Thursday, August 13, 2009

The Probability of God

I had a revelation this morning while I was swimming.

I had been thinking of something I read regarding a common atheist response to the anthropic principle (see my post from June 1). There is so much design apparent in the universe, it would be a monumental coincidence for all of this to happen by chance. But the atheists say that that does not prove God exists, because no matter how improbable our universe is, the statistical probability of God existing is even more remote. (Apparently, this is one of Richard Dawkins' arguments in The God Delusion; I haven't read it, but I was reading a book that critiqued it.)

The more I thought about it, the more absurd that argument seemed. First, my response was, "Who sez?" Or perhaps more eloquently, "How can you demonstrate that?" It seems to me that there is a fallacy of burden of proof here. After a statement like Dawkins', proponents of the Christian faith may be left sputtering, trying to figure out how to show that the probability of God existing is not as remote as Dawkins says it is. However, that is overlooking two things. First, perhaps Dawkins should be proving that it is remote, rather than the other way around. Looking at it that way, Dawkins' counterargument here to the anthropic principle seems to rest on an unprovable assumption. But even more intriguing is that this is a red herring fallacy as well. Simply saying that God, as an explanation for the improbability of this universe, is Himself improbable does not explain how we got here. It's just a negative argument against one explanation of how we got here. What we're left with in Dawkins' treatment of the situation is that we are here because we are here. Which doesn't answer any of the big questions at all.

So that's two logical fallacies Dawkins seems to be committing here. (And please forgive me if I'm not representing his case very well. I really should do my homework before critiquing someone else's position.) First is shifting the burden of proof, and second is the red herring fallacy.

But it gets worse.

The second thing I realized this morning is that the statement "no matter how improbable this universe is by chance, the probability of God is even less" is tantamount to saying "the probability of God existing is zero." Think about it. The only non-negative number that is guaranteed to be smaller than all positive numbers is zero. This is quite a strong statement. It goes far beyond saying God doesn't exist. It says that God cannot exist. In other words, Dawkins is using an assumption that God cannot exist to try to prove that God does not exist. It is a completely circular argument.

Here's another way to think about it, for the more math-oriented folks. In probability and statistics, the proof we're trying to make is something called a conditional probability. We see an improbable universe around us. What is the probability that God exists given we live in an improbable universe (ie, what's P(G|U))? Using Bayesian inference, we can easily come up with:

P(G|U) = P(G)/(epsilon + P(G)).

Here P(G) is the prior probability that God exists, and "epsilon" is the small chance that this universe came together by coincidence (all scientists would agree that epsilon is very small...something like 10^-50 or less). When doing Bayesian inference, you often have to bring in some a priori assumptions to assign prior probabilities (hence the name), so we have to guess at what P(G) is. But you never outright assume that P(G) is identically zero (or one). That would be the same as saying "no matter what our studies tell me, I will choose to believe X." (That's called blind faith.) Usually, when you don't know, you simply set your prior probabilities equal to 1/2. It's easy to see that P(G|U) (the probability that God exists given the universe we live in) would be extremely close to one for any reasonable choice of P(G). The only choice that makes P(G|U) small is P(G) = 0. Which is apparently what Dawkins wants to say.

Perhaps I'll write more on this later, but for now, I'd appreciate any comments.