Ernst von Glasersfeld:

"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.


09/04/2012 9:24am

Hey Brian, this is a great explanation of Platonic forms. A perfect circle in the physical world is never perfect, but rather participates more in the idea of a perfect circle. The best parts of math and science (and philosophy) are in the mysteries. Perfection, nothingness, existence, all the physical and metaphysical paradoxes we seek out in order to better our understanding. I might argue that math and science describe reality better than we can experience it. I don't know if we are organizing our reality, or measuring nature with these disciplines. I am not sure the two are mutually exclusive either.

I love reading your blog. I always sit and stare at the comment I am writing thinking about math and metaphysics and end up thinking way more than I write. It's intellectually stimulating and I appreciate it.

09/05/2012 8:56pm

Thanks Jay. I was interested by your argument that "math and science describe reality better than we can experience it." I might counter that with the idea that we might never truly "know" reality, but we can only construct ideas based on our experiences and interactions with our environment. Not sure what the implications of that might be, but it at least acknowledges math (and other forms of knowledge) as something that is constructed rather than discovered. To me, that is a human aspect that is often overlooked or ignored in teaching. Thanks for your thoughts.

09/06/2012 10:21am

I believe that knowledge can only really be qualified as human knowledge and truth only as subjective truth. As humans we have our 5 senses with which to experience existence, plus our intellect which we use to describe and express it. As beings we have but out own perspective from which we comprehend the world. It is unique for everyone. Similar, but no the same, to everyone else.

I would say that because we can never really "know" reality, the sciences we have created to measure the world create a more accurate portrayal of existence. I would also say that; our environments interact and affect us. And we discover as much as we create. We create math, and discover the Fibonacci sequence. We create optics and discover that the sun is the center of the solar system. I suppose what happens is that we create knowledge and then we discover subjective truths. I suppose the questions I am getting at is this: Does that fact that we have created these intellectual ideas make them any less real, or true?

09/08/2012 9:02am

You pose some interesting ideas and questions here. I think my initial thoughts are that to each individual, their current way of knowing is always regarded as real (or true). I believe that, as thinking humans, part of our fundamental being is to maintain internal consistency...I'm not sure we could live without it.


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