A Quoran asks: “Is the Uncertainty Principle in QM primarily a consequence of the mathematics involved (ie. derived from it) or of empirical evidence (which the mathematics has then been constructed to describe)?”
I answered “Both”.
The original idea of the uncertainty principle was motivated by the apparent empirical facts that in order to determine the position of an object very precisely it seems necessary to illuminate it with light of a very short wavelength, and that the empirically observed spectrum of black body radiation suggests that short wavelengths interact with matter not continuously but in discrete ‘quanta’ which transfer more and more momentum as the wavelengths get shorter. In the mathematical theories that were developed to describe these phenomena (but not so much motivated by the uncertainty principle itself) the uncertainty principle (and in fact a more general form of it) can be derived from the fact that measurements of position and momentum correspond to non-commuting operators on the Hilbert space that is used to represent states of the system.