Date: Wed, 9 Sep 1998 21:37:51 -0400
From: Alex Bogomolny
Dear Colin:
Do not even doubt whether your question is meaningful or not. The question of infinity, of the infinitely small or infinitely large, is absolutely profound and does not admit a simple answer.
I can't go to any length about this but will only suggest a few points to ponder.
- The question is actually at least 2,000 years old. Please look into Zeno's paradoxes. Probably any encyclopedia will have them covered.
- I do not believe that the definition of the point as a something with no dimensions is universally acceptable nowadays. Please think of it, the notion is absolutely counterintuitive. Have you ever seen an electron not to mention a location without dimensions? What kind of a definition is it then?
- Can you define a dimension independently of points and lines that you are ready to accept a location without dimensions?
- If lines do not consist of points then what is the intersection of two lines? So there are points on a line. For any two points, there is a middle point in between. Is there anything else on the line?
- There is a profound difference between finite
and infinite. So drawing parallels may not be always
appropriate.
Nowadays, points and lines are defined through axioms simultaneously as two kinds of objects such that
- through two distinct points passes a single line
- two lines intersect in at most one point
- etc.
There is no circularity. What you do on the blackboard, kind of visual, intuitive geometry, is a model of axioms that endows points and lines with some intuitive meaning. However, there are other models.
As I said, these are things to think about. I must add that I applaud your pausing to raise this question.
Sincerely,
Alexander Bogomolny
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