# A Coordinate Is Not A Point

> The introduction of numbers as coordinates is an act of violence
>
> Hermann Weyl

A Coordinate is not a point. No, (1,2) is just a coodinate, and no
t a point itself. A coordinate system is useful for science, since
you have to calculate quantities. But they have no geometric meani
ng. Although you can express a point as (1,2), that means nothing
in geometry. Things having geometric meanings are independent of c
oordinate systems. Length, distance, angle, volume, &c. are all in
dependent of coordinate systems.

A Cartesian space is not an Euclidean space. An Euclidean space is
a space which holds Euclidean axioms. You don't even need set theo
ry to do Euclidean geometry. You need only Euclidean axioms and lo
gics. A Cartesian space is R^n, the set of real n-tuples (x1,...,x
n). Since a Cartesian space meets the axioms, you can interpret it
as an Euclidean space. But it is just a specific case. It's not an
Euclidean space itself. An Euclidean space do not assume a coordin
ate system and the origin point. A Cartesian space is by itself a
coordinate system, and it has the origin point.

For more information, I suggest "On the Hypotheses which lie at th
e Bases of Geometry" by Bernhard Riemann. He said, since a coordin
ate system is dependent on local charts, we should find invariants
that are independent of coordinate systems. An invariant he sugges
ted was a curvature tensor.

I don't say a coordinate system is useless for geometry. But since
it means nothing, it is useful only when you can get invariants by
using it.