geometry question

>>8507175
>but non-euclidean geometry makes no sense. it's built entirely upon the plane being a disk (hyperbolic), or an infinite sphere (elliptical).

>what is affine geometry
>what is projective geometry
>what is inversive geometry

>>8507709
that's just translations

>>8507175
Even straight lines curve in on themselves at infinity, if your measure of signed distance (displacement) is restricted to the one point compactification of the real line, instead of just the real line itself.

So think of non Euclidean geometry as what happens when your measure of displacement is based in a FINITE one point compactification of the reals. As in, one where infinity is a finite number, and any numbers beyond that are undefined because that's where it curves back around.

So yeah, a plane is still flat, it's just that flat has actually been infinitesimally curved all along, and now we're redefining the curvature of flat to not be infinitesimal.

>>8507175
It can be proven quite easily that the fifth postule is not right now wrong, it is completely optional.

The only things in mathematics that have truth values (right or wrong) are propositions. If the fifth postulate was a proposition that could be deduced then changing it would have no effect or it would cause a contradiction with the other 4 postulates.

For example, take a simple theorem like:
>Given any 3 points in any line, one must be between the others

Lets now say that theorem is false and consider that one of the other possibilities is true:
Given any 3 points in a line, none are between the others.

Lets now find what this implies. The definition of a point being inbetween for the three points A, B and C is simply:

B is between A and C if and only if AC = AB + BC

(where AC means the length of the segment AC, AB the lenth of AB and BC the lenth of BC)

Now lets write down: If B is between A and C:
AC = AB + BC
if A is between B and C
BC = BA + AC
If C is between A and B
AB = AC + CB

NOT(AB = AC + CB OR BC = BA + AC OR AC = AB + BC)

=

NOT(AB = AC + CB) AND NOT(BC = BA + AC) AND NOT(AC = AB + BC)

From algebra we know what negating equality means in an ordered set so this equals to:

(AB > AC + CB OR AB < AC + CB) AND (BC > BA + AC OR BC < BA + AC) AND (AC > AB + BC OR AC < AB + BC)

And that statement contradicts the algebra of real numbers, which would imply that adding a measure to lines (a bijection between the points in a line and a set of numbers, most of the time R but Q if you are Wild) causes logical problems with the algebra of that set.

Why? Because if you take any triple of the posible relations, they would contradict with each other.

This same thing would happen if the 5th postulate was actually a theorem that could be proven with the other postulates.