We do not have any experimental reason to believe that "irrational probabilities" have any physical meaning. There is no method to measure them even indirectly, so a theory that operates only with natural numbers is quite plausible.
That's an interesting point. I did wonder, after posting the above comment, whether frequentists, at least, are ipso facto committed to all probabilities being rational, on account of their insistence that they are only meaningful in the context of repeated trials? (FWIW, I lean towards the frequentist viewpoint, insofar as I have a coherent position on the issue.)
But then, what are we to make of physical formulae in which complex, transcendental and other non-rational numbers appear? Are they 'just' tractable approximate models of physical reality? I guess the Maxwell-Boltzmann distribution, at least, could be fairly characterized as such.
As far as i can understand all our measurements would be the same if all equations using real numbers were replaced by equations with finite precision (containing several hundred digits). The only experiment that could be different is quantum computer, because quantum fourier transform on n qubits requires at least n binary digits of precision on probabilities too. This is why some people think quantum computers won't work.
