Are we sure that the mathematics is enough? well, to the extent that, if you make a set of postulates based on some observations which could not be explained by classical physics, this is how quantum mechanics was initially put in, and then you ask what are it's predictions, what are the results of it's calculations and then you observe once again, do those experiments and test it out and so on, you are testing this theory and this formalism and the mathematics associated with it. 