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In a message dated 3/12/01 8:45:11 AM Eastern Standard Time, 
[log in to unmask] writes:


> There are two  books on the subject that I'm aware of, in addition to
> numerous articles in semi-popular mathematical collections.
> 
> 
> The Four-Color Problem: Assaults and Conquest, by Thomas L. Saaty and
> Paul C. Kainen, the authors of the original proof, first published in 1977 
> by
> McGraw-Hill, and reprinted in a slightly expanded edition by Dover 
> Publications
> in 1986.
> 
> 
> The Four-Color Theorem: History, Topological Foundations and Idea of
> Proof, by Rudolf Fritsch and Gerda Fritsch, published by Bibliographisches 
> Institut
> & F.A. Brockhaus in 1994, English translation published by Springer-Verlag, 
> 1998.
> 
> 
> This latter book covers both the original proof, recent advances in the 
> simplification
> of the original proof (although still computer-assisted), as well as the 
> history and
> foundations of the topological (not design) problem.
> 
Thanks, Charles. Saaty's book I have. I'll look for Fritsch. What difference 
are you seeing between a topological problem and a design problem? The only 
two I can see are attitudinal. (a) That mathematicians "own" a design problem 
(with their territory recently invaded by computer programmers), and (b) the 
approach would be different, maybe weighted in favor of the designers in this 
case. I actually had a mathematician from the Courant Institute tell me that 
a program was needed for coloring maps because--he said--it was tremendously 
hard to do and he himself always "made mistakes" when he tried! It seemed to 
me that this said more about his not having had the training to work with 
visual elements than about any presumed "difficulty" in the actual task. He 
probably also would have found it difficult to write a symphony, or more 
difficult than it would be for a person trained in musical composition. In 
any case, I pointed out to him that map makers don't seem to find any 
difficulty in the task, and maybe he ought to ask them how they did it.  His 
answer was that one didn't approach a problem in "pure" mathematics in that 
manner. With a design problem, one doesn't have that limitation. There's no 
barrier to doing anything necessary to solve the problem, including speaking 
to the map-makers if one wants. 

pat 


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<HTML><FONT FACE=arial,helvetica><FONT  SIZE=3 FAMILY="SANSSERIF" FACE="Arial Narrow" LANG="0"><B>In a message dated 3/12/01 8:45:11 AM Eastern Standard Time, 
<BR>[log in to unmask] writes:
<BR>
<BR></FONT><FONT  COLOR="#000000" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></B>
<BR><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">There are two &nbsp;books on the subject that I'm aware of, in addition to
<BR>numerous articles in semi-popular mathematical collections.
<BR>
<BR>
<BR>The Four-Color Problem: Assaults and Conquest, by Thomas L. Saaty and
<BR>Paul C. Kainen, the authors of the original proof, first published in 1977 
<BR>by
<BR>McGraw-Hill, and reprinted in a slightly expanded edition by Dover 
<BR>Publications
<BR>in 1986.
<BR>
<BR>
<BR>The Four-Color Theorem: History, Topological Foundations and Idea of
<BR>Proof, by Rudolf Fritsch and Gerda Fritsch, published by Bibliographisches 
<BR>Institut
<BR>&amp; F.A. Brockhaus in 1994, English translation published by Springer-Verlag, 
<BR>1998.
<BR>
<BR>
<BR>This latter book covers both the original proof, recent advances in the 
<BR>simplification
<BR>of the original proof (although still computer-assisted), as well as the 
<BR>history and
<BR>foundations of the topological (not design) problem.
<BR></FONT><FONT  COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial" LANG="0"></BLOCKQUOTE>
<BR></FONT><FONT  COLOR="#000000" SIZE=3 FAMILY="SANSSERIF" FACE="Arial Narrow" LANG="0"><B>Thanks, Charles. Saaty's book I have. I'll look for Fritsch. What difference 
<BR>are you seeing between a topological problem and a design problem? The only 
<BR>two I can see are attitudinal. (a) That mathematicians "own" a design problem 
<BR>(with their territory recently invaded by computer programmers), and (b) the 
<BR>approach would be different, maybe weighted in favor of the designers in this 
<BR>case. I actually had a mathematician from the Courant Institute tell me that 
<BR>a program was needed for coloring maps because--he said--it was tremendously 
<BR>hard to do and he himself always "made mistakes" when he tried! It seemed to 
<BR>me that this said more about his not having had the training to work with 
<BR>visual elements than about any presumed "difficulty" in the actual task. He 
<BR>probably also would have found it difficult to write a symphony, or more 
<BR>difficult than it would be for a person trained in musical composition. In 
<BR>any case, I pointed out to him that map makers don't seem to find any 
<BR>difficulty in the task, and maybe he ought to ask them how they did it. &nbsp;His 
<BR>answer was that one didn't approach a problem in "pure" mathematics in that 
<BR>manner. With a design problem, one doesn't have that limitation. There's no 
<BR>barrier to doing anything necessary to solve the problem, including speaking 
<BR>to the map-makers if one wants. 
<BR>
<BR>pat 
<BR></B></FONT></HTML>

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