geometry - Basic Geometric intuition, context is undergraduate mathematics

At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object.

For me the first instance that comes to memory was in 7th grade in a inner city USA school district.

Getting to the point, my geometry teacher taught,

"a point has no length width or depth in any dimensions, if you take a string of points and line them up for "x" distance you have a line, the line has "x" length and zero height, when you stack the lines on top of each other for "y" distance you get a plane"

Meanwhile I'm experiencing cognitive dissonance, how can anything with zero length or width be stacked on top of itself and build itself into something with width of length?

I quit math.

Cut to a few years after high school, I'm deep in the math's.

I rationalized geometry with my own theory which didn't conflict with any of geometry or trigonometry.

I theorized that a point in space was infinitely small space in every dimension such that you can add them together to get a line, or add the lines to get a plane.

Now you can say that the line has infinitely small height approaching zero but not zero.

What really triggered me is a Linear Algebra professor at my school said that lines have zero height and didn't listen to my argument. . .

I don't know if my intuition is any better than hers . . . if I'm wrong, if she's wrong . . .

I would very much appreciate some advice on how to deal with these sorts of things.

Answer

This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an infinitesimally thickened point, and it sounds to me like you want to do geometry with with infinitesimals. Whereas Omnomnomnom suggests looking at nonstandard analysis, I would suggest a different approach to infinitesimals, namely smooth infinitesimal analysis. It's pretty intuitive (no pun on intuitionistic logic intended) and easy to use. I personally think in terms of synthetic differential geometry and smooth infinitesimal analysis all the time when working with smooth manifolds and Lie groups/algebras. If you're interested, have a look at John L. Bell's A Primer of Infinitesimal Analysis. He's a prominent philosopher of mathematics and I'm sure you'll find something in common with his views.