For my own reference, I record the following articles.
Igor Pak, How to Write a Clear Math Paper: Some 21st Century Tips, Journal of Humanistic Mathematics, vol. 8 (2018), 301-328. Link: https://www.math.ucla.edu/~pak/papers/how-to-write1.pdf
I enjoy learning languages, so I mostly self-taught Japanese and French (that's why my French is broken). Sure this is written in English so you may add one category. This helps me a bit on research and a lot in studying the history of math. Recall that lots of 18~20C math articles and books were written in French (though not all of them were; I should learn German and perhaps basic Latin someday). This led me to delve into the history of math also.
However, regarding the overall math, learning language gives me an ambivalent feeling.
First, the language barrier is real in mathematics (it's also complex, I know). It burdens me every time I encounter articles written in German, Russian, or other languages I'm not used to. Actually, it doesn't exclude Japanese or French articles, and even English articles: they are all not written in my first language.
I'm not trying to say that this is the authors' fault; writings are best when written in the author's mother tongue. It is also not practical to ask every author to provide a separate version in Korean, in my case. Thus, we have compromised to use English, the de facto Lingua Franca of contemporary math, in both writing and reading. However I propose an alternative way: I claim that every scholar should be free to write in whatever language they prefer, and we should call for a proper, stable translation system. This may sound a bit weird, so I'll add some more.
First, we should not ask everyone to become a multiplayer. If one becomes fluent in English and provides English writings, that person is automatically contributing to this translation system. However, is the system asking every single person to perform these dual role sustainable? In my opinion, people have different strengths and flavors; some might prefer pushing the frontier of mathematics further and further, while others might prefer spreading these discoveries to colleagues and neighbors. I don't think language skill (especially toward "English") is mandatory for all those steps.
This may complicate the system, and indeed I'm claiming that. The current math system requires each person to first excel in research, but also contribute to the institutions and society to some degree. This demands high research productivity, but also makes not to focus only on that. Given this situation, the best strategy is to first optimize every skill to research (i.e., promoting one's English to the moderate level) and keep that status. One can look for other aspects of math community only after being recognized for research.
This provokes a personal dilemma for me as a graduate student. I'm also into spreading the math using my language skills, but as I explained above, I cannot concentrate on this before I get recognized my research ability. Being not sure whether I'm suitable for this procedure, I'm afraid my skills are discarded.
(There are two more reasons I'd add later; if you are in hurry, please refer to the Korean version)
On top of this, the math environment in Korea requires more communicators. There are two ways a math society can grow nowadays. The first is to provide nice and in-depth materials in that society's language so that students learn with those. The second is to urge everyone to become versatile. I'm doubting the second system. Thus I think a system involving good math translators/communicators should be in place.
As a result, I'm trying to learn as much as I can and perform research to test myself. At the same time, I'm trying to convey math ideas to the community in Korean. This effort gave birth to my cartoons and forces me to produce notes in Korean.
P. S. And actually I can add one more word: if the de facto Lingua Franca was Latin (indeed!), I would have toned down significantly. Not because I favor Latin (I can't write/read/speak Latin), but because I don't favor English! If you speak English as a native speaker and want to experience the language barrier, try writing down some math proofs in Coq. And imagine arXiv accepting Coq-readable arguments only... (this still also favors English, right?)