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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

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    It is not so clean and clear cut, and there are multuiple controversies as to how exactly mathematical models relate to reality between realists and anti-realists, foundationalists and holists, etc. One does not first perform experiments, to make sense of experiments one needs some prior scheme already in place, then it is adjusted based on the outcomes, etc. It is more of an iterative chicken/egg dynamic. For general information see e.g. SEP Scientific Method, as is the question is too broad for us, I am afraid. – Conifold Jun 7 at 2:35
  • You may like to read Unreasonable effectiveness of math – Rusi Jun 7 at 6:04
  • Conifold's comment is accurate. Your question is quite enormous in scope. Some pragmatist philosophers of physics (Richard Healey), for example, think that there are no true statements, representations, or models in quantum mechanics. Other philosophers of QM and physics (Shan Gao) think that Schrodinger's wave function is literally real somehow, as part of the ontology of quantum mechanics. Catholic philosopher Bas Van Fraassen founded constructive empiricism, according to which scientific theories and models are not true, but only empirically adequate. It's a variety of instrumentalism. – Bruce Long Jun 7 at 9:37
  • One thing that most philosophers of science agree upon is that representations are necessarily only partial: they cannot in principle capture all of the information in the phenomenon they are used to represent. There are many ways of characterising representation and how it works, and lots of disagreement about it. James Ladyman and Don Ross think that scientific theories and representations are based upon information acquired through channels from natural phenomenon and experimentation ('Everything Must Go: Naturalising Metaphysics'). Ian Hacking emphasises experiment over theory. Big debate. – Bruce Long Jun 7 at 9:43
  • As for structure. There are multiple kinds of structural realism, and one thing that makes your question difficult to answer is that the two main competing approaches - epistemic structural realism and ontic structural realism - disagree about what structure is at a fundamental level. Moreover, there is mathematical structuralism, which can overlap with structural realism, but is also importantly different. According to ontic structural realists, structure has been variously defined in mathematical and physical terms: as relations in equations, as Ramsey sentences, and as physical statistics. – Bruce Long Jun 7 at 9:47

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