> (4) Won't all mathematics teachers recognize the phenomenon of having
> someone grasp how to use an algorithm by way of a few examples? Or at
> least how to use it except for exceptional cases of some kind?
Certainly every engineer will, because that is often the way they teach.
I met a civil engineer who had enormous skill in implicit function
calculations around singularities, but he had never heard of the
Implicit Function Theorem: he had learned it by sitting down with his
thesis advisor and working through a vast number of examples, until he
had a good, intuitive sense of what works. The key word here is _vast_
-- it was very labour-intensive on both sides.
When he was introduced to a formal test for when his manipulations were
legitimate, he was fascinated.
It is possible spend equally vast development of axioms found by
twitching existing axiom schemes, but "The future does not lie in the
theory of generalised left pseudo-heaps". The good axiom schemes, like
the good sets of theorem hypotheses, are developed as sets of statements
that are often true together about the same object. When the
consequences are rich, as in Euclidean geometry or in calculus in
several variables, one culture has a tool of great power that another
culture (in the above example, civil engineering) lacks. When they are
solving the same problems, they can be compared (like making glasses
with and without geometric optics). Sometimes one does better,
sometimes the other. Sometimes the goals are not comparable, but
sometimes, in some respects, one mathematical culture is objectively
more effective.
My "pure" mathematical training gave me a far more superior attitude to
engineering ethnomathematics than to Chinese, and I now see the
provincialism of that. But the attitude that a judgement of superiority
is never appropriate is an equally limited one, and often flavoured with
the subtext "We must never make explicit judgements of another culture,
because of course our superiority would show, and that would be
impolite." This is like using the English name Cologne for Koeln, but
following every name change in Third World countries out of a
condescending 'politeness' we do not show to developed countries.
Mathematicians do not spare the feelings of those they respect. (My
wife once said "That student must be very very stupid, you were being so
nice to her!" With good students, I have vigorous debates, and enjoy it
when I lose.)
It is important to analyze where different mathematical cultures (in
place, time, and "subject") are comparable, and where they are
comparable, to work through the comparison without fear of reaching
conclusions. Otherwise, mathematical history fails to consider a
central characteristic of mathematics: its power.
Tim Poston