This is a question that people often ask and find themselves answering that yes, mathematics is indeed a language, because it uses particular symbols to convey information and carries meaning in itself. From the definition of language we have:
A language is a system that consists of the development, acquisition, maintenance and use of complex systems of communication, particularly the human ability to do so; and a language is any specific example of such a system.
source: Wikipedia
But, is this really fully compatible with the definition of mathematics? Is language the only thing mathematics is about, or does it have a deeper reality? From the definition of the same source:
Mathematics includes the study of such topics as quantity, structure, space, and change. Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof.
From this we see that mathematics is not a language per se. Even though it uses rigorous formalisms and logic to formulate its postulates, those are only the methods of conveying information more than its underlying ontological reality: mathematics is the abstract study of patterns.
In a sense, it is a bit like music. Some people say music is a language, but it is not. Music is a reality deeper than the symbols it uses to try to frame and categorize its own concepts.
A language is an end in itself because it has a grammar and rules with the only purpose of communicating ideas, but it doesn’t carry any underlying reality. Music and mathematics use a type of language, but are not languages themselves.
musimathician.com 