> When I'm talking about labeled vs unlabeled, what I really mean is that we have some intuition for how the labeled dataset might behave. E.g. "an increase in money supply causes an increase in inflation" is a better causal hypothesis than "an increase the president's body temperature causes an increase in inflation". We can make that judgement having never seen data, based on our understanding of the system.

What you have here is a circular argument. You are arguing that we can label variables with labels that are theory driven and so we can infer causality between those labels. You have already theorised causality without the data. So, the data is not the source of explanation it is merely a means to, rhetorically, assert that causality is the explanation. You have a causal explanation in mind and label the data informed by that causal explanation and then you carry out a mathematical operation on the numbers labelled and so, because you have labelled them you infer a causal explanation.

So you are correct: you can make a judgement without seeing the data. The data adds nothing to your understanding of the system because you have started from a theory, a model, and your activities with the causal relationship in mind. The data does not "contain causal knobs".

> Having made that hypothesis, we can look back to see if the data support it. The combination of a reasonable causal mechanism, plus correlated data, is typically seen as evidence of causation.

I would argue that what you are doing here is establishing rules for a rhetoric. Let us assume that we both accept mathematics is a kind of unbiased source of knowledge. This is a broad and possibly unwarranted assumption that would need refining, but accept it, broadly, for now.

You have a set of data which you recognise as x and y values. You have no theoretical labels to add them. But you list them and you are lazy. So you use a spreadsheet to tell you that the y column can be derived from the x column by

  f(x) = x^2   with R^2  = 1

So you are happy. The coefficient of determination ( R^2 ) tells you that the data "100% supports" the y=x^2 hypothesis. You are happy until someone comes along and says, have you considered

 f(x) = x * x
 f(x) = sqrt(g(x)), g(x) = x * x
 f(x) = (x * x * x) / x
 f(x) = (x * x * x * x) / (x * x)
 f(x) = (x^n) / (x^n-1) forall(n)>2

You object that this is all just messing about with variations on squaring things. I agree. But I point out that all I am doing is showing that there is more than one way to express a relationship of x to y but, generally, avoiding the use of y as a label.

So when you have f(x) = sqrt(g(x)), g(x) = x * x it is an awful circumlocution but it demonstrates that you can have a whole range of things "happening" to avoid using y. Which raises an interesting point about your notions of labelling data.

For a moment, pretend x can be relabelled "money supply" and y can be relabelled "inflation". We have the data set, as before {(1,1),(2,4),(3,9), ..., ( n,n^2 )} and we are supposing that the relationship is f(x) = sqrt(g(x)), g(x) = x * x or it is f(x) x * x. The first things first,

    f(x) is clearly to be relabelled as inflation.
    g(x) is also inflation (see your point^1 below)
    sqrt(g(x)) is money supply

Your point is that labelling clarifies causality. So, in mathematics it is permissible to rearrange a formula. But you are inferring causality so the only symbol in common in all of the formulations is the equals symbol. Which you might be holding in place of "causes". Which does correspond to your notion of Directed Acyclic Graphs but then places a huge constraint onto what you can actually say with labels.

So, because we have two formulations that you definitely agree on - the ones in the footnote - you can, rhetorically, say that we cannot tell if the causal case is

 y=x^2 .................... y is caused by x^2     
 x=sqrt(y) ................ x is caused by sqrt(y)

which is then translated into

 inflation is caused by squaring the money supply
 the money supply is caused by square rooting inflation

What this highlights is that you now actually need, back in the labels, some meaningful understanding of what "squaring the money supply" is and what "square rooting inflation" is. Because, to be causally coherent, these cannot just be vacuous utterances. This example is incredibly simple.

Just imagine what would happen if your chosen econometric methodology dictates the use of linear regression. You then have a philosopical need to explain x and y in terms of a lot of mathematical structuring around squares, roots, differences, and so on.

Which might boil down to me saying, "I do not think that the equals sign is a synonym for causality". But it might also be saying that "data adds nothing to causal explanation in economics".

Quite literally, you have show two possible formulae for a simple relationship. Which suggests that, at best, a 1 in 2 chance (50% probability p=0.5) that you randomly select the "correct" relationship - where, here, "correct" requires that the relationship expresses something causal. This becomes worse when you realise that it is possible to express x^2 in an infinite variety of ways (rendering p=0, effectively true). This means that you are never really talking about causation.

Which leaves you in the position that econometrics is a good source of rhetorical support for causation but only really provides evidence of correlation: that there is, indeed, a pattern in the data. That pattern in the data does not, in any way, vouchsafe your theoretical causal explanation with certainty. Even if you label it.

^1 E.g. in your x->x2 example, if all you had were a list of Xs and Ys, you couldn't tell if the operation was y=x2 or x=sqrt(y). Without any knowledge of what the Xs and Ys refer to, you're stuck.