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.