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Eugine_Nier comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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I'm not sure about that. You and komponisto seem to be using 'philosophically sophisticated' to contrast with Platonism. This use strikes me as similar to how arguing that 'death is good' is sophisticated, i.e., showing of your intelligence by providing convincing arguments for a position that violates common sense. In this case arguing that mathematical statements don't have inherent truth value.
Remember just because you can make a sophisticated sounding argument for a preposition doesn't mean its true.
Mathematica statements do have inherent truth value, but that value is relative to the axioms. And as far as the axioms go, the most you can say is that a system of axioms is consistent, and beyond that you get into non-mathematical statements. What exactly is sophisticated about this?
Yes, which agrees with my complaint quoted above. Neither of us is a Platonist, so we both assume that philosophically sophisticated people won't be Platonists, although we derive different things thereafter.
I'm certainly not trying to show off my intelligence. I just think that the idea of inherent truth value for abstract statements about completed infinities violates common sense!
If that's so, what accounts for your intuition that ZF and other systems for reasoning about completed infinities are consistent?
To the extent that I have this intuition, this is mostly because people have used these systems without running into inconsistencies so far. (At least, not in the systems, such as ZF, that people still use!)
But strictly speaking, ‘ZF is consistent.’ is not a statement with an absolute meaning, because it is itself a statement about a completed infinity. I have high confidence that no inconsistency in ZF has a formal proof of feasible length, but I really have no opinion about whether it has an inconsistency of length 3^^^3; we haven't come close to exploring such things.
(Come to think of it, I believe that my Bayesian probability as to whether ZF is consistent to such a degree ought to be quite low, for essentially the same reason that a random formal system is likely to be inconsistent, although I'm not really sure that I've done this calculation correctly; I can think of at least one potential flaw.)
I cannot speak for komponisto about any of this, of course.
I'm mostly with you.
These feasibility issues are definitely interesting. Another possibility is that there is a formal proof of feasible length, but no feasible search will ever turn it up. (Well, unless P = NP). Yet another possibility is that a feasible search will turn it up, I certainly regard it as more likely than most people do.
I agree that this counts as evidence, but it's possible to overestimate it. Foundational issues hardly ever come up in everyday mathematics, so the fact that people are able to prove astonishing things about 3-manifolds without running into contradictions I regard as very weak evidence in favor of ZF. There have been a lot of man-hours put into set theory, but I think quite a bit less than have been put into other parts of math.
JoshuaZ and I had a discussion about this a while ago, starting here.
This reminds me of people who argue that, because P != NP, we will never prove this. (The key to the argument, IIRC, is that any proof of this fact will have very high algorithmic complexity.) I'm not sure how to find this argument now. (There is something like it one of Doron Zeilberger's April Fools opinions.)
Yes, these results should be formalisable in higher-order arithmetic (indeed nth order for n a single-digit number). It is the set theorists' work with large cardinals and the like that provides the only real evidence for the consistency of such a high-powered system as ZF.