Answer:
Why is it important to study the conventions of musical language? It only is so if you want to create and share music.
So there are two real questions here: 1) Why is it important to study math at all and 2) Why does language and notation seem so important in mathematics?
Math is important because it captures elements of the human mind that are really common. Very few other places in human experience can you be guaranteed that if you take a given set of thoughts, and use them in the most natural way, as long as you do it carefully, any two humans will agree. What mathematical thinking captures is the idea that we share an underlying logic, even if rigid logic does not appeal equally to all of us. Every other discipline relies upon that faith, but most of them cannot build it as effectively as mathematics. There is also a texture to that shared mental space, a feel for when facts are being treated badly, that can be useful everywhere, but I'd developed most clearly by learning what one can and cannot do using only that shared mental furniture.
Mathematics has not always been done with a language of its own. There is a lot of math preserved in older cultures that is much more discursive, just using natural language. But mathematics makes a huge step forward every step it takes away from natural language. When the Greeks went from long form descriptions in Geometry to diagrams and notes that just marked pieces with letters, the discipline exploded. When the Arabs contrived the notation of the variable (al jabra, Algebra) and its combinations they succeeded at vastly outpacing those around them. The Alchemists were so impressed that they tried to produce a symbolic tool like it for all natural phenomena. But it didn't help as much. We still write chemical equations, but we don't expect them to capture as much about chemistry as algebra and other notations capture about math.
So it seems that the specific roots of our deeply common thoughts probably lie in the domain of language. We even call such things 'logic’, even when we abstract away the words (logoi). So if you are going to do math, it seems to “want to be done" with specialized, stripped down and suped up language.
