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 'cra ...

, one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as

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 and Lie group theory;
*Boolean algebra, which is widely used for the study of the logical structure of computers.
The study of types algebraic structures as mathematical objects is the object of universal algebra and

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 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 Zermelo–Fraenkel set theory, 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, 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),

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 'cra ...

, 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 ...

, and 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 ...

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 (c. 287–212 BC) of Syracuse, Italy, Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the Series (mathematics), summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD).
The Hindu–Arabic numeral system 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 Indian mathematics, India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.
During the Islamic Golden Age, Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra
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 ...

. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Muhammad ibn Musa al-Khwarizmi, Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.
During the early modern period, mathematics began to develop at an accelerating pace in 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 ...

by Isaac Newton and Gottfried Wilhelm Leibniz, Gottfried Leibniz in the 17th century revolutionized mathematics. 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 Friedrich Gauss, 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 ...

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

s and their proofs."

an

"math, ''n.3''"

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

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 ...

in cryptography.
This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what the physicist Eugene Wigner has named "The Unreasonable Effectiveness of Mathematics in the Natural Sciences, the unreasonable effectiveness of mathematics". The philosopher of mathematics 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 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 , , and . Computer science ...

.
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 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 ...

's proof that there are infinitely many

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

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 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 assistants 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

online

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

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 'cra ...

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 ...

), 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 ...

), 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 ...

), 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 ...

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 ...

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 , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

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, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the the ...

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, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrat ...

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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

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 ...

'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 Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer ...

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

, medicine
Medicine is the science
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 ...

, 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 ...

, 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 , , and . Computer science ...

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 ...

.
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 ...

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 pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Frie ...

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 ...

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 s sharing resources located on or provided by . The computers use common s over to communicate with each other. These interconnections are made up of technologies, based on physically wired, optical, and wire ...

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 Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...

, 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 ...

'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 ...

, 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 theRenaissance
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 ...

, 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 'cra ...

, 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 ...

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 ...

, 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
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contrad ...

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 ...

, 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 ...

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 ...

, 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
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 cel ...

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

, which were often considered as mathematics, are now considered as belonging to physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

. 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 ...

(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 ...

. 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 ...

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 ...

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 , , and . Computer science ...

) 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 that studies , 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 , is mostly concerned with those that are relevant to ...

, 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 ofnumber
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 ...

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
$(\backslash mathbb),$ later expanded to integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s $(\backslash Z)$ and rational 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 ...

s $(\backslash Q).$ 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 'cra ...

, 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
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Republic (fren ...

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 aca ...

, 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 ...

(more specifically scheme 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). It h ...

), 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 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument
...

, algebraic number theory
Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...

, 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 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 and transcendence theory (problem oriented).
Geometry

Geometry is, withlines
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
...

, 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 c ...

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 ...

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 surveyo ...

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 ...

.
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, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the the ...

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 ...

around 300 BC in his book '' Elements''.
The resulting 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 consists in assuming a sma ...

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
...

, planes 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 ...

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 . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

.
Euclidean geometry was developed without a change of methods or scope until the 17th century, when René Descartes
René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

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'') ...

. 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 ...

), 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 ...

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 ...

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; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

, 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 ...

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 ...

). 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:
' ...

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 (mathematics), space. This results in a number of subareas and generalizations of geometry that include:
*Projective geometry, introduced in the 16th century by Girard Desargues, it extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by avoiding to have a different treatment for intersecting and parallel lines.
*Affine geometry, the study of properties relative to parallel (geometry), parallelism and independent from the concept of length.
*Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions
*Manifold theory, the study of shapes that are not necessarily embedded in a larger space
*Riemannian geometry, the study of distance properties in curved spaces
*Algebraic geometry, the study of curves, surfaces ,and their generalizations, which are defined using polynomials
*Topology, the study of properties that are kept under continuous deformations
**Algebraic topology, 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, the study of finite configurations in geometry
*Convex geometry, the study of convex sets, which takes its importance from its applications in convex optimization, optimization
*Complex geometry, the geometry obtained by replacing real numbers with complex numbers
Algebra

Algebra may be viewed as the art of manipulating equations andformula
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 ...

s. Diophantus (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 word that he used for naming one of these methods in the title of The Compendious Book on Calculation by Completion and Balancing, his main treatise.
Algebra began to be a specific area only with François Viète (1540–1603), who introduced the use of letters ( variables) for representing unknown or unspecified numbers. This allows describing consisely the Arithmetic operation, operations that have to be done on the numbers represented by the variables.
Until the 19th century, algebra consisted mainly of the study of linear equations that is called presently linear algebra, 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 (algebra), unknown, 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 matrix (mathematics), matrices, modular arithmetic, modular integers, and geometric transformations), on which some operation (mathematics), operations can operate, which are often generalizations of arithmetic operations. For dealing with this, the concept of algebraic structure was introduced, which consist of a set (mathematics), 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, the latter term being still used, mainly in an educational context, in opposition with elementary algebra 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;
*field (mathematics), field theory;
*vector spaces, whose study is essentially the same as linear algebra;
*ring theory;
*commutative algebra, which is the study of commutative rings, includes the study of polynomials, 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 ...

;
*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 (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 spaces; this particular area of application is called algebraic topology.
Calculus and analysis

Calculus, formerly calledinfinitesimal 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 Isaac Newton, Newton and 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, with the introduction of the concept of a function (mathematics), function, and many other results. Presently "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.
Analysis is further subdivided into real analysis, 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 where variables represent complex numbers. Presently there are many subareas of analysis, some being shared with other areas of mathematics; they include:
* Multivariable calculus
* Functional analysis, where variables represent varying functions;
* Integration (mathematics), Integration, measure theory and potential theory, all strongly related with Probability theory;
* Ordinary differential equations;
* Partial differential equations;
* 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, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians. Before the study of infinite sets by Georg Cantor, mathematicians were reluctant to consider collections that are actual infinite, actually infinite, and considered infinity as the result of an endless enumeration. 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) and the existence of mathematical objects that cannot be computed, and not even be explicitly described (for example, Hamel bases of thereal 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
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 ...

s). This led to the controversy over Cantor's theory, 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", "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 ...

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

s, proofs, etc.) as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems 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 that excludes the law of excluded middle.
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, 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. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, computer program, program certification, proof assistants 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 , , and . Computer science ...

, contributed in turn to the expansion of these logical theories.
Applied mathematics

Applied mathematics 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 ...

, business, and Industry (manufacturing), industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. 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 withprobability 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 and with randomized design of experiments, experiments; 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 study, observational studies, statisticians "make sense of the data" using the art of statistical model, modelling and the theory of statistical inference, inference—with model selection and estimation theory, estimation; the estimated models and consequential Scientific method#Predictions from the hypothesis, predictions should be statistical hypothesis testing, tested on Scientific method#Evaluation and improvement, new data.
Statistical theory studies statistical decision theory, decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a statistical method, procedure in, for example, parameter estimation, hypothesis testing, and selection algorithm, selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or 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 mathematical optimization, optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.:
Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis (mathematics), analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization, discretisation with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-numerical linear algebra, matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.History

The history of mathematics can be seen as an ever-increasing series of abstraction (mathematics), abstractions. 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 tally sticks, tallies found on bone, in addition to recognizing how to counting, count physical objects, 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 Babylonians and Egyptians began usinggeometry
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 ...

for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Ancient Egypt, Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division (mathematics), division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.
Beginning in the 6th century BC with the Pythagoreans, with Greek mathematics
Greek mathematics refers to 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 ...

the Ancient Greeks began a systematic study of mathematics as a subject in its own right. Around 300 BC, 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 and topology, differential geometry, matrix theory, 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 ...

, 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 ...

. In the early 20th century, Kurt Gödel transformed mathematics by publishing his Gödel's incompleteness theorems, incompleteness theorems, 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 Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well as an invention. ...

, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the ''Bulletin of the American Mathematical Society'', "The number of papers and books included in the ''Mathematical Reviews'' 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 Etymology

The word ''mathematics'' comes from Ancient Greek ''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 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 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
Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contrad ...

" (or sometimes "astronomy") 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's warning that Christians should beware of ''mathematici'', meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural form in English, like the French plural form (and the less commonly used singular Morphological derivation, derivative ), goes back to the Latin Neuter (grammar), neuter plural (Cicero), based on the Greek plural ''ta mathēmatiká'' (), used by 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 that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

'' and ''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 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 and projective geometry, 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, and Formalism (mathematics), 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 (1870): "the science that draws necessary conclusions." In the ''Principia Mathematica'', Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic. An example of a logicist definition of mathematics is Russell's (1903) "All Mathematics is Symbolic Logic."Intuitionist definitions

Intuitionist 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 (i.e., $P\; \backslash vee\; \backslash neg\; P$). While this stance does force them to reject one common version of proof by contradiction as a viable proof method, namely the inference of $P$ from $\backslash neg\; P\; \backslash to\; \backslash bot$, they are still able to infer $\backslash neg\; P$ from $P\; \backslash to\; \backslash bot$. For them, $\backslash neg\; (\backslash neg\; P)$ is a strictly weaker statement than $P$.Formalist definitions

Formalism (mathematics), Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". A formal system 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 referred to mathematics as "the Queen of the Sciences". More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery". The philosopher Karl Popper observed that "most mathematical theories are, like those ofphysics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and biology, hypothesis, hypothetico-deductive: 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 deductive reasoning, exploration of the logical consequences of assumptions. Intuition (knowledge), Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics 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.
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; 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.
Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences pose problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four Fundamental interaction, 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 betweenpure 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. 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 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 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, such as the nature of mathematical proof.
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 "set theory
Set theory is the branch of that studies , 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 , is mostly concerned with those that are relevant to ...

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 mathematicsNotes

References

Bibliography

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

* * * * * * – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, anonline

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