fidesquaerens (marta_bee) wrote,

What Philosophers Cant(or) Refute

I was on the train today and it was empty enough I could take a seat and get out a book. The only book I happened to have on me was Robinson's "God" anthology (, and so I was skimming William Lane Craig's article "Philosophical and Scientific Pointers to Creation ex Nihilo."

Basically he's trying to make the case that the best scientific and philosophical evidence suggests the universe came into being at some point. And that means that either it had to come into existence spontaneously (which Craig thinks is implausible) or it had to be created by something. It's kind of a version of the cosmological argument, but updated to take modern math + physics into account. I've never found the cosmological argument particularly convincing, since even if the argument worked it would only show that something exists that kicks off the whole chain of cause-and-effect not that this something is just + merciful or a kind of loving father or whatever you want to believe about God's nature.

Anyway, the first step of the argument was to show that the universe couldn't have existed forever - that it had to have a beginning. And to do this, Craig talks about infinites (yes, there's more than one). He tries to argue that something called an actual infinity - an infinite group of objects in reality and not just our minds is impossible, so you couldn't have an infinite world without a beginning. To do this he proposes a thought experiment:

Imagine a library with an infinite number of books, all with red spines. Then imagine a second library that also has an infinite number of books, but where half have red spines and the other half have black spines. Are there as many red books in the second library as ther are in the first one? Most of us would say there's twice as many - I mean, the libraries are the same size and in the second one only half the books are red. But Dr. Craig argues that, according to higher mathematics, there are the same number of red books in each group. But this is obviously rubbish (quoth Craig), so the whole idea of having actually infinite sets of stuff --not just the idea of infinity, but actual infinities-- is also rubbish.

This is a problem mathematicians have struggled with. We say that the set of even numbers ({2, 4, 6, 8, .... }) is infinite - there's no limit to the number of its members. The set of whole positive numbers ({1, 2, 3, 4, ....}) also has an infinite number of members. But our intuition tells us that the second group should be twice as big as the first one; after all, every other member in that second list isn't in the first list. But the way I learned about infinities (yes, there's more than one infinity) it's wrong to say either is bigger than the other. Technically they're the same size. But the second list is more dense.

As odd as it might seem, there's an infinite number of numbers between every two numbers. Take one and two. Between those two numbers we would have:

1.01, 1.001, 1.0001, etc.
1.01, 1.011, 1.0101, etc.
1.02, 1.002, 1.0002, etc.

... and on down the list. Now, with even numbers, there's that odd, whole number exactly halfway between each member: 1.000, 3.000, etc. There's also room for half-numbers, quarter-numbers, numbers with the remainder .017, whatever you like. There's a lot of space between each number of the {2, 4, 6, 8, ....} set. There's also a lot of space between the different members of the {1, 2, 3, 4, ....} set. So in Dr. Craig's example, the library full of red books has the same number of books as the library with alternating red and black books. The first group is just more dense than the second.

Why? Because infinite numbers are just screwy like that. They simnply don't behave like you've come to expect using normal numbers.

So, three takeaway lessons from this adventure in number theory, at least for me:

#1. Number theory rocks. Cantor numbers rock. I really wish I remembered more of this because I'd like to nail down more precisely just why what Dr. Craig is saying jumps out at me as mathmatical nonsense.

#2. Scientists really don't make good philosophers. That I knew. But apparently, philosophers don't make good scientists either.

#3. Blogging about math + the late-night hour = maybe something to avoid in the future. Hope this isn't too dense!
Tags: math, philosophy, rl
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