There has been lots of debate on the scientificity of the do operator and causal inference in general. When Pearl explains why he thinks do(x) is scientific, i.e. empirically relevant in terms of operational procedures that are -at least in principle- feasible, his discussion is concerned mostly with systems in which there is a legitimate fundamental problem of causal inference, as in the case of medicine.
However it is in other domains –like in physics- that we have theories purporting to accurately describe most of the phenomena we care about. In that context, whatever the operational definition of do(X=x), where X is some physical quantity, the relevant operations carried out to make the variable X take the value x should obey the relevant theoretical constraints.
Whether our physical theories allow the implementation of the do() operator, and to what extent, is not just an academic question. While the lack of an operational definition corresponding to a theoretical construct can be brushed away as irrelevant to the empirical value of the final outcome of a procedure for which the construct is an intermediate step –like complex numbers for solving third-degree equations- it may still be an indication that the theory can be improved by excluding that concept –as in the case of absolute space for relativity. So there is definitely some practical value in asking this question.
The answer to our question can be either that the do-operator is universally implementable -at least in principle- and the notion of causality it formalises is compatible with our physical theories; or that -in some circumstances- our physical theories predict that causal reasoning as formalised by the do-operator is not empirically implementable. If the latter is the case, then there is a limited domain of applicability of the causal inference machinery based on the do operator: conditions apply.
It should be noted that clarifying which variables we are working with is crucial. Conservation laws and the symmetries they imply will ensure that certain choices of coordinates result in variables that are in principle un-manipulable. For instance if X is the position of the center of mass of a closed system, it does not make sense to write do(X=x), because it is not possible to implement this intervention within the closed system: the position of the center of mass is un-manipulable for fundamental physical reasons.
In an isolated system composed of two equal-mass particles with coordinates X1 and X2, X = (X1 + X2)/2. X must be conserved, at least in the appropriate reference frame. Since do(X1 = x1) assumes that X2 is left undisturbed, and X1 = 2X - X2, then do(X1=x1) is equivalent to do(2X - X2 = x1), that is do(X = (x1 + X2)/2). But this is not implementable in general, because X is un-manipulable. We have seen that we cannot just take any choice of coordinates for our system and ask what the causal effect of manipulating one is, because this would lead us to implicitly attempt manipulating a conserved quantity –which cannot be done from within the system, even in principle.
Un-manipulable degrees of freedom should be excluded from causal discovery or inference if we want our do-operators to be physically implementable: in the two-particle system’s case the only manipulable coordinate is the distance between the two particles S = X1 – X2, for which we can happily write do(S = s). Alternatively, general applicability of the do operator can be achieved by opening the system., which lets the center of mass move by pushing onto something else.
In general, conserved quantities –if they exist- are not manipulable unless an interaction with something that is outside the system occurs. For instance, in a conservative system, where energy E is a constant of motion, do(E = e) is not implementable from within the system, i.e. there is no sequence of operations that could be carried out by a subset of the system capable of setting E to a pre-defined value other than the one set by the system’s initial conditions. An external intervention, which requires opening the system so that it can interact with an energy source or sink, is needed to implement do(E = e) for a generic value of e.
If the system cannot be opened, for instance because it is the whole Universe, then we should accept that certain global, un-manipulable properties of the system have no causal significance.
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MMH… maybe in the two particle case you can consider X2 (the position of particle 2) to be a causal child of X1 (position of particle 1). So you can do(X1 = x1) and this will affect, downstream, X2, forcing it to become equal to some x2 such that the center of mass does not move. However here the situation is completely symmetrical if we switch particle 1 with particle 2 so X1 should also be a causal child of X2! But then the system is cyclic.
Basically this: https://www.nature.com/articles/ijo200882 but you cannot change BMI in isolation because of physical constraints (e.g. BMI + exercise is a conserved quantity of the system).