I meant "both rational and irrational numbers".  Had problems with that
insert / delete mode on the keyboard.

Robert

	-----Original Message-----
	From:	Meyer Robert K GS-9 99 CES/CECT
[SMTP:[log in to unmask]]
	Sent:	Monday, March 12, 2001 2:07 PM
	To:	[log in to unmask]
	Subject:	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