# 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 Euc
lidean space is a thing in which Euclidean axioms hold. It can be
anything if that meets the axioms. n-tuples of real numbers are ty
pical though. The important thing is that you need a whole space t
o meet the axioms. Only after that, you can call the elements of t
he space "points".

A type of relations called "geometric relations" are what only exi
st 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 ax
ioms without our intuition. That's nothing but a measure of simila
rity between the elements. And with some topology, we can know tha
t a distance is sufficient to induce a topology, i.e., the skeleto
ns of a space. Hence if there is a well-defined similarity, a spac
e 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 po
int. If they hold a geometric relation, it becomes a space. Only a
fter that, the elements become points by being the atomic componen
ts of the geometric structure.

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