Are dimensions needed because of Infinity?

If its any help, the "number" of points inside a cube is the same as found on one of its defining edges (lines).

Reply to jgill

I don't mean number of points in a math object. Lets say miles or representation of number of countable things.

Quoting TiredThinker

If there were a finite number of things on 3 axises, could that same information be represented in a single line? Do higher dimensions exist only for when values go on forever in an axis?

It sounds like you'd want an invertible function [math]f : S \subset \mathbb{R}^3 \to\mathbb{R} [/math], with [math]S[/math] finite. That should be easy (an infinity of choices.) One boring but easy approach is ordering the points in the domain set ('alphabetically') and just counting them off like [math] 0 ,1, ... , n - 1 [/math].

I think what you are really interested in is bases for vector spaces.
https://en.wikipedia.org/wiki/Basis_(linear_algebra)

"If there were a finite number of things on 3 axises, could that same information be represented in a single line?"

Yes indeed. You might read up on Georg Cantor, who did a lot of work in this area.

"If its any help, the "number" of points inside a cube is the same as found on one of its defining edges (lines)."

Can you develop that a bit further? Points in any continuum are infinitely many.

Reply to alan1000 There's a one to one correspondence between points on such a line and points in the interior of the cube. You could look it up, it's a simple arithmetic trick involving manipulating the digits of the numbers representing the points.