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I often wonder if learning a dialect of a certain field has a similar effect. For example math terms or a complex programming language with things like monads

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I would think so. I often find myself cross-germinating the concepts I've learned from new programming languages, paradigms, and mathematics to other problems and discussions. Most recently I learned TLA+ which plays heavily with set theory; in particular the construction of logical "filters" on infinite sets. Ever since, those ways of thinking have become part of my mental toolbox that I interject into other areas.

I think in mathematics and programming, you often are working with an abstract idea or algorithm, projected onto the tools you are using. Be that the syntax/style/limitations of a specific language, or a branch of mathematics.

In that way, it is the same phenomenon of trying to express an abstract idea, projected onto the language in which you are speaking, with similar issues of syntax/style/limitations inherent to that language.

However, a human language opens a whole dimension of culture and history that is somewhat truly inaccessible from the outside. In this way, math and programming don't appear to me to have an obvious analog.

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