How the concept of entropy arises from injectivity problem

The injectvity problem I call the problem that the reals are bigger than the integers. Or the uncountable sets and countable sets:

If you don’t have the initial uncountable position of a particle, you can only know it countably by making a measurement. So if you make a measurement you can only point at a box in space where you can say it is. You can not point at a position in space and say it is there. As a result, you ‘bin’ the particle and QM-like phenomena and the phenomenon of entropy arise. If you do know the position uncountably then the entropy phenomenon disappears and you get the concept of Laplaces demon and determinism comes back.

This is also related why it is so hard to quantize gravity, because it is curvature of space(time).