Is there any theory or field of study that concerns itself with modeling causation rather than correlation? My understanding is that probability (at least from a frequentist viewpoint) is a mathematical tool for modeling correlations. So, for example, we can say that two events $X$ and $Y$ are defined to be independent if $P(X\cap Y) = P(X)P(Y)$, or equivalently $P(X|Y) = P(X)$, and so on.
However, something like whether or not $P(X|Y) = P(X)$ tells us nothing about causation.
This leads me to the question that this post is about.
Is there any sort of theory or mathematical field of study that concerns itself with modeling causation?
I suspect there are answers in two possible forms. The first is that there might be specialized models specific to a given domain of study (biology, physics, economics).
The second might be some generalized abstract theory akin to constructor theory that honestly only a mathematician would think up.
An answer in any possible form would be appreciated.

There are two main approaches to the New Causal Revolution. One is the graphical approach (as in, directed acyclic graphs), championed by Judea Pearl. The other is the potential outcome framework, championed by Donald Rubin.
For the graphical approach, I recommend these books in this order:

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*The Book of Why, by Pearl and MacKenzie. Prerequisite: introductory statistics.

*Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell. Prerequisites: the full calculus sequence, followed by mathematical statistics.

*Causality: Models, Reasoning, and Inference, by Pearl. This book is extremely difficult (I have not read it, though I have skimmed in parts) but has almost everything in there. One recent development not in this book is the maximal ancestor graph approach. Prerequisites: first mathematical statistics, then Bayesian statistics (I would recommend first Bayesian Statistics for Beginners: a step-by-step approach and then Bayesian Data Analysis, but the first book might be sufficient; also note that a pdf of Bayesian Data Analysis can be obtained for free at the website of one of the authors), and finally Bayesian networks. This book has accompanying homework sets on its webpage: go to CAUSALITY, then 7. Viewgraphs and homeworks for instructors. Indeed, Pearl's UCLA webpage has errata for all three of the above books.

For the potential outcomes framework, the main book appears to be Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction, by Rubin and Imbens. I have not read it; but the prerequisites appear to be mathematical statistics at least - analysis and design of experiments wouldn't hurt, nor would Bayesian statistics (see #3 above for recommendations there). One weakness of this book is that it has no exercises or errata page.
The two approaches have different strengths and weaknesses, but as noted by Carlos in his comment, they are theoretically unified via Structural Causal Modeling - addressed in several of the books above.
