The language of mathematics or mathematical language is an extension of the

natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...

(for example English) that is used in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and in

science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

for expressing results (

scientific law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow ...

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

s, logical deductions, etc.) with concision, precision and unambiguity.

Features

The main features of the mathematical language are the following. * Use of common words with a derived meaning, generally more specific and more precise. For example, " or" means "one, the other or both", while, in common language, "both" is sometimes included and sometimes not. Also, a " line" is straight and has zero width. * Use of common words with a meaning that is completely different from their common meaning. For example, a mathematical ring is not related to any other meaning of "ring".

Real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s and

imaginary number An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...

s are two sorts of numbers, none being more real or more imaginary than the others. * Use of

neologism In linguistics, a neologism (; also known as a coinage) is any newly formed word, term, or phrase that has achieved popular or institutional recognition and is becoming accepted into mainstream language. Most definitively, a word can be considered ...

s. For example

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

. * Use of

symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...

s as words or phrases. For example,

A=B

and

\forall x

are respectively read as "

A

equals

B

" and * Use of

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

s as part of sentences. For example: "''represents quantitatively the

mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstei ...

.''" A formula that is not included in a sentence is generally meaningless, since the meaning of the symbols may depend on the context: in "", this is the context that specifies that is the

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

of a

physical body In natural language and physical science, a physical object or material object (or simply an object or body) is a wiktionary:contiguous, contiguous collection of matter, within a defined boundary (or surface), that exists in space and time. Usual ...

, is its

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

, and is the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

. * Use of phrases that cannot be decomposed into their components. In particular, adjectives do not always restrict the meaning of the corresponding noun, and may change the meaning completely. For example, most

algebraic integer In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...

s are not

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s and integers are specific algebraic integers. So, an ''algebraic integer'' is not an ''integer'' that is ''algebraic''. * Use of

mathematical jargon The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...

that consists of phrases that are used for informal explanations or shorthands. For example, "killing" is often used in place of "replacing with zero", and this led to the use of '' assassinator'' and '' annihilator'' as technical words.

Understanding mathematical text

The consequence of these features is that a mathematical text is generally not understandable without some prerequisite knowledge. For example, the sentence "''a

free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...

is a module that has a basis''" is perfectly correct, although it appears only as a grammatically correct nonsense, when one does not know the definitions of ''basis'', ''module'', and ''free module''. H. B. Williams, an electrophysiologist, wrote in 1927:

Features

Understanding mathematical text

See also

References

Further reading

Linguistic point of view

In education