I learned as well that the uncertainty principle comes from the math, but a different one, as it suits Heisenbergs (and Plancks) approach to quantum physics more: linear algebra. It is that observables are obtained by applying state operators on the state vectors of a quantum system, and if the order of applying operators do not commute those operators have a relationship expressed by Heisenbergs principle (replace x and p by any other pair of not-commuting operators). Location and momentum are just a special case of this. The spatial components of angular momentum do not commute either, regardless of which pair you are choosing, thus leaving you only with one certain spatial component and the scalar value of that momentum.

However, one way or the other, don't be fooled by "it is the math, stupid". It is the nature as well, as has been verified countless times.