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The generic triangle lives in the collection of languages which axiomatizes that you can do with a triangle.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 addition is a functor from the category of for each (semi)rings semi)ring $R$ to subsets R$, addition is specified as some subset of $R^{R\times R}$.

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 "$n$". Come on, it 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. One might say that the functor described above gives a realization of the addition of (semi)rings.

But

In a critic can say similar way, for each constant curvature plane, axioms that this does not give what generic addition govern reflection, axioms that provide congruence criteria, Pappus' proof is generated from these axioms. He

As a final refinement, one might say that addition also consider triangles in vector other non-constant curvature spacesdifferent from . Just as addition in rings (and in ordered can be considered for vector spaces or groupsetc.) This forces me to refine my , not just rings. In this case, the full answer : Addition should is that for any language in which triangles can be a functor from languages ${\mathcal{L}}$ containing the addition $+$ symboltalked about, to "reasonable" axiomatizations there is a corresponding axiomatization of additionwhat can be done with triangles.For instance

Eg, the functor might send ${\mathcal{L}}={0,+}$ to Pappus proof is for the axioms first-order axiomatization of abelian groups, ${\mathcal{L}}={0,1,+,\times}$ to (non)-Euclidean geometry. There are different proofs for the axioms of rings plane as a Riemannian manifold (via integration) etc.

So finallyFor each of these axiomatizations, what is the generic we have a different notion of triangle? (Not finished).
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The generic triangle lives in the collection of languages which axiomatizes that you can do with a triangle.

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 addition is a functor from the category of (semi)rings $R$ to subsets of $R^{R\times R}$.

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 "$n$". Come on, 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. One might say that the functor described above gives a realization of the addition of (semi)rings.

But a critic can say that this does not give what generic addition is. He might say that addition in vector spaces different from addition in rings (and in ordered groups etc.) This forces me to refine my answer: Addition should be a functor from languages ${\mathcal{L}}$ containing the addition $+$ symbol, to "reasonable" axiomatizations of addition. For instance the functor might send ${\mathcal{L}}={0,+}$ to the axioms of abelian groups, ${\mathcal{L}}={0,1,+,\times}$ to the axioms of rings etc.

So finally, what is the generic triangle? (Not finished)