The central question involved in discussing mathematics as a language can be stated as follows:
A secondary question is:
To answer the first question, we need some definitions of language:
These definitions describe language in terms of the following components:
To expand on the concept of mathematics as a language, we can look at each of these components within mathematics itself.
Like any other profession, mathematics also has its own brand of technical terminology. In some cases, a word in general usage has a different and specific meaning within mathematics—examples are group, ring, field, category.
In other cases, specialist terms have been created which do not exist outside of mathematics—examples are tensor, fractal, functor. Mathematical statements have their own moderately complex taxonomy, being divided into axioms, conjectures, theorems, lemmas and corollaries. And there are stock phrases in mathematics, used with specific meanings, such as "if and only if", "necessary and sufficient" and "without loss of generality". Such phrases are known as mathematical jargon.
When mathematicians communicate with each other informally, they use phrases that help to convey ideas. Examples of some of the more idiomatic phrases are "kill this term", "vanish this interval" and "grow this variable".
Diagrams are used informally on blackboards, as well as in published work. When used appropriately, diagrams display schematic information more easily. Diagrams also help visually and aid intuitive calculations. Sometimes, as in a visual proof, a diagram even serves as complete justification for a proposition. A system of diagram conventions may evolve into a mathematical notation - for example, the Penrose graphical notation for tensor products.
The grammar that determines whether a mathematical argument is or is not valid is mathematical logic. In principle, any series of mathematical statements can be written in a formal language, and an algorithm can apply the rules of logic to check that each statement follows from the previous ones.
Various mathematicians (most notably Frege and Russell) attempted to achieve this in practice, in order to place the whole of mathematics on an axiomatic basis. Gödel's incompleteness theorem shows that this ultimate goal is unreachable: any formal system that is powerful enough to capture mathematics will contain undecidable statements. Nevertheless, the existence of undecidable statements (relative to this or that formal system) is not a serious obstacle to practical mathematics, and the vast majority of statements one is likely to come across in practice in pure mathematics are decidable on the basis of Zermelo set theory (and usually in much more elementary theories, such as second order arithmetic or Peano arithmetic). The statements which are not are then either equivalent to or intimately related to nebulous statements of a purely set-theoretical character (usually the axiom of choice), or are equal to the consistency statement of Zermelo set theory.
Mathematics is used by mathematicians, who form a global community. It is also used by students of mathematics. As mathematics is a part of primary education in almost all countries, almost all educated people have some exposure to pure mathematics. It is interesting to note that there are very few cultural dependencies or barriers in modern mathematics. There are international mathematics competitions, such as the International Mathematical Olympiad, and international co-operation between professional mathematicians is commonplace.
Mathematics is used to communicate information about a wide range of different subjects. Here are three broad categories:
Mathematics can communicate a range of meanings that is as wide as (although different from) that of a natural language. As German mathematician R.L.E. Schwarzenberger says:
Some definitions of language, such as early versions of Charles Hockett's "design features" definition, emphasize the spoken nature of language. Mathematics would not qualify as a language under these definitions, as it is primarily a written form of communication (to see why, try reading Maxwell's equations out loud). However, these definitions would also disqualify sign languages, which are now recognized as languages in their own right, independent of spoken language.
Other linguists believe no valid comparison can be made between mathematics and language, because they are simply too different: