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

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}

1. 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?

• How do you define a geometry? – Asaf Karagila Mar 18 at 19:16
• That's pretty much what I am trying to illustrate here. Do we actually have a set definition on what is a geometry,similar to how we have a set definition on what is a group? It seems when we call something geometry in mathematics, it is largely motivated by intuition, thus I am curious if there is a minimal definition in order for a mathematical object to become a geometry? – Secret Mar 18 at 19:20
• There is a somewhat intersting case, where I think axioms came before the geometry. There is something called a median algebra, which was defined in the 40 or 50's as generalizing some operation in Boolean algebra, but relatively recently it has been shown that there is a lot of interesting geometry involved. For example connection to CAT(0) cube complexes – Paul Plummer Mar 18 at 19:34
• Take a look at this very recent question math.stackexchange.com/q/3153006 asked some dealing with an axiomatic approach to "betweenness" (even if it is not said). My answer is that the system IMHO hasn't the right axioms by exhibiting a model of the system of axioms where the looked for property isn't valid. – Jean Marie Mar 18 at 21:17