The wager and the machine
[For context, read this post.]
Apart from his mathematical work, Blaise Pascal is most well known for his infamous “wager,” an argument for belief in God which plays a central role in his Pensées (that is, “Thoughts”; his notes toward a defense of the Christian religion). Perhaps only Anselm’s ontological argument has a worse reputation. But–as with the ontological argument, for that matter–there’s actually a lot to be said in the wager’s favor; and the things to be said for it open out onto the system of the Pensées as a whole. So the simplest reason to read the Pensées is this: it will clarify how concepts like chance, certainty, infinity, and self-interest, play a role in our understanding of belief (or lack thereof) in God.
Pascal never puts it this way–though (or because) he was a mathematician, his prose style put elegance above clarity–but the wager boils down to a complex practical syllogism:
- When truth is not available, one should seek happiness.
- The return in happiness on various actions can be compared.
- So one should do what has the greatest expected return.
- Some course of action B presents itself, with probability of success R>0, such that the return on the action depends on some event G, with probability P>0;
- And such that it has the following expected returns: ex(not-B)=(1-P)*u+P*x, and ex(B)=(1-R)*((1-P)*v+P*y)+R*((1-P)*w+P*z); that is,
no attempt | B fails (1-R) | B succeeds (R) Not-G (1-P) u v w G (P) x y z
- And such that u, v, and w are finite, z is positively infinite, and x and y are either finite or or negatively infinite.
- So no matter the specific values of P, R, u, v, w, x, or y, the expected return on attempting B is (infinitely) greater than that on not attempting it.
Practical conclusion: One should (attempt to) B.
The argument certainly seems valid. Pascal, of course, takes G=”God exists” and B=”believe in God,” and declares it sound. Is it?
In isolation, this move invites many objections, which seem to me to be of three basic types:
- Why should G=”God exists”? And why the Catholic God? This is the most obvious point of attack. Other possibilities, after all, present themselves: the Muslim God; the Hindu God; Zeus; Odin; the Flying Spaghetti Monster; and no one said P need be large, so long as it’s greater than zero.
- Why should B=”believe in God”? The usual objection is to point out the strangeness of choosing to believe; we can’t belief because it will make us happy, we have to believe because we think it’s true! It’s less frequently noticed that, just as with (1), there are other possible B’s that gum up the works: for example, flip a coin, and if it lands heads, believe in God; or, wait till your deathbed, then believe in God; or, do nothing, and perhaps wind up believing in God anyway. It doesn’t matter that these have a lower chance of success than an immediate conversion; after all, no one said R need be large, so long as it’s greater than zero.
- Why suppose z can be infinite? Why allow infinite returns in the first place? Not only do we have a hard time imagining infinite returns (Heaven always sounds rather boring), but allowing it causes numerous problems. (1) and (2) are just warm-ups. Consider: what if we acknowledge the possibility (no matter how slim) both of “negative infinite returns, no matter what strategy one adopts” and “positive infinite returns, no matter what strategy one adopts” (e.g. predestination)? This situation would contradict the second theoretical premise: it would be impossible to compare expected returns with one another, for every course of action would include infinity minus infinity, which is not zero, but rather undefined.
In response to objection (1), I believe, Pascal would ask another question: Why suppose P can be greater than 0? It’s easy to take for granted that any logically possible event, no matter how implausible, has at least some positive probability; but are we really justified in doing so? We might (I would) instead require that, before we grant that there is a chance an event will happen, we demonstrate that it is not only possible, but probable, in the original sense of tending toward a proof; demonstrate that there be reasons for believing it to be the case. So for any G, we must ask: why think G? Pascal spends much of the Pensées arguing that “the Catholic God exists” is the only G for which we can offer plausible reasons.
In response to objection (2), a similar question arises: Why suppose R can be greater than 0? What reasons do we have for thinking that we are able to B? Pascal agrees that one cannot choose to believe. One can, however, choose to “immediately put oneself in a position which will bring one to believe in the Catholic God.” That this does not involve “dishonest” belief is is easier to accept if you think of “belief in God” as less like “belief that there’s a chair” and more like “belief in reading”. We cannot choose to believe, any more than we can choose to know how to read; but we can choose to seek a teacher, and, Pascal spends much time arguing, this is the only course of action which we have any reason to think will succeed.
Objection (3) does establish a paradox. However, it holds only if we allow for more than one G. It is not, then, a weakness of the argument, but a strength. If you really, truly find multiple incompatible religious systems plausible, then Pascal’s wager gives you reason to feel paralysis. If, on the other hand, you accept Pascal’s arguments for the only plausible G and B, then your only rational option is to try to B. This will make game-theoretic reasoning in the future somewhat useless: B will win out every time, in every situation. So perhaps exclude infinite returns when deciding among strategies all of which are compatible with trying to B. But none of this is reason to dismiss infinite returns a priori. And Pascal gives many reasons for believing infinite returns to be plausible.
So the application Pascal would make of the wager commits him to three further claims:
- Our actions can have infinite consequences.
- Catholicism is the only plausible account of an adequate response to these consequences.
- Immediately and wholeheartedly immersing oneself in Catholicism is the only plausible strategy for becoming Catholic.
The wager makes no sense if we do not take into account his arguments for them. He does not pull them from thin air. All three follow from the second pillar of the Pensées: the Machine.
By the Machine, Pascal means the World seen as a mechanism. It is to establishing this point of view that most of the Pensées devotes itself. It is, for this reason, less argument than rhetoric; or, perhaps, not rhetoric, but poetics.
(1) The mechanism is of infinite size; indeed, as Pascal writes in the most famous sentence in French literature, “The silence of these infinite spaces terrifies me.” It is as if, for Pascal, this aesthetic sublimity requires an ethical absolute to counter-balance it. Either our actions can have infinite consequences, or they do not matter in the least, overshadowed as they are by the vast cosmic void.
(2) The mechanism includes humanity. We are selfish animals, acting according to a shallow (that is, finite) conception of our self-interest. Catholicism–in any case, Christianity–is the only instance of a way of life spreading throughout the world that is not based on selfishness, but rather on charity. Pascal has much more to say about the virtues of Christianity, but this alone, he believes, is enough to single it out as potentially divine: it indicates that an outside force tampered with the mechanism.
(3) This outside force is able to affect us because, though we are part of the mechanism, we, alone in the universe, are capable of thought: “Man is but a reed, the most feeble thing in nature, but he is a thinking reed.” He is not, however, a willing reed; he inclines where the wind blows him. Thought offers a fulcrum outside the universe, but does not offer an unmoved motive force. We cannot, in fact, supply that force; it must come from outside. We can only consider–from within the mechanism–how best to position ourselves for the outside blow. This involves not doing something, but being something: adopting the appropriate habit, making it part of the fabric of our lives. So the death-bed conversion strategy doesn’t work; in fact, no strategy will work. For it is not neutral to put belief off till tomorrow. Resolving to believe tomorrow is not resolving to believe August 21th, but rather resolving not to believe now. If we put belief off till tomorrow, we are putting on unbelief today.
Postscript: I suspect that one reasons Pascal left the Pensées the way he did–fragmentary and disheveled–was that, though he had the argument, the wager, down, he could not get the poetics, the machine, quite right. He could not find an order of words which would not just lay out the idea for the reader to take or leave, but would operate like an Audenesque poetic machine, forcing the reader to adopt the perspective of infinity. Of course, such a machine is impossible; grace cannot be externally compelled. But that does not stop perfectionists from making the attempt. This applies, I suppose, to Wittgenstein as well. People often say that the two have much in common.