> Of course, having a proof of a valid algorithm does not create its
> validity. However, ....
>
> How do we know the algorithm WILL always work? A proof is the
> assurance that the algorithm will work. I will call any VALID
> assurance that an algorithm will always work a "proof".
I don't want to sound too foundational, because this is a history list.
But from a certain point of view, 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.
That's all pretty moot anyway, because (as far as I know) at the time
these first 'assuring' proofs of these algorithms surfaced, no one had
bothered to investigate the nature and status of proofs.
What I'm trying to get at is that I think it's a matter of psychology (or
perhaps simply culture) what one finds 'assuring'. I was suggesting that
the original users of the algorithm might have found a large body of
successful examples much more assuring than an abstract proof, even though
from a certain point of view neither of them really assures us of
anything. To say that there was a 'need' for proof that was somehow
fulfilled by the Greeks (or whoever) is, in my opinion, a reflection of
our own cultural bias.
-Jeremy Smith