You wrote
>a 'proof' in our sense of the word
>offers no assurance that the algorithm will always work. One might object
>that we can never be sure that the proof doesn't contain a mistake, or
>(more seriously) that there is an inconsistency in whatever mathematical
>system we're talking about.
I do find proofs reassuring. However, should the algorithm actually fail,
we would not just sit there and say
Oh well, proofs are just a cultural bias anyway.
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.
What is your evidence for expecting inconsistencies in our mathematical systems?
An example: The four color theorem. It was very soon proved with a faulty
proof that took ten years to discover. To me that shows that
mathematicians err, but it does not show that no correct proof is possible.
Erdos was always looking for the proof in the (God's) book. There is
nothing wrong with that.
Best wishes,
Sam Kutler