[solomatov]

It's possible to argue that there's no such a thing as infinite precision floats. I.e. all objects have limited "size", and real numbers is just a nice abstraction to simplify discrete world. (https://en.wikipedia.org/wiki/Finitism)

Turing machine is not such thing. At each moment in time, only finitely many cells of the tape are used. (The same applies to natural numbers, there are infinitely many of them, but each one of them has a finite description).