> We would investigate the proof and our algorithm to find where our error
> lies. We will find out what is causing the problem. As a result my trust
> in the mathematical community and its system of demonstrations would not
> decrease. The mere fact that we examine a situation in which there is a
> failure shows that we expect our proofs to be meaningful.
I'm not sure what is means to 'expect proofs to be meaningful'. But I
agree that if a proof says that a certain algorithm will always 'work',
then we would expect it to always work. But it still might not, since
there might be a mistake (in which case I guess we wouldn't really have a
'proof'), or because our mathematical system is inconsistent.
> What is your evidence for expecting inconsistencies in our mathematical
> systems?
My only evidence is that I've been persuaded by the logicians that we have
no way of telling whether or not our systems (provided they're
sufficiently interesting) are consistent, unless we actually *find* an
inconsistency (e.g. a certain algorithm fails, but a certain proof says it
won't).
All I'm saying is that a certain type of person might feel, because
of the incompleteness theorems, that proofs are not very reassuring. One
is reassured by proofs only if they secretly assume that mathematics is
consistent, i.e. that you won't get contradictory statements as theorems.
So, in my opinion, the question of weather a proof is more reassuring than
a large number of successful cases depends upon where you come from and
what you happen to find reassuring. From an objective standpoint, there's
no good reason that either one should convince anyone of anything.
-Jeremy