My language learning problem is slightly different and quite under-served because my motivation is not the most common one.

I want to learn so that I can read/understand publications in mathematics in a foreign language, mostly Swedish, French, German. (*) For this exercise, the typical apps do not help much.

(*) I would have liked to add Latin and Greek too but that's mostly a pipe dream.

Reading old mathematicians and scholars I have realized something that runs quite counter to the common perception we have, especially in my country.

That common perception is school kids, especially in mid and high school are overwhelmingly burdened by sheer volume of subject matter to learn at a very young age. But then I look at educated teenagers from 17th - 18th centuries, who went on to become mathematicians or scholars, they were so immensely well read at a very young age. I understand this is a biased sample, but many of these people, Newton, for example, were ordinary folks (socio-economically speaking)

Hamilton (I concede that one cannot compare Hamilton with a typical modern teenager) was already quite fluent in thirteen languages in his pre-teens. Apart from the usual suspects, he knew Arabic, Hebrew, Farsi, Sanskrit, Hindi, Marathi.

This might sound atypical but this was not unheard of. One of the poets in my language was fluent in Hebrew, Greek, Italian, French, Latin, Sanskrit, Telugu, Tamil, Bengali, English.

That's quite different than what most folks are looking for when learning a new language, but I guess some techniques can apply just fine. You would have to take the lead find and prepare your own study material.

Something like collecting phrases from these books, loading them into SRS, collecting youtube videos of natives discussing the material you are into, extracting the sound and listening several hours of it for immersion... That is basically the way I learn but focusing on different material.

With LLMs, it is much easier to create your own study material nowadays, as you can ask to translate, break down and explain things as you go.

I find your problem to be much easier than the problem faced by a typical language learner. Mathematics literature uses quite a small subset of a language, and in many languages math authors use a lot of equations, which are quite language agnostic.

One way to tackle this problem is to just get started with an LLM. You ask ChatGPT for example, to translate for you, and then you try to figure out what word correspond to what word and keep going. After a while you will need the LLM help less and less.

Who am I to tell you this? I only read math in my native language, and in French and English. But once I wanted to do some calculations using Gauss's Theorema Egregium, so, out of curiosity, I picked up both the English translation of Gauss's original publication, and the original Latin text. I was able to understand sufficient Latin to figure out what Gauss was saying and to find out that the English translation has a bug.

what makes Latin difficult in your context? My focus isn't Math and fwiw found many very good, free, entry-level[1] self-study[2] books (Hans Orberg and others), and even Latin podcasts. There is even a fun Latin track on some of the popular language learning apps.

[1] https://archive.org/details/conspectus-grammaticus-familia-r...

[2] https://latinitium.com/best-books-for-learning-latin/

Good luck! This kind of reminds me of how Bobby Fischer at a relatively young age learned Russian for the explicit purpose of being able to keep up with the best Chess manuals and periodicals - a great deal of which were coming out of the Soviet Union.

I understand both your historical question and the more concrete practical one. Separating reading comprehension as a skill from all the other discrete functions of a language is very straight-forward [1] and, in fact, there are some good analog resources for this.

For French: Dandberg and Tatham, French for Reading

For German: Jannach, German for Reading Knowledge

I've used both and swear they're magic, especially if you're trying to learn to read in a scientific domain that you're already a specialist in (versus literature).

Once you've sort of "learned the game" it isn't very hard to do a similar process for other languages on your own. Then, my main recommendation is to take a text you're deeply familiar with in your native language or English that exists in X other language and just go ahead and start reading it with a dictionary. It starts slow, but progress is very very fast if you stick with it, especially compared to learning to speak or even just listen to a language.

For life reasons, I've found myself having to learn Danish, so I'll let you know if I figure out any good resources for Scandinavian languages.

[1] The only downside I've encountered is trying to later learn to speak a language I had been reading for a while where overcoming the sort of "fictitious phonetics" that existed in my head proved problematic.

Interesting, you are learning languages for math publications but you didn't include Russian ? unless of course you are native (or you are from ex soviet)