🚀 go-pugleaf

In article <7521@boring.cwi.nl> jurjen@cwi.nl (Jurjen N.E. Bos) writes:
>In article <1009@sdcc13.ucsd.EDU> ln63wgq@sdcc13.ucsd.edu.UUCP (Keith Messer) writes:
>>You know, that's my logic too, Bob.  A mathematical (even if we have to call
>>it mathematical) property is useful whether or not it is proven.  Supposing
>>I find some regularity in a mathematical system by induction (but in a case
>>where mathematical induction is not adequate proof) and decide to write a
>>program to do analysis based on that regularity.  Either the program will
>>succeed and be useful to me or it will fail, disproving my hypothesis.  The
>>point is that I win either way.
>
>I think you're missing something here. Your program NEVER will succeed, 
>because there is an infinity of cases. It can only fail.

I think you're missing something here.  Keith is talking about a result
being USEFUL not about PROVING it.

> If you want to
>prove something, you need a proof, not a 'convincing' number of tries.

Sure, but we don't, do we?  After all, we're computer scientists,
not mathematicians  :-)

Bob and Jurjen are of course right that a proper mathmatical proof is
intellectually satisfying and essential to make a hypothesis respectable
no matter how many examples in favour of it you find.  But that isn't what
Steven Tate and Keith were talking about.  If I am 99.99% confident in
the validity of the Rieman hypothesis and write a program which depends
on it to work, then I can be *reasonably* confident in my program even
without a "proof".  Hell, even if the hypothesis were proven there would
still be a > 0.01% (to say the least!) chance of a bug in my program, so
in a practical sense the proof of the hypothesis does nothing to increase
confidence in my program.  OK, so next I try to prove my program is correct,
but then where's the proof that my "proof" is correct ... ?

Humans are fallible beings, so we have to accept the fact that in principle
any hypothesis/program/aeroplane engine etc may fail in some situation.
While this may not be very satisfying, it doesn't stop us making use of
the thing concerned.  We learn to live with this possibity of failure
(though of course this doesn't prevent us from trying to keep the probability
of failure down as low as possible).

-Rolf

P.S.  Apologies if this article is less than topical for most people.  Our
      news connection is abysmally slow.