambiguous between:

(A) There is a person who was in the hotel yesterday but isn't today, and no one has taken his place.

(B) The number of people in the hotel today is smaller by one than the number of people that were in it yesterday.

When we're dealing with finite hotels, (A) is true iff (B) is true. But one of the funny things about infinite hotels is that (A) can be true even if (B) isn't! That's not a *contradiction*; it's just one of the funny things about infinite sets.

Anonymous said...

>>If you have the set of primes and then kick out the smallest prime, there's a sense in which there is one less member in the first set than the second (i.e., 2 is contained in the first set only). There is a sense in which there is not one less member in the first set than the second because if you wanted to have a dance you could put the members of the sets into 1-1 correspondence.>>

So, according to you, there's one sense in which there is one less member, and another sense in which there is not one less member.

Which word has these dual senses? Surely not "one," right? And not "member" either, right? Those words are both univocal in these sentences.

So I suspect it must be that you think "less" has multiple senses. That comes as a surprise to me. What exactly are these two senses of "less" at work in this case?

I'm having a hard time hearing any ambiguity. "Less" fails all the standard ambiguity tests I can think of...

If there actually is no ambiguity, then there IS one less member and also there is NOT one less member in exactly the same sense. But then we have a genuine paradox, which is what Craig (and Hilbert before him) were trying to point out.

Anonymous said...

This is a reply to exapologist above:

>>Does Craig bring up these replies to his readers (in his popular apologetics books) or his listeners (in his debates)? Not so much.>>

As far as I know, Craig has replied to Draper and Morriston in print, and I seem to recall that these replies are available on his website.

As to why Craig doesn't bring up this higher-level dialectic in his popular apologetics books and debates, isn't the answer already in the question? Those books and debates are for a popular level. Perhaps he doesn't have the space to include the entire dialectic, and perhaps he thinks that the general public wouldn't appreciate these technical higher-level back and forths.

Clayton said...

Anon,

Interesting points. You wrote:

Which word has these dual senses? Surely not "one," right? And not "member" either, right? Those words are both univocal in these sentences.

So I suspect it must be that you think "less" has multiple senses. That comes as a surprise to me. What exactly are these two senses of "less" at work in this case?

If there actually is no ambiguity, then there IS one less member and also there is NOT one less member in exactly the same sense. But then we have a genuine paradox, which is what Craig (and Hilbert before him) were trying to point out.


I'd say that there's a sense in which one set has less members and one sense in which there is less in one set than the other. Suppose you have the set containing The Beatles {John, Paul, Ringo, George} and then the set of surviving members of The Beatles {Paul, Ringo}. If someone said that there's less in the second than the first because you can't pair off all of the members of the second with the first that seems right. If someone said that there's less in the second than the first because there are elements of the second that aren't of the first but not vice versa.

Obviously those two tests for less don't come to the same thing when we are dealing with infinite sets since such sets have proper subsets where the members of the set and its proper subset can be put into 1-1 correspondence.

Anyway, I could be wrong about what Craig wants but I had thought that what he wanted to say was that there cannot be in the world infinite collections of things like (non-infinitesimal) moments of time but that the mathematical claims we all want to come out true come out as being true. I think I've given him one way of saying coherently what we want to say about sets of numbers but I don't then see how he's shown the incoherence or impossibility of something like an infinite past.

[Yes, he has his aposteriori arguments for the finitude of the past but that's another matter.]

Clayton said...

Anon,

Sorry, I was interrupted. I meant to add that if I'm just wrong about the ambiguity thesis, then I don't see what's wrong with just saying that this is mistaken:

If there actually is no ambiguity, then there IS one less member and also there is NOT one less member in exactly the same sense.

Suppose A has less members than B = the members of A cannot be put into 1-1 correspondence with the members of B. Then, there's no ambiguity and there's no question that it's true to say "There is not one less member".

exapologist said...

As far as I know, Craig has replied to Draper and Morriston in print, and I seem to recall that these replies are available on his website.


If he has, I'd greatly appreciate it if you could point me to at least one of Craig's journal articles or books that addresses them. (note: again, I'm not talking about Morriston's criticisms of Craig's causal premise, or of his criticisms of Craig's big bang arguments -- just the a priori arguments about the possibility and traversability of actual infinites).

As to why Craig doesn't bring up this higher-level dialectic in his popular apologetics books and debates, isn't the answer already in the question? Those books and debates are for a popular level. Perhaps he doesn't have the space to include the entire dialectic, and perhaps he thinks that the general public wouldn't appreciate these technical higher-level back and forths.

Um, no. Craig discusses arcane details about theoretical physics, bubble universes, special relativity, quantum mechanics, string theory, etc., in his popular-level apologetics books. He also discusses other technical criticisms from Morriston of the sort I gestured to in my original comment. In fact, the criticisms I'm referring to from Draper and Morriston aren't nearly as complicated and technical as the criticisms from Morriston that he does address. So it's not an adequate reply to say the criticisms are too technical.

Lewis said...

I would have thought the standard view (in the case of the primes) is to say that the set of primes greater than 2 does not have fewer members than the set of all primes.