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.
 A: 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.
A: Causal inference is a part of statistical inference, so it falls within the field of statistics.  Causal inference generally requires inference of statistical associations under an appropriate experimental structure that limits statistical associations to certain structures.  This is dealt with in specialist books that look at the interaction of causality and probability, most notably the excellent works of Judea Pearl (see e.g., Pearl 2009, Pearl 2015, and Pearl, Glymour and Jewell 2016) and the potential-outcome framework of Donald Rubin (see e.g., Holland 1986, Rubin 1991, Rubin 2005).  Pearl has criticised the statistical profession for paying insufficient attention to this material (see related question here), but his works nevertheless fall within the field of statistics and they can properly be regarded as contributions in the interface of probability, causality, and statistics.
As you will see from reading these works, it is possible to augment traditional probability theory by adding an operator to represent an action/intervention in the system (called the "do" operator), which allows causality to be built into the analysis at an axiomatic level.  This is a useful extension of traditional probability theory.  Presently, the statistics curriculum for students does not incorporate much of this material, except in some specialist classes that occur late in the program.  It is my hope that this extension to probability theory will eventually be built into the statistical curriculum in a more cohesive manner, so that students become fluent in causal reasoning earlier on in their statistical studies, rather than seeing it as an add-on that they encounter only later in their career.
A: Simple answer. Yes - the scientific method is there to help FIND and confirm causal relationships. It requires an hypothesis, then a controlled experiment, then a (statistically verified) conclusion. As data are gathered to support the hypothesis, it becomes more and more certain that "x" causes "y".
A: 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)
