I was recently listening to a lecture by Richard Feynman where he talks about how there are different formulations of theories in physics. The different formulations start from different axioms but they are all mathematically equivalent.
He talks about how these mathematically equivalent formulations are mentally far from equivalent. When we are thinking of extending a theory one set of axioms may be easier to think about than others.
So I was wondering if this is true for mathematical systems such as Euclidean geometry.
An example are Birkhoff's axioms. These are axioms for Euclidean geometry which are build upon the real numbers.
There are also Tarki's axioms. They don't cover the whole Euclidean geometry, but they do cover a substantial part of it.