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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.
<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. 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? 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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