I want to understand if there is truly a rigorous definition for the degrees of freedom in a system. Say all of a system's physical states are contained in some set $S$. A seemingly acceptable (and I think mostly referred to definition) is the number of real numbers needed to describe it or, more rigorously,
A system $S$ has $n$ degrees of freedom if there exists a bijective function $f:\mathbb{R}^n\rightarrow S$.
This however doesn't seem to uniquely define the degrees as there are bijective functions between $\mathbb{R}^n$ and $\mathbb{R}^m$ for any $n$ and $m$.
This may seem like pushing too hard into rigor but there are systems where the degrees of freedom have direct physical consequences like in statistical physics especially so I suspect there is a true and better definition out there.
My thoughts: My guess is it has something to do with topology (I am severely undertaught in this field) of the system's physical states as that is the only thing I can think of that distinguishes these different spaces (again, I have no idea). Also, this case is seemingly trivial for cases like a particle moving in 3 dimensions which of course should have 3 DOF but physical dimensions have so many more features like for e.g. scaling that can uniquely define dimension which I don't see as the case for other general systems.
