Mathematics as a language

I have been reading a Machine Learning textbook for the past 3 days. It uses a lot of probability theory, something I know the basics of (up to Probability theory II course). I managed to get the gist of what the book was trying to say, but am still somewhat struggling to understand how the proof of the main theorem works and what its implications are.

At some point, my roommate (who is a psychology major and very fond of languages as well) asked me to show her the proof I was reading.. and then instantly turned away when she saw the formulae. So I tried to explain to her what my problem was, in terms she could understand. And then I stumbled upon a pretty interesting and deep analogy:

Mathematics is just a set of symbols used to very precisely say things that would take a lot of plaintext to explain without the special notation. Just a means of precisely expressing yourself. Like the alphabet. However, each branch of mathematics has its own key concepts and terminology (the words) as well as diffent proof techniques (syntax). Different branches of mathematics can, in this sense, be likened to different languages. And different languages can be either very close to eachother (Statistics and Machine learning seem quite like Estonian and Finnish, for instance - they have quite a lot of common ground both in terms of key concepts and proof methods, but they are nevertheless quite distinct, with quite a few things having nearly opposite meaning)

My mother tongue in mathematics is Cryptology, with a slight accent from complexity theory. Reading a Machine Learning textbook for me is akin to reading something in a language I have studied a little (a few years, maybe) but which I am not comfortable in yet - I can make out the meaning of most that is said, but I do not grasp the nuances of the more complex words (ideas) and sometimes get their meaning completely wrong (which happens if you are forced to deduce their meaning just from the context). However, this is a pretty good way of learning the new language, as it forces you into a very deep mode of processing - you have to build up the abstract "big picture" as fast as possible and constantly try to deduce the missing details back from it. Hard, but quite enjoyable, as it makes me feel I am actually learning and very rapidly.

For those who find mathematics courses hard, this analogy might help understand what is going on: you are usually expected to learn the rudiments of a new language, often up to the point where you need to be able to not only understand it but actually "speak" it to a degree by the end. It is not harder than learning normal languages, and as with normal languages, the more different languages you learn, the easier it becomes to acquire a new one.