Re: Hofman and Diaby talk about P=NP at INFORMS 2007

In article <1170533683.121534.99940@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<moustapha.diaby@xxxxxxxxxxxxxxxxxx> wrote:
I don't think it is reasonable for you to ask that we simply ignore
all this.

I am not ignoring it. I am simply suggesting that we do these things
in the opposite order. That is, first locate the violated constraint,
and then return to the more abstract mathematical argumentation.

I think you/Hofman/Moews should first properly address the arguments I
have already posted on this issue.

This possibility has been tried, and we have already seen that it has not
succeeded. A good teacher should be willing to try alternative approaches
if the first approach he has attempted is not understood by the student.

Students often find abstract argumentation difficult to follow. Concrete
calculation with numbers is more tangible to them. Once someone came into
my office, excitedly claiming he had proved Fermat's Last Theorem. When I
saw his "proof," I noticed two things right away: (1) his argument could
equally be applied to prove that there are no solutions when the exponent
is 2; (2) his argument could also be applied to prove that there are no
*real* solutions. I tried to point this out to him, but these arguments
were too abstract for his level of mathematical competence. Therefore, I
went through the more tedious process of going through his proof line by
line, until I found the step that was wrong. At one point he argued that
because a < b and c < d, it must follow that ac < bd. When I suggested
that he consider a = -2, b = -1, c = 1, d = 4, he understood immediately.
He was disappointed, but at least we reached "convergence" quickly. If I
had insisted on my (absolutely correct) abstract arguments, he would have
gone away confused and frustrated.

You're the professor here, right? Shouldn't you be trying to explain your
ideas in a way that your students can understand, rather than insisting on
one particular pedagogical approach? By *your own account*, Hofman and
Moews lack the expertise in mathematical programming to follow the kind of
abstract arguments you have been presenting. So shouldn't you adjust to
their level? The beauty of mathematics is that we can always settle
disputes by boiling down to something absolutely obvious and concrete.
In this case, exhibiting the violated constraint would settle the dispute
conclusively, in a way that even Hofman and Moews could understand.

Frankly, this principle applies not just to teacher-student interactions,
but interactions between scholars. Frequently I give a talk, and someone
in the audience says, "That theorem doesn't look like it could be right,
because isn't such-and-such a counterexample?" I always stop and try to
compute whether the counterexample is correct, because it's a win-win
situation: if the counterexample is correct then I've detected a mistake
in my theorem (usually easily fixed by a slight restatement); if it's
wrong then the speaker's understanding has been improved.

You are asking that we stray away from mathematical argumentation and
logic in favor of "chasing" through things that number in the millions!

Millions of variables and constraints are trivial to deal with in the
modern age of computers. You're not doing any complicated computations;
you're just reading in some variables and plugging them into your
constraints. This takes linear time; any computer today can zip through
this computation in a blink. The only real cost is a couple of hours of
programming. I repeat: This is *your* research. You are paid to be a
teacher. You could probably even get a student to do the grunt work for
you. How can you possibly regard those few hours as being not worth your
while?
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.

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