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

A Cartesian space is not an Euclidean space. An Euclidean space is
a space which holds Euclidean axioms. You don't need even set theory
to do Euclidean geometry. You need only Euclidean axioms and logics.
A Cartesian space is R^n, the set of real n-tuples (x1,...,xn).
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 coordinate
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
the Bases of Geometry" by Bernhard Riemann. He said, since a
coordinate system is dependent on local charts, we should find
invariants that are independent of coordinate systems. An invariant
he suggested was the 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.