Last night, aided by a great deal of coffee and a strange mood,
hansandersen, jrpseudonym and I had a discussion about Pascal's Wager and the mathematics of belief. The results were... well, somewhat strange, but some of you may find them amusing or interesting, so here's a brief summary. It's incomplete - guys, you want to add in some comments with your own notes and thoughts from last night? I know there's plenty.So, without further ado:
The original Pascal's Wager: (The short version) You want to decide whether or not to believe in God. You can do it with a simple probability calculus. Say that there is some fixed probability p that God does, in fact, exist. p is therefore a number between 0 and 1. Now, if you believe in God and act accordingly, following God's laws, you pay some fixed price for it in this life -- call it F -- but then after death, if God exists, you receive infinite reward, and if God does not exist, nothing happens. Thus the expected total reward, in this life and the next, for following God's laws is -F+p*∞ + (1-p)* 0, which is infinite.
Conversely, if you don't believe in God and follow his laws, you pay no penalty in this life, but in the afterlife you receive infinite torment with probability p and nothing with probability 1-p. Thus the expected total reward for not following God's laws is p*(-∞)+(1-p)*0, which is negative infinity. From this one can conclude that, no matter how small p is, as long as there is a nonzero probability that God exists, one should obey his laws, and therefore be a good Christian.
The flaw in this argument, of course, is that it doesn't answer the question of precisely which set of God's laws one ought to follow. I mean, there's a finite probability that the Catholics are right, but there's also a finite probability that the Mormons are right, and the Hindus, and so on and so on... we therefore set out last night to try to improve a bit on Pascal's Wager and see what we can learn.
The setup: We want to figure out which God's laws, if any, we ought to follow, based on a similar probability calculus. So we first need to define what a set of God's Laws consist of.
Note: The following will describe the mathematical calculation we did, and it may get slightly technical at times. Feel free to skip and head down to our conclusions at the end.
For these purposes, then, we can define a religion to be a set of precepts - rules to follow, with associated rewards and punishments. Technically speaking, a religion is a function from the set of all behaviors you may have onto the set of all possible rewards.
To avoid one unneeded complexity: Everyone has to do this calculation for themselves -- one man's heaven is another man's hell. A person's individual preferences can be thought of as a map from the set of all possible rewards to the set of real numbers, where people basically rate things by preference. (If reward A is preferable to reward B, A should map to a higher value than B, et cetera.) There may be subtleties in this, but this is really the sort of thing that economists do all the time when discussing "free-market value." So we can say that, for any individual, a religion can be thought of as a function from the set of all behaviors onto the reals. (Or, to accomodate religions that have infinite rewards and punishments, the reals plus infinity. If, like me, you feel a bit uncomfortable with having infinities inside the sums below, you can map the range [-∞,∞] to [-1,1] using the hyperbolic tangent.)
Following Pascal's construction, then, the expected reward for any particular behavior is then given by the sum over all religions of the probability of that religion times the reward function for that religion. One should therefore choose the behavior which maximizes this expectation value.
Note, however, that for every religion there exists a corresponding antireligion, whose ruleset is identical but whose reward function is multiplied by -1. (As a side effect - this means that a religion whose reward function is identically zero is self-dual, and in some sense "special.")
Now, the hard part of this process is the assignment of probabilities so we can perform the infinite sum. The problem appears to be that we're operating in the complete absence of any information about which religion is actually correct; we simply know that the sum of all the probabilities is equal to 1. In the absence of a priori information, we can therefore do no better than to assign equal probability to each religion. (More on this in a moment)
Now, however, the sum is easy to compute: If any religion appears with probability p, its antireligion appears with probability p as well, and the sum of these two terms is zero. Since religions occur in pairs, we can therefore conclude that - so long as every religion has the same probability as its antireligion - the Pascal's Wager sum is identically zero, for any behavior. ♠
Conclusions: Oddly enough, this statement isn't content-free. In plain English, what it means is that, absent any information about the afterlife, any set of actions in this life is equally likely to lead to anything after death. In any calculation of expected consequences of an action, in particular, the set of possible consequences in the afterlife sums to zero. So Pascal's Wager, generalized to consider the set of all possible religions, tells you that when trying to determine how to act in this life, you shouldn't think about the afterlife: you simply don't know what the consequences of any given action in the afterlife will be.
Not quite what Pascal was hoping to prove, I'm certain, but interesting nonetheless.
Possible changes to the argument: The big assumption we had to insert into this calculation was that all religions are equally probable to be correct. Clearly, the only way out of the conclusion above would be if there were some symmetry-breaking process which could meaningfully assign different probabilities to different religions. (Or at least, distinguish between a religion and its antithesis) Hans and Robert both had some interesting ideas on this point.
Side note: Contemplatable Religions This is a side theorem we came up with, which turned out not to be necessary for the final analysis, but is somewhat interesting nonetheless. We consider a religion to be "contemplatable" if its rules can be recognized as such. This is a very weak requirement: all we require is that it is always possible to tell whether a given statement is or is not a rule of the religion. To be uncontemplatable would require some very strange rules - for instance, rules which require infinitely long sentences to state.
Strangely enough, though, one can easily show that almost all religions cannot be contemplated. The proof is basically the same as the proof that the set of real numbers which can be computed is of measure zero in the reals. (i.e., a real number chosen at random is never computable)
First of all, the set of contemplatable rules is no more than countably infinite. This is because the set of contemplatable rules is a subset of the set of all Turing machines (since if a rule is contemplatable, then by definition, there exists a Turing machine that reads in a sentence and outputs "YES" if that sentence is the rule, and "NO" otherwise). Since every Turing machine can be coded as an integer (cf. Gödel's first incompleteness theorem), the set of all Turing machines is a subset of the set of integers, and thus the set of all rules is a subset of the set of integers.
From this we can conclude that the set of all collections of contemplatable rules (i.e., the set of all subsets of the set of contemplatable rules) is uncountably infinite - it is equinumerous with the real numbers. And since for a ruleset to be contemplatable, each of its rules must individually be contemplatable, we conclude that the set of all contemplatable religions is equinumerous with the real numbers. (And not, as one may have supposed, with the grains of sand in the universe)
But on the other hand, the set of all rules - contemplatable or not - is clearly at least uncountably infinite. As an idiotic example to prove this point, consider rules of the form "You must eat exactly X fishes per day," for any real number X. There are uncountably many such rules possible. (But most of them aren't contemplatable, because most real numbers are not computable - one can show through an argument basically identical to the one above that the set of all real numbers which can be specified meaningfully to arbitrary precision, like 3 or 1/3 or π, is actually only countably infinite) From this we can conclude that the set of all possible rulesets is at least equinumerous with the power set of the reals.
And finally, since the power set of the reals is strictly larger than the set of reals, we can conclude that the set of contemplatable religions is actually of measure zero in the set of all religions. ♠
Thus, oddly enough, a religion picked at random is never contemplatable.
Conclusion: Even given the probability calculus above, and assuming that one could resolve the probabilities question to satisfaction, the odds appear to be overwhelming - in fact, certain - that the "right" religion cannot be comprehended by the mind of man in any meaningful sense. Would this count as a "misanthropic principle?"
(End of digression)
