Geometry is unnatural

Why learning (and teaching) the subject is hard.

Our tutor works along with several others at his office, each with particular strengths and styles.  There is enough variety that almost every subject is covered (though not Latin, as yet).  Several tutors help students with Geometry.  All of them, however, dislike teaching two-column proofs: the rigorous application of logic and previous results to derive new ones.

Largely this is because most students find it very difficult.  In part this is because they must learn previous theorems well enough to apply them; and not only that, but to choose those that might be useful.  The greater difficulty lies in the fact that a proof is a creative act: there is no sure-fire procedure that a student can follow to do what’s required.  In fact there may be several ways to prove a certain result, all equally valid.  (The great mathematician Leonhard Euler was fond of discovering new proofs for things he had already proved, which shows something of how the mathematical mind works.)  It’s this combination of rigor and creativity that makes proofs hard.

And it’s not the way we normally work.  The idea of the rigorous proof comes from Classical Greece, no doubt connected with the philosophical search for absolutes.  As a practical matter, there’s no justification for it.  The Babylonians and Egyptians were capable calculators, quite able to work out dimensions of pyramids and positions of planets to their required accuracy.  Other cultures did without it also.  What’s the point of knowing, say, pi to such accuracy that you can calculate the circumference of a circle ten miles in diameter to within a fraction of an inch?  It would be hard enough to make a circle that accurately circular in the first place.

And this rigorous moving from step to step, only doing what can be justified and always examining oneself for unstated assumptions, is not the way we normally argue.  Our hunter-gatherer forebears had to go where the edible plants and game were likely to be found, and farmers had to plant what was likely to grow.  If they’d required mathematical proof before acting, they’d never have done anything.  Strict deductive logic is not something humans do naturally.

So it has to be taught.  That, as we’ve mentioned before, is probably the main reason for teaching Geometry at all: to inculcate the idea of a rigorous proof.

Now, many of our scientific colleagues are fold of saying that science is a very human activity.  We naturally collect and classify things, look for connections and similarities, formulate hypotheses in the search for explanations.  This much is true, of course noting that different people do it with different levels of consciousness and organization.

But the hard part of science it the testing of hypotheses.  This is the deductive part of the process, working what what must be the results if it’s true, and deciding without pity.  That’s not a natural human activity, and it’s what Geometry brings to us.  If we can learn it.