Philosophy:Language of mathematics

The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves, and is distinct from natural languages in that it aims to communicate abstract, logical ideas with precision and unambiguity. This language consists of a substrate of some natural language (e.g., English), using technical terms and grammatical conventions that are peculiar to mathematical discourse (see mathematical jargon). It is also supplemented by a highly specialized symbolic notation for mathematical formulas.

Similar to natural languages, discourse using the language of mathematics can employ a scala of registers. Research articles in academic journals are sources for detailed theoretical discussions about ideas concerning mathematics and its implications for society.

What is a language?

Here are some definitions of language:

• A systematic means of communicating by the use of sounds or conventional symbols
• A system of words used in a particular discipline
• A system of abstract codes which represent antecedent events and concepts
• 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 'syntax' or propositional structure, which places the symbols in linear structures.
• A 'discourse' or 'narrative,' consisting of strings of syntactic propositions
• A community of people who use and understand these symbols
• A range of meanings that can be communicated with these symbols

Each of these components is also found in the language of mathematics.

The vocabulary of mathematics

Mathematical notation has assimilated symbols from many different alphabets (e.g., Greek, Hebrew, Latin) and typefaces (e.g., cursive, calligraphic, blackboard bold). It also includes symbols that are specific to mathematics, such as

[math]\displaystyle{ \forall \ \exists \ \vee \ \wedge \ \infty. }[/math]

Mathematical notation is central to the power of modern mathematics. Though the algebra of Al-Khwārizmī did not use such symbols, it solved equations using many more rules than are used today with symbolic notation, and had great difficulty working with multiple variables (which through symbolic notation can be simply denoted as [math]\displaystyle{ x, y, z }[/math], etc.).

Sometimes, formulas cannot be understood without a written or spoken explanation, but often they are sufficient by themselves. In other occasions, they can be difficult to read aloud or information is lost in the translation to words, as when several parenthetical factors are involved or when a complex structure like a matrix is manipulated.

In other cases, specialist terms, such as "tensor", "fractal" and "functor", have been created exclusively for the use in mathematics. Mathematical statements have their own moderately complex taxonomy, being divided into axioms, conjectures, propositions, 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.

The vocabulary of mathematics also has visual elements. Diagrams are used informally on blackboards, as well as more formally in published work. When used appropriately, diagrams display schematic information more easily. Diagrams can also help visually and aid intuitive calculations. Sometimes, as in a visual proof, a diagram can even serve as complete justification for a proposition. A system of diagram conventions may evolve into a mathematical notation, such as the case of the Penrose graphical notation for tensor products.

The grammar of mathematics

The mathematical notation used for formulas has its own grammar, not dependent on a specific natural language, but shared internationally by mathematicians regardless of their mother tongues. This includes the conventions that the formulas are written predominantly left to right, even when the writing system of the substrate language is right-to-left, and that the Latin alphabet is commonly used for simple variables and parameters. A formula such as

[math]\displaystyle{ \sin x + a\cos 2 x \ge 0 }[/math]

is understood by Chinese and Syrian mathematicians alike.

Such mathematical formulas can be a part of speech in a natural-language phrase, or even assume the role of a full-fledged sentence. For example, the formula above, an inequation, can be considered a sentence or an independent clause in which the greater than or equal to symbol has the role of a symbolic verb. In careful speech, this can be made clear by pronouncing "≥" as "is greater than or equal to", but in an informal context mathematicians may shorten this to "greater or equal" and yet handle this grammatically like a verb. A good example is the book title Why does E = mc²?; here, the equals sign has the role of an infinitive.

Mathematical formulas can be vocalized (i.e., spoken aloud). The vocalization system for formulas has to be learned, and is dependent on the underlying natural language. For example, when using English, the expression "ƒ(x)" is conventionally pronounced "eff of eks", where the insertion of the preposition "of" is not suggested by the notation per se. The expression "[math]\displaystyle{ \tfrac{dy}{dx} }[/math]", on the other hand, is commonly vocalized like "dee-why-dee-eks", with complete omission of the fraction bar, which in other contexts are often pronounced as "over". The book title Why does E = mc²? is said aloud as Why does ee equal em see-squared?.

Characteristic for mathematical discourse – both formal and informal – is the use of the inclusive first person plural "we" to mean: "the audience (or reader) together with the speaker (or author)".

Typographical conventions

As is the case for spoken mathematical language, in written or printed mathematical discourse, mathematical expressions containing a symbolic verb, like [math]\displaystyle{ =,\ \in,\ \exists }[/math], are generally treated as clauses (dependent or independent) in sentences or as complete sentences, and are punctuated as such by mathematicians and theoretical physicists. In particular, this is true for both inline and displayed expressions. In contrast, writers in other disciplines of natural science may try to avoid using equations within sentences, and may treat displayed expressions in the same way as figures or schemes.

As an example, a mathematician might write:

If [math]\displaystyle{ (a_n) }[/math] and [math]\displaystyle{ (b_n) }[/math] are convergent sequences of real numbers, and [math]\displaystyle{ \lim_{n\to\infty} a_n=A }[/math], [math]\displaystyle{ \lim_{n\to\infty} b_n=B }[/math], then [math]\displaystyle{ (c_n) }[/math], defined for all positive integers [math]\displaystyle{ n }[/math] by [math]\displaystyle{ c_n = a_n+b_n }[/math], is convergent, and

[math]\displaystyle{ \lim_{n\to\infty} c_n=A+B }[/math].

In this statement, "[math]\displaystyle{ (a_n) }[/math]" (in which [math]\displaystyle{ (a_n) }[/math] is read as "ay en" or perhaps, more formally, as "the sequence ay en") and "[math]\displaystyle{ (b_n) }[/math]" are treated as nouns, while "[math]\displaystyle{ \lim_{n\to\infty} a_n=A }[/math]" (read: the limit of [math]\displaystyle{ a_n }[/math] as n tends to infinity equals 'big A'), "[math]\displaystyle{ \lim_{n\to\infty} b_n=B }[/math]", and "[math]\displaystyle{ \lim_{n\to\infty} c_n=A+B }[/math]" are read as independent clauses, and "[math]\displaystyle{ c_n = a_n+b_n }[/math]" is read as "the equation [math]\displaystyle{ c_n }[/math] equals [math]\displaystyle{ a_n }[/math] plus [math]\displaystyle{ b_n }[/math]".

Moreover, the sentence ends after the displayed equation, as indicated by the period after "[math]\displaystyle{ \lim_{n\to\infty} c_n=A+B }[/math]". In terms of typesetting conventions, broadly speaking, standard mathematical functions such as sin and operations such as +, as well as punctuation symbols including the various brackets, are set in roman type, while Latin alphabet variables are set in italics. On the other hand, matrices, vectors and other objects made up of components are sometimes set in bold roman (mostly in elementary texts), and sometimes in italic (mostly in advanced texts).

(There is some disagreement as to whether the standard constants, such as e, π and i = (–1)^1/2, or the "d" in dy/dx should be italicized. Upper case Greek letters are almost always set in roman, while lower case ones are often italicized.)

There are also a number of conventions, or, more exactly traditions, for the part of the alphabet from which variable names are chosen. For example, i, j, k, l, m, n are usually reserved for integers, w and z are often used for complex numbers, while a, b, c, α, β, γ are used for real numbers. The letters x, y, z are frequently used for unknowns to be found or as arguments of a function, while a, b, c are used for coefficients and f, g, h are mostly used as names of functions. These conventions are not hard rules, but instead are suggestions to be met to enhance readability, and to provide an intuition for the nature of a given object, so that one has neither to remember, nor to check the introduction of the mathematical object.

Definitions are signaled by words like "we call", "we say" or "we mean", or by statements like "An [object] is [word to be defined] if [condition]" (e.g., "A set is closed if it contains all of its limit points."). As a special convention, the word "if" in such a definition should be interpreted as "if and only if".

Theorems have generally a title or label in bold type, and might even identify its originator (e.g., "Theorem 1.4 (Weyl)."). This is immediately followed by the statement of the theorem, which in turn is usually set in italics. The proof of a theorem is usually clearly delimited, starting with the word Proof, while the end of the proof is indicated by a tombstone ("∎ or □") or another symbol, or by the letters Q.E.D.

The language community of mathematics

Mathematics is used by mathematicians, who form a global community composed of speakers of many languages. 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. 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.

Concise expression

The power of mathematics lies in economy of expression of ideas, often in service to science. Horatio Burt Williams took note of the effect of this compact form in physics:

Textbooks of physics of seventy-five years ago were much larger than at present. This in spite of the enormous additions since made to our knowledge of the subject. But these older books were voluminous because of minute descriptions of phenomena which we now recognize as what a mathematician would call particular cases, comprehended under broad general principles.

In mathematics per se, the brevity is profound:

In writing papers which will probably be read only by professional mathematicians, authors not infrequently omit so many intermediate steps in order to condense their papers that the filling in of the gaps even by industrious use of paper and pencil may become no inconsiderable labor, especially to one approaching the subject for the first time.

Williams cites Ampère as a scientist that summarized his findings with mathematics:

The smooth and concise demonstration is not necessarily conceived in that finished form...We can scarcely believe that Ampère discovered the law of action by means of the experiment which he describes. We are led to suspect, what indeed, he tells us himself, that he discovered the law by some process which he has not shewn us, and that when he had afterwards built up a perfect demonstration, he removed all traces of the scaffolding by which he raised it.

The significance of mathematics lies in the logical processes of the mind have been codified by mathematics:

Now mathematics is both a body of truth and a special language, a language more carefully defined and more highly abstracted than our ordinary medium of thought and expression. Also it differs from ordinary languages in this important particular: it is subject to rules of manipulation. Once a statement is cast into mathematical form it may be manipulated in accordance with these rules and every configuration of the symbols will represent facts in harmony with and dependent on those contained in the original statement. Now this comes very close to what we conceive the action of the brain structures to be in performing intellectual acts with the symbols of ordinary language. In a sense, therefore, the mathematician has been able to perfect a device through which a part of the labor of logical thought is carried on outside the central nervous system with only that supervision which is requisite to manipulate the symbols in accordance with the rules.

Williams' essay was a Gibbs Lecture prepared for scientists in general, and he was particularly concerned that biological scientists not be left behind:

Not alone the chemist and physicist, but the biologist as well, must be able to read mathematical papers if he is not to be cut off from the possibility of understanding important communications in his own field of science. And the situation here is worse than it is in the case of inability to read a foreign language. For a paper in a foreign language may be translated, but in many cases it is impossible to express in ordinary language symbols the content of a mathematical paper in such a way as to convey a knowledge of the logical process by which the conclusions have been reached.

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 applied 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 England 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.

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."

See also

• Formulario mathematico
• Formal language
• History of mathematical notation
• Mathematical notation
• Linguistics
• Philosophy of language
• Rewriting

Further reading

• Knight, Isabel F. (1968). The Geometric Spirit: The Abbe de Condillac and the French Enlightenment. New Haven: Yale University Press. 
• R. L. E. Schwarzenberger (2000), The Language of Geometry, published in A Mathematical Spectrum Miscellany, Applied Probability Trust.
• Kay O'Halloran (2004) Mathematical Discourse: Language, Symbolism and Visual Images, Continuum ISBN 0826468578
• Charles Wells (2017) Languages of Mathematics from abstractmath.org

External links

• What is Language
• Mathematics and the Language of Nature - essay by F. David Peat.
• Mathematical Words: Origins and Sources (John Aldrich, University of Southampton)
• Communicating in the Language of Mathematics by Dr. David Moursund
• Handbook of Mathematical Discourse by Charles Wells.
• One Mathematical Cat, Please! Exploring the Language of Mathematics by Dr. Carol JVF Burns