A Problem on a Formal Science

In the similar manner that a theory of physics can be verified by the correspondence to a physical phenomenon, can mathematics, more generally formal sciences be verified?
In order for the verification, a specific group of phenomena may be presumed for each formal science. For example, the Peano arithmetic as a theory of mathematics can be verified by the correspondence to a sort of phenomena that we intuitively have on the operations of natural numbers.
Also some of naive questions on a formal science can be simulated through a computer so that we can observe and analyse phenomena tangibly. In this sense, a computer simulation can be seen as a generator of phenomena.
The central limit theorem may be a good example for this: consider a distribution of sample means of independent random variables with finite variances as a naive question, a simulation of it is straightfoward and the distribution seemingly turns out to be the normal distribution; then the proof of the CLT may give us a theory which corresponds to the phenomenon actually generated by the simulation.
A problem related to this argument we can easily imagine is that some theories of a formal science precede corresponding phenomena or be utterly immune from phenomena. Once we build a theory or a formal model X for a specific phenomenon, then we can crank out theories by additions or omissions of parameters of X, which potentially might not correspond to any phenomena. Given that the pursuit of such theories is a commonplace in formal sciences, how can the pursuit be justified? 
One way of the justification can be an appeal to the potential usefulness of such a theory, which may be predicted by additions or omissions of specific parameters.