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Re: physics before math????

Hi sll-

On Fri, 21 Jan 2000, John Denker wrote:

1) The world of knowledge is highly nonsequential. Facts are related to
other facts in highly multi-dimensional ways. One of the hard parts of
teaching (or indeed any communication) is finding a way to serialize the
information. Serialization is necessary because almost all the
communication channels are basically one-dimensional. Alas, there is a
theorem (the flower-pressing theorem) that states that you cannot change
dimensions in a way that is one-to-one and continuous. Therefore there
cannot be any easy solution to the serialization problem. Any simple
solution is wrong.
How true!

2) Another fact to keep in mind is that physics requires math, and not vice
versa.
Partially true, I think. Math is required to make quantitative
predictions, certainly. But every pair of football players who complete
a forward pass is demonstrating the solution of a complicated problem in
physics. Some of those guys are not math-literate.

For example, even if you believe that geometry was invented to
solve certain problems in Egyptian geodesy, the origin has been irrelevant
for thousands of years. The topic we call geometry has an axiomatic basis
quite independent of its applications to geodesy, physics, or anything
else. IMHO the most important goal of high-school geometry is to introduce
students to the notion of proofs and rigorous thinking in general.

3) The usual approach to serializing math and physics is to do a little
math, then do a little physics, then more math, then more physics, and so
forth. Under this scheme, the math runs a little ahead of the
physics. Exactly how much ahead is a question of taste and depends on
details of the local conditions. But we can pin down a few points:

3a) One extreme would be to teach lots and lots of math before starting any
physics, which is possible but would unduly delay the physics.

3b) Another possibility would be to teach physics and the relevant math
topics concurrently. This has advantages and disadvantages. One advantage
is that it is easier to teach math if the students can see immediate
physical applications. The corresponding disadvantage is that it might put
too much emphasis on applied math, to the detriment of the abstract and
rigorous thinking that is the hallmark of real math. Math and physics are
not the same. Physics is validated by experiment. Math is validated by
rigor.
I have problems with the parallelism of the last two sentences. I
think that they have subtly different meanings of "validated". Rigor in
mathematics was a much later development than the use of mathematics in
physics. I think that many mathematical ideas can be broached and
implemented with only modest doses of "rigor". See, e.g.
http://gate.hep.anl.gov/jlu/index.html and click on chap3 (either the
ps or pdf versions), where we implement Leibniz's notion of the
infinitesimal. The approach was not made "rigorous" until the '50's by
Abraham Robinson (Non-Standard Analysis, 1966), which is probably why it
is not used more often.


3c) One might imagine attempting to teach physics before math, but this is
absurd. The whole point of physics is to build quantitative models of the
real world. This attempt becomes a double-or-nothing proposition: either
the physics teacher covers physics *and* math [in which case this reduces
to the previous case, (3b)] or the syllabus becomes so watered-down that
the students emerge knowing neither physics nor math.
I think that high school students in the '30's learned some
physics (and physical chemistry) concepts with little math exccept a
bit of algebra. This seemed to be enough of a framework for many of us
to build upon.

Bottom line: It is 100% guaranteed that there are no easy solutions -- but
"physics first" (i.e. before math) sounds like asking for trouble.

Bottom line: Whatever works, works.

Regards,
Jack