Subject: Re: Dimensions of a line
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.

1. The question is actually at least 2,000 years old. Please look into Zeno's paradoxes. Probably any encyclopedia will have them covered.
2. 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?
3. Can you define a dimension independently of points and lines that you are ready to accept a location without dimensions?
4. 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?
5. 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

1. through two distinct points passes a single line
2. two lines intersect in at most one point
3. 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