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Mathematics (from Greek: ) includes the study of such topics as numbers (
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
), formulas and related structures (
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
), shapes and spaces in which they are contained (
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
), and quantities and their changes (
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 generalizations ...

calculus
and
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
). There is no general consensus about its exact scope or . Most of mathematical activity consists of discovering and proving (by pure reasoning) 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 Plant, plant ...
. These objects are either
abstraction Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...

abstraction
s from nature (such as
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s or "a
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
"), or (in modern mathematics) abstract entities that are defined by their basic properties, called
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s. A
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
consists of a succession of applications of some deductive rules to already known results, including previously proved
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration. The result of a proof is called a ''theorem''. Contrary to
physical law Scientific laws or laws of science are statements, based on repeated experiment An experiment is a procedure carried out to support or refute a , or determine the or of something previously untried. Experiments provide insight into by d ...
s, the validity of a theorem (its truth) does not rely on any
experimentation File:Mirror baby.jpg, 160px, Even very young children perform rudimentary experiments to learn about the world and how things work. An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likeliho ...
but on the correctness of its reasoning (though experimentation is often useful for discovering new theorems of interest). Mathematics is widely used in
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

science
for
modeling In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
phenomena. This enables the extraction of quantitative predictions from experimental laws. For example, the movement of planets can be predicted with high accuracy using
Newton's law of gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
combined with mathematical computation. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model for describing the reality. So when some inaccurate predictions arise, it means that the model must be improved or changed, not that the mathematics is wrong. For example, the
perihelion precession of Mercury Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury (planet), Mer ...
cannot be explained by Newton's law of gravitation, but is accurately explained by
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...

Einstein
's
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. This experimental validation of Einstein's theory shows that Newton's law of gravitation is only an approximation (which still is very accurate in everyday life). Mathematics is essential in many fields, including
natural sciences Natural science is a branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biology. A botanist, plant scientist or ph ...
,
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 ...

engineering
,
medicine Medicine is the science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (proced ...

medicine
,
finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in the left corn ...

finance
,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
and
social sciences Social science is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a branch of biol ...

social sciences
. Some areas of mathematics, such as
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
and
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
, are developed in direct correlation with their applications, and are often grouped under the name of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
. Other mathematical 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
), but practical applications are often discovered later. A fitting example is the problem of
integer factorization In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
which goes back to
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
but 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 An acronym is a word or name formed from the initial components of a longer name or p ...
(for the security of
computer network A computer network is a set of computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operati ...
s). Mathematics has been a human activity from as far back as written records exist. However, the concept of a "proof" and its associated "
mathematical rigour Rigour (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...
" first appeared in
Greek mathematics Greek mathematics refers to mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geomet ...
, most notably in
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
's '' Elements''. Mathematics developed at a relatively slow pace until the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

Renaissance
, when algebra and
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
were added to arithmetic and geometry as main areas of mathematics. Since then the interaction between mathematical innovations and
scientific discoveries Discovery is the act of detecting something new, or something previously unrecognized as meaningful. With reference to sciences and Discipline (academia), academic disciplines, discovery is the observation of new Phenomenon, phenomena, new action ...
have led to a rapid increase in the rate of mathematical discoveries. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. This, in turn, gave rise to a dramatic increase in the number of mathematics areas and their fields of applications; a witness of this is the
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 ...
, which lists more than sixty first-level areas of mathematics.


Areas of mathematics

Before the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

Renaissance
, mathematics was divided into two main areas,
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, devoted to the manipulation of
number A number is a 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 deduct ...

number
s, and
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
, devoted to the study of shapes. There was also some
pseudo-science Pseudoscience consists of statements, belief A belief is an Attitude (psychology), attitude that something is the case, or that some proposition about the world is truth, true. In epistemology, philosophers use the term "belief" to refer to ...
, such as
numerology Numerology is the pseudoscientific belief in a divine or mysticism, mystical relationship between a number and one or more Coincidence#Interpretation, coinciding events. It is also the study of the numerical value of the letters in words, names, ...
and
astrology Astrology is a pseudoscience that claims to divination, divine information about human affairs and terrestrial events by studying the movements and relative positions of Celestial objects in astrology, celestial objects. Astrology has be ...
that were not clearly distinguished from mathematics. Around the Renaissance, two new main areas appeared. The introduction of
mathematical notation Mathematical notation is a system of symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or prag ...
led to
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, which, roughly speaking, consists of the study and the manipulation of
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

formula
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 generalizations ...

Calculus
, a shorthand of ''infinitesimal calculus'' and ''integral calculus'', is the study of
continuous functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which model the change of, and the relationship between varying quantities ( variables). This division into four main areas remained valid until the end of the 19th century, although some areas, such as
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motion (physics), motions of celestial object, objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical obje ...
and
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point part ...
, which were often considered as mathematics, are now considered as belonging to
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
. Also, some subjects developed during this period predate mathematics (being divided into different) areas, such as
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, which only later became regarded as autonomous areas of their own. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
led to an explosion in the amount of areas of mathematics. The
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 more than 60 first-level areas. Some of these areas correspond to the older division in four main areas. This is the case of 11:
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
(the modern name for higher arithmetic) and 51:
Geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

Geometry
. However, there are several other first-level areas that have "geometry" in their name or are commonly considered as belonging to geometry.
Algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

Algebra
and
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 generalizations ...

calculus
do not appear as first-level areas, but are each split into several first-level areas. Other first-level areas did not exist at all before the 20th century (for example 18:
Category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
;
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
, and 68:
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
) or were not considered before as mathematics, such as 03:
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 formal sys ...
and foundations (including
model theory In 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 p ...
,
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
,
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
,
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 In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
, and
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
).


Number theory

Number theory started with the manipulation of
number A number is a 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 deduct ...

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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s (\mathbb), later expanded to
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s (\Z) and
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s (\Q). Number theory was formerly called
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, but nowadays this term is mostly used for the methods of calculation with numbers. A specificity of number theory is that many problems that can be stated very elementarily are very difficult, and, when solved, have a solution that require very sophisticated methods coming from various parts of mathematics. A notable example is
Fermat's Last theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
that was stated in 1637 by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of suc ...

Pierre de Fermat
and proved only in 1994 by
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national a ...
, using, among other tools,
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
(more specifically
scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicity (mathematics), multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebra ...
),
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
. Another example is
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in 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 a ...
, that 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 (; ; March 18, 1690 – November 20, 1764) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
it remains unproven despite considerable effort. In view of the diversity of the studied problems and the solving methods, number theory is presently split in several subareas, which include
analytic number theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
,
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
,
geometry of numbersGeometry 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 (group), lattice in \mathbb R^n, and the study of these lattices provides fundame ...
(method oriented),
Diophantine equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and transcendence theory (problem oriented).


Geometry

Geometry is, with
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s and
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

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

surveying
and
architecture upright=1.45, alt=Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted), Plan of the second floor (attic storey) of the Hôtel de Brionne in Paris – 1734. Architecture (Latin ''archi ...

architecture
. A fundamental innovation was the elaboration of proofs by
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
: it is not sufficient to verify by measurement that, say, two lengths are equal. Such a property must be ''proved'' by abstract reasoning from previously proven results (
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s) and basic properties (which are considered as self-evident because they are too basic for being the subject of a proof (
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s)). This principle, which is foundational for all mathematics, was elaborated for the sake of geometry, and was systematized by
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
around 300 BC in his book '' Elements''. The resulting
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
is the study of shapes and their arrangements constructed from
lines Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat A long terminal repeat (LTR) is a pair of identical sequences of DNA ...

lines
,
planes Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes come in a va ...
and
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
s in the
Euclidean plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(
plane geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a small ...
) and the (three-dimensional)
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. Euclidean geometry was developed without a change of methods or scope until the 17th century, when
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

René Descartes
introduced what is now called
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
. This was a major change of paradigm, since instead of defining
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s as lengths of line segments (see
number line In elementary mathematics 300px, Both groups are equal to 5. Apples are frequently used to explain arithmetic in textbooks for children. Elementary mathematics consists of mathematics Mathematics (from Ancient Greek, Greek: ) include ...

number line
), it allowed the representation of points using numbers (their coordinates), and for the use of
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
and later,
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 generalizations ...

calculus
for solving geometrical problems. This split geometry in two parts that differ only by their methods,
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
, which uses coordinates systemically. Analytic geometry allows the study of new shapes, in particular
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s that are not related to circles and lines; these curves are defined either as (whose study led to
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
), or by
implicit equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, often
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s (which spawned
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
). Analytic geometry makes it possible to consider spaces dimensions higher than three (it suffices to consider more than three coordinates), which are no longer a model of the physical space. Geometry expanded quickly during the 19th century. A major event was the discovery (in the second half of the 19th century) of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, which are geometries where the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

parallel postulate
is abandoned. This is, besides Russel's paradox, one of the starting points of the foundational crisis of mathematics, by taking into question the ''truth'' of the aforementioned postulate. This aspect of the crisis was solved by systematizing the
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, 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 are invariant under specific transformations of the
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
. This results in a number of subareas and generalizations of geometry that 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, proj ...
, introduced in the 16th century by
Girard Desargues Girard Desargues (; 21 February 1591 – September 1661) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( ...
, it extends Euclidean geometry by adding
points at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
at which
parallel lines In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

parallel lines
intersect. This simplifies many aspects of classical geometry by avoiding to have a different treatment for intersecting and parallel lines. *
Affine geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, the study of properties relative to parallelism and independent from the concept of length. *
Differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, the study of curves, surfaces, and their generalizations, which are defined using
differentiable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

differentiable function
s * Manifold theory, the study of shapes that are not necessarily embedded in a larger space *
Riemannian geometry#REDIRECT Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and m ...
, the study of distance properties in curved spaces *
Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

Algebraic geometry
, the study of curves, surfaces ,and their generalizations, which are defined using
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s *
Topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

Topology
, the study of properties that are kept under continuous deformations **
Algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, the use in topology of algebraic methods, mainly
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
*
Discrete geometry Discrete geometry and combinatorial geometry are branches of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are re ...
, the study of finite configurations in geometry *
Convex geometryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, the study of
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, 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 o ...

convex set
s, which takes its importance from its applications in
optimization File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alter ...
*
Complex geometry In mathematics, complex geometry is the study of complex manifolds, Complex algebraic variety, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this cate ...
, the geometry obtained by replacing real numbers with
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s


Algebra

Algebra may be viewed as the art of manipulating
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
s and
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...

formula
s.
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 autho ...
(3d century) and Al-Khowarazmi (9th century) were two main precursors of algebra. The first one solved some relations between unknown
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s (that is, equations) by deducing new relations until getting the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term ''algebra'' is derived from the
Arabic Arabic (, ' or , ' or ) is a Semitic language The Semitic languages are a branch of the Afroasiatic language family originating in the Middle East The Middle East is a list of transcontinental countries, transcontinental region ...

Arabic
word that he used for naming one of these methods in the title of his main treatise. Algebra began to be a specific area only with
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
(1540–1603), who introduced the use of letters ( variables) for representing unknown or unspecified numbers. This allows describing consisely the
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
that have to be done on the numbers represented by the variables. 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+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

linear equation
s that is called 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 matrix (mat ...
, and
polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s in a single
unknown Unknown or The Unknown may refer to: Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), a silent boxing film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film) * The Unknown (1927 film), ''The Unknown'' (19 ...
, which were called ''algebraic equations'' (a term that is still in use, although it may be ambiguous). During the 19th century, variables began to represent other things than numbers (such as
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, modular integers, and
geometric transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s), on which some
operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...
can operate, which are often generalizations of arithmetic operations. For dealing with this, the concept of
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
was introduced, which consist of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. So, the scope of algebra evolved for becoming essentially the study of algebraic structures. This object of algebra was called ''modern algebra'' or
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, the latter term being still used, mainly in an educational context, in opposition with
elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...
which is concerned with the older way of manipulating formulas. Some types of algebraic structures have properties that are useful, and often fundamental, in many areas of mathematics. Their study are nowadays autonomous parts of algebra, which include: *
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
; * field theory; *
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 matrix (mat ...
; *
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
; *
commutative algebra Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
, which is the study of
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
s, includes the study of
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s, and is a foundational part of
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
; *
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
*
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
and
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
theory; *
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, which is widely used for the study of the logical structure of
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
s. The study of types algebraic structures as mathematical objects is the object of
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...
and
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. The latter applies to every
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(not only the 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s; this particular area of application is called
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
.


Calculus and analysis

Calculus, formerly called
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
, was introduced in the 17th century by
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

Newton
and
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

Leibniz
, independently and simultaneously. It is fundamentally the study of the relationship of two changing quantities, called '' variables'', such that one depends on the other. Calculus was largely expanded in the 18th century by
Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Euler
, with the introduction of the concept of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, 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 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
, where variables represent
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
where variables represent
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. Presently there are many subareas of analysis, some being shared with other areas of mathematics; they 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 variables, functions of several variables: the Differential calculus, different ...
*
Functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, where variables represent varying functions; * Integration,
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
and
potential theoryIn 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 gravity ...
, all strongly related with
Probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
; *
Ordinary differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s; *
Partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s; *
Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications of mathematics.


Discrete mathematics


Mathematical logic and set theory

These subjects belong to mathematics since the end of the 19th century. Before this period, sets were not considered as
mathematical objects A mathematical object is an abstract concept arising in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
, and
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
, although used for
mathematical proof A mathematical proof is an inferential argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
s, belonged to
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

philosophy
, and was not specifically studied by mathematicians. Before the study of
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...
, mathematicians were reluctant to consider collections that are actually infinite, and considered
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

infinity
as the result of an endless
enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), s ...
. Cantor's work offended many mathematicians not only by considering actually infinite sets, but also by showing that this implies different sizes of infinity (see
Cantor's diagonal argument 250px, An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. In set theory, Cantor's diagonal argument, also cal ...
) and the existence of mathematical objects that cannot be computed, and not even be explicitly described (for example,
Hamel bases In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s). This led to the controversy over Cantor's set theory. In the same period, it appeared in various areas of mathematics that the former intuitive definitions of the basic mathematical objects were insufficient for insuring
mathematical rigour Rigour (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...
. Examples of such intuitive definitions are "a set is a collection of objects", "
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
is what is used for counting", "a point is a shape with a zero length in every direction", "a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
is a trace left by a moving point", etc. This is the origin of the foundational crisis of mathematics. It has been eventually solved in the mainstream of mathematics by systematize the
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 people, Italian mathematician Giuseppe Peano. These axioms have been ...
, the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s are defined by "zero is a number", "each number as a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even as many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and finding proofs. This approach allows considering "logics" (that is, sets of allowed deducing rules),
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, 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 Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (nu ...
assert, roughly speaking that, in every theory that contains the natural numbers, there are theorems that are true (that is provable in a larger theory), but not provable inside the theory. This approach of the foundations of the mathematics was challenged during the first half of the 20th century by mathematicians leaded by L. E. J. Brouwer who promoted an
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 of ...
that excludes the
law of excluded middle In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
. These problems and debates led to a wide expansion of mathematical logic, with subareas such as
model theory In 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 p ...
(modeling some logical theories inside other theory),
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 In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
,
type theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
,
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...
and
computational complexity theory 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 by a computer. A computation problem is solvable by ...
. Although these aspects of mathematical logic were introduced before the rise of
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

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

compiler
design, program certification,
proof assistant An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of i ...
s and other aspects of
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
, contributed in turn to the expansion of these logical theories.


Applied mathematics

Applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
concerns itself with mathematical methods that are typically used in science,
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 ...

engineering
,
business Business is the activity of making one's living or making money by producing or buying and selling products (such as goods and services). Simply put, it is "any activity or enterprise entered into for profit." Having a business name A trad ...

business
, and
industry Industry may refer to: Economics * Industry (economics) In macroeconomics, an industry is a branch of an economy that produces a closely related set of raw materials, goods, or services. For example, one might refer to the wood industry ...
. Thus, "applied mathematics" is a
mathematical science The mathematical sciences are a group of areas of study that includes, in addition to mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
with specialized
knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to e ...
. The term ''applied mathematics'' also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, ''applied mathematics'' focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
.


Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
. Statisticians (working as part of a research project) "create data that makes sense" with
random sampling In statistics, quality assurance, and Statistical survey, survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a population (statistics), statistical population to estimate characteristics o ...
and with randomized
experiments An experiment is a procedure carried out to support or refute a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that on ...
; the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). When reconsidering data from experiments and samples or when analyzing data from
observational studies In fields such as epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants In mathematics, the determinant is a Scalar (mathematics), scalar value that is a function (mathematics ...
, statisticians "make sense of the data" using the art of
modelling In general, a model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. ...
and the theory of
inference Inferences are steps in reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic ...
—with
model selection Model selection is the task of selecting a statistical model A statistical model is a mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act ...
and
estimation Estimation (or estimating) is the process of finding an estimate, or approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived ...
; the estimated models and consequential
predictions frame, '' The Old Farmer's Almanac'' is famous in the US for its (not necessarily accurate) long-range weather predictions. A prediction (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the ...
should be
tested ''Tested'' is a live album by punk rock band Bad Religion. It was recorded in the United States, 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 cro ...
on new data.
Statistical theoryThe 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 theory, statistical-decision p ...
studies decision problems such as minimizing the
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to ...

risk
(
expected loss Expected may refer to: *Expectation (epistemic) In the case of uncertainty, expectation is event that considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advant ...
) of a statistical action, such as using a procedure in, for example,
parameter estimation Estimation theory is a branch of statistics that deals with estimating the values of Statistical parameter, parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such ...
,
hypothesis testing A statistical hypothesis is a hypothesis A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method The scientific method is an Empirical evidence ...
, and selecting the best. In these traditional areas of
mathematical statistics Mathematical statistics is the application of probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related ...
, a statistical-decision problem is formulated by minimizing an
objective functionIn mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event (probability theory), event or values of one or more variables onto a real number intuitive ...
, like expected loss or
cost In production Production may be: Economics and business * Production (economics) * Production, the act of manufacturing goods * Production, in the outline of industrial organization, the act of making products (goods and services) * Production ...

cost
, 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 File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt= Mathematical optimization (alter ...
, the mathematical theory of statistics shares concerns with other
decision science Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent (economics), agent's choices. Decision theory can be broken into two branches: Norm (philosophy), normative decision theory, which analyzes the ...
s, such as operations research,
control theory Control theory deals with the control of dynamical system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...
, and
mathematical economics Mathematical economics is the application of mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country locate ...
.:


Computational mathematics

Computational mathematics Computational mathematics involves mathematics, mathematical research in mathematics as well as in areas of science where computation, computing plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computa ...
proposes and studies methods for solving
mathematical problem A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ...
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 computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
studies methods for problems in
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
using
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
and
approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler function (mathematics), functions, and with Quantitative property, quantitatively characterization (ma ...
; numerical analysis broadly includes the study of
approximation An approximation is anything that is intentionally similar but not exactly equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a ...
and discretisation 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
algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

algorithm
ic-
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
-and-
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. 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 expression (mathematics), ...
and
symbolic computation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.


History

The history of mathematics can be seen as an ever-increasing series of
abstractions Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, ma ...
. Evolutionarily speaking, the first abstraction to ever take place, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely the quantity of their members. As evidenced by tallies found on bone, in addition to recognizing how to
count Count (feminine: countess) is a historical title of nobility Nobility is a social class normally ranked immediately below Royal family, royalty and found in some societies that have a formal aristocracy (class), aristocracy. Nobility ...
physical objects,
prehistoric Prehistory, also known as pre-literary history, is the period of human history Human history, or world history, is the narrative of Human, humanity's past. It is understood through archaeology, anthropology, genetics, and linguistics, ...

prehistoric
peoples may have also recognized how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 , when the
Babylonia Babylonia () was an and based in central-southern which was part of Ancient Persia (present-day and ). A small -ruled state emerged in 1894 BCE, which contained the minor administrative town of . It was merely a small provincial town dur ...
ns and Egyptians began using
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
,
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, and
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
for taxation and other financial calculations, for building and construction, and for
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
. The oldest mathematical texts from
Mesopotamia Mesopotamia ( grc, Μεσοποταμία ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in th ...

Mesopotamia
and
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country This is a list of countries located on more than one continent A continent is one of several large landmasses. Generally identi ...

Egypt
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

Pythagorean theorem
seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') denotes the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Bab ...
that
elementary arithmetic Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is sig ...
(
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
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 ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
) 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 A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ...
numeral system which is still in use today for measuring angles and time. Beginning in the 6th century BC with the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
, with
Greek mathematics Greek mathematics refers to mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geomet ...
the
Ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
began a systematic study of mathematics as a subject in its own right. Around 300 BC,
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
introduced the
axiomatic method In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
still 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 (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
(c. 287–212 BC) of
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily Syracuse ( ; it, Siracusa , or scn, Seragusa, label=none ; lat, Syrācūsae ; grc-att, wikt:Συράκουσαι, Συράκουσαι, Syrákousai ; grc-dor, wikt:Συράκο ...
. 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 Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
to calculate the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
under the arc of a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

parabola
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 (or simply conic) is a curve obtained as the intersection of the Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, the par ...

conic sections
(
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος ''Apollonios o Pergeos''; la, Apollonius Pergaeus; ) was an Ancient Greece, Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the ...
, 3rd century BC),
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

trigonometry
( Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (
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 autho ...
, 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 Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
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 Hindi (Devanagari: , हिंदी, ISO 15919, ISO: ), or more precisely Modern Standard Hindi (Devanagari: , ISO 15919, ISO: ), is an Indo-Aryan language spoken chiefly in Hindi Belt, ...
and were transmitted to the
Western world The Western world, also known as the West, refers to various regions, nations and state (polity), states, depending on the context, most often consisting of the majority of Europe, Northern America, and Australasia.
via
Islamic mathematics Mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Other notable developments of Indian mathematics include the modern definition and approximation of
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

sine
and
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
, and an early form of
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. During the
Golden Age of Islam The Islamic Golden Age ( ar, العصر الذهبي للإسلام , al-'asr al-dhahabi lil-islam), was a period of cultural, economic, and scientific flourishing in the history of Islam The history of Islam concerns the political, social, ...
, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of
Islamic mathematics Mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
was the development of
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
. Other achievements of the Islamic period include advances in
spherical trigonometry Spherical trigonometry is the branch of spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (plan ...
and the addition of the
decimal point A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...
to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as ,
Omar Khayyam Omar Khayyam (; fa, عمر خیّام ; 18 May 1048 – 4 December 1131) was a Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, Persian people, the majority ethni ...
and
Sharaf al-Dīn al-Ṭūsī Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرف‌الدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian peoples, Iranian Islamic mathematics, mathematician and Islamic as ...
. During the
early modern period The early modern period of modern history Human history, or world history, is the narrative of Human, humanity's past. It is understood through archaeology, anthropology, genetics, and linguistics, and since the History of writing, adve ...
, mathematics began to develop at an accelerating pace in
Western Europe Western Europe is the western region of Europe Europe is a continent A continent is any of several large landmasses. Generally identified by convention (norm), convention rather than any strict criteria, up to seven geographical r ...

Western Europe
. The development of
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 generalizations ...

calculus
by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
and
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...

Gottfried Leibniz
in the 17th century revolutionized mathematics.
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician
Carl Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princ ...

Carl Gauss
, who made numerous contributions to fields such as
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
,
analysis Analysis is the process of breaking a complex topic or substance Substance may refer to: * Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes * Chemical substance, a material with a definite chemical composit ...
,
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
,
matrix theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
. In the early 20th century,
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
transformed mathematics by publishing his
incompleteness theorems Complete may refer to: Logic * Completeness (logic) In mathematical logic and metalogic, a formal system is called complete with respect to a particular property (philosophy), property if every Well-formed formula, formula having the property can ...
, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will 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 enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

science
, 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 Bulletin or The Bulletin may refer to: Periodicals (newspapers, magazines, journals) * The Bulletin (Australian periodical), ''The Bulletin'' (Australian periodical), an Australian magazine (1880–2008) ** Bulletin Debate, a famous dispute from ...
'', "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
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s and their proofs."


Etymology

The word ''mathematics'' comes from
Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
''máthēma'' ('), meaning "that which is learnt," "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its
adjective In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most langu ...
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 the teachings and beliefs held by Pythagoras Pythagoras of Samos, or simply ; in () was an ancient and the eponymous founder of . His political and religious teachings were wel ...
was known as the ''mathēmatikoi'' (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. In Latin, and in English until around 1700, the term ''mathematics'' more commonly meant "
astrology Astrology is a pseudoscience that claims to divination, divine information about human affairs and terrestrial events by studying the movements and relative positions of Celestial objects in astrology, celestial objects. Astrology has be ...
" (or sometimes "
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example,
Saint Augustine In religious belief, a saint is a person who is recognized as having an exceptional degree of holiness Sacred describes something that is dedicated or set apart for the service or worship of a deity A deity or god is a supernatural being ...

Saint Augustine
's warning that Christians should beware of ''mathematici'', meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians. The apparent
plural The plural (sometimes abbreviated An abbreviation (from Latin ''brevis'', meaning ''short'') is a shortened form of a word or phrase, by any method. It may consist of a group of letters, or words taken from the full version of the word or ph ...

plural
form in English, like the French plural form (and the less commonly used singular
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
), goes back to the Latin neuter plural (
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Ancient Rome, Roman statesman, lawyer, scholar, philosopher and Academic skepticism, Academic Skeptic, who tried to uphold optimate principles during crisis of ...

Cicero
), based on the Greek plural ''ta mathēmatiká'' (), used by
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

Aristotle
(384–322 BC), and meaning 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 Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

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

metaphysics
'', which were 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).


Philosophy of mathematics

There is no general consensus about the exact definition or of mathematics.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

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 the study of mathematics increased in rigor and began to address abstract topics such as
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
and
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, proj ...
, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. 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."


Three leading types

Three leading types of definition of mathematics today are called logicist,
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...
, and formalist, each reflecting a different philosophical school of thought. All have severe flaws, none has widespread acceptance, and no reconciliation seems possible.


Logicist definitions

An early definition of mathematics in terms of logic was that of
Benjamin Peirce Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

Benjamin Peirce
(1870): "the science that draws necessary conclusions." In the ''
Principia Mathematica Image:Principia Mathematica 54-43.png, 500px, ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...
'',
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
and
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
advanced the philosophical program known as
logicism In the philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is ...
, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of
symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. ...
. An example of a logicist definition of mathematics is Russell's (1903) "All Mathematics is Symbolic Logic."


Intuitionist definitions

Intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...
definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other." A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the
law of excluded middle In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
(i.e., P \vee \neg P). While this stance does force them to reject one common version of
proof by contradiction In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
as a viable proof method, namely the inference of P from \neg P \to \bot, they are still able to infer \neg P from P \to \bot. For them, \neg (\neg P) is a strictly weaker statement than P .


Formalist definitions

Formalist definitions identify mathematics with its symbols and the rules for operating on them.
Haskell Curry Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
defined mathematics simply as "the science of formal systems". A
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 for ...
is a set of symbols, or ''tokens'', and some ''rules'' on how the tokens are to be combined into ''formulas''. In formal systems, the word ''axiom'' has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.


Mathematics as science

The German mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Carl Friedrich Gauss
referred to mathematics as "the Queen of the Sciences". More recently,
Marcus du Sautoy Marcus Peter Francis du Sautoy (; born 26 August 1965) is a British mathematician and author of popular science books. In 1996, he was awarded the Title of Distinction of Professor of Mathematics at the University of Oxford , mottoeng = ...

Marcus du Sautoy
has called mathematics "the Queen of Science ... the main driving force behind scientific discovery". The philosopher
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian-British philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as tho ...

Karl Popper
observed that "most mathematical theories are, like those of
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
and
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

biology
, hypothetico-
deductive Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Popper also noted that "I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience." Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions.
Intuition Intuition is the ability to acquire knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is ...
and experimentation also play a role in the formulation of
conjecture In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s in both mathematics and the (other) sciences.
Experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the ...
continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. Several authors consider that mathematics is not a science because it does not rely on
empirical evidence Empirical evidence for a proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is s ...
. The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven
liberal arts Liberal arts education (from Latin "free" and "art or principled practice") is the traditional academic program in Western higher education. ''Liberal arts'' takes the term ''Art (skill), art'' in the sense of a learned skill rather than spec ...
; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is ''created'' (as in art) or ''discovered'' (as in science). In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in
philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bra ...
.


Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce,
land measurement Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land surveyo ...
, architecture and later
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
; today, all sciences pose problems studied by mathematicians, and many problems arise within mathematics itself. For example, the
physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at leas ...

physicist
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation The path integral formulation is a description in quantum mechanics Quantum mech ...

Richard Feynman
invented the
path integral formulation The path integral formulation is a description in quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic p ...
of
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
using a combination of mathematical reasoning and physical insight, and today's
string theory In physics, string theory is a Mathematical theory, theoretical framework in which the Point particle, point-like particles of particle physics are replaced by Dimension (mathematics and physics), one-dimensional objects called String (physic ...

string theory
, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics. Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
. However pure mathematics topics often turn out to have applications, e.g.
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
in
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

cryptography
. This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what the physicist
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his cont ...
has named " the unreasonable effectiveness of mathematics". The
philosopher of mathematics A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...
Mark Steiner has written extensively on this matter and acknowledges that the applicability of mathematics constitutes “a challenge to naturalism.” For the philosopher of mathematics Mary Leng, the fact that the physical world acts in accordance with the dictates of non-causal mathematical entities existing beyond the universe is "a happy coincidence". On the other hand, for some anti-realism, anti-realists, connections, which are acquired among mathematical things, just mirror the connections acquiring among objects in the universe, so there is no "happy coincidence". As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest
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 ...
runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
. For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the ''elegance'' of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
, such as
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

Euclid
's proof that there are infinitely many
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, and in an elegant numerical method that speeds up calculation, such as the fast Fourier transform. G. H. Hardy in ''A Mathematician's Apology'' expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization (mathematics), characterization of an object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments have been published in ''Proofs from THE BOOK''. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. At the other social extreme, philosophers continue to find problems in
philosophy of mathematics The philosophy of mathematics is the branch The branches and leaves of a tree. A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bra ...
, such as the nature of
mathematical proof A mathematical proof is an inferential argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
.


Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, limiting mathematical discovery. Leonhard Euler, Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more ''abstract'' and more ''encrypted'' than those of natural language. Unlike natural language, where people can often equate a word (such as ''cow'') with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog. Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas. Language of mathematics, Mathematical language can be difficult to understand for beginners because even common terms, such as ''or'' and ''only'', have a more precise meaning than they have in everyday speech, and other terms such as ''open set, open'' and ''Field (mathematics), field'' refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as ''homeomorphism'' and ''Integral, integrable'' that have no meaning outside of mathematics. Additionally, shorthand phrases such as ''iff'' for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor". Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "
theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s", based on fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a notable cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous. On the other hand,
proof assistant An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of i ...
s allow for the verification of all details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the Feit–Thompson theorem. Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has independence (mathematical logic), undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless, mathematics is often imagined to be (as far as its formal content) nothing but
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.Patrick Suppes, ''Axiomatic Set Theory'', Dover, 1972, . p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."


Mathematical awards

Arguably the most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was instituted in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field. A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now 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, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. Currently, only one of these problems, the Poincaré conjecture, has been solved.


See also

* International Mathematical Olympiad * List of mathematical jargon * Outline of mathematics * Lists of mathematics topics * Mathematical sciences * Mathematics and art * Mathematics education * National Museum of Mathematics * Philosophy of mathematics * Relationship between mathematics and physics * Science, technology, engineering, and mathematics


Notes


References


Bibliography

* * * * * * . * * * * *


Further reading

* * * * * *  – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, an
online
. * * * {{Authority control Mathematics, Formal sciences Main topic articles