“The assumption that (in a triangle) the sum of the three angles is less than 180° leads to a curious geometry, quite different from ours, but thoroughly consistent, which I have developed to my entire satisfaction. “
– Gauss the troublemaker.https://www.cut-the-knot.org/triangle/pythpar/Drama.shtml
If I do, will you then describe it in 2 dimensional terms?
You will describe them as arcs, for we are using a 2 dimensional representation system (our screens)
I cannot hand you a ball that you can then hold and look at and watch the shapes change from lines to arcs and back again.
If I draw a triangle on a flat piece of paper and I then bend the paper such that the triangle no longer has 3 angles adding to 180 degrees, what then?
Can you prove the deformation without the paper?
I’ve decided that you are a Euclidian and that is ok. I can’t prove it to you in this 2D space we have available, and while I could find a video showing how a triangle stays a triangle even when on a curving and flattening piece of paper, I also know you can validly describe that triangle via different means while it is shifting form.
It changes identity as the paper shifts. That’s actually a fascinating outlook and I respect that.
a fascinating morphism in category theory is https://en.wikipedia.org/wiki/Zero_morphism as
“If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category. “
” But you have no reasons to assume that from Euclidean geometry, and yet you do.
You and I don’t have a common ontology. However your ontology is a special case of mine. So I can understand you, but you cannot understand me.
There’s actually a good reason and it’s one I envy:
Once you have adopted an ontology as truth or fact or self-evident, you can get to work.
Ontology is a framework/foundation you can build upon, and Euclid is powerful, productive and in use for how many centuries now?
Non-Euclidean is more my kind of fun but I _know_ that Chris is far more productive for holding this as truth.
Meanwhile, I’m still building my ontology.