Conversations on the mathematics of belief

(or, The Gods Must Be Crazy)

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)

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Published in: on June 19, 2003 at 11:40  Comments (12)  
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12 Comments

  1. Frankly, this was my favorite part of Philosophy, and a good deal of what my personal believe is based on. The flaw you mention in the original Pascal’s Wager is called the Many Gods Theorem. My favorite poke at this was that if there was a god which rewarded you for not believing in him the whole thing falls apart.
    Unfortunately, the flaw I see in the reasoning you have above is that you assume that there is a direct anti-religion for each non-zero religion. There are two possible alternate cases I’d like to bring up. What if there was a triplet pair, such that there was no direct anti-religion to one religion, but rather a pair of anti-religions. Alternately, we could consider a religion which does not have a direct opposite but which is not in itself it’s own direct opposite.
    The second case I was about to suggest, I just realized, is perfectly well handled by your theory. I was thinking of taking the system out of a two dimensional belief/anti-belief construct where faith could be calculated on a line between the two poles (believe and belief in the exact opposite) and putting it into a three dimensional belief-space. However, you’ll always be multiplying all three coordinates by -1, so regardless of number of dimensions you’ll always have a corresponding negative belief point.

  2. I think our model handles that case appropriately. Since we’ve identified a religion with a function from the set of behaviors onto the reals, we can simply define the antireligion to be the religion whose function is the negative of the first religion. Thus each religion has a unique antireligion, and the only way a religion can be its own antireligion is if its function is identically zero. This seems to rule out triplets or other complicated structures, and it guarantees the cancellation in the sum, since the sum can be broken into pairs of the form p f(B) – p f(B) = 0.

  3. The most important Zen things you can do are not provable determinable within the system of Zen.
    Mmm, g\”odel.

  4. Simon
    This goes to prove that super intelligent human beings should never be allowed to not work. A idle mind is a very very very dangerous one. Specially Yonatan’s mind. Thank god its only idle til monday.

  5. Actually, I believe the triplet of anti-religions or the non-opposite anti-religion are possible, but only in the realm of the non-contemplatables. Since a non-contemplatable can’t be expressed as a function (as it cannot be formally expressed by definition), there’s no reason that occurs to me you’d need religion-antireligion pairs in that space. Thus, the non-contemplatable religions could give a non-zero term to our sum.
    However, since they aren’t contemplatable, it is impossible to evaluate the value of following any given non-contemplatable, let alone the expectation value of all of them. So, we are given that there is an extra, and by definition, unknowable term in here.
    I like to call this “God’s Grace”, or alternatively, the “You’re all screwed” factor.

  6. Yes, but that’s circular reasoning. You say you handle the the case because you defined that you can handle the case. In other words, since you’ve reduced the religion down to a function you can just multiply it by -1. However, the process of reducing a religion down to an equation is left as an excercise for the reader. This means that you haven’t established that doing so produces an ordinary equation. It is, in fact, possible that a particular religion’s equation contains unsigned numbers or other values for which simply multiplying by -1 is insufficent for negation (sadly my math skills are lacking to the point where I may be talking complete bollocks, but I’m going to soldier on regardless).
    In essence, I’m pondering the case where p f(b) – p f(b) != 0. I strongly suspect that this violates at least one mathmatical law, but then again we’re pondering the divine, so anything is really possible.

  7. Another path to wander down would be to question if religions are vectors. If you follow 90% of the Catholic church and 10% Zen Buddhism, do you end up an Episcopalian?

  8. Well, this comes down to the question of whether the model is valid at all. For the purpose of this argument, a religion has been modelled by a function from the set of behaviors onto the reals, which I believe is the correct generalization of Pascal’s original construction, and for the specific purpose of a probability calculus seems to encapsulate all of the “relevant” information.
    The choice of the reals as the target space for these functions is a bit arbitrary, but I believe it is correct in that we follow the economist’s usual assumption that people can rank-order their preferences, and these are quantifiable by some sort of universal medium of exchange. (e.g., “how much money will you give me for this sack of oranges” determines the value of a sack of oranges, and “how much money do I have to pay you to let me hit you over the head” determines the value of a blunt trauma.)
    So the model is that any religion R is uniquely specified by its reward-function fR(B), and the set of all such functions is isomorphic to the set of religions. So for any R, we can define the antireligion R’ by fR'(B)=-fR(B). There’s nothing novel in this definition; I’m simply observing that if f is a function, so is -f, and so the set of all functions can be broken into pairs. (They’re always pairs and not triplets or something else, since -(-f)=f) Then the sum over all functions can be broken into the sum over these pairs, and the pairs sum to zero, since both p and f(B) are real numbers and so obey the usual laws of arithmetic.
    So I don’t think this is circular — simply the nature of the model.

  9. Well, I think our idea was that non-contemplatables could still be expressed as functions, just not computable ones. But the argument of pairs should still apply, since for any function f, –f is still a function, whether or not it is computable.

  10. A somewhat more practical poke at this argument:
    It is useless, for the original basis of the wager, to consider the set of all *possible* religions. They are not all accessible, and thus one cannot place bets upon them. One must consider the set of *accessible* religions, which would be a) religions that currently or previously existed on Earth, and b) religions that one could entirely conduct on one’s own from scratch (which would sum out to zero, per the previous argument). I don’t believe that (a) would, however.

  11. Well, (a) almost certainly wouldn’t, but why would it be interesting? In the wager, we’re trying to calculate the expected actual reward given a behavior, which means we need to average over possible maps from behavior to reward. Since the only thing that affects the results is the actual correct religion, the set of already-known religions doesn’t enter into it. (b) is close to the set of “contemplatable religions” described above, but as you point out, the cancellation argument applies to that set.

  12. Oh my god?
    What really horrifies me is that I have taken just enough discrete math and probability for that all to make PERFECT SENSE. I’m going to go soak my head in Epsom salts for a little bit to ease the pain.
    Fortunately for me, this is pretty much the argument I have been living my life by since about 10 anyways (minus the math, I had only gotten as far as geometry at that point), coupled with an ornery state of mind that indicated if god(ess)(e)(s) didn’t like me the way I like me, he/she/it could fuck off.


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