On Thu, 12 Nov 1998, Jeremy Smith wrote:
> Hello,
>
> 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