LW is pretty much the only site I visit where I feel significantly intimidated about commenting. I've left a couple of comments, but I seem to be more self-conscious about exposing my ignorance here than I am elsewhere – probably because I know that the chances of such ignorance being noticed are higher. It occurs to me that this is completely backwards and ridiculous, but there you have it.

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.

That is what A and B are: a randomly wandering variable A and its rate of change B.

Maybe I'm not quite understanding, but it seems to me that your argument relies on a rather broad definition of "causality". B may be dependent on A, but to say that A "causes" B seems to ignore some important connotations of the concept.

I think what bugs me about it is that "causality" implies a directness of the dependency between the two events. At first glance, this example seems like a direct relationship. But I would argue that B is not caused by A alone, but by both A's current and previous states. If you were to transform A so that a given B depended directly on a given A', I think you would indeed see a correlation.

I realize that I'm kind of arguing in a circle here; what I'm ultimately saying is that the term "cause" ought to imply correlation, because that is more useful to us than a synonym for "determine", and because that is more in line (to my mind, at least) with the generally accepted connotations of the word.