Swami’s post on Truth and Provability has generated some interesting comments. My thoughts on the subject are rather longish, and I thought they deserve a post here.
Truth is different from provability.To explain why I think so, I need to resolve the surprise quiz paradox, which goes like this:
On friday, the professor tells his students that he was going to set a surprise quiz the next week. A student thinks “If on friday, the professor has not yet set the quiz, I’ll come to know that the quiz is on friday. So the quiz has be held on or before thursday.
But then if he doesn’t hold a quiz monday to wednesday, I’ll know that the quiz is on thursday. So the quiz can’t be on thursday either.”
So by elimination, Swami (for that was the student’s name) decides that the prof was bluffing about the quiz, and there wasn’t going to be one.
Thereupon the professor surprises Swami with a quiz on Monday morning, jolting him out of his weekend blues.
Where’s the catch? Did Swami prove that there wouldn’t be a quiz? Think again. The professor had said that he would set a surprise quiz. Model this statement logically and you have two premises:
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I will set a quiz this week.
You will not be able to predict the day on which the quiz will be held.
Given these two conditions, could Swami have proved on friday morning that the quiz would be on that day? Any candidate proof would immediately violate condition 2. But would he know on thursday night that the quiz would be held on friday? A computer would hang if we tried to model a set of impossible conditions and asked it to solve for the result. Swami, who is presumably a human (unless he is an automated blogger trying to pass the Turing test) would find it possible to get at the ‘truth’ beyond provability.