Given axioms, how do we know it defines a geometry?

Question

It is known that besides using coordinates and algebra, there are axiomisation of geometry such as Tarski, Hilbert and Euclid.

1. However looking at the axioms of Tarski for example:

Betweeness $B(\cdot,\cdot,\cdot)$ satisfy e.g.:

\begin{align} Bxyz &\to x=y\\ (Bxuz \land Byvz) &\to \exists a(Buay \land Bvax) \end{align}

Congruence $\equiv$ satisfy e.g.: \begin{align} xy\equiv zz &\to x=y\\ & \equiv \text{is reflexive and transitive} \end{align}

2. Now look at the axioms of a group: \begin{align} &\text{There exist $e \in G$ such that for all $x \in G$: }xe=ex=x\\ &\text{There exist $x^{-1} \in G$ such that for all $x\in G$: }xx^{-1}=x^{-1}x=e\\ &\text{For all $x,y\in G$: $xy \in G$}\\ &\text{For all $x,y\in G$: $(xy)z=x(yz)$} \end{align}

If these are the two sets of axiomatic systems given and no other context is given, then in some way, they prescribe how primitive mathematical objects should behave under some logical rules. The reason we knew that $(1)$ is related to geometry is because we build $(1)$ motivated by the need to describe about triangles and planes.

Is there a way to tell whether an axiomatic system defines a geometry just by looking at it and start deriving and exploring the theorems it resulted from without context. Is there some kind of axioms or schema that all notions of geometry must obey?

Answer 1

It depends on your definition of a geometry. And usually, such a definition would be "A geometry is something that satisfies the following axioms." Of course, when we talk about non-Euclidean geometries, we know what we mean, namely, things that satisfy all axioms for a Euclidean geometry except for the parallel axiom. But would something satisfying all axioms except some other axiom still be a geometry? It depends on what you mean with "geometry". Probably not, if you want your definition to be interesting to the mathematical community.

But more to the point, you might be interested in the fact that when we prove things based on the Hilbert axioms except the parallel axiom we are proving things about absolute geometries, i.e., things that are true in both Euclidean and non-Euclidean geometries. And it is remarkable that you lose very few theorems from Euclidean geometry. In this sense, I guess that absolute geometry is the notion that you are looking for.

EDIT: It is relevant whether you are only interested in planar geometry or not. Things might be more complicated in higher dimensions, but I think that there are Hilbert's axioms for 3D absolute geometry as well.

EDIT: I am assuming here that you are interested in "classical" geometry, and not in more complicated things like differential geometry, even though parts of what I have said are still relevant (think of hyperbolic manifolds)

Answer 2

No. The term "geometry" is informal and it is entirely subjective what qualifies as a "geometry" or not. (I'm sure someone somewhere has given some precise definition of what a "geometry" is, but such definitions are usually not intended as a definitive answer to your question (rather they are technical definitions being used in a certain restricted context) and certainly are not broadly accepted as such.)