"I have a vivid memory of how our teacher started off in geometry. Chalk in hand, he made a small circular splotch on the blackboard and said, 'This is a point.' He hesitated for a moment, looked at the splotch once more, and added, 'Well, it isn't really a point, because a point has no extension.'
Then he went on to lines and other basic notions of geometry. We were left uneasy. We thought of grains of sand or specks of dust in the sunlight, but realized that, small though they were, they still had some extension. So, what was a point?
The question was buried in our struggle to keep up with the lessons, but it was not forgotten. It smouldered unresolved under whatever constructs came to cover it, and did not go away. In the course of the next few years it was joined by some other bubbles of uneasiness. When we came to infinite progressions, limits, and calculus, we were tacitly expected to think that there was a logically smooth transition from very small to nothing. We were told that Zeno's story of Achilles and the tortoise was a playful paradox, an oddity that did not really matter.
I did not like it, but I had decided to love mathematics anyway. Some of my schoolmates, however, concluded that mathematics was a silly game. Given the way some of it was presented, their reaction was not unjustified.
In retrospect, decades later, I realized that there had been quite a few occasions where the teacher could have resolved all those perplexing questions by one explanation. Shortly after the point episode in that geometry class, the teacher introduced the term 'equilateral triangle.' It was in the days when wooden rulers and triangles were used to draw on the blackboard. The teacher picked up one of these contraptions and showed it to the class. 'This is an equilateral triangle because its three sides have the same length.' As he was holding it up, he noticed that one of the corners was broken off. 'It's a little damaged,' he said, 'it would be an equilateral triangle, if you imagine the missing corner.' He missed a most appropriate occasion to explain that all the elements of geometry, from the point and the line to conic sections and regular bodies, have to be imagined. He could have explained that the points, lines, and perfect triangles of geometry are fictions that cannot be found in the sensorimotor world, because they are concepts rather than things. He could have told us that, no matter exactly a physical triangle is machined, it is clear that, if ones raises the standard of precision, its sides will be found to be not quite straight and their length not quite what it was supposed to be. He could have gone on to explain that mathematics - and indeed science in general - is not intended to describe reality but to provide a system for us to organize experience. I do not think that many students would be unable to understand this - and once it was understood, the domains of mathematics and science would seem a little more congenial."
Something to think about as we start a new school year.