On Nov 5, 12:18 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> I don't think that captures the essence of the paradox.
> You wouldn't 4 to be heap, would you? Well, maybe you would.
Oh yes, absolootly-bootly! The smallest possible heap. If I saw 4 stones arranged as I said, I would have NO hesitation in refrerring to "that little heap of stones there". And you?
> I resolve it in a different way, in that I would say that if I'm > looking at a bunch of stones, there is a certain *probability* > that I would call it a "heap".
Oh no, this is far far too subjective an approach, IMHO. And in any event probability is a weaselly way out of logic matters.
> In the case of mathematics, I think that most people would agree > that the axioms of PA are obviously true (forget about induction, > and just concentrate on the axioms defining plus and times and > characterizing successor).
> And if A is obviously true, and A implies B > is obviously true, then B would be obviously true, right?
Yes, if. But IF you have tiny doubts about whether P(13) is true, but you have no tincture of doubt about P(1) being true, THEN there is some (specific!) number between 1 and 12 where you (first!) have tinctural doubts about P(n)-->P(n+1).
And the same applies to ANY such chain of deduction.
> But that principle means that anything provable is obviously true, > which is obviously not true.
So I think we can safely discard this (as you say) silly conclusion.