the value of [mathematical] language is not only communication, but also reasoning. It is remarkable, and sometimes unfortunate, how much mathematical reasoning you can do without knowing what you're doing, by pattern matching. Notation doesn't just allow you to communicate. It also allows you to manipulate e.g. truths that you know into truths that you do not yet know. In doing this abstract manipulation, you postpone (when necessary, and hopefully not indefinitely) thinking about semantics, while acting only syntactically.
A lot of people say something like "math is a universal language", and it sounds like we both agree that's nonsense. But I think we disagree on the extent to which math is a natural language. Your comment does not, in my opinion, recognize the symbolic manipulation (which doesn't have to be mostly-linear sequences of symbols, but can be e.g. diagrams too) that make math math.
A lot of people say something like "math is a universal language", and it sounds like we both agree that's nonsense. But I think we disagree on the extent to which math is a natural language. Your comment does not, in my opinion, recognize the symbolic manipulation (which doesn't have to be mostly-linear sequences of symbols, but can be e.g. diagrams too) that make math math.