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Re: Pure mathematics and problems for physicists too?



Hi all-
K. Lee Lerner asks, making my blood boil:
******************************************************
I sometimes wonder if we as scientists and teachers of science become too
concerned only with "prerequisite math" that enables students to solve
problems. During the course of the 19th century there was a dramatic
increase in the number of dedicated pure mathematicians with minimal
interest in applications to science and technology. Although, at that time,
this divergence was some concern to the scientific and mathematics
communities, subsequent technological leaps made the divergence less real
(i.e., some seemingly "pure" mathematical concepts were brought into the
realm of applied mathematics). I would suggest that the math experience
bears only superficial resemblance to the divergence of theoretical and
experimental physics but I'd appreciate any thoughts in this regard.

Would there be any value of having physics students develop their logic and
reasoning skills by taking some pure mathematics in addition to the standard
fare of applied courses? Should they at least take a peek at the lay of the
mathematical landscape? Perhaps math departments should be encouraged to
develop some math for scientists courses structured along the lines of math
for liberal arts majors courses. Although this option might not help lower
level undergraduates or be accessible to brighter high school students in
any formal way, what topics would be the best to steer motivated and capable
students with an interest in both science and math toward?

As a divergent question or discussion suggestion: Is anyone aware of a
Hilbert-style listing of great problems for physicists to tackle during the
next century? If not, besides the quest for a GUT what would the members
think belonged on such a list? If such a list isn't available perhaps we
could discuss the parameters (theoretical vs. technological problems) and
composition of such a list. It might make an interesting archive entry.
**********************************
If K. Lee Lerner can point me to some current math texts that
enable students "only to solve problems" he would make me very happy.
The trouble with modern math teaching IMO is that math teachers have
totally forgotten that most math was invented for the purpose of making it
easier (or possible) to solve problems. Then, during the past century, the
Bourbaki collaboration buried three millenia of progress in one mighty burst
of obfuscation. Now we get students who know some mathematics' vocabulary
but are incapable of starting on simple problems with the proposition:
"Let x stand for the unknown quantity ______." The job of the physics, or
chemistry, or engineering teacher must then include the teaching of math
from elementary algebra on up.
Calculus teachers around the country seem to shrug their shoulders
when they find themselves "teaching" calculus to students who can barely
do the most elementary algebraic operations. They can, however, use the
phrase "solution set" instead of the word "solution" in a complete sentence.
Lerner seems to have the history of modern mathematics backwards.
Most modern mathematics had originated from attempts to solve physics and
engineering problems, not the other way around. Many of these problems,
at the end of the 19th and beginning of the 20th century were concerned
with astronomy. It was from the background of such investigations that
the towering figure of Poincare` emerged. And in recent years the
Field medal (the "Nobel Prize Equivalent" for mathematicians) was awarded
to Witten for work connected with string theory.
The brutal fact is, that in the pursuit of a solution to a practical
problem it is easy enough to get waylaid by a mathematical puzzle or a
computer-connected project. Lerner's proposal is a distraction, not an
aid, to mathematics education.
Regards,
Jack




"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography