Axioms vs Models in projective and hyperbolic geometry

Question

I'm studying Projective Geometry.

The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. Eventually we get to the axiomatic definition of Projective Spaces.

But in the next chapter the author begins by (re)defining Projective Spaces in terms of Vector Spaces. I'm not comfortable with that, since we've defined projective spaces axiomatically this doesn't seem like a definition to me, but more like a construction.

$\textbf{Question:}$ Do we lose anything when we decide to use only one model for our axioms?

For example, the only example of Euclidean Space I see people talking about is $\mathbb{R}^n$, but we study a lot of different models for Hyperbolic Geometry. In the book I'm studying the only example of Projective Spaces are Vector Spaces. We know that all Real Vector Spaces with the same (finite) dimension are isomorphic, but we don't fix $\mathbb{R}^n$ as the unique example of vector spaces.

Why it is (is it?) a good idea to "fix" models for our axioms?

Answer 1

There is another common model where the points of the projective plane consist of the points of the Euclidean plane and equivalence classes of lines for the equivalence relation "is parallel to".

The mental image is that the latter types of points are to be thought of as the "point at infinity" that the class of parallel lines intersects at.

(and the lines in this model are the lines of the Euclidean plane, together with one additional line, the "line at infinity" that passes through all of the points at infinity)

The vector space version of projective space is popular for the same reason that the vector space version of Euclidean space is popular: they give a fairly simple way to translate between algebra and geometry. Thus, it makes it easy to use algebra to study geometric problems, and conversely makes it easy to use geometry to study algebraic problems.

(disclaimer: I am not an expert on hyperbolic geometry and its history) We use a lot of different models for hyperbolic geometry partly due to history; it was a rather mysterious thing when it was first discovered and people were floundering around for good ways to understand it. We also use different models for practicality: to my knowledge, unlike with Euclidean and projective geometry, there isn't really a system of coordinates that fits hyperbolic geometry very nicely, which means you have much more reason to want to switch between models to whichever is more convenient for whatever you're doing. And the models are complicated enough that it is uncommon that a "naturally" occurring problem has a simple translation to hyperbolic geometry.

Answer 2

The approach you seem to favor is called the synthetic approach and it was very popular in the 19th century. It is all a question of being practical; to present the material in an efficient manner it is often useful to work in a specific model rather than start from axioms of projective geometry. Also proving things from axioms may be trickier than proving things in a model, even though from a purist viewpoint it may be preferable. But if you wish to tackle questions such as "which axioms are independent of which" for example Pappus and Desargues properties, you obviously have to follow the synthetic route. This sheds more light on the foundations of projective geometry but may prove to be tedious in the classroom. When I teach projective geometry I combine both approaches.