So euclidian distance is defined (in space $R^m$) for points $A(a_1, 1_2,...,a_m)$ and $B(b_1, b_2,...,b_m)$ as $d(A,B)=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+...+(a_m-b_m)^2}$.
It also can be thought of as the norm of the euclidian space.
Now:
- If it is axiomatic, then how can it be if Pythagorean theorem, which states basically the same thing, has multiple proofs? But then, if the definition of the distance is fundamental for euclidian geometry, aren't those proofs circular?
- If it is provable, how do you do that? (without making making circular argument)
*(Or maybe it is axiomatic as a concept but this certain formula isn't?, whatever would mean that...)
I understand that my question may come from the fact that I don't get yet how are the fundamentals of mathematics build. If so, please explain anyway what are the gaps in my reasoning/knowledge.