🚀 go-pugleaf

> Your program NEVER will succeed because there are an infinite number of
> cases, it can only fail.

Ahh!  This points up the exact problem we're having in communicating here.
I'm not a mathematician, I don't write papers, and I feel no urge to know
for sure whether my hypothetical regularity is universal or not.  My point
is that, although a hypothesis may or may not be true, if it's a little bit
useful then it's worth implementing in an algorithm.  So, perhaps one day
the algorithm flails.  I can take note of that and know that the hypothesis
is incorrect, or at least needs refinement.  By "winning either way," I don't
mean proving the hypothesis, I mean getting some value out of using it to
develop algorithms that may be valid in the specific cases I'm interested in,
though they may bomb if taken generally. :-)

There seem to be a lot of mathematically-minded people here, so I'll ask
some questions that have been on my mind.  If a general result of
quantificantion theory is that set theory and number theory contain contra-
dictions (ie. It's possible to construct a proof for a statement and one for
its negation for certain statements), how can we expect to get meaningful
results from these things?  Essentially what this says is that number theory
and set theory are not valid, so that proofs in number and set theory may
not be true.  To some extent, this problem extends into other mathematical
fields.  On one hand I want to know how we can put so much emphasis on proof,
if it is known that our postulates are contradictory (and therefore not
representative of reality either), and on the other hand I'm curious as to
why, assuming that our math is formal enough to be used mechanically, one of
you sharp mathematicians out there don't write a proof machine to construct
and prove every theorem in every field of mathematics.  Doubtless there are
a lot of them, but I'd think you could narrow the range of output with some
reasonable formal criterion like the number of mathematical "words" in the
theorem and the number of postulates and logical inferences used to construct
the shortest possible proof.  Perhaps this would take a lot of cpu time, but
it might well be worth tying up someone's cray for ever to learn more about
the formalistic "coding" of reality.  Is there a reason this isn't possible?

						Keith Messer
						messer%emlab@sdcsvax.ucsd.edu