Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of

an

"math, ''n.3''"

. ''Oxford English Dictionary,'' on-line version (2012).

online

(registration required). * * * – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, an

online

. * * * {{Authority control Formal sciences Main topic articles

number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...

, geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, and analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline
An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, higher learning (and generally also research or honorary membershi ...

.
Most mathematical activity involves the discovery of properties of abstract objects
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...

and the use of pure reason
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...

to prove them. These objects consist of either abstraction
Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or " concrete") signifiers, first principles, or other methods.
"An abst ...

s from nature orin modern mathematicsentities that are stipulated to have certain properties, called axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

s, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of the theory under consideration.
Mathematics is essential in the natural science
Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...

s, engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, medicine
Medicine is the science and practice of caring for a patient, managing the diagnosis, prognosis, prevention, treatment, palliation of their injury or disease, and promoting their health. Medicine encompasses a variety of health care pr ...

, finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...

, computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

and the social sciences
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of so ...

. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

and game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

, are developed in close correlation with their applications and are often grouped under applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...

. Other areas are developed independently from any application (and are therefore called pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

), but often later find practical applications. The problem of integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are s ...

, for example, which goes back to Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

in 300 BC, had no practical application before its use in the RSA cryptosystem
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publi ...

, now widely used for the security of computer network
A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are ...

s.
Historically, the concept of a proof and its associated mathematical rigour
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...

first appeared in Greek mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...

, most notably in Euclid's '' Elements''. Since its beginning, mathematics was essentially divided into geometry and arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

(the manipulation of natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

s and fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

), until the 16th and 17th centuries, when algebra and infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...

were introduced as new areas. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...

led to the systematization of the axiomatic method
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...

, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification
The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH ...

lists more than 60 first-level areas of mathematics.
Etymology

The word ''mathematics'' comes fromAncient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...

''máthēma'' ('), meaning 'that which is learnt', 'what one gets to know', hence also 'study' and 'science'. The word came to have the narrower and more technical meaning of 'mathematical study' even in Classical times
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...

. Its adjective
In linguistics, an adjective ( abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun.
Traditionally, adjectives were considered one of the ...

is ''mathēmatikós'' (), meaning 'related to learning' or 'studious', which likewise further came to mean 'mathematical'. In particular, ''mathēmatikḗ tékhnē'' (; la, ars mathematica) meant 'the mathematical art'.
Similarly, one of the two main schools of thought in Pythagoreanism
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...

was known as the ''mathēmatikoi'' (μαθηματικοί)which at the time meant 'learners' rather than 'mathematicians' in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

and geometry. By the time of Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

(384–322 BC) this meaning was fully established.
In Latin, and in English until around 1700, the term ''mathematics'' more commonly meant 'astrology
Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Di ...

' (or sometimes 'astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

') rather than 'mathematics'; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine
Augustine of Hippo ( , ; la, Aurelius Augustinus Hipponensis; 13 November 354 – 28 August 430), also known as Saint Augustine, was a theologian and philosopher of Berber origin and the bishop of Hippo Regius in Numidia, Roman North Afr ...

's warning that Christians should beware of ''mathematici'', meaning 'astrologers', is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural
The plural (sometimes list of glossing abbreviations, abbreviated pl., pl, or ), in many languages, is one of the values of the grammatical number, grammatical category of number. The plural of a noun typically denotes a quantity greater than the ...

form in English goes back to the Latin neuter plural (Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...

), based on the Greek plural ''ta mathēmatiká'' () and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective ''mathematic(al)'' and formed the noun ''mathematics'' anew, after the pattern of ''physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

'' and ''metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...

'', inherited from Greek. In English, the noun ''mathematics'' takes a singular verb. It is often shortened to ''maths'' or, in North America, ''math''."maths, ''n.''"an

"math, ''n.3''"

. ''Oxford English Dictionary,'' on-line version (2012).

Areas of mathematics

Before theRenaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...

, mathematics was divided into two main areas: arithmeticregarding the manipulation of numbers, and geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, regarding the study of shapes. Some types of pseudoscience
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contradictory, exaggerated or unfalsifiable claim ...

, such as numerology
Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in ...

and astrology, were not then clearly distinguished from mathematics.
During the Renaissance, two more areas appeared. Mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...

led to algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...

which, roughly speaking, consists of the study and the manipulation 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. Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

, consisting of the two subfields ''differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...

'' and ''integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

'', is the study of continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...

, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areasarithmetic, geometry, algebra, calculusendured until the end of the 19th century. Areas such as celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...

and solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...

were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.
At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...

led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification
The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH ...

contains no less than first-level areas. Some of these areas correspond to the older division, as is true regarding number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

(the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

and foundations.
Number theory

Number theory began with the manipulation ofnumber
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...

s, that is, natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

s $(\backslash mathbb),$ and later expanded to integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s $(\backslash Z)$ and rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s $(\backslash Q).$ Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. Number theory dates back to ancient Babylon
''Bābili(m)''
* sux, 𒆍𒀭𒊏𒆠
* arc, 𐡁𐡁𐡋 ''Bāḇel''
* syc, ܒܒܠ ''Bāḇel''
* grc-gre, Βαβυλών ''Babylṓn''
* he, בָּבֶל ''Bāvel''
* peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru''
* elx, 𒀸𒁀𒉿𒇷 ''Babi ...

and probably China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...

. Two prominent early number theorists were Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

of ancient Greece and Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

and Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

. The field came to full fruition with the contributions of Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...

and Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

.
Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

, and homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...

. Another example is Goldbach's conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...

, which asserts that every even integer greater than 2 is the sum of two prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. Stated in 1742 by Christian Goldbach
Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling ...

, it remains unproven despite considerable effort.
Number theory includes several subareas, including analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...

, algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...

, geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informa ...

(method oriented), diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...

s, and transcendence theory (problem oriented).
Geometry

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines,angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...

s and circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

s, which were developed mainly for the needs of surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ...

and architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing buildings ...

, but has since blossomed out into many other subfields.
A fundamental innovation was the ancient Greeks' introduction of the concept of proofs
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 ...

, which require that every assertion must be ''proved''. For example, it is not sufficient to verify by measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...

that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

s) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates), or are part of the definition of the subject of study (axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book '' Elements''.
The resulting Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

is the study of shapes and their arrangements constructed from lines, planes
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

and circles in the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

(plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

) and the three-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

.
Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...

introduced what is now called Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

. This constituted a major change of paradigm: Instead of defining real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s as lengths of line segments (see number line), it allowed the representation of points using their ''coordinates'', which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...

, which uses purely geometrical methods, and analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and enginee ...

, which uses coordinates systemically.
Analytic geometry allows the study of curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

. They can also be defined as implicit equation
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...

s, often polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...

s (which spawned algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.
In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segmen ...

. By questioning that postulate's truth, this discovery has been viewed as joining Russel's paradox in revealing the foundational crisis of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...

. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...

.
Today's subareas of geometry include:
*Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...

, introduced in the 16th century by Girard Desargues
Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are ...

, extends Euclidean geometry by adding points at infinity at which parallel lines
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or int ...

intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
*Affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...

, the study of properties relative to parallelism and independent from the concept of length.
*Differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

, the study of curves, surfaces, and their generalizations, which are defined using differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

s.
* Manifold theory, the study of shapes that are not necessarily embedded in a larger space.
*Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

, the study of distance properties in curved spaces.
*Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the study of curves, surfaces, and their generalizations, which are defined using polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

s.
*Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, the study of properties that are kept under continuous deformations.
**Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, the use in topology of algebraic methods, mainly homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...

.
*Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...

, the study of finite configurations in geometry.
*Convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of ...

, the study of convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s, which takes its importance from its applications in optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

.
*Complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...

, the geometry obtained by replacing real numbers with complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s.
Algebra

Algebra is the art of manipulatingequation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...

s and formulas. Diophantus (3rd century) and al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...

(9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term ''algebra'' is derived from the Arabic
Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...

word ''al-jabr'' meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise.
Algebra became an area in its own right only with François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

(1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.
Until the 19th century, algebra consisted mainly of the study of linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

s (presently ''linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...

''), and polynomial equations in a single unknown
Unknown or The Unknown may refer to:
Film
* ''The Unknown'' (1915 comedy film), a silent boxing film
* ''The Unknown'' (1915 drama film)
* ''The Unknown'' (1927 film), a silent horror film starring Lon Chaney
* ''The Unknown'' (1936 film), a ...

, which were called ''algebraic equations'' (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

, modular integers, and geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...

s), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

addresses this, consisting of a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called ''modern algebra'' or abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

, as established by the influence and works of Emmy Noether. (The latter term appears mainly in an educational context, in opposition to elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...

, which is concerned with the older way of manipulating formulas.)
Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:
*group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

;
* field theory;
*vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

s, whose study is essentially the same as linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...

;
*ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

;
*commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

, which is the study of commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

s, includes the study of polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

s, and is a foundational part of algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

;
*homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topolo ...

;
*Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

and Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...

theory;
*Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

, which is widely used for the study of the logical structure of computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...

s.
The study of types of algebraic structures as mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...

s is the purpose of universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...

and category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...

. The latter applies to every mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additiona ...

(not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

s; this particular area of application is called algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

.
Calculus and analysis

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton andLeibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...

. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...

with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.
Analysis is further subdivided into real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...

, where variables represent real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, and complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, where variables represent complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s. Analysis includes many subareas shared by other areas of mathematics which include:
* Multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with Function of several real variables, functions of several variables: the Differential calculus, di ...

* Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

, where variables represent varying functions;
* Integration, measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...

and potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

, all strongly related with probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

on a continuum;
* Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s;
* Partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...

s;
* Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.
Discrete mathematics

Discrete mathematics, broadly speaking, is the study of individual,countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

sespecially their implementation
Implementation is the realization of an application, or execution of a plan, idea, model, design, specification, standard, algorithm, or policy.
Industry-specific definitions
Computer science
In computer science, an implementation is a real ...

and computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...

play a major role in discrete mathematics.
The four color theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...

and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.
Discrete mathematics includes:
* Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

s of a given set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...

s
* Graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

and hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, an undirected hypergraph H is a pair H = (X,E) w ...

s
* Coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...

, including error correcting codes and a part of cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adv ...

* Matroid theory
* Discrete geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...

* Discrete probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

s
* Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

(although continuous game
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the playe ...

s are also studied, most common games, such as chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

and poker
Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game wa ...

are discrete)
* Discrete optimization, including combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combi ...

, integer programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective ...

, constraint programming
Mathematical logic and set theory

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, andlogic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

, although used for mathematical proofs, belonged to philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...

and was not specifically studied by mathematicians.
Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...

to be the result of endless enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...

. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

. This led to the controversy over Cantor's set theory.
In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...

. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.
This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...

, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism
Formalism may refer to:
* Form (disambiguation)
* Formal (disambiguation)
* Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
* Formalism (linguistics)
* Scien ...

, as founded by David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

around 1910.
The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinionsometimes called "intuition"to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phil ...

assert, roughly speaking that, in every consistent formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A fo ...

that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'.
Brouwer
* Adriaen Brouwer (1605–1638), Flemish painter
* Alexander Brouwer (b. 1989), Dutch beach volleyball player
* Andries Brou ...

, who promoted intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...

, which explicitly lacks the law of excluded middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

.
These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...

(modeling some logical theories inside other theories), proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...

, type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

, computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...

and computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...

. Although these aspects of mathematical logic were introduced before the rise of computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...

s, their use in compiler
In computing, a compiler is a computer program that translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily used for programs tha ...

design, program certification, proof assistant
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof edi ...

s and other aspects of computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, contributed in turn to the expansion of these logical theories.
Statistics and other decision sciences

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especiallyprobability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

. Statisticians generate data with random sampling
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attemp ...

or randomized experiments. The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...

s and the theory of inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and in ...

, using model selection and estimation
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...

. The models and consequential predictions should then be tested
''Tested'' is a live album by punk rock band Bad Religion. It was recorded in the USA, Canada, Germany, Estonia, Denmark, Italy and Austria, in 1996, and released in 1997. It is Bad Religion's second live album. Instead of using crowd microphon ...

against new data.
Statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics.
The theory covers approaches to statistical-decision problems and to statistica ...

studies decision problems such as minimizing the risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...

(expected loss Expected loss is the sum of the values of all possible losses, each multiplied by the probability of that loss occurring.
In bank lending (homes, autos, credit cards, commercial lending, etc.) the expected loss on a loan varies over time for a nu ...

) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...

, and selecting the best. In these traditional areas of mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...

, a statistical-decision problem is formulated by minimizing an objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...

, like expected loss or cost
In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one of acquisition, in whic ...

, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, the mathematical theory of statistics overlaps with other decision science
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...

s, such as operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decis ...

, control theory
Control theory is a field of mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive ...

, and mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference ...

.
Computational mathematics

Computational mathematics is the study ofmathematical problem
A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...

s that are typically too large for human, numerical capacity. Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

studies methods for problems in analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

using functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

and approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...

; numerical analysis broadly includes the study of approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

and discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerica ...

with special focus on rounding error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...

s. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

-and-graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

. Other areas of computational mathematics include computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

and symbolic computation
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...

.
History

Ancient

The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are of them. As evidenced by tallies found on bone, in addition to recognizing how tocount
Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York ...

physical objects, prehistoric
Prehistory, also known as pre-literary history, is the period of human history between the use of the first stone tools by hominins 3.3 million years ago and the beginning of recorded history with the invention of writing systems. The us ...

peoples may have also known how to count abstract quantities, like timedays, seasons, or years.
Evidence for more complex mathematics does not appear until around 3000 , when the Babylonia
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c ...

ns and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia
Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the ...

and Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a List of transcontinental countries, transcontinental country spanning the North Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via a land bridg ...

are from 2000 to 1800 BC. Many early texts mention Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

s and so, by inference, the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic
The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type ...

(addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...

, subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

, multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

numeral system which is still in use today for measuring angles and time.
In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...

such as the Pythagoreans
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...

appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, '' Elements'', is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

(c. 287–212 BC) of Syracuse
Syracuse may refer to:
Places Italy
* Syracuse, Sicily, or spelled as ''Siracusa''
* Province of Syracuse
United States
*Syracuse, New York
**East Syracuse, New York
** North Syracuse, New York
* Syracuse, Indiana
*Syracuse, Kansas
*Syracuse, M ...

. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...

to calculate the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

under the arc of a parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...

with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

(Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributio ...

, 3rd century BC), trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

( Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD).
The Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...

and were transmitted to the Western world
The Western world, also known as the West, primarily refers to the various nations and states in the regions of Europe, North America, and Oceania.

via Islamic mathematics
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics ( Aryabhata, Brahmagupta). Important progress was made, such as ...

. Other notable developments of Indian mathematics include the modern definition and approximation of sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...

and cosine, and an early form of infinite series.
Medieval and later

During theGolden Age of Islam
The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...

, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...

and Sharaf al-Dīn al-Ṭūsī. The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.
During the early modern period, mathematics began to develop at an accelerating pace in Western Europe
Western Europe is the western region of Europe. The region's countries and territories vary depending on context.
The concept of "the West" appeared in Europe in juxtaposition to "the East" and originally applied to the ancient Mediterranean ...

, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

(1646–1716) in the 17th century. Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

, matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...

, number theory, and statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

. In the early 20th century, Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...

transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic systemif powerful enough to describe arithmeticwill contain true propositions that cannot be proved.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence ...

, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the ''Bulletin of the American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society.
Scope
It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

'', "The number of papers and books included in the ''Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
The AMS also pu ...

'' database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."
Symbolic notation and terminology

Mathematical notation is widely used in science andengineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

for representing complex concept
Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
They play an important role in all aspects of cognition. As such, concepts are studied by ...

s and properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy an ...

in a concise, unambiguous, and accurate way. This notation consists of symbols
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different co ...

used for representing operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...

s, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

or Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...

letters, and often include subscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...

s. Operation and relations are generally represented by specific symbols
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different co ...

or glyph
A glyph () is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A g ...

s, such as (plus
Plus may refer to:
Mathematics
* Addition
* +, the mathematical sign
Music
* ''+'' (Ed Sheeran album), (pronounced "plus"), 2011
* ''Plus'' (Cannonball Adderley Quintet album), 1961
* ''Plus'' (Matt Nathanson EP), 2003
* ''Plus'' (Martin Ga ...

), (multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

), $\backslash int$ (integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

), ( equal), and (less than
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different ...

). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrase
In linguistics, a noun phrase, or nominal (phrase), is a phrase that has a noun or pronoun as its head or performs the same grammatical function as a noun. Noun phrases are very common cross-linguistically, and they may be the most frequently oc ...

s and formulas play the role of clause
In language, a clause is a constituent that comprises a semantic predicand (expressed or not) and a semantic predicate. A typical clause consists of a subject and a syntactic predicate, the latter typically a verb phrase composed of a verb wit ...

s.
Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...

that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...

. Through a series of rigorous arguments employing deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...

, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), ...

. A proven instance that forms part of a more general finding is termed a corallary.
Numerous technical terms used in mathematics are neologism
A neologism Ancient_Greek.html"_;"title="_from_Ancient_Greek">Greek_νέο-_''néo''(="new")_and_λόγος_/''lógos''_meaning_"speech,_utterance"is_a_relatively_recent_or_isolated_term,_word,_or_phrase_that_may_be_in_the_process_of_entering_com ...

s, such as ''polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

'' and ''homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...

''. Other technical terms are words of the common language that are used in an accurate meaning that may differs slightly from their common meaning. For example, in mathematics, " or" means "one, the other or both", while, in common language, it is either amiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...

"). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fiel ...

is flat" and "a field is always a ring".
Relationship with sciences

Mathematics is used in mostscience
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence ...

s for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

's general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, which replaced Newton's law of gravitation as a better mathematical model.
There is still a philosophical
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable
Falsifiability is a standard of evaluation of scientific theories and hypotheses that was introduced by the philosopher of science Karl Popper in his book '' The Logic of Scientific Discovery'' (1934). He proposed it as the cornerstone of a so ...

, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...

. Similarly as in science, theories and results (theorems) are often obtained from experimentation
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a ...

. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not on empirical evidence.
Pure and applied mathematics

Until the 19th century, the development of mathematics in the West was mainly motivated by the needs oftechnology
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, scien ...

and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

s with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece.
In the 19th century, mathematicians such as Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...

and Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...

increasingly focused their research on internal problems, that is, ''pure mathematics''. This led to split mathematics into ''pure mathematics'' and ''applied mathematics'', the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.
The aftermath of World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

led to a surge in the development of applied mathematics in the US and elsewhere. Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".
An example of the first case is the theory of distributions, introduced by Laurent Schwartz
Laurent-Moïse Schwartz (; 5 March 1915 – 4 July 2002) was a French mathematician. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function. He was awarded the Fields Medal in ...

for validating computations done in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressi ...

, a problem of pure mathematics that was proved true by Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

, with an algorithm that is impossible to implement because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.
In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". However, these terms are still used in names of some university
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United Stat ...

departments, such as at the Faculty of Mathematics at the University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...

.
Unreasonable effectiveness

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicistEugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...

. It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the prime factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are s ...

of natural numbers that was discovered more than 2,000 years before its common use for secure internet
The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, p ...

communications through the RSA cryptosystem
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publi ...

. A second historical example is the theory of ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

s. They were studied by the ancient Greek mathematicians
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

as conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...

s (that is, intersections of cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...

s with planes). It is almost 2,000 years later that Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

discovered that the trajectories
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...

of the planets are ellipses.
In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...

developed the theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

that uses fundamentally these concepts. In particular, spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

of the special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...

is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collide ...

and the baryon
In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...

$\backslash Omega^.$ In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of a unknown particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...

, and to search these particles. In both cases, these particles were discovered a few years later by specific experiments.
Philosophy

Reality

The connection between mathematics and material reality has led to philosophical debates since at least the time ofPythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...

. The ancient philosopher Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...

. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in ...

summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
...

, Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...

and Albert Einstein that support his views.
Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.
Proposed definitions

There is no general consensus about a definition of mathematics or its epistemological statusthat is, its place among other human activities. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study. Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. In the 19th century, when mathematicians began to address topicssuch as infinite setswhich have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task. Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theoremassertions whose validity relies on a proof, that is, a purely-logical deduction. Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.Rigor

Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and theproof
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 c ...

s must be reducible to a succession of applications of inference rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...

s, without any use of empirical evidence and intuition
Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...

. Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' concision
Concision (also called brevity, laconicism, or conciseness) is a writing principle of eliminating redundancy.UNT Writing Lab. "Concision, Clarity, and Cohesion." Accessed June 19, 2012Link./ref> For example, this:
* "It is a fact that most argum ...

, rigorous proofs can require hundreds of pages to express. The emergence of computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.
Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a ...

s has allowed proof lengths to further expand, such as the 255-page Feit–Thompson theorem In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by .
History
conjectured that every nonabelian finite simple group has even order. suggested using t ...

. The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.
The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.
At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstr ...

) and contradictions (Russel's paradox). This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm
Pleonasm (; , ) is redundancy in linguistic expression, such as "black darkness" or "burning fire". It is a manifestation of tautology by traditional rhetorical criteria and might be considered a fault of style. Pleonasm may also be used for em ...

. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.
Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.
Training and practice

Education

Mathematics has a remarkable ability to cross cultural boundaries and time periods. As ahuman activity
Human behavior is the potential and expressed capacity ( mentally, physically, and socially) of human individuals or groups to respond to internal and external stimuli throughout their life. Kagan, Jerome, Marc H. Bornstein, and Richard M. ...

, the practice of mathematics has a social side, which includes education
Education is a purposeful activity directed at achieving certain aims, such as transmitting knowledge or fostering skills and character traits. These aims may include the development of understanding, rationality, kindness, and honesty ...

, careers, recognition
Recognition may refer to:
*Award, something given in recognition of an achievement
Machine learning
*Pattern recognition, a branch of machine learning which encompasses the meanings below
Biometric
* Recognition of human individuals, or biomet ...

, popularization
In sociology, popularity is how much a person, idea, place, item or other concept is either liked or accorded status by other people. Liking can be due to reciprocal liking, interpersonal attraction, and similar factors. Social status can be ...

, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM
Stem or STEM may refer to:
Plant structures
* Plant stem, a plant's aboveground axis, made of vascular tissue, off which leaves and flowers hang
* Stipe (botany), a stalk to support some other structure
* Stipe (mycology), the stem of a mushro ...

academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, statistician
A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, and statisticians may w ...

, actuary
An actuary is a business professional who deals with the measurement and management of risk and uncertainty. The name of the corresponding field is actuarial science. These risks can affect both sides of the balance sheet and require asset man ...

, financial analyst
A financial analyst is a professional, undertaking financial analysis for external or internal clients as a core feature of the job.
The role may specifically be titled securities analyst, research analyst, equity analyst, investment analyst, ...

, economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this ...

, accountant
An accountant is a practitioner of accounting or accountancy.
Accountants who have demonstrated competency through their professional associations' certification exams are certified to use titles such as Chartered Accountant, Chartered Certifi ...

, commodity trader
A trader is a person, firm, or entity in finance who buys and sells financial instruments, such as forex, cryptocurrencies, stocks, bonds, commodities, derivatives, and mutual funds in the capacity of agent, hedger, arbitrageur, or speculator. ...

, or computer consultant.
Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East
The ancient Near East was the home of early civilizations within a region roughly corresponding to the modern Middle East: Mesopotamia (modern Iraq, southeast Turkey, southwest Iran and northeastern Syria), ancient Egypt, ancient Iran ( Elam, ...

and then for the Greco-Roman world
The Greco-Roman civilization (; also Greco-Roman culture; spelled Graeco-Roman in the Commonwealth), as understood by modern scholars and writers, includes the geographical regions and countries that culturally—and so historically—were dir ...

starting around 300 BCE. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE in Eygpt. Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition
Oral tradition, or oral lore, is a form of human communication wherein knowledge, art, ideas and Culture, cultural material is received, preserved, and transmitted orally from one generation to another.Jan Vansina, Vansina, Jan: ''Oral Traditio ...

since the Vedic period
The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betwe ...

(). In Imperial China
The earliest known written records of the history of China date from as early as 1250 BC, from the Shang dynasty (c. 1600–1046 BC), during the reign of king Wu Ding. Ancient historical texts such as the '' Book of Documents'' (early chapt ...

during the Tang dynasty
The Tang dynasty (, ; zh, t= ), or Tang Empire, was an Dynasties in Chinese history, imperial dynasty of China that ruled from 618 to 907 AD, with an Zhou dynasty (690–705), interregnum between 690 and 705. It was preceded by the Sui dyn ...

(618–907 CE), a mathematics curriculum was adopted for the civil service exam
Civil service examinations are examinations implemented in various countries for recruitment and admission to the civil service. They are intended as a method to achieve an effective, rational public administration on a merit system for recruit ...

to join the state bureaucracy.
Following the Dark Ages, mathematics education in Europe was provided by religious schools as part of the Quadrivium
From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...

. Formal instruction in pedagogy
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and psychological development of learners. Pedagogy, taken ...

began with Jesuit
, image = Ihs-logo.svg
, image_size = 175px
, caption = ChristogramOfficial seal of the Jesuits
, abbreviation = SJ
, nickname = Jesuits
, formation =
, founders ...

schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was '' L'Enseignement Mathématique'', which began publication in 1899. The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core componentinitially for its military applications. While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.
During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics. Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as math anxiety or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.
Psychology (aesthetic, creativity and intuition)

The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by acomputer program
A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. Computer programs are one component of software, which also includes software documentation, documentation and oth ...

. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.
Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving puzzle
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together ( or take them apart) in a logical way, in order to arrive at the correct or fun solution of the puzzl ...

s. This aspect of mathematical activity is emphasized in recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

.
Mathematicians can find an aesthetic
Aesthetics, or esthetics, is a branch of philosophy that deals with the nature of beauty and taste, as well as the philosophy of art (its own area of philosophy that comes out of aesthetics). It examines aesthetic values, often expressed t ...

value to mathematics. Like beauty
Beauty is commonly described as a feature of objects that makes these objects pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, together with art and taste, is the main subject of aesthetics, o ...

, it is hard to define, it is commonly related to ''elegance'', which involves qualities like simplicity
Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something is simple or complex depending on the way we ...

, symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

, completeness, and generality. G. H. Hardy in '' A Mathematician's Apology'' expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic. Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book ''Proofs from THE BOOK
''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each math ...

'', inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in ...

for harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...

.
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts
Liberal arts education (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as La ...

. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are ''created'' (as in art) or ''discovered'' (as in science). The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...

, which primarily identifies mathematics with certain creative processes in the mind. Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle.
In the end, neither constructivism nor intuitionism displaced classical mathematics In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory. It stands in contrast to other types of mathematics such as constructive ...

or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.
Awards and prize problems

The most prestigious award in mathematics is theFields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award h ...

, established in 1936 and awarded every four years (except around World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

) to up to four individuals. It is considered the mathematical equivalent of the Nobel Prize
The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...

.
Other prestigious mathematics awards include:
* The Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Pri ...

, instituted in 2002 and first awarded in 2003
* The Chern Medal for lifetime achievement, introduced in 2009 and first awarded in 2010
* The AMS Leroy P. Steele Prize, awarded since 1970
* The Wolf Prize in Mathematics
The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts ...

, also for lifetime achievement, instituted in 1978
A famous list of 23 open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...

s, called "Hilbert's problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...

", was compiled in 1900 by German mathematician David Hilbert. This list has achieved great celebrity among mathematicians, and, , at least thirteen of the problems (depending how some are interpreted) have been solved.
A new list of seven important problems, titled the " Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...

, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. To date, only one of these problems, the Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured ...

, has been solved.
See also

*List of mathematical jargon
The language of mathematics has a vast 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 ...

* Lists of mathematicians
Lists of mathematicians cover notable mathematicians by nationality, ethnicity, religion, profession and other characteristics. Alphabetical lists are also available (see table to the right).
Lists by nationality, ethnicity or religion
* Lis ...

* Lists of mathematics topics
* Mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

* Mathematical sciences
* Mathematics and art
* Mathematics education
In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge.
Although re ...

* Outline of mathematics
* Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...

* Relationship between mathematics and physics
* Science, technology, engineering, and mathematics
Science, technology, engineering, and mathematics (STEM) is an umbrella term used to group together the distinct but related technical disciplines of science, technology, engineering, and mathematics. The term is typically used in the context of ...

Notes

References

Bibliography

* * * * * * * . * * * * * *Further reading

* * Availablonline

(registration required). * * * – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, an

online

. * * * {{Authority control Formal sciences Main topic articles