>I think the notion of continuity (and calculus) could not even exist without the idea of infinitely precise numbers.
Oh, I'm certain they have the "idea" of it - but what does that idea lead to in reality? Actions (like proving theorems, or calculating digits of pi) that have a distinctly computational aspect to them. If you have time look up the Curry Howard correspondence (just for example, its not the whole kit-n-kaboodle as these musings go...)
Having an idea of God, does not prove that God exists, though some may disagree.