Question

What determines the fact that a dimension actually exists?

This entire question is based off the assumption that a Point (0-dimensions) is nonexistent in the fact that no matter how infintesimally close you get to a point, it still does not exist other than the representation that it symoblizes a location.

Considering this assumption, I propose my question, which you can view here
http://www.ted.com/conversations/12291/what_determines_the_fact_that.html

Answer 1

electrons are points and they exist.

Answer 2

Nothing.

It's an assumption you are forced to make to be able to define anything. Axioms work the same way. Our understanding of geometry is based on assumptions, which we then expand on by proofs. Euclid himself states "a point is that which has no part" and "a line is breadthless length". That's his definition of a point and line, which work with the axioms he set and which then he built his geometry on.

Today, we generalize and say these notions are undefined (this is due to the fact that there are many types of geometry, Euclid was only interested in working with one). As in, a point or a line can be anything, provided they satisfy the axioms you're working under. Also, there is no concept of "getting close to a point" or "stretching of a point" if the point doesn't exist.

Have you taken a 300 or 400 level undergraduate course in Geometry? This kind of discussion is usually covered within the first month or two. I can also suggest to you a book on Geometry if you'd like to read more about it.

Answer 3

I have not taken any courses in geometry, nor much mathematics past single-variable calculus. I would love to read a book on geometry if you are able to provide a name.

Answer 4

I really liked the book "Geometry: Euclid and Beyond" by Robin Hartshorne. It starts you out by explaining axioms and defining Euclidean Geometry, then gradually moving you into other geometries. The exercises are really proof focused, so be prepared for that if you're interested in working on problems in the book too. But, even if you just read it, it gives you a pretty good look into what geometry is about.