Beamer?

This is mathematics goes far, far beyond just counting. I'll give you that originally math was invented to serve a real life purpose. The more important and surprising mathematical implications over physics have come from abstract mathematical work that was done without trying to explain anything around them. Citation: http://en.wikipedia.org/wiki/Group_theory#History Group theory has some extremely far reaching implications over physics in the fields of quantum mechanics, general relativity, particle physics, etc.

How does it have anything to do with Heisenberg's uncertainty principle? That relation is a direct result of the properties of Fourier transforms. The randomness in quantum mechanics being described here is the inability to know to which of the eigenstates the wave function will collapse to after being measured beyond the probability of obtaining each state.

Mathematics is not, in general, refined to describe reality. Mathematics is refined by taking every logical rule to its farthest reaching implication. This goes far, far beyond anything that we currently see as based in our reality (though, as the current argument is about, it has the uncanny tendency to end up describing our physical reality extremely often). Physics however IS refined to describe our reality. It is precisely physics that ties the mathematical underpinnings to the reality that we observe.