Pascal's wager

At The Pearly Gates:

Well, Mr. God, Sir, I'm sorry I didn't believe in you back when I was alive. But I believe in you now. So I'm forgiven for my past sins, right?

Are You Truly Sorry?

Well, no, not really. I'm damn proud that I only believed in what was logical and rational. I was wrong, but I was wrong because I thought. The people who were right were right because they didn't. I'm proud that I wasn't an idiot. If I have to be truly sorry that I was smart in order to get into Heaven then I'll just go to Hell I won't sell out to anyone, even God. Even for eternal bliss.

"I consider it a compliment rather than an insult to be called an agnostic. I do not pretend to know where so many ignorant men claim to be sure." -- Clarence Darrow

Live in virtue, no desire.
Heaven must be Hell in the sky.

As anybody who has studied probability theory from the point of view of measure theory knows, you cannot have a probability measure without first defining the sample space. Unfortunately, this is very difficult to do when relating to probabilities of the Real World being the way it is (although Bayesians try to do this, at least for empirical events). For doing metaphysics, your sample space will determine the outcome, as Pascal's wager shows.

So what are sample spaces, probability measures, and measure theory?

If you don't want to read up on them directly, here's a short primer...

A probability measure is a function P(⋅) which assigns a real number 0≤P(⋅)≤1 to events. So events are anything for which you can measure a probability. Typically, you'll want to measure probabilities for a huge range of events, so you need to make them live together nicely. The solution is to talk of a sample space, which essentially represents "everything that can happen". Events are subsets of the sample space.

For instance, if I want to talk about the throw of 2 dice, my sample space could be {1,2,3,4,5,6}×{1,2,3,4,5,6} (the set of all possible results for each die separately); it could also be much larger, e.g. by describing the set of every molecule. In the first formulation, the event "the sum is 6" would be the set {(1,5),(2,4),(3,3),(4,2),(5,1)}. And I could assign a probability of 5/36 to this event.

To make it possible to do interesting things, some properties are required from probability functions; technically, these make P(⋅) into a measure.

It turns out that often not everything is measurable; the reasons go deep into the foundations of mathematics and are often technical in nature. But there are also excellent practical reasons for ensuring that this remain the case.

Say I play poker with you. In order to place bets, we each try to calculate or estimate probabilities of events:

• Probability that I'll have a full house or better after swapping 3 cards. (*)
• Probability that you'll have a full house or better after swapping 2 cards. (**)
• ...

That's because the only important probability -- P(you'll beat me if I swap 3 cards) -- is measurable in terms of these. Note, however, that (due to lack of mirrors etc.) I measure P(*) and P(**) entirely differently from the way you do!

What's going on is that we're using different measure spaces, since each of us knows different things. Conditional probability discusses this; its formalization makes extensive use of different measures on the same sample space.

Is probability true? The pragmatic view is that we apply it to the world because "it works". For instance, when playing poker many people will apply probability on the sample space which describes a perfectly shuffled deck. Persi Diaconis, however, knows better, and may well use a different sample space (when he doesn't think you'll cheat) or no sample space (which he thinks you will and he won't be able to catch you out). As always, application of mathematics to the Real World™ is an empirical thing, and responsibility for it lays in the hands of who applies it.

Now, back to God(s), or lack thereof. Pascal wants to employ a technical tool -- probability -- to argue for the existence of God. Therefore he is obligated to show that it is applicable. In other words, he must give a sample space, a measure space, and a probability measure, and argue that they are either correct or a close approximation. After that, he gets to apply expectation and other tools of probability theory, maybe showing that maximal utility is gained by believing that God exists.

And I get stuck on the first -- sample space. If I don't admit the possibility that God exists, my sample space won't have God. Clearly, that won't do. So say I'm trying to decide if He Does or Doesn't. To apply the probabilistic method, I need to know about the sample space. Presumably, it will be set up so that both "God exists" and "no He doesn't" will be measurable. The sample space may well be bigger, but it's not clear what it should be -- it will have to encompass not only my entire metaphysical beliefs, but also all other "possible" metaphysical beliefs! The same goes for every other part.

How bad is not defining every part of your probabilistic model?? For an example, just see the two envelope paradox...

The problem with Pascal's Wager, it seems to me, is the possibility that you might choose the wrong god. Suppose I decide to believe in Jehovah and when I die it turns out that, say, the Hindus were right all along (disclaimer: I know jack about Hinduism). I could be in deep trouble.

Works like this:

|-------|--------------|-------------|
|       | Belief       | Non-Belief  |
|------ |--------------|-------------|
| Right |Go To Heaven  | no loss     |
|       |Big Gain      |             |
|-------|--------------|-------------|
| Wrong |Small Loss    |10 Die       | 
|       | (some time)  |20 GOTO Hell | 
|-------|--------------|-------------|

So, you basically may lose a trifle amount of time in religious services if you're wrong about believing God exists, but you have a lot to gain if you're right, which, if you were a gambling type, it would make more sense to bet on God existing, because that comes with the biggest pay off.

Initally, this was offered as an initial motivation for a person to join the Church, and later it was supported by internal Dogma. This was the First Step, so to speak.

The french writer Cavanna has an extension of the wager :

1. If You choose to worship a god and he does indeed exist, you will be saved.
2. If You choose not to worship a god and he does indeed not exist, you lose nothing, not even a second of your precious time.
3. If You choose to worship a god and he does in fact not exist, you lose some of your precious time.
4. If You choose not to worship a god and he does in fact exist, three possibilities :
   
   1. He does not care if you worship him, that's basically the same as 3.
   2. He is a good god, and He knows he made you like you are, i.e. inperfect and irreverencious. He cannot blame you not worshipping Him, it's his very own fault. He will send you to heaven anyway, because He is good and forgiving.
   3. He is an evil god, and no matter what you do, you are fucked. Even more, if you worship him, he will do the evil action, and be evil to you in the first place. In fact you're better off with disrespecting him, he'll find that cool.

"Live a life you love
Use a god you trust
Don't take it all too seriously"
–Everything New Nodes

A Wager is the title of an article written by Blaise Pascal in a bare room while on drugs. Most people (unsurprisingly) call the philosophical argument formulated in the said article Pascal's Wager.

The argument is addressed to Pascal's former libertine friends. The French libertines of Pascal's 17th century prided themselves on their rationality so Pascal provides them with a rational argument.

The Mathematical Hope of God

Let's start by playing a little game : coin toss. Heads, you win 10 (of your preferred currency), tails you win 20. Any experienced gambler will tell you that coin tosses hardly have even odds but since this is a mathematical game of coin toss, you've got exactly 1/2 chance of earning 10 and 1/2 chance of earning 20. So your mathematical hope (H), i.e. what you can hope to win for this game is the odds multiplied by the gains is :

H = 1/2 * 10 + 1/2 * 20
  = 15

Now of course this is a mathematical figure : even though your hope is 15, you'll never win 15, either 10 or 20. But if someone offers you to pay 15 to play this game, then it's a fair game. But if he (or she!) offers 17, you're getting ripped off.

Now, says Pascal, let's play The God Game!

Let's say that a is the probability of God's existence and b is the probability of God's non-existence. If God exists, what you can hope out of life is infinite (∞) : paradise. But if He doesn't, all you can hope out of life is life itself, say Λ (that's not an A that's a capital lambda).

Now, this is what the libertine thinks : God doesn't exist, so a = 0. Therefore his mathematical hope (H) for life goes like this :

H = a * ∞ + b * Λ
  = 0 * ∞ + 1 * Λ
  = Λ

All he can hope out of life is life itself, so he boozes up, gambles, has premarital hanky-panky and other such sins. But, Pascal says, that's not reasonable! How can you say that God doesn't exist? You can't know whether God exists or not. You have to at least entertain the possibility of God's existence. The probability of God's existence therefore becomes ε (small epsilon) such as ε > 0. So, Pascal says, the mathematical hope for life goes like this :

H = ε * ∞ + b * Λ

And since infinity is so, well, infinite, Pascal argues, it overshadows everything. No matter how small ε is, the infinity renders everything else irrelevant :

H = ε * ∞ + b * Λ
  = ∞

So basically, what Pascal says is that no matter how remote the possibility of God's existence is, you can't not take it into account. Too much is at stake -- the possibility of an eternal life, of a loving God... Of course you can't know that there is a God (and you can't know that there's not, either), and that's why he says : wager!

On the one hand, you can't know that there is a God, and on the other hand, you can't ignore the question, because the stakes are so high, so, you libertine gambler, take a bet!

Which God?

A common argument against Pascal's Wager is that it seems to not take into account the fact that there are actually several religions out there : it's all fine and dandy to wager there is a God, but what if you pick the wrong God? You're still screwed -- this is called the many-gods objection. However, there are several things to take into account here.

First of all, since Pascal was a friend of the Jansenists and therefore dangerously flirted with heresy in the eyes of the Catholic Church, he was well-aware of the differences between religions. This supposed flaw in his argument can't be attributed to some Christian superiority complex according to which there can be no other interpretation than his.

But more importantly, the many-Gods objection overlooks an important part of Pascal's wager: the argument is actually two-fold. The first part is the idea of a wager : even though you can't rationally know whether there is a God the stakes are so high that you should bet on Him anyway. The objection to that is infallible : sure, bet on God, but which distro? This is where the second part of the argument comes into play.

Let's look back at our mathematical formula. The potential reward of worship for God is infinite. But in mathematics there are several infinites, some bigger than others. Likewise, the Muslim paradise is different to the Christian one* (Note: I am making no qualitative judgement, it's up to you which one you like best). So you're left with a set numbers of infinites to choose from, as many as there are religions : ∞₁, ∞₂, ... ∞_n. And therefore, the bigger infinite value you pick, the bigger H becomes (since, according to the first part of the argument, H = ∞). This is the rebuttal the proponents of Pascal's Wager give to the many-gods objection, that it tackles only a part of the issue and not the main point : the idea that even though you can't know God you have to bet on Him.

An other pertinent objection is the Intellectualist objection : the idea that choosing to believe in (a) God isn't enough to make you believe. If I offered to pay you $1000 for believing the sky is green, could you sincerely adopt this belief simply by wishing to? 'Course not. But then again, do you do anything by just wishing to? You can't node by just wishing to. You have to get at your computer, go to E2 (if it doesn't lag), pick a subject, research it, etc. etc. You don't just choose to get a degree, to get married or to become a rockstar : you have to take steps toward that. But you will never do it if you don't make a willful choice in that direction.

Take a Bet

All of Pascal's religious philosophy is permeated with the idea of choice. He postulates (a century before Kant proved it) that we simply can't know whether there is a God or who He is, and even that we shouldn't. If the answer to the question of God is out of our reach, the question itself is not. Then only one thing is left to do : choice.

Pascal's Wager is often seen as a rational argument by Pascal to push his own religion on the contemporary libertines. Sure, that's what it's meant to do. But the article of the wager isn't an appeal to convert. By laying out the alternatives like he does -- by picking God, you have nothing to lose, etc. -- he's not telling you what to bet on. Pascal is simply playing bookmaker! He's giving you the odds.

That is the real point of Pascal's Wager. Pascal isn't telling you "You must believe in God because so-and-so," he's simply telling you "Wager! This is the most important question of all. The stakes are too high for you not to decide to answer it. This is like a coin toss, you have no way of finding out the outcome, so bet." Almost like you would with a purebred horse, find the God you like best and bet on Him.

Pascal's Wager is a rational argument, but its pertinence goes far beyond the limits of its rationality.

Note : I gave up maths in high school. If you know more than I do on probability calculus (and that's likely) and I made mistakes in this w/u (and that's likely), feel free to correct me via /msg.

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* The rest of Pascal's argument will be to convince the libertines that the biggest infinite is his particular brand of Christianity and therefore that they should bet on that one, but that isn't part of the Wager.