Thoughts from
San Diego on Law, Politics, and Culture

 
Infinity, etc., etc.
By Tom Smith

My kids get a magazine from the Discovery channel, and the most recent issue was on space. It announced in a little blub that astronomers had discovered that we were part of a multiverse, and that in fact there were an infinite number of universes, and so therefore there must be some that had children in it just like those reading the magazine (but presumably slightly different). I must have missed the announcement of this important discovery.

In fact, the possibility or impossibility of the "actual infinite" is a fascinating topic in the philosophy of mathematics and indirectly in philosophical theology. Curiously, some of the better arguments for the existence of God rely on the impossibility of the actual infinite. The "actual infinite" may not be what you think it is. You are familiar with the set of positive, whole numbers, (1, 2, 3 . . . ). You can always go one higher in any ennumeration of them, so they are potentially infinite. That is not the same thing as actually infinite. An actually infinite set would be a set like the set of rooms in a hotel that actually had an infinite number of rooms. While the potentially infinite makes sense, the existence of the actual infinite gives rise to all sorts of paradoxes, the most famous of which is Hilbert's Hotel, named after David Hilbert, the great German mathematician. Suppose a hotel had an infinite number of rooms and every room was full. A new guest shows up and wants a room. You say, no problem, and move the guest in room 1 to room 2, the guest in room 2 to room 3, and so on, putting the new guest in room 1. So you can make room for one more. But how is this possible, given that all your rooms were full, ex hypothesi? After you consider this and many more such paradoxes, you may conclude, along with probably the majority of mathematicians and philosophers, that while actually infinite sets are beautiful and useful in mathematics, they could not really exist. If you want to read up this sort of thing, here is one of the best collections of stuff, which includes Hilbert's seminal essay, "On the Infinite."

What does all this have to do with the Big Guy? The argument goes that an infinite succession of past moments of time would be an actual infinite, which is impossible, so time must have had a beginning, and that beginning must have been caused by God. It seems to me like a version of the Aquinas first mover argument, put in the form of modern mathematics. William Lane Craig deftly presents the math and the philosophy in this book. (Interestingly, really committed atheists, such as Stephen Hawking, really hate the idea of a universe with a beginning, sensing correctly that it implies, at least intuitively, a beginner, if you will. He contorts himself amazingly to come up with the baffling idea of "imaginary time," a temporal dimension that is at a right angle to our familiar time (don't ask me), in order to avoid the conclusion the universe had a beginning. The lengths some people will go to . . . !)

But to get back to the Discovery channel, whether an infinite past would be an instance of an actual infinite or not, an infinite collection of actually existing universes certainly would be. So maybe it's just an infinite number, minus one. (Just kidding!) But even if there were an infinite number of universes, it does not follow that there must be another one with a Discovery magazine in it. In fact, this could be the only universe with that rather lame magazine in it, and still we could have an infinite number of universes. You could have a universe just like this one, but with no magazine (let's hope so), and me standing 5 feet from my chair, then one where I'm half that distance, then another where I'm half that distance, and so on, ad infinitum. You could, that is, have an infinite number of universes without having all possible universes represented. That is, assuming you could actually have an infinite number of anything, which I doubt.
posted by thomas at 11/12/2003 04:38:00 PM