Mathematics as a language
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The central question involved in discussing '''mathematics as a language''' can be stated as follows :
:''What do we mean when we talk about the '''language of mathematics''' ? To what extent does mathematics meet generally accepted criteria of being a language ?''
A secondary question is :
:''If it is valid to consider mathematics as a language, does this provide any new insights into the [[history of mathematics| origins of mathematics]], the [[mathematical practice| practice of mathematics]] or the [[philosophy of mathematics]] ?''
==What is a language?==
To answer the first question, we need some definitions of [[language]]:
* ''a systematic means of communicating by the use of sounds or conventional symbols'' [http://www.cogsci.princeton.edu/~wn/ WordNet]
* ''a system of words used in a particular discipline'' [http://www.cogsci.princeton.edu/~wn/ WordNet]
* ''the code we all use to express ourselves and communicate to others'' [http://www.nlg.nhs.uk/SpeechLanguage/glossary.htm Speech & Language Therapy Glossary of Terms]
* ''a set (finite or infinite) of sentences, each finite in length and constructed out of a finite set of elements'' [[Noam Chomsky]]
These definitions describe language in terms of the following components:
* A [[vocabulary]] of symbols or words
* A [[grammar]] consisting of rules of how these symbols may be used
* A [[community]] of people who use and understand these symbols
* A range of [[Meaning (semiotics)|meanings]] that can be communicated with these symbols
To expand on the concept of mathematics as a language, we can look at each of these components within mathematics itself.
==The vocabulary of mathematics==
[[Mathematical notation]] has assimilated [[table of mathematical symbols|symbols]] from many different [[alphabet]]s and [[typeface|fonts]]. It also includes symbols that are specific to mathematics, such as
:<math>\forall \ \exists \ \nabla \ \wedge \ \infty</math>
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 (mathematics)|group]], [[ring (mathematics)|ring]], [[field (mathematics)|field]], [[category theory|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 [[axiom]]s, [[conjecture]]s, [[theorem]]s, [[lemma (mathematics)|lemma]]s and [[corollary|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 in 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 of mathematics==
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 a [[finite state automaton]] can apply the rules of logic to check that each statement follows from the previous ones.{{Fact|date=March 2008}}
Various mathematicians (most notably [[Gottlob Frege|Frege]] and [[Bertrand Russell|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.
==The language community of mathematics==
Mathematics is used by [[list of mathematicians|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 mathematics==
Mathematics is used to communicate information about a wide range of different subjects. Here are three broad categories:
* '''Mathematics describes the real world''': many areas of mathematics originated with attempts to describe and solve real world phenomena - from measuring farms ([[geometry]]) to falling apples ([[calculus]]) to gambling ([[probability]]). Mathematics is widely used in modern [[mathematical physics|physics]] and [[engineering]], and has been hugely successful in helping us to understand more about the universe around us from its largest scales ([[physical cosmology]]) to its smallest ([[quantum mechanics]]). Indeed, the very success of mathematics in this respect has been a source of puzzlement for some philosophers (see [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]] by [[Eugene Wigner]]).
* '''Mathematics describes abstract structures''': on the other hand, there are areas of pure mathematics which deal with [[abstract structure]]s, which have no known physical counterparts at all. However, it is difficult to give any categorical examples here, as even the most abstract structures can be co-opted as models in some branch of physics (see [[Calabi-Yau spaces]] and [[string theory]]).
* '''Mathematics describes mathematics''': mathematics can be used reflexively to describe itself—this is an area of mathematics called [[metamathematics]].
Mathematics can communicate a range of meanings that is as wide as (although different from) that of a natural language. As [[Germany|German]] mathematician [[R.L.E. Schwarzenberger]] says:
:''My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the [[Fundamental Theorem of Algebra]] using the English language.'' - Schwarzenberger (2000)
==Alternative views==
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 language]]s, 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:
: ''Mathematics would appear to be both more and less than a language for while being limited in its linguistic capabilities it also seems to involve a form of thinking that has something in common with art and music.'' - Ford & Peat (1988)
==References==
* R. L. E. Schwarzenberger (2000), ''The Language of Geometry'', published in ''A Mathematical Spectrum Miscellany'', Applied Probability Trust.
* Alan Ford & F. David Peat (1988), ''The Role of Language in Science'', Foundations of Physics Vol 18.
==See also==
* [[Linguistics]]
* [[Philosophy of language]]
==External links==
*[http://www.aucegypt.edu/academic/anth/anth352/what_is_language.htm What is Language]
*[http://www.fdavidpeat.com/bibliography/essays/maths.htm ''Mathematics and the Language of Nature''] - essay by F. David Peat
[[Category:Philosophy of mathematics]]