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In a message dated 3/11/01 7:27:31 PM Eastern Standard Time, 
[log in to unmask] writes:


> In Dublin I read part of the book on the history of the map-colouring 
> problem and how it was solved, which was really interesting to read - it 
> was one of the first proofs in which computers played a great part and 
> hence it was and still is very controversial. 

Arwin, 

What's the book? The reason the proof was controversial is that it isn't 
actually a "proof" in any logical or mathematical sense. Just a brute force 
examining of all examples the computer could think of, and no proof (or way 
of proving) that the program was actually sufficient to test "every" possible 
example. 

Personally, I think the map-coloring problem is a design problem, not a 
mathematical problem, and that's precisely why problems arose in trying to 
formulate a solution in mathematical terms. One question is whether there are 
criteria for identifying a mathematical problem as opposed to some other kind 
of problem. If these criteria exist, I sure don't see where they are, and the 
envelope can't be pushed to include any question with a numerical answer. 
There's no provable mathematical formula, for example, that allows anyone to 
determine what numbers are used on the license plate for my car. And the 
question isn't actually a mathematical question, even though the answer can 
be expressed in numbers. One can get an answer by checking the motor vehicle 
records. But looking up a number on a list isn't the same thing as 
constructing a mathematical proof...unless mathematicians are willing to 
broaden their standard of what constitutes a mathematical proof so that it 
includes all the "trivial" proofs that seem to be ignored at present.

I think Cantor developed some proof that there can't be any highest possible 
number or any end to the number series...which tends to validate the common 
sense observation that no matter how large a number is, one can always add 1 
to it. So why should we believe a computer that erroneously tells us that 
there actually is a highest possible number...this being the highest number 
that this particular computer can express given its limitations of memory and 
technology?

In any case, I'd like to see the actual program for supposedly "proving" the 
four color theorem, which I think came from U of Illinois. Was it reprinted 
in the book you read? It's supposed to be teribbly long, rather than a 
relatively short program that's used recursively. I'd like to see whether 
what made it so long is an attempt to define "every possible" situation.  

There was a period, you know, when an attempt was made to demonstrate that 
computers can do all kinds of things they aren't actually capable of doing. 
One engineer was claiming a computer could make Mondrian paintings, and 
another started that stupid "the Mona Lisa is Leonardo in drag" stuff. Here, 
the arguments were set forth in relatively brief articles. And when one 
finished reading the articles, one realized each was written by a person who 
didn't know a thing about art, and therefore didn't know how to interpret the 
data.

In the 4-color theorem, too, it seems to me from what I've read that the 
programmers skipped too many steps. Starting at square one, the first job is 
to program a computer to construct mathematical proofs. Until that's been 
accomplished, a computer isn't able to construct a mathematical proof of 
anything. And this of course is the borderline question that I hope is being 
considered in AI. What's the limit to what a computer can do? And what 
problems can only be solved by a sentient being?

All that to-do about computers playing chess is far from persuasive, because 
chess has a finite number of possible moves.  What about programming a 
computer to write a recipe book, with some assurance that the receipes will 
be superior to those a human being might develop?  Here, it's a tougher 
problem, because more is involved than shuffling numbers, which is all 
computers can do. But, as above, I think the main problem with any 
computerized "mathematical proof" of the four-color theorem is that one can't 
skip the necessary step of developing a program capable of constructing 
mathematical proofs. 

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 3/11/01 7:27:31 PM 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></FONT><FONT  COLOR="#0000ff" SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0"><BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">In Dublin I read part of the book on the history of the map-colouring 
<BR>problem and how it was solved, which was really interesting to read - it 
<BR>was one of the first proofs in which computers played a great part and 
<BR>hence it was and still is very controversial. </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>
<BR>Arwin, 
<BR>
<BR>What's the book? The reason the proof was controversial is that it isn't 
<BR>actually a "proof" in any logical or mathematical sense. Just a brute force 
<BR>examining of all examples the computer could think of, and no proof (or way 
<BR>of proving) that the program was actually sufficient to test "every" possible 
<BR>example. 
<BR>
<BR>Personally, I think the map-coloring problem is a design problem, not a 
<BR>mathematical problem, and that's precisely why problems arose in trying to 
<BR>formulate a solution in mathematical terms. One question is whether there are 
<BR>criteria for identifying a mathematical problem as opposed to some other kind 
<BR>of problem. If these criteria exist, I sure don't see where they are, and the 
<BR>envelope can't be pushed to include any question with a numerical answer. 
<BR>There's no provable mathematical formula, for example, that allows anyone to 
<BR>determine what numbers are used on the license plate for my car. And the 
<BR>question isn't actually a mathematical question, even though the answer can 
<BR>be expressed in numbers. One can get an answer by checking the motor vehicle 
<BR>records. But looking up a number on a list isn't the same thing as 
<BR>constructing a mathematical proof...unless mathematicians are willing to 
<BR>broaden their standard of what constitutes a mathematical proof so that it 
<BR>includes all the "trivial" proofs that seem to be ignored at present.
<BR>
<BR>I think Cantor developed some proof that there can't be any highest possible 
<BR>number or any end to the number series...which tends to validate the common 
<BR>sense observation that no matter how large a number is, one can always add 1 
<BR>to it. So why should we believe a computer that erroneously tells us that 
<BR>there actually is a highest possible number...this being the highest number 
<BR>that this particular computer can express given its limitations of memory and 
<BR>technology?
<BR>
<BR>In any case, I'd like to see the actual program for supposedly "proving" the 
<BR>four color theorem, which I think came from U of Illinois. Was it reprinted 
<BR>in the book you read? It's supposed to be teribbly long, rather than a 
<BR>relatively short program that's used recursively. I'd like to see whether 
<BR>what made it so long is an attempt to define "every possible" situation. &nbsp;
<BR>
<BR>There was a period, you know, when an attempt was made to demonstrate that 
<BR>computers can do all kinds of things they aren't actually capable of doing. 
<BR>One engineer was claiming a computer could make Mondrian paintings, and 
<BR>another started that stupid "the Mona Lisa is Leonardo in drag" stuff. Here, 
<BR>the arguments were set forth in relatively brief articles. And when one 
<BR>finished reading the articles, one realized each was written by a person who 
<BR>didn't know a thing about art, and therefore didn't know how to interpret the 
<BR>data.
<BR>
<BR>In the 4-color theorem, too, it seems to me from what I've read that the 
<BR>programmers skipped too many steps. Starting at square one, the first job is 
<BR>to program a computer to construct mathematical proofs. Until that's been 
<BR>accomplished, a computer isn't able to construct a mathematical proof of 
<BR>anything. And this of course is the borderline question that I hope is being 
<BR>considered in AI. What's the limit to what a computer can do? And what 
<BR>problems can only be solved by a sentient being?
<BR>
<BR>All that to-do about computers playing chess is far from persuasive, because 
<BR>chess has a finite number of possible moves. &nbsp;What about programming a 
<BR>computer to write a recipe book, with some assurance that the receipes will 
<BR>be superior to those a human being might develop? &nbsp;Here, it's a tougher 
<BR>problem, because more is involved than shuffling numbers, which is all 
<BR>computers can do. But, as above, I think the main problem with any 
<BR>computerized "mathematical proof" of the four-color theorem is that one can't 
<BR>skip the necessary step of developing a program capable of constructing 
<BR>mathematical proofs. 
<BR>
<BR>pat
<BR>
<BR>
<BR></B></FONT></HTML>

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