I was just wondering about the formal foundations of analytic geometry, I mean axiomatically. I've noticed along my course of linear algebra that the axioms of vectorial space already include the fact of getting the distance between two points by just giving the definition of norm. So my question was how can one be sure that such a definition makes sense. And the answer is obvious seen from the point of view of analytic geometry and the pythagorean theorem. But how can it be formally explained right on the field of analytic geometry. In my perception it comes from its foundations. The think is that I don't know what they are. I think that there is no need to have any axioms if we have just defined the foundations of plane geometry and found a correspondence between the real numbers and the line. So maybe the thing that is not clear is the notion of distance between to points, is it given by definition? how can it be explained?. Or maybe is not that simple, so any comments about this are welcome. Also if you have any good book about the subject it will be very helpful.
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Something close to what you ask about was done by Hilbert, in his Foundations of Geometry(1899). He gave a purely geometric set of basic axioms, and showed that, up to isomorphism, the only model is the usual Cartesian plane.
A flaw, inevitable at the time, is that the formulation was second-order. Tarski gave a first-order formulation. Categoricity is lost, but there are useful things gained, such as the existence of an algorithm for determining the truth of sentences of the theory. Some information can be obtained from this Wikipedia article, and much more can be found by searching.