I agree with you in the sense that *I* find our proofs to be more
reassuring than examples. But that's not the point. It seems to me that
all of your arguments for the superiority of proofs could be applied
mutatis mutandis to proof-by-example.
You argue that ZF has 'stood the test of time', in the sense that people
have tried to find inconsistencies and failed. You feel that this somehow
makes it more likely that ZF is consistent. Suppose a person has an
algorithm that has worked for all the cases anyone has ever tried, even
when people have been specifically searching for failure. This strikes me
as a perfectly analogous situation. In both cases, there are countably
many (hence impossibly many) cases to check, and the people are gaining
confidence in their thing based on a finite collection of cases.
As for the *proof* that ZFC is consistent if ZF is consistent, I should
remind you that this, too, could be false in the same way that a theorem
*in* ZF could be false. Namely, whatever meta-system we're using could be
inconsistent.
Incidentally, one argument that someone might make is that a savage
believer in unproved algorithms would, in general, be more likely to be
reassured than a believer in proofs. It is easy to construct algorithms
that work for billions of cases and then fail. For such an algorithm, our
savage will be reassured of its correctness while people as superior as we
would not be.
This is in fact true, but it doesn't generalize. There are, I agree,
cases where one reassured by examples would be tricked while one reassured
by theorems would not. On the other hand, if ZF in inconsistent, there
are also cases where a prover would be assured but an examplemeister
wouldn't. For example, there may be an algorithm or a formula that is
horrendously difficult to compute, and for which there exists a
correctness proof. If the proof is in error (in either of the ways I've
mentioned) then the prover would be in the position of the savage from the
last paragraph, and vice versa.
Hope you can make sense of the mess above.
-Jeremy