What are "axioms"?
They are "properties that characterize something";
  abstractions from a concrete example.

EG, polynomials and integers are kinda similar: you can add them,
multiply, subtract -- you can't divide (then you get rational
numbers/rational functions! which is a good analogy), but you can do
division with remainder (euclidean algorithm).
...so we abstract the notion to "ring".

EG, norms are abstractions of "absolute value"

It's often unclear -what- axioms you should take: what are -useful-
properties that "capture an essential aspect"
(not: capture -the- essence: there's no one particular essence)

An EG from physics:
a "tensor" is covarient/contravarient depending on how it
"transforms", i.e., how it -behaves-: what -axioms- it satisfies
under transformation.

