Your real question should probably be “why are you asking this?” The reason why I feel that the question of mathematics as language needs to be answered has to do with theories published in 2000 by a mathematician (Devlin, K., The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip) and a linguist and cognitive psychologist (Lakoff, G. and Nunez, R. E., Where mathematics comes from: how the embodied mind brings mathematics into being). In both cases, language becomes the underlying explanation for mathematical cognition. Devlin posits that mathematics piggy-backed on language. Lakoff and Nunez use linguistics constructs to explain the cognitive processes underlying mathematics.
So let's decide if mathematics is a language using a general definition of language. Language is a complex system so no simple definition will cover all the nuances, but we can for the time simplify as much as possible. A definition of language includes producing speech, analyzing the speech we hear, the vocabulary and its symbolic references we use, grammar, and syntax. (Deacon, 1998, p.40) Deacon offers a generic definition for language as “a mode of communication based upon symbolic reference (the way words refer to things) and involving combinatorial rules that comprise a system for representing synthetic logical relationships among these symbols.” Deacon also states that within this definition mathematics “might qualify as having the core attributes of language.” (Deacon, 1998, p.41)
Let's see what the "core attributes" of language might be. You can say that language consists of basic sound units, phonemes that combine to form morphemes. These phonemes and morphemes when combined according to predefined rules become abstract symbols which we ascribe meaning to and understand. In other words, they become words. Language also allows various combinations of these symbols by following syntactic and pragmatic rules - sentences and paragraphs. (See Matlin, 2005, p.298) The result is communication that describes an object, event, or action, which need not be present or even exist. In other words, language symbolizes and creates meaning using a series of abstract symbols (letters or sounds) which have no particular connection to a concrete object other than those connections we agree exist.
Presented in these terms, the phonemes of mathematics are the basic digits, zero through nine. These digits are combinable following a few simple rules and form mathematics morphemes, numbers such as 123 or 3.1416 or even 4/5. The branch of mathematics in question defines the particular syntax used. For example, Algebra’s syntax determines how to write an equation and the order of operations used to solve the equation. Each digit, number, or equation symbolically refers to a quantity or describes a system, event, or form. The actual quantity need not be present nor does a quantity or an equation need to refer to a concrete object or collection of objects. Many of the systems, forms, and events described by mathematics are themselves abstract concepts such as equations which describe the multidimensional shape of the universe.
Do these similarities between language and mathematics provide a sufficient condition to call mathematics language? In some instances, mathematical and language processing take place in areas of the brain that are generally similar. Experiments done on bilingual individuals indicate that exact calculations occur in the left inferior frontal lobe of the brain. This area controls linguistic representations of exact numerical values. However, approximations of numbers occur in the left and right intraparietal sulci in areas associated with visuo-spatial tasks. (Butterworth, 1999; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999)
Given this, my answer to the question is “yes, there exists a well defined language of mathematics used to communicate mathematical knowledge.” However, I believe that it is a mistake to confuse the act of communicating mathematics with the cognitive processes taking place while doing mathematics so my answer to a mathematical language that defines the cognitive process is “no.”
Butterworth, B. (May 7, 1999). A Head for Figures. Science, 284, 928-929.
Deacon, T. W. (1998). The Symbolic Species: The Co-evolution of Language and the Brain. New York: W.W. Norton and Company.
Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (May 7, 1999). Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence. Science, 284, 970-974.
Devlin, K. (2000). The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. Basic Books.
Matlin, M. W. (2005). Cognition (Sixth). Hoboken, NJ: John Wiley & Sons, Inc.
Lakoff, George; Nunez, Rafael E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. Basic Books.