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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 ?

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    $\begingroup$ Shouldn't you be asking a physicist? The reason for the goodness of the approximation is, ultimately, not mathematical. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 30, 2012 at 7:09
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    $\begingroup$ I believe that this is a bit off-topic, and actually not a real question. $\endgroup$
    – DamienC
    Commented Aug 30, 2012 at 7:13
  • $\begingroup$ @Damien: I also thought people may find it off-topic, what do you mean by a real question ? $\endgroup$
    – Pritam Majumder
    Commented Aug 30, 2012 at 7:21
  • $\begingroup$ Maybe I would add the word "locally"... and from a mathematical point of view, I would say that $\mathbb{R}$ is not only an algebraic construction (that would maybe be true for algebraic numbers), but you are strongly considering it's topology. $\endgroup$
    – user47274
    Commented Aug 30, 2012 at 7:25
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    $\begingroup$ Maybe this has something to do with the fact that $\mathbb{R}$ is the unique complete ordered field up to isomorphism. $\endgroup$
    – Qfwfq
    Commented Aug 30, 2012 at 12:09

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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 by Lee Smolin; a alternative approach is promoted 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?

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    $\begingroup$ I do not see why this interesting question is closed. Poincare wrote a lot about various aspects of this. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 30, 2012 at 23:37

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