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AlephNeil comments on About addition and truth - Less Wrong
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Comments (25)
It immediately strikes me that what you're asserting is either necessarily true or necessarily false, and whichever it is it could not be otherwise.
That's fine, but it's not at all the same thing.
Why is the difference relevant? I honestly can't imagine how someone could be in the position of 'feeling as though 2+2=4 is either necessarily true or necessarily false' but not 'feeling as though it's necessarily true'.
(FWIW I didn't downvote you.)
That seems to imply you think it would feel different than how you felt at first looking at my sum. Why, besides the fact that it's much simpler?
I sort of agree, in the sense that "2+2 = 4" is a huge cliche and I have a hard time imagining how someone could not have memorized it in grade school, but that's part of the reason why I regard the "self-evidence" of this kind of claim as an illusion. We take shortcuts on simple questions.
I believe that "2+2=4 is either necessarily true or necessarily false". I believe 2+2=4 is necessarily true (modulo definitions). I don't believe it's necessarily true that "2+2=4 is necessarily true".
There's some pretty strong evidence that the proof that 2+2=4 doesn't have a mistake in it (heckuva lot of eyeballs). I have good reasons (well, reasons anyway) to believe that mathematical truths are necessary. Thus most of my mass is on "2+2=4 is necessarily true". Yet, even if it's necessarily true that "2+2=4 is either necessarily true or necessarily false", and 2+2=4 is true, it still needn't be necessarily true that "2+2=4 is necessarily true", even though 2+2=4 is necessarily true.
If your eyes have glazed over at this point, I'll just say that Provable(X) doesn't imply Provable(Provable(X)), and if you think it does, it's because your ontology of mathematics is wrong and Gödel will eat you.
That's exceptionally unlikely for more reasons than one might think.
Not sure what work "necessarily" is doing, but mostly I'm with you. Still, I think this is mistaken:
Though it is true and important that Unprovable(X) does not imply Provable(Unprovable(X)).