Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space? EDIT: update, I found that Euclid's axioms are not considered rigorous.  David Hilbert did a full axiomatization of Euclidean Geometry (1899 in his book Grundlagen der Geometrie--tr. The Foundations of Geometry).  To do so, he required 6 primitive terms which were undefined, including points, lines, and planes.  Lines and planes are both spaces.  Therefore, a true axiomatization of Euclidean Geometry does in fact require space.   


source: http://userpages.umbc.edu/~rcampbel/Math306/Axioms/Hilbert.html
http://en.wikipedia.org/wiki/Hilbert%27s_axioms#The_axioms

I left the rest of the post as it was below, but I believe ^^this is the answer.
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  aside: Here's idea: Euclidean geometry is done using points and lines (0 and 1 dimensional spaces embedded in 2 dimensional space.)  Why stop there?  Why not do geometry with plane, line, point constructions embedded in 3 space?  Or 3-space constructions in 4-space?  The intersection of two lines is a point, the intersection of two planes would be a line, and the intersection of two "non parallel" 3-spaces would be a plane.


Here are the axioms of Euclidean Geometry:


A straight line segment can be drawn joining any two points.
Any straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
All right angles are congruent.


That's without even including the parallel postulate.

Here are the axioms: http://mathworld.wolfram.com/EuclidsPostulates.html


  EDIT: here's a possible first postulate that I came up with. Timax's postulate: There exists a 2-dimensional space which has distance-preserving maps.  (maybe some linear algebra or topology would be necessary to define it fully)


But how can you even say "points exist" without first saying "space exists"? If points are defined as being mathematically represented by co-ordinates, (eg (23, 1, 9) is a point in 3-space,) then it seems the first axiom would have to be "3-space exists", and "real numbers exist" in order to create co-ordinates.

It seems pretty obvious that space must come before points.

I guess you could say a point exists, but to get beyond a zero dimensional space, and to get more than 1 point, you would need to establish that space exists.  You can't have 2 points without at least 1 dimension.  To get more than 1 line, you have to have 2 dimensions, to get more than 1 plane, you have to have 3 dimensions.

This is probably why the concept of "space" always seems so mysterious, BECAUSE THE EXISTENCE OF SPACE IS AN AXIOM THAT WAS NEVER STATED!

Here's another idea: Do Euclid's Elements-style constructions in 3 dimensional space or even N dimensional space, instead of only 2 dimensional space.  Has that been done?

2015 M. Wanzek
Why would you have to say “space exist” before you can say “points exist”? In order to have something that contains all those points? But then by the same reasoning, you can't say “space exist” without first postulating the existence of some entity that can contain spaces. Before you know it, you require an infinite regress of existence statments.

In short, “points exist” is a fine start. It was good at Euclid's time, and it's good now.
You've got to remember that this was done thousands of years ago. Euclid probably didn't realize the need for such an axiom. Euclid's Elements is one of the first attempts at making math logically rigorous. That took mathematicians hundreds of years to do and only really got going around the 1600s.