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Robert,

You'd better not say this in the presence of a friend of mine who's got
degrees both in mathematics and in artificial intelligence. Even his
part-time job (he's still finishing his AI thesis) is training people to use
math-where which is pretty well capable of solving pretty complex problems
all by itself.

A computer program is nothing more than separating the knowledge and skill
of one or more human beings from an actual physical being, but also nothing
less. Let's not underestimate that too much. I'm rather worried that a
rather big and successful AI project of several years ago, "CYC", has gone
underground and now apparently belongs to type a classified material. Or at
least, that's what I thought because I couldn't find anything about it
anymore, but that was over a year ago and now that I checked again, it could
also be that now this: http://www.cyc.com/ company is cashing in on it's
success.

Ayway, best not get me started on Artificial Intelligence. One common
definition of "Intelligence" outside the field but formulated of course
inside the field is that "Intelligence is what computers can't do". Best
slow down on those comments, or else we'll lose our 'homo sapiens' status.

Arwin

> -----Oorspronkelijk bericht-----
> Van: [log in to unmask]
> [mailto:[log in to unmask]]Namens Meyer Robert K GS-9 99
> CES/CECT
> Verzonden: maandag 12 maart 2001 23:07
> Aan: [log in to unmask]
> Onderwerp: RE: (VERY) OFF TOPIC - Map coloring
>
>
> You're on the right track, Pat, especially with your second point.
> Computers can't prove anything.  Basically, because they can't do
> real math
> (excuse the pun).  They only deal with finite numbers, very big finite
> numbers maybe, but still finite.  And there are two types of
> infinite sets,
> countable (like rationals that can be put into 1-to-1 correspondence to
> integers) and uncountable (like irrationals that can't, or reals
> that are a
> set with both rational and rational numbers).
>
> Robert
>
> 	-----Original Message-----
> 	From:	[log in to unmask] [SMTP:[log in to unmask]]
> 	Sent:	Monday, March 12, 2001 1:33 PM
> 	To:	[log in to unmask]
> 	Subject:	Re: (VERY) OFF TOPIC - Map coloring
>
> 	In a message dated 3/12/01 12:14:55 PM Eastern Standard Time,
> 	[log in to unmask] writes:
>
>
>
>
> 		  It is the job of the mathematicians is to determine what
> is
> 		mathematically
> 		meant by "every case". That's done all the time. For
> example, in poker, the
> 		"odds" of getting five cards of the same suit is determined
> by calculating
> 		the NUMBER of ways five randomly drawn cards can be of the
> same suit, and
> 		then dividing that number by the TOTAL NUMBER OF WAYS any
> five cards can be
> 		drawn from a 52 card deck.
>
>
>
> 	Yes, and if n colors are going to be used to color n shapes, the
> number of
> 	possibilities is n to the nth power. This really includes all
> possibilities,
> 	including using only one of the n colors for all of the n shapes.
>
> 	Most of these color arrangments are not going to meet the criteria
> for what
> 	Saaty calls correct k-coloring--that is, no two contiguous countries
> are the
> 	same color. For his correct k-coloring, the limit is n!. That is, n
> 	factorial.
>
> 	Thus, for four color used on a basic four-shape module, there are
> 256
> 	possible combinations of which only 24 are "correct."
>
> 	Now here are my questions.
>
> 	1) This sounds to me like it's leading into a problem in what I
> believe is
> 	called combinatorial algebra. So why does one need a computer?
>
> 	2) For each n in the number series, the nth power of n is an
> infinite series.
> 	So is n factorial.  So the computer, if using brute force, is going
> to run
> 	out of steam at some point, as it can't deal with an infinite series
> by
> 	testing "every" possibility.
>
> 	3) What was wrong with de Morgan's proof, which was rejected? It
> seems to me
> 	that he was on the right track, much more so than what followed
> after.
>
> 	4) Maybe you should rethink your poker example. The number is finite
> because
> 	there's a limit to the number of cards in a pack. There's no limit
> to the
> 	number of shapes that can be on a map. That's why one has to do
> something
> 	along the line of breaking a large map down into, say, units of four
> or fewer
> 	shapes.  How, exactly, did they compute the number of possible
> combinations,
> 	and what did the number turn out to be? You keep assuring me it was
> 	"gazillions," which I don't find very clear. I'm actually more
> interested in
> 	how it was computed than what it turned out to be.
>
> 	pat
>
>
>
>