Why the "Bet on God" is a Bad Bet

Many people believe in "God," I suspect, to be safe. If He/She/It, exists, well then great! Their belief will "get them into Heaven," they believe, and they'll reap infinite reward. And, if not, there's no loss. This bet is formally known as "Pascal's Wager," after Blaise Pascal, a famous 17th century mathematician/physicist/philosopher who discussed it in his book Pensées (French for "Thoughts").  The bet, though, is not a good bet. The problem is that it assumes that if 'God" exists, then "God" necessarily will open the doors to "Heaven." "Heaven" is also assumed to exist, of course. The bet also assumes there is no downside in living one's life in a potentially false belief. If the only possible assumption is that believing in "God" will open the doors to "Heaven," the bet isn't necessary: the idea that it's good to get an infinite reward is jejune. Yet if one purports to know the assumption is true, then the bet is a tautology; you've assumed the conclusion. But those aren't the only possible outcomes.

Let us suppose that we do not know if the bet's assumptions are true, as Pascal properly acknowledged he did not,¹and that there are multiple possible outcomes -- that's why it's a wager.² But, then, we have a problem. In addition to postulating a god that rewards belief, we might equally postulate a god exists who punishes belief in god, indeed, who punishes belief in any God.³ Moreover, there might be multiple such gods, and gods of different kinds.

The options for existence of "God" related to the bet include some of the following sets: (a) there are no gods; (b) there are one or more gods who neither reward nor punish belief; (c) there are one or more gods who reward belief if you happen to believe in the right one(s); (d) there are one or more gods who punish belief in the wrong god(s); (e) there are one or more gods who reward belief in any god; (f) there are one or more gods who punish belief in any god. These sets are not mutually exclusive; they may overlap or be subsets of each other.⁴ Thus:

Among the Christians historically have been disputations, excommunications, executions, massacres, and wars over minor differences in belief. Muslim beliefs aren't that far from Christian beliefs -- they believe in the divine inspiration of the Hebrew and Christian bibles, consider themselves children of Abraham and Moses, and believe Jesus was a prophet -- does a Christian god accept Muslims into heaven? Does Allah accept Christians? What if the set of beliefs one happens to gamble on happen to be wrong?

Tens of billions of people throughout history have believed in multiple gods and never heard of Christianity or followed anything similar to Christian beliefs. Pascal, despite having converted from traditional Catholicism to the sect Jansenism (the key doctrines of which were condemned by the Pope during Pascal's lifetime), dismissed the idea that there would be any "God" other than the god he believed in or that one would wager on any god besides his own. From what little we know he seems to have never heard of Hindus, Buddhists, Confucians, Incans ....

If there were only one possibility of "God" and that "God" would reward belief with infinite return then we might state (where p is the probability that "God" exists with a return, if right, of  ∞⁺):⁵

 ∞⁺ * p + cost of belief  > cost of doing nothing 

∞ > 0

If one assumes, however, there's an infinite penalty for wrongfully believing, then (where p₁ is the probability that Pascal's god exists and p₂ is the probability that the "wrong" god exists -- one that punishes belief or wrong belief):

 ∞⁺ * p₁  + ∞^- * p₂ + cost of belief  > cost of doing nothing 

cost of belief  > cost of doing nothing ?

If one could "normalize" the opposite infinities we're left with the cost of belief, which could be substantial. In fact, the relative probabilities of different gods existing doesn't matter as long as the consequences are infinite. That is because anything times infinity equals infinity. As it is, though -- normalization being out of the question -- we can see that the bet on a "God" is simply a meaningless exercise. Of course it cannot tell one whether "God" exists, yet it is not even valuable in telling one how to live one's life. It is sad if it is true, as I speculate, that many people believe in their gods out of fear for the fate of their "souls."

On the other hand, many people, such as myself, believe there is overwhelming evidence that no "gods" exist. That is because the physical predictions regarding such gods have repeatedly been shown incorrect and the superstitious structure under such belief ridiculous. That should not be surprising as those predictions and superstitions reflect ancient storytelling and are premised on ignorance of the Universe.

_______________________________
¹  Pascal's statement in the Pensées was:

       If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is. This being so, who will dare to undertake the decision of the question? Not we, who have no affinity to Him.
       Who then will blame Christians for not being able to give a reason for their belief, since they profess a religion for which they cannot give a reason? They declare, in expounding it to the world, that it is a foolishness, stultitiam; and then you complain that they do not prove it! If they proved it, they would not keep their word; it is in lacking proofs, that they are not lacking in sense. "Yes, but although this excuses those who offer it as such, and takes away from them the blame of putting it forward without reason, it does not excuse those who receive it." Let us then examine this point, and say, "God is, or He is not." But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.
       Do not then reprove for error those who have made a choice; for you know nothing about it. "No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all."
       Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.--"That is very fine. Yes, I must wager; but I may perhaps wager too much."--Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. It is all divided; wherever the infinite is and there is not an infinity of chances of loss against that of gain, there is no time to hesitate, you must give all. And thus, when one is forced to play, he must renounce reason to preserve his life, rather than risk it for infinite gain, as likely to happen as the loss of nothingness.

Pensées § 233 (Translation from French by W.F. Trotter) (significant parts highlighted in bold). The Pensées is hardly exhaustive on the "wager" as surely people have had this view long before Pascal, and many have had it since with no knowledge of him.
² Pascal's wager, as he states it, is sort of like playing Blackjack blindly and saying after the deal, "I have the option of taking a card or not, and if I take a card I assume it will definitely be the card I want; therefore, I conclude I should get a card." A skeptic might point out that you will not definitely get the card you want. In fact, based on observation, that card isn't even in the deck.
³ This can be made a positive statement as the assertion there may be a God who rewards only those who do not believe in any God by sending them to "Heaven."
⁴ Strictly, sets (a) and (b) can be omitted since they represent null sets of the others -- indeed, for our purposes sets (a) and (b) are equivalent even though they seem so different. Our question, though, is not whether "God" exists, it is, having put that question to the side, whether it is wise to believe in a "God."
⁵ As an alternative formula (adapted from the Stanford Encyclopedia of Philosophy article on Pascal's Wager) take p as the probability of Pascal's "God" existing, 1-p the probability of that no such "God" exists, r(b+) = ∞ the return on that belief, r(b-) the finite return/loss if there is no such "God," r(nb+) the finite return/loss if there's no bet and there is such a "God," and r(nb-) the finite return/loss if there's no bet and there is no such "God," then ∞⁺ * p + r(b-) * (1 − p) > r(nb+) * p + r(nb-) * (1 - p), and ∞⁺ > finite.