Why are we justified in using the real numbers to do geometry?

Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also very useful, since I can then use geometry whenever I have a vector space at hand.

However, some of that structure is superfluous, and I'm afraid that we can prove things that are not true in the more modest axiomatic geometry (say in axiomatic euclidian geometry versus the similar geometry in $\mathbb{R}^3$).

My questions are thus, in the context of plane geometry in particular:

1. Can we deduce, from some axiomatic geometries, an algebraic structure?

2. Are some axiomatic geometries equivalent, in some way, to their more algebraic counterparts?

(Note that by « more algebraic » geometry, I mean geometry in a vector space. The « more algebraic » counterpart of axiomatic euclidian geometry would be geometry in $R^2$ with the usual lines and points, and where we might restrict in some way the figures that we can build.)

I think it is useful to know when the two approaches intersect, first to be able to use the more powerful tools of algebra while doing axiomatic geometry, and second to aim for greater generality.

Another use for this type of considerations could be in the modelisation of geometry in a computer (for example in an application like Geogebra). Even though exact symbolic calculations are possible, an axiomatic formulation could be of use and maybe more economical, or otherwise we might prefer to do calculations rather than keep track of the axiomatic formulation. One of the two approaches is probably better for the computer, thus the need to be able to switch between them.

Hilbert's Foundations of Geometry did more or less precisely what you are asking for. Starting from an extension of Euclid's Axioms, Hilbert proves that any model of the axioms is isomorphic to $\mathbb{R}^2$ with the usual definition of line.

Later, Tarski gave a first-order axiomatization of plane geometry. Because of built in restrictions in the first-order approach, one cannot get isomorphism with the natural geometry of $\mathbb{R}^2$. But one can get isomorphism to the natural geometry of $F^2$, for some real-closed field $F$.

I'll be repeating some stuff that has already been said, but I have my own spin on it.

Can we deduce, from some axiomatic geometries, an algebraic structure?

Yes. As you can find in Hartshorne's book, or in Hilbert's Foundations, the idea is an "algebra of segments" for any ordered Desarguesian plane in which you construct an Archimedian, ordered division ring $D$ with certain geometric operations such that the points and lines in $D\times D$ are coordinatized in exactly the way we're taught for $\Bbb R \times \Bbb R$. So, they are a natural starting point for ordered geometry.

Now, every Archimedian, ordered division ring embeds in $\Bbb R$. Since $\Bbb R$ is also an Archimedian, ordered field, you can see it is the maximal such field for such a geometry. In fact, needing the reals to coordinatize a plane is equivalent to very strong "completeness" of lines in that geometry. This completeness/maximality property makes it very attractive to study geometry with it.

Ordered geometry is great, but reexamination of the ideas shows that division rings in general are exactly what you need to coordinatize Desaguesian planes. They imbue their lines with exactly the translation and scaling behavior one would expect in a geometry according to the Erlangen program.

Are some axiomatic geometries equivalent, in some way, to their more algebraic counterparts?

Let me continue briefly along the lines above. The amazing theorem that a Desaguesian plane is Pappian iff the division ring is commutative has already been mentioned. There are also theorems about equivalence of certain types of fields and constructability criterion in the geometry.

Desarguesian projective planes enjoy the same coorditinization theorem with division rings. Hyperbolic planes require one more unique axiom before they can be coordinatized by a division ring. The division ring you get is an analogue of the "algebra of segments" called the "field of ends". So as far as these theorems go, you have a really strong connection between synthetic geometry and analytic geometry. Geometry with vector spaces over division ring captures a large part, but not the whole of synthetic geometries.

In fact, there are even more general coordinatization theorems for projective planes using 'ternary rings'. As you add more special properties to the projective plane, the ternary ring gets closer to being a division ring.

More posts you may like:

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which axiom(s) are behind the Pythagorean Theorem