Jay,
The question you raise about the (un)importance to pedagogy of the
ontological status of mathematical forms (plato was comfortable with that
term which could include both what we think of as objects, as well as
relations) and how mathematicians relate to those forms is well taken. A
similar argument is the starting point in Jerry King's "The Art of
Mathematics" but he follows it to a different conclusion. Rather than
concluding that the mathematician's take on math is irrelevant he argues
that it is fundamental to reshaping the math curriculum. He points out that
the calculus, the gateway to all higher math, is basically taught from the
point of view of its applications to physics and engineering. Consequently
it remains out of reach and irrelevant for most folks. It also assumes a
terribly dry aspect. He makes the case that what mathematics tune into in
mathematical forms is in fact more akin to the beautiful than the practical
and he argues for a curriculum which approaches it from this direction.
He also addresses why this is important with an interesting example drawn
from reactions to the Three Mile Island accident among mathematical and
non-mathematical faculty at t he University where he taught at the time.
The "gateless gate" of the Fundamental Theorem has the practical effect of
restricting participation in any discourse that presumes an understanding of
those forms which, he argues, does not require an understanding of their
application to physics and engineering BUT that is the only way in at
present. Some weirdos actually take math because they just like it (eg,
amateur mathematician) and are willing to put up with the extraneous (and
really irrelevant) applications through which it is taught.
The book is rather old (70s?) so maybe there has been some experimentation
with curriculum in the direction he proposed but I haven't heard about it..
Paul H. Dillon
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