Pascal’s Wager for God

"Pascal," by Mitch Francis
Our selections come from the W. F. Trotter translation (introduced by T. S. Eliot) (New York: Dutton, 1958) of Pascal's Pensées.

"Pascal," by Mitch Francis
French genius Blaise Pascal (1623 – 1662) presented belief in God as a rational wager. (Image: “Pascal,” by Mitch Francis.)

Blaise Pascal
Blaise Pascal

The mathematician, geometer, physicist, inventor, theologian, and philosopher Blaise Pascal lived from 1623 until 1662 – just after he turned 39. At 16, his first serious work in an essay “On Conics” (about the Mystical Hexagram), states what is now called “Pascal’s Theorem.” The prodigy Pascal shaped economics, probability theory, and computing, among other disciplines. He even invented and created over 20 mechanical calculators – two of which are still on display in Paris and Dresden. After a mystical experience in 1654, he mostly devoted himself to theology.

Pascal never called section 233 of the Pensées “The Wager” – although it’s easy to see why it has been thusly named: It approaches the question about whether to believe in God not based primarily on the logical evidences that “God exists,” but on an assessment of the practical pay-offs of belief and non-belief. In short, belief – Pascal argues – is the “best bet.”

The significance of Pascal’s work in this domain extends well beyond theology into game theory and decision theory. It forms the basis of what modern probability theorists call “expected utility.” One poignant application: It provides a mathematical assessment of how good a bet is. If you gamble in order to “win big” – as opposed to “for fun” – then you want to know when doing so is the action with the highest expected utility.

Some "expected utility" math.
General definition of expected value:

EU(Bet) = [Pr(Win) * V(Win)] + [Pr(~Win) * V(~Win)]

In English, that reads something like, “The expected utility of a bet = (the probability of winning multiplied by the value of winning) plus (the probability of losing multiplied by the value of losing).”

Example 1: What’s the EV of accepting this bet on a 10-number roulette wheel?: I’ll give you $5 if the ball lands on any number 1 through 4; you give me $2 if it lands in the 5-10 zone.

[Pr(Win) * V(Win)] + [Pr(~Win) * V(~Win)]

[.4 * $5] + [.6 * -$2] =
$2 + -$1.20 = $.80

According to the expected utility formula, this is arguably a favorable gamble.

Example 2: What’s EV of buying a $1 ticket to win a $1001 lottery when Pr(Win) = .0005?

[Pr(Win) * V(Win)] + [Pr(~Win) * V(~Win)]
[.0005 * $1000 (cost-of-ticket factor)] + [.9995 * -$1] =
$.50 + -$.9995 = -$.4995

According to the expected utility formula, this is arguably a lousy gamble.

WikiPedia lists numerous items of importance named for Blaise Pascal:

  • Pascal (unit) (Pa), the SI unit of pressure (equivalent to one newton per square meter)
  • Pascal (programming language), a programming language developed by Niklaus Wirth between 1968 and 1969
  • Pascal distribution, a special case of the negative binomial distribution
  • Pascal’s triangle, a geometric arrangement of the binomial coefficients in a triangle
  • Pascal’s law, a physics principle relating pressure at various points in an incompressible fluid
  • Pascal (crater), a lunar crater
  • Pascal’s theorem, also known as the Hexagrammum Mysticum Theorem
  • Pascal’s Wager, a philosophical argument for belief in God

As you read Pascal’s thought here, ask yourself what position Pascal is specifically defending. How does he assess the expected utility of betting on God? That is, is it worth “betting” a believing life in the hope of “winning” heaven? How do you assess his approach?

“The Wager”

233. Infinite- nothing. — Our soul is cast into a body, where it finds number, dimension. Thereupon it reasons, and calls this nature necessity, and can believe nothing else.

Unity joined to infinity adds nothing to it, no more than one foot to an infinite measure. The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice. There is not so great a disproportion between our justice and that of God as between unity and infinity.

The justice of God must be vast like His compassion. Now justice to the outcast is less vast and ought less to offend our feelings than mercy towards the elect.

We know that there is an infinite, and are ignorant of its nature. As we know it to be false that numbers are finite, it is therefore true that there is an infinity in number. But we do not know what it is. It is false that it is even, it is false that it is odd; for the addition of a unit can make no change in its nature. Yet it is a number, and every number is odd or even (this is certainly true of every finite number). So we may well know that there is a God without knowing what He is. Is there not one substantial truth, seeing there are so many things which are not the truth itself?

We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because He has neither extension nor limits.

But by faith we know His existence; in glory we shall know His nature. Now, I have already shown that we may well know the existence of a thing, without knowing its nature.

Let us now speak according to natural lights.

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; [I Cor. 1. 21.] 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; where-ever 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.

For it is no use to say it is uncertain if we will gain, and it is certain that we risk, and that the infinite distance between the certainly of what is staked and the uncertainty of what will be gained, equals the finite good which is certainly staked against the uncertain infinite. It is not so, as every player stakes a certainty to gain an uncertainty, and yet he stakes a finite certainty to gain a finite uncertainty, without transgressing against reason. There is not an infinite distance between the certainty staked and the uncertainty of the gain; that is untrue. In truth, there is an infinity between the certainty of gain and the certainty of loss. But the uncertainty of the gain is proportioned to the certainty of the stake according to the proportion of the chances of gain and loss. Hence it comes that, if there are as many risks on one side as on the other, the course is to play even; and then the certainty of the stake is equal to the uncertainty of the gain, so far is it from fact that there is an infinite distance between them. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain. This is demonstrable; and if men are capable of any truths, this is one.

What if one is convinced that “in theory” betting on God is a “smart bet”?

“I confess it, I admit it. But, still, is there no means of seeing the faces of the cards?” Yes, Scripture and the rest, etc. “Yes, but I have my hands tied and my mouth closed; I am forced to wager, and am not free. I am not released, and am so made that I cannot believe. What, then, would you have me do?”

Here, Pascal provides practical direction in the form of “fake-it-till-you-make-it” advice…

True. But at least learn your inability to believe, since reason brings you to this, and yet you cannot believe. Endeavour, then, to convince yourself, not by increase of proofs of God, but by the abatement of your passions. You would like to attain faith and do not know the way; you would like to cure yourself of unbelief and ask the remedy for it. Learn of those who have been bound like you, and who now stake all their possessions. These are people who know the way which you would follow, and who are cured of an ill of which you would be cured. Follow the way by which they began; by acting as if they believed, taking the holy water, having masses said, etc. Even this will naturally make you believe, and deaden your acuteness. “But this is what I am afraid of.” And why? What have you to lose?

But to show you that this leads you there, it is this which will lessen the passions, which are your stumbling-blocks.

The “cost” of living as Pascal proposes.

The end of this discourse. — Now, what harm will befall you in taking this side? You will be faithful, humble, grateful, generous, a sincere friend, truthful. Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others? I will tell you that you will thereby gain in this life, and that, at each step you take on this road, you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognise that you have wagered for something certain and infinite, for which you have given nothing.

“Ah! This discourse transports me, charms me,” etc.

If this discourse pleases you and seems impressive, know that it is made by a man who has knelt, both before and after it, in prayer to that Being, infinite and without parts, before whom he lays all he has, for you also to lay before Him all you have for your own good and for His glory, that so strength may be given to lowliness.