Why are points considered to be zero dimensional?

Question

Aside

This is a silly observation born out of my own curiosity. I'd love some perspective from more mathematically minded folks.

Please try to evaluate my reasoning, and expand on which assumptions are incorrect or misguided.

General question

Why are points considered to be zero dimensional? Is there a distinction between zero dimensional and dimensionless that I am missing?

Assumptions

• A single point cannot exist in the absence of at least one dimension.
• The point has no meaning beyond its distance from zero along that dimension.

Conclusion

• A single point is not dimensionless.
• Every point is coupled to some dimension(s), and its meaning is generated via its distance to zero along those dimension(s).
• In this sense, every point has content, as it assumes a value on a dimension. A dimensionless point would have no capacity to assume a value.

Answer 1

Here's just one viewpoint.

You may be interested in the difference between affine dimension and linear dimension. For flat things:

1. The affine dimension is the least number of points needed to specify the thing.
2. The linear dimension is the number of orthogonal directions available to someone living on the thing.

For example, since there's a unique line through any two points, ergo a line has affine dimension $2$. But if you're living on a line, there's only one direction along which you can move back and forth, so it has linear dimension $1$.

In general, the linear dimension is always $1$ less than the affine dimension.

Since a point takes $1$ point to specify, its affine dimension is $1$. Thus its linear dimension is $0$. Since "dimension" usually means "linear dimension", hence a point can be considered $0$-dimensional.