Jay,
While it is true that one question is:
> How abstract and formal a view of the nature of mathematics should we
> present to some/all students at different stages of their educations?
I think that the crucial point doesn't If I teach the Fibonacci series when
studying series in general in the Calculus as applied to the study of
springs, for example, that's one thing. If I teach it as an emergent
property in the sectioning of the nautilus shell or the proportion present
in "the most perfect shape", the golden rectangle of Greek architecture,
that's another.
You seem to recognize this somewhat when you write:
"It's not a question of formalist or informalist from my perspective.It's a
technical
> skill that requires an unusual form of intuition to do well, and to
> initiates what is done has a special kind of beauty somewhat akin to that
> of great music. I would treat its place in the curriculum in much the same
> way: some practical fundamentals of music theory for all, some music
> appreciation for all, the option to pursue the matter further for all who
> are interested, and almost certainly advanced training for very, very few.
The analogy can be pushed a little further. The way math is taught now is
sort of like teaching everyone the scales before one is exposed to beautiful
music itself. Your distinction between "applied" and "pure" leads away from
the point I'm trying to make. Pure math doesn't mean anything other than
"not immediately applied" and it has little to do with the development of
the intuition. I don't know if you believe that intuition can be developed
but I think it can, just as taste can be cultivated. One can start
listening to Mile's "Kind of Blue" and come to appreciate Coltrane's
"Impressions" or Mingus' "Better 'Git it in your Soul" though at first the
latter two might sound like pure noise to the uninitiated.
The question for me isn't between "pure" and "applied" at all. One
shouldn't overlook the fact that what is pure now becomes applied later.
The most famous case in mathematics being the fate of the attempts to prove
that parallel lines don't meet at infinity. How "pure": is that? It
certainly had no application to anything practical at all. But I'm sure you
know that this seemingly pointless pursuit led to the non-euclidean
geometries (how abstract) that were "applied" in Einstein's theories.
Those who pursued this demonstration, and later the equally non-applicable
development of geometries where lines both meet and diverge, certainly were
attracted by something and it certainly wasn't just "pure".
Is it elitist to develop an intuition for the beautiful rather than teach
the practical? I prefer to think along the lines of Bourdieu in "Le
Distincion" that the way taste and appreciation for the beautiful is
cultivated has everything to do with the maintenance of class structures.
Paul H. Dillon
p.s. Charles Morse, utopian socialist who designed my favorite type of
chair, used to say "fill your house with practical and beautiful things" so
I don't think the distinction is exclusive.
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