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## Pascal’s Wager(s)

Pascal’s Wager is a famous argument created by 17th Century polymath Blaise Pascal, urging us, even if we’re not convinced of it, to bet on God’s existence.

In its basic form, it goes something like this: If you believe in God and he doesn’t exist, you incur a small loss (you’re wrong, and you have given up some of your freedom and time worshiping something that isn’t real). If you believe in God and he does exist, you win an infinite reward (eternity in Heaven). If you don’t believe in God, and he doesn’t exist, you get a small gain (you’re right, and you get to live life without religious strictures). If you don’t believe in God, and he does exist, you are infinitely punished (eternity in Hell). The argument is usually presented with a 2×2 matrix, like this:

Viewed this way, the Belief column looks more attractive than the No Belief column! And so, Pascal says, you should put your metaphysical chips there. Case closed!

Of course, you might be thinking: Obviously, you can’t simply come to believe something because it seems like a good bet. Pascal has an answer for this, and claims that if you’ll only continue to act as if God exists, sooner or later, you’ll actually come to believe it.

Even if he’s right about this cognitive-behavioral outcome, it turns out that the wager isn’t as simple as Pascal would have it. For one thing, there are thousands of Gods one could believe in, each with its own infinitely good Heaven. (Of course there are thousands more one could worship with no heavens at all, or with only finitely good heavens. But we’ll leave that aside for now.)

Now, if all of these possible gods really do offer up an infinitely good Heaven, but only one of them actually exists, the math says you should still bet on one of them. Lord knows (ha) how you’d decide, but Pascal’s wager is still reasonable here.

However, if there are actually an infinite number of gods that could exist and offer infinitely good Heavens, things change. How could there be infinitely many gods like this, you ask? It doesn’t seem unreasonable to me. Take one of the versions of the Christian God. Now say that he prefers people who own one bunny. (I’m saying this is possible, not necessarily likely!) Now take that God and say he prefers people who own two bunnies. Well, this can’t be the same God, because there are contradictory beliefs involved. So there are two possible gods generated this way. Well, keep going: A god that prefers people who own three bunnies; four bunnies; etc., ad infinitum.

I have no idea how the math would work in such a case: An infinite number of gods with infinitely good heavens, and only one of those gods actually exists… How do you choose which to bet on? Any mathematicians out there, please let me know what you think! My mathematical intuition says that choosing one item in an infinite bucket is an infinitesimally small gesture; that is, that it’s actually a zero-probability bet to choose one god out of an infinite number of them.

In any event, we’re not even done tweaking Pascal’s wager at this point. There are a lot more possible gods we have to consider. There are, of course, thousands (if not infinitely many) “normal” gods that promise an infinitely good Heaven if you believe and behave, and an infinitely bad hell if you don’t believe and don’t behave. But there are also possible bizarro gods in the mix: Gods that promise infinitely good heavens for bad behavior and lack of belief, and infinitely bad hells for those who believe and act well. (Again, I’m not saying this is likely — just that it’s possible. I don’t see why it’s any less likely than the normal god scenario, but that’s a topic for another day.)

But wait! There are also possible nice gods, who reward believers and non-believers alike with an infinitely good heaven! And there are possible mean gods, who punish believers and non-believers with an infinitely bad hell, no matter what.

What are the odds now? Especially if any/all of these categories allow for infinitely many potential gods.

The math is completely beyond me. Which isn’t to say that it’s undoable, of course! But the simplistic math that Pascal’s wager generally relies on is completely inadequate, once you plumb its depths a bit.