But I would argue that B is not caused by A alone, but by both A’s current and previous states.
Consider not the abstract situation of B = dA/dt, but the concrete example of the signal generator. It would be a perverse reading of the word “cause” to say that the voltage does not cause the current. You can make the current be anything you like by suitably manipulating the voltage.
But let this not degenerate into an argument about the “real” meaning of “cause”. Consider instead what is being said about the systems studied by the authors referenced in the post.
Lacerda, Spirtes, et al. do not use your usage. They talk about time series equations in which the current state of each variable depends on the previous states of some variables, but still they draw causal graphs which do not have a node for every time instant of every variable, but a node for every variable. When x(i+1) = b y(i) + c z(i), they talk about y and z causing x.
The reason that none of their theorems apply to the system B = dA/dt is that when I discretise time and put this in the form of a difference equation, it violates the precondition they state in section 1.2.2. This will be true of the discretisation of any system of ordinary differential equations. It appears to me that that is a rather significant limitation of their approach to causal analysis.
Consider not the abstract situation of B = dA/dt, but the concrete example of the signal generator. It would be a perverse reading of the word “cause” to say that the voltage does not cause the current. You can make the current be anything you like by suitably manipulating the voltage.
But you can make a similar statement for just about any situation where B = dA/dt, so I think it’s useful to talk about the abstract case.
For example, you can make a car’s velocity anything you like by suitably manipulating its position. Would you then say that the car’s position “causes” its velocity? That seems awkward at best. You can control the car’s acceleration by manipulating its velocity, but to say “velocity causes acceleration” actually sounds backwards.
But let this not degenerate into an argument about the “real” meaning of “cause”. Consider instead what is being said about the systems studied by the authors referenced in the post.
But isn’t this really the whole argument? If the authors implied that every relationship between two functions implies correlation between their raw values, then that is, I think, self-evidently wrong. The question then, is do we imply correlation when we refer to causation? I think the answer is generally “yes”.
I think intervention is the key idea missing from the above discussion of which of the the derivative function and the integrated function is the cause and which is the effect. In the signal generator example, voltage is a cause of current because we can intervene directly on the voltage. In the car example, acceleration is a cause of velocity because we can intervene directly on acceleration. This is not too helpful on its own, but maybe it will point the discussion in a useful direction.
Consider not the abstract situation of B = dA/dt, but the concrete example of the signal generator. It would be a perverse reading of the word “cause” to say that the voltage does not cause the current. You can make the current be anything you like by suitably manipulating the voltage.
But let this not degenerate into an argument about the “real” meaning of “cause”. Consider instead what is being said about the systems studied by the authors referenced in the post.
Lacerda, Spirtes, et al. do not use your usage. They talk about time series equations in which the current state of each variable depends on the previous states of some variables, but still they draw causal graphs which do not have a node for every time instant of every variable, but a node for every variable. When x(i+1) = b y(i) + c z(i), they talk about y and z causing x.
The reason that none of their theorems apply to the system B = dA/dt is that when I discretise time and put this in the form of a difference equation, it violates the precondition they state in section 1.2.2. This will be true of the discretisation of any system of ordinary differential equations. It appears to me that that is a rather significant limitation of their approach to causal analysis.
But you can make a similar statement for just about any situation where B = dA/dt, so I think it’s useful to talk about the abstract case.
For example, you can make a car’s velocity anything you like by suitably manipulating its position. Would you then say that the car’s position “causes” its velocity? That seems awkward at best. You can control the car’s acceleration by manipulating its velocity, but to say “velocity causes acceleration” actually sounds backwards.
But isn’t this really the whole argument? If the authors implied that every relationship between two functions implies correlation between their raw values, then that is, I think, self-evidently wrong. The question then, is do we imply correlation when we refer to causation? I think the answer is generally “yes”.
I think intervention is the key idea missing from the above discussion of which of the the derivative function and the integrated function is the cause and which is the effect. In the signal generator example, voltage is a cause of current because we can intervene directly on the voltage. In the car example, acceleration is a cause of velocity because we can intervene directly on acceleration. This is not too helpful on its own, but maybe it will point the discussion in a useful direction.