There's also a broader picture -- that this effect arises because we use discrete, independent words to describe inter-dependent phenomena. The concepts of momentum and position are not good 'factorizations' of reality, so trying to talk about it in those terms leads to structural problems when you increase precision.
It's the same kind of problem you get when you try to talk about objective/subject morality, or whether science can prove things.
Another example I give:
The Sun revolves around the Earth -- True from Earth's perspective.
The Earth revolves around the Sun -- True from Sun's perspective.
They both revolve around their center of mass -- True in Newtonian model.
The both follow straight geodesics in spacetime -- True in Einsteinian model.
Everything is true in some approximation. The idea is to increase your precision so that you can find truths that incorporate more information and thus provide greater insight and generality. If your language doesn't have enough 'bandwidth' to carry the information of -- that is, mimic the structure of -- your experiences, you have to develop more structured abstractions, or you'll lose clarity and expressive power.
So, for instance, you wouldn't have to prove that unicorns don't exist: you just have to show that providing more specific information doesn't result in a lack of clarity in a general model. A theory of three-legged unicorns offers no advantages over a model of n-legged unicorns because unicorns don't exist.
But now I've gone off the deep end...