# A Space Exists Before Its Points

> 1.1 The world is the totality of facts, not of things.
>
> Ludwig Wittgenstein, "Tractatus Logico-Philosophicus"

People often think that points make a line, lines make a plane,
&c. Therefore a point should be the most primitive thing in geometry.
Well, not really.

For normal people, I'll explain it with an Euclidean space. An
Euclidean space is a thing in which Euclidean axioms hold. It can
be anything if that meets the axioms. n-tuples of real numbers are
typical though. The important thing is that you need a whole space
to meet the axioms. Only after that, you can call the elements of
the space "points".

A type of relations called "geometric relations" are what only
exist in geometry. A space is not a fixed and concrete thing. It's
an abstraction. For example, think of what a distance means in the
axioms without our intuition. That's nothing but a measure of
similarity between the elements. And with some topology, we can
know that a distance is sufficient to induce a topology, i.e., the
skeletons of a space. Hence if there is a well-defined similarity,
a space exists.

Similarily, for an algebraic structure, a corresponding geometric
structure exists. And in that geometric structure, a point can be
literally anything. A function, an ideal, even a space can be a
point. If they hold a geometric relation, it becomes a space. Only
after that, the elements become points by being the atomic components
of the geometric structure.

This is similar to the ontology of Tractatus. The world is the
totality of facts, not of things. A space is the totality of
geometric relations, not of points. A space exists before its
points.