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.
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