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?
------------------------------------------------- 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.")