Mathematics is a language

The language of mathematics is the system used by s to communicate ideas among themselves. This language consists of a of some  (for example ) using s and grammatical conventions that are peculiar to mathematical discourse (see ), supplemented by a highly specialized symbolic notation for s.

Like natural languages in general, discourse using the language of mathematics can employ a scala of s. s in are sources for detailed theoretical discussions about ideas concerning mathematics and its implications for society.

What is a language?
Here are some definitions of :
 * 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 -.

These definitions describe language in terms of the following components:
 * A of symbols or words
 * A 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 of people who use and understand these symbols
 * A range of that can be communicated with these symbols

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

The vocabulary of mathematics
has assimilated from many different s and s. It also includes symbols that are specific to mathematics, such as


 * $$\forall \ \exists \ \vee \ \wedge \ \infty.$$

Mathematical notation is central to the power of modern mathematics. Though the of  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 using symbolic notation can simply be called $$x, y, z$$, etc.). Sometimes formulas cannot be understood without a written or spoken explanation, but often they are sufficient by themselves, and sometimes they are 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 is manipulated.

Like any other profession, mathematics also has its own brand of. In some cases, a word in general usage has a different and specific meaning within mathematics—examples are, , , , , and. For more examples, see.

In other cases, specialist terms have been created which do not exist outside of mathematics—examples are, ,. Mathematical statements have their own moderately complex taxonomy, being divided into s, s, s, s and. And there are stock phrases in mathematics, used with specific meanings, such as "", "" and "". Such phrases are known as.

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 also help visually and aid intuitive calculations. Sometimes, as in a, a diagram even serves as complete justification for a proposition. A system of diagram conventions may evolve into a mathematical notation – for example, the for tensor products.

The grammar of mathematics
The mathematical notation used for formulas has its own, 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, even when the writing system of the substrate language is right-to-left, and that the is commonly used for simple s and s. A formula such as
 * $$\sin x + a\cos 2 x \ge 0$$

is understood by Chinese and Syrian mathematicians alike.

Such mathematical formulas can be a in a natural-language phrase, or even assume the role of a full-fledged sentence. For example, the formula above, an, can be considered a sentence or an independent clause in which the symbol has the role of a symbolic. 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 has the role of an.

Mathematical formulas can be  (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 "$$\tfrac{dy}{dx}$$", on the other hand, is commonly vocalized like "dee-why-dee-eks", with complete omission of the, in other contexts often pronounced "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   "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 $$=,\ \in,\ \exists$$, 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 natural sciences disciplines 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 $$(a_n)$$ and $$(b_n)$$ are convergent sequences of real numbers, and $\lim_{n\to\infty} a_n=A$, $\lim_{n\to\infty} b_n=B$ , then $$(c_n)$$, defined for all positive integers $$n$$ by $$c_n = a_n+b_n$$, is convergent, and
 * $$\lim_{n\to\infty} c_n=A+B$$.

In this statement, "$$(a_n)$$" (in which $$(a_n)$$ is read as "ay en" or perhaps, more formally, as "the sequence ay en") and "$$(b_n)$$" are treated as nouns, while "$\lim_{n\to\infty} a_n=A$ " (read: the limit of $$a_n$$ as n tends to infinity equals 'big A'), "$\lim_{n\to\infty} b_n=B$ ", and "$\lim_{n\to\infty} c_n=A+B$ " are read as independent clauses, and "$$c_n = a_n+b_n$$" is read as "the equation $$c_n$$ equals $$a_n$$ plus $$b_n$$". Moreover, the sentence ends after the displayed equation, as indicated by the after "$\lim_{n\to\infty} c_n=A+B$ ". In terms of typesetting conventions, broadly speaking, standard mathematical functions such as $$ and operations such as $+$ as well as punctuation symbols including the various s are set in $$ while Latin alphabet variables are set in $$. Matrices, vectors, and other objects made up of components are set in $bold roman$. (There is some disagreement as to whether the standard constants (e.g., $e$, π, 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 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 s to be found or as arguments of a function, while $a$, $b$, $c$ are used for s and $f$, $g$, $h$ are mostly used as names of functions. These conventions are not hard rules. Instead these suggestions are met to enhance readability and to provide an intuition for of what kind a given object is, 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]" (for example, "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 "".

have generally a title or label in bold type, and possibly identify the originator (for example, "$Theorem 1.4 (Weyl).$"). This is immediately followed by the statement of the theorem, 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 ("∎") or another symbol, or by the letters.

The language community of mathematics
Mathematics is used by, 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, 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. 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 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 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 with only that supervision which is requisite to manipulate the symbols in accordance with the rules.

Williams' essay was a 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 to falling apples  to gambling . Mathematics is widely used in modern  and, and has been hugely successful in helping us to understand more about the universe around us from its largest scales  to its smallest . Indeed, the very success of mathematics in this respect has been a source of puzzlement for some philosophers (see  by ).
 * Mathematics describes abstract structures: on the other hand, there are areas of pure mathematics which deal with 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 and ).
 * Mathematics describes mathematics: mathematics can be used reflexively to describe itself—this is an area of mathematics called.

Mathematics can communicate a range of meanings that is as wide as (although different from) that of a natural language. As mathematician  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 using the English language.

Alternative views
Some definitions of language, such as early versions of '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 out loud). However, these definitions would also disqualify 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)