Why is physical space equivalent to $\mathbb{R}^3$?

Question

Why is physical space equivalent to $\mathbb{R}^3$, as opposed to e.g. $\mathbb{Q}^3$?

I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$ .

The set of reals $\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. For reference see here http://en.wikipedia.org/wiki/Construction_of_the_real_numbers. Now my question is what is the reason behind our approximation of the physical space by this abstract set. Why is this approximation assumed to be most suitable or good approximation?

Answer 1

The reals are simply chosen so that you don't have to worry about the existence of coordinates of points if you do geometry. If you use Q instead you get into a lot of trouble. Remember that geometry also provided one the first reasons to think about irrational numbers.

If you wonder whether "nature" uses reals to compute its evolution or if it's something else then this question is close to meaningless. Our models will very likely come to a point where we won't think of space as a manifold at all, so specifically not as a real manifold. Or in other words, there is no fundamental physical meaning to real numbers. They're just a convenience for us.