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 $\mathbb{R}$. 


$\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the set of rationals. 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 ? 



Why is the physical space equivalent to $\mathbb{R}^3$
this is basically a question on the granularity of space, which is an active topic of research in physics: space appears to be continuous, but does it actually come in discrete chunks on some very small length scale (Planck length)? there are some attempts to formulate (quantum) mechanics in discrete space-time; loop-quantum-gravity is one approach, described [here](http://www.phys.lsu.edu/faculty/pullin/sciam.pdf) by Lee Smolin; a alternative approach is [promoted](http://arxiv.org/pdf/1204.4926.pdf) by Gerard 't Hooft:


*In modern science, real numbers play such a fundamental role that it is difficult to imag- ine a world without real numbers. Nevertheless, one may suspect that real numbers are nothing but a human invention. By chance, humanity discovered over 2000 years ago that our world can be understood very accurately if we phraze its laws and its symmetries by manipulating real numbers, not only using addition and multiplication, but also subtraction and division, and later of course also the extremely rich mathematical machinery beyond that, manipulations that do not work so well for integers alone, or even more limited quantities such as Boolean variables.
Now imagine that, in contrast to these appearances, the real world, at its most fundamental level, were not based on real numbers at all. We here consider systems where only the integers describe what happens at a deeper level. Can one understand why our world appears to be based on real numbers?*


