> 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.
[snip]
> 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.
Consider the following propositions:
For every natural number n, there is a prime p > n.
There do not exist positive integers p and q such that pp = 2qq.
For every natural number n there are natural numbers u, v, x, y such
that n = uu + vv + xx + yy.
I claim that these are *absolutely certainly true*. In fact, you cannot
find any proposition that has ever been or ever will be stated in any human
culture that is more certain than the above propositions. If you doubt
this, go and look at the proofs. It you then still doubt it, you have not
understood the proofs. These propositions are true: they have been proved.
The theory in which they are proved is in no need of being proved
consistent. It is consisent by virtue of the meaning of the terms used in
it. (You only need a consistency proof for a formal theory whose meaning is
in some doubt.)
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