Art is too materialistic—all that charcoal and clay. Science is too materialistic—all that glassware and goo. Geometry is the most spiritual of all school subjects. Point, line, and plane are pure ideas; only their representations and approximations can be found in the material world—wall meets floor, and we “see” a line. But it is rough, imperfect, tile and wood. It suggests a definition, something of one dimension, length.
Definitions lead to postulates or axioms, propositions that cannot be proved or disproved, but must be accepted as true—parallel lines never meet, for example. And postulates lead to theorems, statements that can be proved to be true by deduction. Present a simple proof of a few steps to a class of tenth graders, however, and some will “see it” immediately. Others may labor over it for days before light dawns.
Because proofs cannot be taught. A teacher can demonstrate, explain, repeat, offer examples… but students must make the intuitive leap, see the pure idea, grasp the truth of the proof for themselves. For that split second, they are entirely on their own, in an undeniable and objective world of ideas. (This makes geometry the perfect subject for fifteen year-olds, who begin to feel entirely on their own but buffeted by the deniable and non-objective demands of a grown-up world for which they are not quite ready.)
Intuitive objectivity? Perhaps for this reason Plato prescribes fifteen years of geometry for the education of the Guardians of his Republic. How common it is, too, that what was holy and reserved for the leading minds of an ancient society is commonplace and mastered by children today.
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