Pascal’s Wager and the Problem of Computability

I was on a cable car yesterday morning and I was intellecually tickled by an advertisement posted on one of the walls in the car. It was an ad for a unitarian church, and in it they quoted Blaise Pascal:

It is incomprehensible that God should exist, and it is incomprehensible that He should not exist.

Pascal was a philosopher and mathematician, and he was trying to understand the question of whether or not one should believe in the existence of a God (specifically the Christian god, but that’s not important). He concluded that since the benefits of belief are supposedly enormous (or infinite), and one cannot actually know, one should wager in favor of there being a God, since then the benefit of being right is maximized and the cost of being wrong is minimized.

But Pascal made an error in his premises, which touches on computability theory. He sets out assuming that the statement “God exists” is either true or false. This is an unwarranted premise. Pascal was a rationalist, so he assumes the dichotomy of truth, that every statement is either true or false. But any programmer can tell you that this isn’t the case.

You see, Aristotle understood that not every statement is either true or false. The law of excluded middle says that a thing either is or is not. And truth or falsehood is an attribute only of statements that refer to things which are. For this reason, some statements are simply absurd, or arbitrary. Such statements cannot be examined for truth or falsehood. They can only be dismissed out of hand, as if nothing had been said. Because, in a very strict sense, nothing really has.

In computation, this is equivalent to the fact that not every program has an answer. Some programs simply bottom (crash or hang). There are types that are nonsensical and have no implementation except for programs whose answer is bottom. And there are questions that have no answer because they are nonsensical, and statements that are neither true nor false because they do not refer to any attributes or configurations of things which exist.

But it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine.

- Aristotle

Not only is it not right, it’s not even wrong!

- Wolfgang Pauli

On two occasions I have been asked,—”Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?” … I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

- Charles Babbage

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26 thoughts on “Pascal’s Wager and the Problem of Computability

  1. I take your point, but the way you’ve expressed it makes it sound as though you’re claiming that Aristotle rejected excluded middle, which would be strange given that he’s typically credited with formulating it for the first time (in the Metaphyics).

    To put thing in more contemporary (well, 20th century, anyhow) terms it would be better to say that Aristotle understood that every proposition is either true or false, but that not every utterance expresses a proposition (eg. by being meaningless, ambiguous or whatever).

  2. You can see the absurdity by applying the Pascal wager reasoning to the religion I just made up – in my religion, you must worship a fleet of invisible purple dinosaurs orbiting the earth – by doing so you will be granted an eternity of happiness, otherwise you’ll have to deal with an eternity of suffering, etc. Clearly you should believe in this religion – if it happens to be true, you’ll be rewarded, if not, you’ll have an eternity of suffering! But by the same reasoning, you should also follow any religion that is invented… including the religion which specifically rejects the existence of invisible orbiting purple dinosaurs…

    • That’s the point exactly. You can apply this as a general principle, to dismiss anything out of hand that is just made up.

      But it’s even more general than that. You can dismiss anything that is arbitrary. I.e. anything that isn’t arrived at by a process of abstraction from experience and experiment.

      It’s an invalid thought process to begin with a comforting or pleasant thought and then rationalize why it must be true, or to dismiss an idea because its consequences are unpleasant.

      How many things do we accept on faith like that? If you meet the Buddha in the road, kill him.

      • This general principle seems very useful (and a bit obvious). I’m sure it must have a name? Who is credited with formulating it first?

      • I believe Aristotle formulated it first. It really is a corollary of the law of identity. You could call it “dismissal of the arbitrary” or “insistence on causation”. I would just call it “the scientific method”.

      • Pirsig and Hofstadter used the expression “mu” to “unask” a question in the seventies, it stems from a Zen Buddist Koan.
        I think “not even wrong” is a very catchy formulation. A bit more clear is “dismissal of the arbitrary”. I’m not sure that the “scientific method” is relevant, it’s obviously not applicable to these kinds of arbitrary propositions, but the fact that a method is not applicable does not really convey a lot of meaning.
        Often people seem to accept that the existence of a god cannot be proven, but that seems to leave them thinking that that means that it is “unknown”, and thus either true or not, but you don’t know which. This of course gives it much too high a status, rather than “unknown” it should be called absurd (or stupid?) or something like meaningless if you want to be more kind.

  3. Oh that’s a coincidence. I saw a talk by Lee Naish on Tuesday about 4-value logic, where the third and fourth values are “undefined” (crash or bottom), and “inadmissable” (garbage in, garbage out – the argument simply doesn’t make sense). You might find the paper interesting: http://ww2.cs.mu.oz.au/~lee/papers/sem4lp/

  4. “Aristotle understood that not every statement is either true or false. The law of excluded middle says that a thing either is or is not. And truth or falsehood is an attribute only of statements that refer to things which are”.
    Pascal’s wager is not merely about the truth value of a proposition, but about the existence of God: whether God is or is not. In the context of the law of the excluded middle, Pascal’s question makes sense.
    Also (without intending to commit a magister dixit) I would not think it is illogical to ask the question. Even Kurt Gödel produced a proof of the existence of God.

    • I don’t think you know what Kurt Gödel proved or did not prove. He produced a proof that:

      1. Every type system will be incapable of producing types for some programs.
      2. If a system allows recursion, then there are bottoms. I.e. if programs can refer to themselves then there are some programs that don’t terminate.

      • Gödel’s proof matters more than you think. The question does not make sense precisely because its answer is bottom.

      • In your post you accept the validity of the law of the excluded middle: a thing either is or is not. Now take “thing” = God. “Is God or Is God Not?” Why isn’t that a valid question in the context of the law of the excluded middle?

      • Because a thing may only be arrived at by a process of induction from experience. There is nothing which we experience that necessitates such a concept. The question does not have an answer because the subject of the question is not a concept that has any actual referent.

      • So you are saying that it doesn’t make sense to ask whether something that we don’t experience exists?

      • Yes. Not only that we don’t experience, but experience no part of. We can ask about things that have actual referents, or whose parts have actual referents.

        It’s crucial to understand the question of “does x exist?” as “what is the cause of x”? Like if you ask “do pink elephants exist?”, you are really asking “what would cause an elephant to be pink?”

      • It does to an extent. All the elements of a unicorn have referents in reality (e.g. horses and horns).

  5. I am not sure we would be able to formulate a concept (be it unicorns, pink elephants, or God) if it didn’t have a referent in “reality”. In the idea of God, for example, we are refering to the concepts of creation, supremeness, etc.
    But anyways, thanks for the conversation. I arrived to your blog looking for a way to learn about scalaz, and I’ve got to get back to that. I think you’ve gained a reader!

  6. No, that is not right, Pauli notwithstanding. He and others confuse a material logic, a logic based in ontology, with an epistemologically based logic. A thing either exists or not, the truth or falsity of statements is an entirely separate matter. Now if go to the semantic or grammatica level then we have things like moral, immoral, and amoral. That is more what you are talking about. Pascal is correct.

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