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 origins of mathematics, the practice of mathematics or the philosophy of mathematics ?

1 What is a language?
2 The vocabulary of mathematics
3 The grammar of mathematics
4 The language community of mathematics
5 The meanings of mathematics
6 Alternative views
7 References
8 See also
9 External links

[edit] 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 WordNet
a system of words used in a particular discipline WordNet
the code we all use to express ourselves and communicate to others 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 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.

[edit] The vocabulary of mathematics

Mathematical notation has assimilated symbols from many different alphabets and fonts. It also includes symbols that are specific to mathematics, such as

$\forall \ \exists \ \nabla \ \wedge \ \infty$

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 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.

[edit] 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.^{[citation needed]}

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.

[edit] The language community of mathematics

Mathematics 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.

[edit] 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 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 structures, 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 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)

[edit] 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 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:

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)

[edit] 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.