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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 (http://bit.ly/RJn53M), 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.
[etc.]
1.02, 1.002, 1.0002, etc.
[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!

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Comments

( 6 comments — Leave a comment )
lindahoyland
Oct. 19th, 2012 07:19 am (UTC)
Fascinating stuff!
marta_bee
Oct. 19th, 2012 01:30 pm (UTC)
Glad you liked it!
dwimordene_2011
Oct. 19th, 2012 12:35 pm (UTC)
I don't know if your post is denser than anything else with infinite numbers between the letters, but... yeah.

I only have an intuitive grasp of numbers, so when people start going on about multiple infinities and how one is denser, but not bigger, and whether we're talking about a set or an actual infinity, I just do the nod-and-smile-fixedly routine until it's over.

Re: philosophers who make crappy scientists: there's a lot of this in environmental ethics, it seems. That interface, science and philosophy, is really fraught, because to master either of them takes so much time and investment that it's really hard for any one person with finite resources to do.
dwimordene_2011
Oct. 19th, 2012 12:41 pm (UTC)
P.S.
Since I do totally suck at numbers, and don't read in these debates, what is your take on the actual versus idea of infinity thing Craig has going?

(The only infinity I deal with is Totality and Infinity, and I try to just smile and move away from that usage, too...)*



*Okay, that's also not true, there are others, but none of them are mathematical infinities, is my point.

Edited at 2012-10-19 12:42 pm (UTC)
marta_bee
Oct. 19th, 2012 01:24 pm (UTC)
Re: P.S.
Thanks for bearing with my inner math geek. I don't usually let her out to play, but I can see how that "nod-and-smile" mechanism would kick in quite easily.

As for actual infinities, I think they're at least possible. I know there are some mathematicians who think they're impossible, so perhaps I shouldn't try to play scientist here and leave that question to people who know what they're talking about better than I do. But I'm certainly not conviniced actual infinities are nonsense.

Here's the thing, though: while I think they're possible, in practice I'm not sure they'd ever happen. Any time you add two finite numbers togethr you'll always get another finite number. big one, to be sure, but still finite. This is Kant's basic point in the Religion book: working on our own in space-time, we just can't become good enough to overcome radical evil. Because each individual effort is limited, no string of efforts will ever become infinite. So if you start with a bit of space-dust that's (say) one cubic meters and then it grows to ten and then a thousand, etc., and you added them all together, you'd eventually get a very, very big sum. But it would still be finite. The only was to get around that is to have one point in the chain that's infinite all by itself.

The bottom line? My gut instinct is that yes, actual infinities are possible, but it's a moot point because i just don't see the universe actually adding up to an infinity, based on what I know about physics. (Keep in mind I know as much about astrophysics and origin theories as Krauss does about what Aristotle + Aquinas meant by nothing; also, this is me before coffee...)

Edited at 2012-10-19 01:32 pm (UTC)
marta_bee
Oct. 19th, 2012 01:29 pm (UTC)
Re: P.S.
To boil it all down a bit more: I think people like Craig actually make a convincing case elsewhere. Craig writes that, thermodynamics being as it is, the universe resembles a clock that was at one time wound up and is now winding down. Which would mean the universe isn't infinite. And based on the little bit of understanding I have, what he says there makes sense.

That would mean the universe wasn't infinite - there's some point where it had to pop into existence. I'm also willing to but that every change requires something to change it (Aquinas's point seems intuitively true), so I'll buy that we need some kind of nonchanging causer - what Aquinas calls God. What I don't buy is that this God has to look anything like what religious people expect. A blind, physical force (something like gravity) would work just fine as "God."
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