This isn't a good analogy, because if you have a human brain then you already understand all the concepts that Japanese is used to represent (except maybe some cultural things that are hard to translate).

In contrast, the language of mathematical notation is used for expressing concepts that are utterly unfamiliar and unintuitive. As a language, mathematical notation is nowhere near the complexity of a full human language, or even a programming language. It's much simpler than that. But the concepts behind it don't come naturally and have to be studied carefully.

While this is true, it doesn't help that postwar mathematics seems to have run out of useful names for things.

Understanding what an algebraic group "really is" (with variations and extensions e.g. Lie groups) is more than one lifetime's work, but if you know what "group" means in English, well, you won't be disappointed. "Ring", even "field" kind of work this way.

If I ever really develop an intuition for what a magma is, and I doubt it, this will not resemble the molten core of the earth. Or will it?

Symbols in math come cheap. Many different books and papers use different symbology for the same idea. Is why a lot of math books have a list at the outset for agreement on what symbol would stand for what idea.

On the other hand, you cannot use non-standard symbols to represent a sound in English. For example, nobody reading an English text will pronounce the letter "k" as "d" whereas mathematicians might do something like that, but they'll let you know in the opening pages of their book.

One of the reasons that Japanese is hard (for a westerner) is that all the symbols used to write Japanese text are unknown. Mathematics is expressed in logic, and that logic is most often expressed in a human language that the reader knows (such as English) plus some symbols. Those symbols are usually either known, or are introduced on a supportive framework of logic, a human language, and known symbols.

> That's hard.