Why is there empirical evidence in physics more than an analytic evidence? In case of mathematics, you can make your own rules and play with them. For example, Euclied made his own rule of Eucledian geometry, and Reiman (or other Non-Eucledian geometers) made their own rule of Non-Eucledian geometry. Almost everything in the system can be derived from the fundamental rule one make, and there (usually) doesn't exists any higher order rule to explain the same system.
In physics (I am extremely amateur learner), I finding the case to be different. In some places, there are laws, in some places - there are theories. 
My main question is that can every theory of physics be precursored and succeeded by another theory/law ? What is the main goal of physics ? Why should we do physics ? 
Take the snells law. It can be derived from Fermat's law, which can be derived from Hyugen's law. Does this type of reasoning follows ad infinitum to converge or something else ? If something else, then what is it ?
 A: Axiomatic theories started with geometry, back at the time of Pythagoras. At that time mathematicians and philosophers were one and the same thing. There is a rumour that even Homer was a mathematician. Education was a one throw business at that time.
How did geometry start? It started in the flat spaces of Egypt and Mesopotamia where it was necessary to decide what field was whose and how large it was , and not have fights over land property. In Greece for example, a mountainous region, there are still contracts stating " the field from the gnarled olive tree to the running brook, to the rock , to the pine tree" to define an approximate rectangle. This does not work in immense flat spaces. The axioms of plane geometry,, Euclidean geometry, were  necessary to start constructing the whole theory, theorems and all. Notice it is an observation of nature, it fits existing field lines, and is predictive. If it were not predictive it would just be a map.
Notice that changing one of the axioms, changes the predictions of the mathematics, it will describe a different earth. And indeed when people were convinced that the earth was round, the parallel axiom was left out, and one got spherical geometry to describe the globe. The axioms of geometry were connected to the observational fact that parallel lines did not meet.  
For centuries, there was no separation of physics and mathematics until the time of Newton. Since then one separates the axiomatic mathematics, from observational laws and postulates. Mathematics is axiomatic; a closed system can be built up where the whole is consistent. Mathematics as a tool for describing nature has to have extra assumptions, to pick up the specific subset of the mathematical theory, that is appropriate and predictive for the problem at hand. In the case of plane geometry it is the parallel lines axiom that is really an observational input, in order to make it a physical theory applicable to measuring fields.
In classical mechanics it is Newton's laws that pick up the set of mathematical solutions that are useful descriptive and predictive for mechanical systems. The mathematical model of mechanics is a subset of a larger mathematical system: a subset decided so that the calculations fit observations.
Thus laws are not derivable mathematically because the only reason they exist is in order to pick up the subset of mathematical solutions which fit the physical problem at hand and predict further behaviours.
