The situation here seems very analogous to that in probability, where there is also a state space $\Omega$ (which is the underlying set of a probability space $(\Omega, {\mathcal B}, {\bf P})$) which is required in the foundations of the subject in order to define everything properly, but is then downplayed as strongly as possible once one starts doing probability. Thus, technically, every random variable $X$ is a function on this state space (e.g., a real random variable would be a function from $\Omega$ to ${\bf R}$), but one tries to avoid explicit mention of this space as much as possible, and in fact every so often one actually exercises the freedom to change the state space or probability space, for instance by adding some new sources of randomness, conditioning to an event (somewhat analogous to your equaliser example), and so forth. One can then view probability theory as the study of objects and concepts that remain invariant under a certain type of change of state space, namely that of extending that space; see these lecture notes of mine for more on this (see also these later notes).

One can adapt this viewpoint to non-probabilistic settings. This brings us back to your proposal to view all mathematical objects as depending on a state space $A$, which is not specified precisely and is in fact downplayed as much as possible. One could view this state space as being somewhat dynamic in nature, for instance it could become larger as one makes more measurements in a physical system or introduces some new variables, or it could shrink as one makes some assumptions or fixes some values of certain observables. If one sets things up properly, these evolutions of the state space should not destroy any mathematical facts and relationships one has already gathered about the existing observables: for instance, if two observables $X,Y$ are known to always obey the relation $Y=X^2$, this fact should be unaffected by any changes to the state space caused by performing a measurement of a new observable $Z$, or by making some hypothesis constraining the known observables. (This suggests also to consider some "quantum" version of this setup where making new measurements can destroy the truth of previously established facts... but I digress.)

Incidentally, information theory, which builds upon probability theory, has a well-developed and quite quantitative theory of dependence: for instance, given two discrete (and finite entropy) random variables $X$ and $Y$, $Y$ is a function of $X$ (almost surely) if and only if the conditional entropy ${\bf H}(Y|X)$ vanishes.

The situation here seems very analogous to that in probability, where there is also a state space $\Omega$ (which is the underlying set of a probability space $(\Omega, {\mathcal B}, {\bf P})$) which is required in the foundations of the subject in order to define everything properly, but is then downplayed as strongly as possible once one starts doing probability. Thus, technically, every random variable $X$ is a function on this state space (e.g., a real random variable would be a (measurable) function from $\Omega$ to ${\bf R}$), but one tries to avoid explicit mention of this space as much as possible, and in fact every so often one actually exercises the freedom to change the state space or probability space, for instance by adding some new sources of randomness, conditioning to an event (somewhat analogous to your equaliser example), and so forth. One can then view probability theory as the study of objects and concepts that remain invariant under a certain type of change of state space, namely that of extending that space; see these lecture notes of mine for more on this (see also these later notes).

One can adapt this viewpoint to non-probabilistic settings. This brings us back to your proposal to view all mathematical objects as depending on a state space $A$, which is not specified precisely and is in fact downplayed as much as possible. One could view this state space as being somewhat dynamic in nature, for instance it could become larger as one makes more measurements in a physical system or introduces some new variables, or it could shrink as one makes some assumptions or fixes some values of certain observables. If one sets things up properly, these evolutions of the state space should not destroy any mathematical facts and relationships one has already gathered about the existing observables: for instance, if two observables $X,Y$ are known to always obey the relation $Y=X^2$, this fact should be unaffected by any changes to the state space caused by performing a measurement of a new observable $Z$, or by making some hypothesis constraining the known observables. (This suggests also to consider some "quantum" version of this setup where making new measurements can destroy the truth of previously established facts... but I digress.)

Incidentally, information theory, which builds upon probability theory, has a well-developed and quite quantitative theory of dependence: for instance, given two discrete (and finite entropy) random variables $X$ and $Y$, $Y$ is a function of $X$ (almost surely) if and only if the conditional entropy ${\bf H}(Y|X)$ vanishes.