It is a known fact that $\mathbb{R}^3$ does not model classical physical space accurately since it includes some non-invariant structure under translations, such as the origin $(0,0,0)$. However, I am unsure as to how one would show that.
My trouble is in understanding what exactly about $(0,0,0)$ is not invariant.
I fear this might belong to a more physics based exchange, but the kind of answer I am looking for is more mathematical than physical. I suppose this is trivial, since there is no one doing it explicitly, but I would appreciate it if someone could write a pseudo-formal proof of this.
The structure of $\mathbb{R}^3$ as a vector space is not preserved by (nontrivial) translations, since the zero vector is mapped to a nonzero vector.
But the structure of $\mathbb{R}^3$ as an affine space is preserved by translations; the origin is no different than any other point.