I'm trying to understand the meaning of using a mathematical structure in order to do physics, what does this really mean? My idea is that first we performs experiments on a physical system in order to find its properties, second we choose the mathematiacl structure that contain all these proprieties, third we choose the set of possible representations. Then sometimes using the representation we find some new proprieties and if we see that they don't depend on the representation we assume they are real
For example: Lets consider the displacements, first we state that two displacement can be summed with the parallelogram rule and we see that every displacement can be obtained as sum of 3 other displacement, second referring on the real world proprieties we say that the set of all the displacements is a 3D vector space, third we say that every displacement can be represented with 3 numbers (obtained by the measurements on 3 orthonormal displacements) and that the sum is represented by the sum of the components. Then we see that in all the possible representations when we compute x^2+y^2+z^2 we obtain the same number, so we state that there must exist some physical entity associated with a displacement that can be represented with a number and indeed is the distance.
Can you help me in clarifying this process? actually I'm really confused about what is a representation, what is a measurement, how do we find physical proprieties in the contest of the example I did