Please try to be lenient with me since I have a hard time formalizing this question. I'm not even a physicist.
Many people are probably familiar with Newton's second law:
$F = ma$
Like other physical laws it expresses a relationship between physical quantities. To make this mathematically precise, I suppose one could say, that we have functions $F,m,a : \mathbb{R} \to \mathbb{R}$ (that are "mapping time" to force, mass and acceleration), s.t. $F = m\cdot a$ in the obvious ring of functions $\mathbb{R}\to \mathbb{R}$. On the other hand this seems fairly arbitrary. One could argue that these quantities have nothing to do with time at all, maybe they have to with something else entirely.
Let's say we work in one-dimension and somebody tells us at some point, that $a$ actually depends entirely on how far our object has travelled, so $a$ "depends" on the "distance" $s$ from the origin. While we could express $s$ in terms of time, I'm not necessarily sure that we should. What makes the choice of time as our "independent variable" more canonical then choosing our "distance" $s$ for example?
This brings me to my question:
Is there a mathematical theory of physical (or other) quantities, that reflects how physicists (in my experience) seem to use them and where we need not choose specific "independent variables"?
Here are some more vague thoughts:
The general idea I have is that it could be like this: We have a field $\mathbb{K}$ (or something like that) and a set of "quantities" $Q$. Then there is another field $\mathbb{K}_Q$ with an injection $\eta_{1,Q} : Q\to \mathbb{K}$ and a field morphism $\eta_{2,\mathbb{K}} : \mathbb{K} \to \mathbb{K}_Q$ (necessarily injective too), s.t. for all other fields $K$ with same set-up there is a unique field morphism $\mathbb{K}_Q \to K$ making "everything" commute, or something like that.