Over at his blog, Clayton Littlejohn has been discussing the Kalam cosmological argument. The basic idea of Kalam arguments is to show that it doesn't make sense for there to be an infinity of past moments. In other words, it's an argument for why the universe has to have a beginning. Certain Christian philosophers have used it, since most forms of Christianity assume there was an absolute beginning to existence. A lot like to appeal to the Big Bang for this belief. William Lane Craig has written extensively on the Kalam arguments. A few years ago he wrote an article in The New Mormon Challenge which appealed to Kalam arguments to argue against the LDS form of Christianity. That's because Mormon theology appears to have as a doctrine the claim that there never was a beginning to existence or the universe.
Now I've discussed this theological point a fair bit here. The most obvious response is to bring up the notion of Linde universes, an idea of late promoted by physicist Lee Smolin. Put simply, it seems that the use by some Christian philosophers of the big bang is problematic because physical theory is much more open than they assert. (Indeed Linde's notions are becoming more and more mainstream as time goes on)
Having said that though the issue of Kalam remains. Blake Ostler has a rather good response to Craig's Kalam arguments. (He addresses a few of Craig's other arguments in some other papers.) Wes Morriston also has up a critique of Craig's Kalam arguments. Put in its simplest terms, the Kalam arguments assume actual infinities are impossible. While that's certainly a popular position historically among philosophers, it is hardly universal and is typically simply an assumption.
The main argument against actual infinities is called Hilbert's Hotel, after the famous mathematician.
Let us imagine a hotel with a finite number of rooms, and let us assume that all the rooms are occupied. When a new guest arrives and requests a room, the proprietor apologises, 'Sorry--all the rooms are full.' Now let us imagine a hotel with an infinite number of rooms, and let us assume that again all the rooms are occupied. But this time, when a new guest arrives and asks for a room, the proprietor exclaims, 'But of course!' and shifts the person in room 1 to room 2, the person in room 2 to room 3, the person in room 3 to room 4, and so on... The new guest then moves into room 1, which has now become vacant as a result of these transpositions. But now let us suppose an infinite number of new guests arrive, asking for rooms. 'Certainly, certainly!' says the proprietor, and he proceeds to move the person in room 1 into room 2, the person in room 2 into room 4, the person in room 3 into room 6, the person in room 4 into 8, and so on... . In this way, all the odd-numbered rooms become free, and the infinity of new guests can easily be accommodated in them.
In this story the proprietor thinks that he can get away with his clever business move because he has forgotten that his hotel has an actually infinite number of rooms, and that all the rooms are occupied. The proprietor's action can only work if the hotel is a potential infinite, such that new rooms are created to absorb the influx of guests. For if the hotel has an actually infinite collection of determinate rooms and all the rooms are full, then there is no more room. (Craig, The Kalam Cosmological Argument, 84-85)
While there is no doubt that actual infinities lead to odd sorts of possible phenomena, I'm not convinced that this is an argument against their existence. Craig's "gut argument" is that this mapping phenomena of infinite sets is simply ludicrous to consider as something actual. The more formal argument is based upon Euclid's Maxim (named after Euclid's fifth axiom). That is, the claim that a whole must be great than any of its parts. (Or in more careful terms, a set must be larger than a proper subset of itself) The problem, and I'll not go through the details, is that the use of Euclid's Maxim rests on some questionable equivocations. Consider for instance Hilbert's hotel. A hotel with the odd numbered rooms and the hotel with all rooms have the same "size" in terms of elements. However clearly in an other sense the hotel with the odd numbered rooms is "smaller" than the other, in that the other hotel has rooms the odd numbered one doesn't. (Think of it in terms of numbers, the set of positive integers and the set of integers have the same cardinality or number of elements. But one set contains elements the other doesn't.) One has to be careful considering infinite sets.
There are other arguments against the Kalam arguments. I'd suggest checking out Blake's papers as well as some of Wes Morriston's.
I think you're a little too kind to Craig. It's apparent to me that Craig either doesn't grasp basic Cantorian set theory, or he conveniently forgets certain properties of infinite sets when it suits his purposes.
His fallacy in the passage you quote lies in the last sentence:
For if the hotel has an actually infinite collection of determinate rooms and all the rooms are full, then there is no more room.
No. To find out out many rooms are available, we have to subtract the number of filled rooms from the number of rooms. If both numbers are infinite, we subtract infinity from infinity, which is indeterminate.
He commits the same fallacy when he argues against beginningless series:
Indeed, the idea of a beginningless series ending in the present seems to be absurd. To give just one illustration: suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished by then.
Again, Craig is applying properties of finite sets to an infinite set. If we try to determine when the man should finish counting, again we come up with an indeterminate. Craig has no basis for claiming that the man should have already finished.
Well, I didn't think I was being kind, but I was trying to avoid being mean. That's why I did the "gut instinct" explanation of his argument with the scare quotes. I know some people I know really like Craig as a philosopher. I'm afraid I'm far less impressed for a variety of reasons. (Same with Moreland) But to each their own.
I've closed comments in order to avoid spam since I don't check this older blog as much anymore.
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