Divisions Between Pure & Applied Mathematics
Date: Around 1993
As I understand it, the mathematics used in physics today is not
the sort of mathematics that interests mathematicians. Yet, Newton was
somewhat of both. Is this division real, and if so, are there any books or
articles that discuss the history of this division?
Yes there is a real division!! You should be able to find some
information if you look in your library under science history or math history.
I recently read a story that illustrates the division as you have outlined it.
There were 3 men/women, an engineer, a physicist, and a mathematician who were
sharing a room while attending a conference. In the middle of the night a
fire broke out. The engineer woke up, grabbed a bucket, filled it with water
and put the fire out. A little later the fire started again. This time the
physicist woke up, calculated the correct amount of water needed to extinguish
the blaze, filled the bucket with just that amount of water, and put the fire
out. Still later the fire started again. This time the mathematician woke up,
calculated the amount of water needed to put out the fire, worked out the
proof that it would work, then went back to bed.
Certainly, in the past, pure and applied mathematicians have
argued at length over which endeavor is nobler. I think that there is not so
much conflict along those lines today. It certainly is the case that
mathematics is much broader in scope than the portion of it which is
traditionally of highest interests to physicists. Physics is only one area of
applications. There are lots of significant applications of mathematics to
fields such as economics, business, biology, computer science, the social
sciences, etc. There seems to be both an old and a new meaning to the term
"applied mathematics". The Mathematical Experience by Philip J. Davis and
Reuben Hersh. Check the table of contents for the appropriate section. This
book also contains a bibliography with additional references.
Robert Allan Chaffer
However, it should be noted that there really is considerable
interplay between physics and mathematics - mathematicians often become
interested in problems raised by physicists (variational calculus is an old
one, but there are newer examples, particularly in calculus extended to "bad"
functions like distributions), and sometimes physicists even invent new ideas
in mathematics (Dirac's spinors, or more recently wavelets (invented by a
geophysicist, I think)). Mathematicians have of course been greatly inspired
by Einstein's successful use of non-Euclidean geometry (although they hate to
admit that they might be doing anything practical, I doubt differential
topology and its extensions would be as fruitful a field today without this
implication of usefulness). Physicists are often discovering that areas of
mathematics previously deemed useless (various things in advanced number
theory, for instance) actually have implications in the real world. And, I am
speaking only from the physicist's perspective, of course. It is true that
most of modern mathematics is pretty far removed from anything the average
physicist would enjoy using in real work, and sometimes it seems that we
physicists should be able to tell mathematicians, "Hey, I really need to
understand this particular aspect of geometry a little better could you work
on it please?"...but communication between fields is always difficult, and
getting somebody else to work on your problem is too...
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Update: June 2012