maxwell equations - Justification of Physical Laws

I'm a maths student, and I've studied quite a lot of mathematical physics. All my courses have a similar style - we state the laws of the system, and then deduce the physical consequences as theorems. It has become more and more apparent to me (especially studying electromagnetism and QM) that I have no idea why you would expect these equations to be true.

What I'm looking for is either a kind of physical justification "this is how object X behaves, here are mathematical statements that say this precisely" (similar to momentum-based arguments in fluid dynamics), or some kind of experimental justification (the more direct the better), for physical laws. I'm particularly interested in this with respect to Maxwell's equations, but I'm interested in any others as well.

Edit for clarification:

Perhaps I worded myself poorly. The axioms of a physical theory (which is what I meant by 'laws'), like any set of axioms, are invented by human beings with the aim of capturing certain intuitive properties of an object. For example, the axioms defining a vector space in mathematics are not completely arbitrary - they aim to capture the properties of various objects ($\mathbb{R}^{n}$, or function spaces for example) that we have an intuitive idea of.

I am asking the equivalent question with respect to physical theories, and especially in terms of Maxwell's equations, e.g.: what are the intuitive properties of charges and currents that lead one to write down Maxwell's equations? Note that the answer 'they match up with experiment' doesn't fit, because many theories could match the same observations and be mutually contradictory (e.g. Newton vs relativity at small velocities). There also is no condition on the correctness of the interpretation - any explanation of, say, Newtonian mechanics, would necessarily not be an absolutely correct picture of the world (or it would be the correct physical theory, too!).

Obviously such a justification would have to be empirical to have any grounding in reality. For instance if we want to know if Newton's second law is true, we can set up an experiment to check it. Less experimentally, most people take the existence of absolute time, or inertial frames of reference, to be intuitively obvious. Similarly, properties like acceleration and momentum have physical interpretations, and a good explanation could be given in terms of such quantities.