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.

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.
