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Mathematics as a languageThe central question involved in discussing mathematics as a language can be stated as follows :
What is a language ?To answer the first question, we need some definitions of language. The Wikipedia article on language contains several definitions. Other definitions are :-
The vocabulary of mathematicsMathematical notation has assimilated symbols from many different alphabets and fonts. It also includes symbols that are specific to mathematics, such as
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". The grammar of mathematicsThe grammar that determine 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 a finite state automaton 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 a axiomatic basis. Gödel's incompleteness theorem shows that this ultimate goal is unreachable - any formal language that is powerful enough to capture mathematics will contain undecidable statements. Nevertheless, the vast majority of statements in mathematics are decidable, and the existence of undecidable statements is not a serious obstacle to practical mathematics. The language community of mathematicsMathematics is used by mathematicians, who form a global community. 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.The meanings of mathematicsMathematics is used to communicate information about a wide range of different subjects. Here are three broad categories :-
Alternative viewsSome definitions of language, such as early versions of Charles Hockett's "design features" definition, emphasise 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 recognised 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 :-
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