The majority of the books listed in the first link in these comments are about whether and when we can determine causality given observational data - data where the system is perturbed, just not by us (and not as we would ideally like).

I have not looked at the links, but aren't there people who use Bode plots to estimate direction-of-causality, given time-series data? IIRC there are basic relationships between wide-frequency phase behavior and, e.g., impulse-response functions. There is apparent phase information beyond correlation and ANOVA.

But many datasets will be full of recorded perturbations already. Can't you usually hunt for the evidence you need in the data you have?

The Michelson-Morley experiment was enough data to get special relativity, for example.

Einstein was inspired to come up with special relativity from that experiment because he had the relevant concepts to draw upon to think through various thought experiments. That and the math to back up his intuition.

Yes, but Einstein was able to learn cause and effect by interacting with the physical world as a child. Without being able to interact with an environment, the I believe the ability to learn causality is limited if not impossible

Alternatively, we need to extend our typical notion of a dataset to include where it came from/how it was perturbed. If you give your best statistician a dataset without column names and a description of the experiment, they aren't going to be able to do causal inference. We need to make those things more available to the machines too, in a structured way.