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.

Adrian Keister provided a great answer. My answer continues his. It took me a while to realize that the 2 different approaches to causal inference (graphical approach and potential outcomes) are complementary. To get the best appreciation for how these 2 approaches to causal inference work together I would recommend reading Morgan & Winship "Counterfactuals and Causal Inference". From this book you will learn that there are 3 main ways to estimate causal effects: 1) backdoor criterion, 2) instrumental variables, and 3) front-door criterion.
While there are 3 methods, the overwhelming majority of causal inference journal articles in econometrics and social science use method#2: instrumental variables. Incidentally, for the instrumental variable approach the DAG is in many ways unnecessary because it always has the same skeleton and can be easily described in words. One may say that for IV a DAG is crucial; yes it is because it must look like the one below (Z is the instrument). But since all IV DAGs must look like this, how crucial is the DAG in IV and what role does it play other than being a nice visualization?  A DAG is crucial for method#1: backdoor criterion. But in practice, it is difficult to argue convincingly that a suitable DAG has been constructed. In econometrics or social science you will hardly find a journal article using this method. And if you do, it is almost certainly not going to contain a DAG. From what I've seen this method is used successfully quite often in the medical field. For method#3: front-door criterion the DAG is usually relatively simple, with only a couple front-door paths, and thus it can be easily described in words. So, at the end of the day, the graphical approach is a nice add-on but unless you are estimating causal effects using backdoor criterion (which I find to be rare outside the medical field) or with an elaborate front-door criterion (also relatively rare) a DAG isn't crucial. In contrast, the potential outcomes framework underlies the very substance of causal inference and, frankly, you can't even define causal inference without it. The 2 books by Angrist and Pischke are somewhat unambiguously the best introduction to the potential outcomes approach (in econometrics and social science); the book by Hernan and Robins seems to be highly valued as well, particularly in public health/medical field (but I haven't read it in full). What I consider to be the most valuable contribution of the graphical approach camp is to raise awareness around collider variables; some of its implications (e.g. endogenous selection bias) are critical, top of mind considerations in causal inference across disciplines.
My favorite resources:

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*Here is a terrific video series by Abadie, Angrist, and Walters at AEA on Cross-Sectional Econometrics, which can serve as an introduction to the potential outcome framework: https://www.aeaweb.org/conference/cont-ed/2017-webcasts

*Here is an equally terrific video introduction to the graphical method - an edX course by Miguel Hernan with some incredibly interesting practical examples (primarily from epidemiology) https://www.edx.org/course/causal-diagrams-draw-your-assumptions-before-your
(diagram credit: https://donskerclass.github.io/EconometricsII/ControlandIV.html)
