dg.differential geometry - Where does the generic triangle live?

The generic triangle lives in a subset of this function space $Axioms^{Languages}$.

What do I mean? First, let's us what is generic addition? Immediately, you would ask me, addition on which number system? The natural numbers, integers or what? Properly then, one can only say that for each (semi)ring $R$, addition is specified as some subset of $R^{Rtimes R}$.

Now I invoke this duality principle: Operations in number theory are analogous to objects in geometry. Objects in number theory are analogous to operations in geometry.

One of the reasons why this question by Six Winged Seraph is so difficult (and so interesting) is that we should think of the object "triangle" in geometry as analogous to the operation "addition" in number theory. We should not think of the "triangle" as analogous to the number "$n$". It is easy to say where the generic number "$n$" lives in: namely the set of natural numbers ${mathbb{N}}$! (or whatever number system you might be considering) This is an easy question just as where does the generic rotations, dilations and reflections live in: namely the Lie group $Isom({mathbb{R}}^2)$? (or ${mathbb{H}}^2$ or ${mathbb{S}}^2$ depending on which constant curvature space you are considering.)

The "triangle" question is hard because what you can pin down about a generic triangle is not what it is, but what you can do with it. That is, axioms. Using the duality principle again, we pin down what addition is by the axioms, not just axioms for addition, but also how multiplication interacts with it. Are you interested in statements like $23+34=57$, $n+n=2n$ etc? Only to the extent that these statements form part of the completed infinity of theorems from which can be generated by the axioms.

In a similar way, for each constant curvature plane, axioms that govern reflection, axioms that provide congruence criteria, Pappus' proof is generated from these axioms.

As a final refinement, one might also consider triangles in other non-constant curvature spaces. Just as addition can be considered for vector spaces or groups, not just rings. In this case, the full answer is that for any language in which triangles can be talked about, there is a corresponding axiomatization of what can be done with triangles.

Eg, the Pappus proof is for the first-order axiomatization of (non)-Euclidean geometry. There are different proofs for the plane as a Riemannian manifold (via integration) etc. For each of these axiomatizations, we have a different notion of triangle.