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Topic: What is mathematics
Replies: 2   Last Post: Jul 19, 1995 1:14 PM

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Posts: 16
Registered: 12/6/04
What is mathematics
Posted: Jul 19, 1995 9:06 AM
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The thread (mostly between Andrei and Ted) on "What is Mathematics" has been
raising some very interesting issues. I tend to agree with Andrei that if it
ain't got a proof, that ain't it [loose rephrasing]. On the other hand, not
everyone who needs to be an effective user of mathematical tools truly needs to
understand proofs and proof methodologies. (Not that it would HURT them).

Can I suggest a different tack for this discussion?

In graduate school at Michigan I took a course (NOT in the math departrment)
which addressed interdisciplinary connectiosn between natural, formal (i.e.
mathematics) and artificial (i.e. engineering) systems. The first assignment
in that course was to develop an axiom system for a natural system, and prove
at least one significant theorem in it.

Part of the lesson of this exercise was that you had to start with the theorem
if you wanted to get anyplace -- people who started with the natural system and
tried to develop interesting axioms didn't get too far.

In Science, there's this mantra called the scientific method. It gets taught as
early as the 2nd grade, and is the organizing principle for almost all the
science work through high school. And indeed, I suspect that most working
scientists can relate what they do to the scientific method, although they
hardly follow it with the literalness that the 2nd graders are taught.

What is the Mathematical Method? Surely, it isn't calculation. OUTSIDE of math
courses, the main use of mathematics is in modelling. Within mathematics, there
are roots all over the place sucking nourishment from the real world, although
the theory sometimes gets way ahead of the systems being modeled.

I like the notion of mathematics being modelling much better than that it is
problem solving. That separates the mathematical content from the computational
more clearly, and at the same time exposes their interrelationship.

Question 1: Is the statement "Mathematics is modelling" -- with sufficient
modificatiosna and definitions -- a useful, uh, model? Does it help to
organize the material taught in the schools? Does it provide asny insights to
students? Teachers? Parents? Society?

Question 2: Is there a "Mathematical Method?" If there is at least a partial
affirmative to Question 1, then it might involve steps of observation,
isolation of significant features, search of independence, consistency,
completeness. Also, parametrization, calculation, [this is beginning to sound
like the gryphon in "Alice"] and checking against the real world.
"Homomorphism" would clearly be another organizing principle, since that's the
basic constraint a model must satisfy.

I'm not proposing a new curriculum. In fact, what I've seen of IMP is that it
does approach much of the mathematical content through modelling.

Question 3: If there is a "mathematical method" (how ever restricted the
definition would have to be to satisfy everyone who has an opinion) is there a
short form that might be taught to children and the public, right next to the
Scientific Method?

Dennis Geller

I know that this is not the way that math is taught in most post-secondary math
programs. (If it's really a new thought, try to dig up a copy of Michael
Arbib's classic "Brains, Machines and Mathematics.")

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