But is this actually 'fooling yourself'. It looks more like an approximation. What is space happens to be discrete but the discrete points are very small compared to every day life? Then for most purposes approximating a space coordinate with real numbers is a pretty good approximation. Never mind that the possible state space of the universe could actually be smaller than such a single real number. The thing is, a human brain can presumably only grasp finitely many things but actually having to reason about finitely many things is difficult. Note that theorems about numbers are easier when they are about 'natural numbers' than about 'all numbers smaller than 2^32' because all theorems then will have preconditions that limit their size in order to exclude integer overflow and hence get somewhat hairy. So as a cognitive trick working with infinite sets works pretty well. However, complicated reasonings about the nature of these infinities turn into navel gazing pretty quickly. The thing to strive for, I think, is extending the finite and the discrete with the infinite and the continuous but but not having the illusion that we can actually say very much about the nature of the infinte.
