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.
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?
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.
It's a misconception commonly held by non-mathematicians that the symbols are somehow the key to mathematics. I often see complaints, particularly on HN, of it being "impossible to read" research mathematics due to "all the dense symbols".
This isn't how it works. You learn the mathematics. That's hard. As you go along, you realize that some things that require a lot of words can be condensed. So you make up symbols, or, more likely, the source that you're using teaches you some commonly-agreed-upon symbols. Now, once you've learned the math at hand, the symbol isn't the issue anymore.
(Sure, it's a bit of a pain when people disagree about symbols for concepts that everyone agrees should be abbreviated, but in the big picture it's minor)