Mathematics

== Areas of mathematics ==
{{anchor|Branches of mathematics}}
Before the [[Renaissance]], mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and [[geometry]], regarding the study of shapes.<ref>{{cite book |last=Bell |first=E. T. |author-link=Eric Temple Bell |year=1945 |orig-date=1940 |chapter=General Prospectus |title=The Development of Mathematics |edition=2nd |isbn=978-0-486-27239-9 |lccn=45010599 |oclc=523284 |page=3 |publisher=Dover Publications |quote=... mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry.}}</ref> Some types of [[pseudoscience]], such as [[numerology]] and astrology, were not then clearly distinguished from mathematics.<ref>{{cite book |last=Tiwari |first=Sarju |year=1992 |chapter=A Mirror of Civilization |title=Mathematics in History, Culture, Philosophy, and Science |edition=1st |page=27 |publisher=Mittal Publications |publication-place=New Delhi, India |isbn=978-81-7099-404-6 |lccn=92909575 |oclc=28115124 |quote=It is unfortunate that two curses of mathematics--Numerology and Astrology were also born with it and have been more acceptable to the masses than mathematics itself.}}</ref>

During the Renaissance, two more areas appeared. [[Mathematical notation]] led to [[algebra]] which, roughly speaking, consists of the study and the manipulation of [[formula]]s. [[Calculus]], consisting of the two subfields ''[[differential calculus]]'' and ''[[integral calculus]]'', is the study of [[continuous functions]], which model the typically [[Nonlinear system|nonlinear relationships]] between varying quantities, as represented by [[variable (mathematics)|variables]]. This division into four main areas{{endash}}arithmetic, geometry, algebra, calculus<ref>{{cite book |last=Restivo |first=Sal |author-link=Sal Restivo |editor-last=Bunge |editor-first=Mario |editor-link=Mario Bunge |year=1992 |chapter=Mathematics from the Ground Up |title=Mathematics in Society and History |page=14 |series=Episteme |volume=20 |publisher=[[Kluwer Academic Publishers]] |isbn=0-7923-1765-3 |lccn=25709270 |oclc=92013695}}</ref>{{endash}}endured until the end of the 19th century. Areas such as [[celestial mechanics]] and [[solid mechanics]] were then studied by mathematicians, but now are considered as belonging to physics.<ref>{{cite book |last=Musielak |first=Dora |author-link=Dora Musielak |year=2022 |title=Leonhard Euler and the Foundations of Celestial Mechanics |series=History of Physics |publisher=[[Springer International Publishing]] |doi=10.1007/978-3-031-12322-1 |isbn=978-3-031-12321-4 |s2cid=253240718 |issn=2730-7549 |eissn=2730-7557 |oclc=1332780664}}</ref> The subject of [[combinatorics]] has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.<ref>{{cite journal |date=May 1979 |last=Biggs |first=N. L. |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |doi-access=free |issn=0315-0860 |eissn=1090-249X |lccn=75642280 |oclc=2240703}}</ref>

At the end of the 19th century, the [[foundational crisis in mathematics]] and the resulting systematization of the [[axiomatic method]] led to an explosion of new areas of mathematics.<ref name=Warner_2013>{{cite web |last=Warner |first=Evan |title=Splash Talk: The Foundational Crisis of Mathematics |publisher=[[Columbia University]] |url=https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf |url-status=dead |archive-url=https://web.archive.org/web/20230322165544/https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf |archive-date=March 22, 2023 |access-date=February 3, 2024}}</ref><ref name=Kleiner_1991/> The 2020 [[Mathematics Subject Classification]] contains no less than {{em|sixty-three}} first-level areas.<ref>{{cite journal |last1=Dunne |first1=Edward |last2=Hulek |first2=Klaus |author2-link=Klaus Hulek |date=March 2020 |title=Mathematics Subject Classification 2020 |journal=Notices of the American Mathematical Society |volume=67 |issue=3 |pages=410–411 |doi=10.1090/noti2052 |doi-access=free |issn=0002-9920 |eissn=1088-9477 |lccn=sf77000404 |oclc=1480366 |url=https://www.ams.org/journals/notices/202003/rnoti-p410.pdf |url-status=live |archive-url=https://web.archive.org/web/20210803203928/https://www.ams.org/journals/notices/202003/rnoti-p410.pdf |archive-date=August 3, 2021 |access-date=February 3, 2024 |quote=The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications.}}</ref> Some of these areas correspond to the older division, as is true regarding [[number theory]] (the modern name for [[higher arithmetic]]) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as [[mathematical logic]] and [[foundations of mathematics|foundations]].<ref name=MSC>{{cite web |url=https://zbmath.org/static/msc2020.pdf |title=MSC2020-Mathematics Subject Classification System |website=zbMath |publisher=Associate Editors of Mathematical Reviews and zbMATH |url-status=live |archive-url=https://web.archive.org/web/20240102023805/https://zbmath.org/static/msc2020.pdf |archive-date=January 2, 2024 |access-date=February 3, 2024}}</ref>

=== Number theory ===
{{Main|Number theory}}
[[File:Spirale Ulam 150.jpg|thumb|This is the [[Ulam spiral]], which illustrates the distribution of [[prime numbers]]. The dark diagonal lines in the spiral hint at the hypothesized approximate [[Independence (probability theory)|independence]] between being prime and being a value of a quadratic polynomial, a conjecture now known as [[Ulam spiral#Hardy and Littlewood's Conjecture F|Hardy and Littlewood's Conjecture F]].]]
Number theory began with the manipulation of [[number]]s, that is, [[natural number]]s <math>(\mathbb{N}),</math> and later expanded to [[integer]]s <math>(\Z)</math> and [[rational number]]s <math>(\Q).</math> Number theory was once called arithmetic, but nowadays this term is mostly used for [[numerical calculation]]s.<ref>{{cite book |last=LeVeque |first=William J. |author-link=William J. LeVeque |year=1977 |chapter=Introduction |title=Fundamentals of Number Theory |pages=1–30 |publisher=[[Addison-Wesley Publishing Company]] |isbn=0-201-04287-8 |lccn=76055645 |oclc=3519779 |s2cid=118560854}}</ref> Number theory dates back to ancient [[Babylonian mathematics|Babylon]] and probably [[ancient China|China]]. Two prominent early number theorists were [[Euclid]] of ancient Greece and [[Diophantus]] of Alexandria.<ref>{{cite book |last=Goldman |first=Jay R. |year=1998 |chapter=The Founding Fathers |title=The Queen of Mathematics: A Historically Motivated Guide to Number Theory |pages=2–3 |publisher=A K Peters |publication-place=Wellesley, MA |doi=10.1201/9781439864623 |isbn=1-56881-006-7 |lccn=94020017 |oclc=30437959 |s2cid=118934517}}</ref> The modern study of number theory in its abstract form is largely attributed to [[Pierre de Fermat]] and [[Leonhard Euler]]. The field came to full fruition with the contributions of [[Adrien-Marie Legendre]] and [[Carl Friedrich Gauss]].<ref>{{cite book |last=Weil |first=André |author-link=André Weil |year=1983 |title=Number Theory: An Approach Through History From Hammurapi to Legendre |publisher=Birkhäuser Boston |pages=2–3 |doi=10.1007/978-0-8176-4571-7 |isbn=0-8176-3141-0 |lccn=83011857 |oclc=9576587 |s2cid=117789303}}</ref>

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is [[Fermat's Last theorem|Fermat's Last Theorem]]. This conjecture was stated in 1637 by Pierre de Fermat, but it [[Wiles's proof of Fermat's Last Theorem|was proved]] only in 1994 by [[Andrew Wiles]], who used tools including [[scheme theory]] from [[algebraic geometry]], [[category theory]], and [[homological algebra]].<ref>{{cite journal |last=Kleiner |first=Israel |author-link=Israel Kleiner (mathematician) |date=March 2000 |title=From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem |journal=Elemente der Mathematik |volume=55 |issue=1 |pages=19–37 |doi=10.1007/PL00000079 |doi-access=free |issn=0013-6018 |eissn=1420-8962 |lccn=66083524 |oclc=1567783 |s2cid=53319514}}</ref> Another example is [[Goldbach's conjecture]], which asserts that every even integer greater than 2 is the sum of two [[prime number]]s. Stated in 1742 by [[Christian Goldbach]], it remains unproven despite considerable effort.<ref>{{cite book |last=Wang |first=Yuan |year=2002 |title=The Goldbach Conjecture | pages=1–18 |edition=2nd |series=Series in Pure Mathematics |volume=4 |publisher=[[World Scientific]] |doi=10.1142/5096 |isbn=981-238-159-7 |lccn=2003268597 |oclc=51533750 |s2cid=14555830}}</ref>

Number theory includes several subareas, including [[analytic number theory]], [[algebraic number theory]], [[geometry of numbers]] (method oriented), [[diophantine equation]]s, and [[transcendence theory]] (problem oriented).<ref name=MSC/>

=== Geometry ===
{{Main|Geometry}}
[[File:Triangles (spherical geometry).jpg|thumb|On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.]]

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as [[line (geometry)|lines]], [[angle]]s and [[circle]]s, which were developed mainly for the needs of [[surveying]] and [[architecture]], but has since blossomed out into many other subfields.<ref name="Straume_2014">{{Cite arXiv|last=Straume |first=Eldar |date=September 4, 2014 |title=A Survey of the Development of Geometry up to 1870 |class=math.HO |eprint=1409.1140 }}</ref>

A fundamental innovation was the ancient Greeks' introduction of the concept of [[mathematical proof|proofs]], which require that every assertion must be ''proved''. For example, it is not sufficient to verify by [[measurement]] that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ([[theorem]]s) and a few basic statements. The basic statements are not subject to proof because they are self-evident ([[postulate]]s), or are part of the definition of the subject of study ([[axiom]]s). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by [[Euclid]] around 300 BC in his book ''[[Euclid's Elements|Elements]]''.<ref>{{cite book |last=Hilbert |first=David |author-link=David Hilbert |year=1902 |title=The Foundations of Geometry |page=1 |publisher=[[Open Court Publishing Company]] |doi=10.1126/science.16.399.307 |lccn=02019303 |oclc=996838 |s2cid=238499430 |url={{GBurl|id=8ZBsAAAAMAAJ}} |access-date=February 6, 2024}} {{free access}}</ref><ref>{{cite book |last=Hartshorne |first=Robin |author-link=Robin Hartshorne |year=2000 |chapter=Euclid's Geometry |pages=9–13 |title=Geometry: Euclid and Beyond |publisher=[[Springer New York]] |isbn=0-387-98650-2 |lccn=99044789 |oclc=42290188 |url={{GBurl|id=EJCSL9S6la0C|p=9}} |access-date=February 7, 2024}}</ref>

The resulting [[Euclidean geometry]] is the study of shapes and their arrangements [[straightedge and compass construction|constructed]] from lines, [[plane (geometry)|planes]] and circles in the [[Euclidean plane]] ([[plane geometry]]) and the three-dimensional [[Euclidean space]].{{efn|This includes [[conic section]]s, which are intersections of [[circular cylinder]]s and planes.}}<ref name=Straume_2014/>

Euclidean geometry was developed without change of methods or scope until the 17th century, when [[René Descartes]] introduced what is now called [[Cartesian coordinates]]. This constituted a major [[Paradigm shift|change of paradigm]]: Instead of defining [[real number]]s as lengths of [[line segments]] (see [[number line]]), it allowed the representation of points using their ''coordinates'', which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: [[synthetic geometry]], which uses purely geometrical methods, and [[analytic geometry]], which uses coordinates systemically.<ref>{{cite book |last=Boyer |first=Carl B. |author-link=Carl B. Boyer |year=2004 |orig-date=1956 |chapter=Fermat and Descartes |pages=74–102 |title=History of Analytic Geometry |publisher=[[Dover Publications]] |isbn=0-486-43832-5 |lccn=2004056235 |oclc=56317813}}</ref>

Analytic geometry allows the study of [[curve]]s unrelated to circles and lines. Such curves can be defined as the [[graph of a function|graph of functions]], the study of which led to [[differential geometry]]. They can also be defined as [[implicit equation]]s, often [[polynomial equation]]s (which spawned [[algebraic geometry]]). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.<ref name=Straume_2014/>

In the 19th century, mathematicians discovered [[non-Euclidean geometries]], which do not follow the [[parallel postulate]]. By questioning that postulate's truth, this discovery has been viewed as joining [[Russell's paradox]] in revealing the [[foundational crisis of mathematics]]. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem.<ref>{{cite journal |last=Stump |year=1997 |first=David J. |title=Reconstructing the Unity of Mathematics circa 1900 |journal=[[Perspectives on Science]] |volume=5 |issue=3 |page=383&ndash;417 |doi=10.1162/posc_a_00532 |eissn=1530-9274 |issn=1063-6145 |lccn=94657506 |oclc=26085129 |s2cid=117709681 |url=https://philpapers.org/archive/STURTU.pdf |access-date=February 8, 2024}}</ref><ref name=Kleiner_1991/> In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that [[Invariant (mathematics)|do not change]] under specific transformations of the [[space (mathematics)|space]].<ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=February 1996 |title=Non-Euclidean geometry |website=MacTuror |publisher=[[University of St. Andrews]] |publication-place=Scotland, UK |url=https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/ |url-status=live |archive-url=https://web.archive.org/web/20221106142807/https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/ |archive-date=November 6, 2022 |access-date=February 8, 2024}}</ref>

Today's subareas of geometry include:<ref name=MSC/>
* [[Projective geometry]], introduced in the 16th century by [[Girard Desargues]], extends Euclidean geometry by adding [[points at infinity]] at which [[parallel lines]] intersect. This simplifies many aspects of classical geometry by unifying the treatments 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 function]]s.
* [[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 [[polynomial]]s.
* [[Topology]], the study of properties that are kept under [[continuous deformation]]s.
** [[Algebraic topology]], the use in topology of algebraic methods, mainly [[homological algebra]].
* [[Discrete geometry]], the study of finite configurations in geometry.
* [[Convex geometry]], the study of [[convex set]]s, which takes its importance from its applications in [[convex optimization|optimization]].
* [[Complex geometry]], the geometry obtained by replacing real numbers with [[complex number]]s.

=== Algebra ===
{{Main|Algebra}}
[[File:Quadratic formula.svg|thumb|The [[quadratic formula]], which concisely expresses the solutions of all [[quadratic equation]]s]]  [[File:Rubik's cube.svg|thumb|The [[Rubik's Cube group]] is a concrete application of [[group theory]].<ref>{{cite book |last=Joyner |first=David |year=2008 |chapter=The (legal) Rubik's Cube group |title=Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys |pages=219–232 |edition=2nd |publisher=[[Johns Hopkins University Press]] |isbn=978-0-8018-9012-3 |lccn=2008011322 |oclc=213765703}}</ref>]]

Algebra is the art of manipulating [[equation]]s and formulas. Diophantus (3rd century) and [[Muhammad ibn Musa al-Khwarizmi|al-Khwarizmi]] (9th century) were the two main precursors of algebra.<ref>{{cite journal |last1=Christianidis |first1=Jean  |last2=Oaks |first2=Jeffrey |date=May 2013 |title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria |journal=Historia Mathematica |volume=40 |issue=2 |pages=127–163 |doi=10.1016/j.hm.2012.09.001 |doi-access=free |eissn=1090-249X |issn=0315-0860 |lccn=75642280 |oclc=2240703 |s2cid=121346342}}</ref>{{sfn|Kleiner|2007|loc="History of Classical Algebra" pp. 3–5}} Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term ''algebra'' is derived from the [[Arabic]] word ''al-jabr'' meaning 'the reunion of broken parts'<ref>{{Cite web |last=Lim |first=Lisa |date=December 21, 2018 |title=Where the x we use in algebra came from, and the X in Xmas |website=[[South China Morning Post]] |url=https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas |url-access=limited |url-status=live |archive-url=https://web.archive.org/web/20181222003908/https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas |archive-date=December 22, 2018 |access-date=February 8, 2024}}</ref> 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 became an area in its own right only with [[François Viète]] (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.<ref>{{cite journal |last=Oaks |first=Jeffery A. |year=2018 |title=François Viète's revolution in algebra |journal=[[Archive for History of Exact Sciences]] |volume=72 |issue=3 |pages=245–302 |doi=10.1007/s00407-018-0208-0 |eissn=1432-0657 |issn=0003-9519 |lccn=63024699 |oclc=1482042 |s2cid=125704699 |url=https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf |url-status=live |archive-url=https://web.archive.org/web/20221108162134/https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf |archive-date=November 8, 2022 |access-date=February 8, 2024}}</ref> Variables allow mathematicians to describe the operations that have to be done on the numbers represented using [[mathematical formulas]].

Until the 19th century, algebra consisted mainly of the study of [[linear equation]]s (presently ''[[linear algebra]]''), and polynomial equations in a single [[unknown (algebra)|unknown]], which were called ''algebraic equations'' (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as [[matrix (mathematics)|matrices]], [[modular arithmetic|modular integers]], and [[geometric transformation]]s), on which generalizations of arithmetic operations are often valid.{{sfn|Kleiner|2007|loc="History of Linear Algebra" pp. 79–101}} The concept of [[algebraic structure]] addresses this, consisting 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. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called ''modern algebra'' or [[abstract algebra]], as established by the influence and works of [[Emmy Noether]].<ref>{{cite book |last=Corry |first=Leo |author-link=Leo Corry |year=2004 |chapter=Emmy Noether: Ideals and Structures |title=Modern Algebra and the Rise of Mathematical Structures |pages=247–252 |edition=2nd revised |publisher=Birkhäuser Basel |publication-place=Germany |isbn=3-7643-7002-5 |lccn=2004556211 |oclc=51234417 |url={{GBurl|id=WdGbeyehoCoC|p=247}} |access-date=February 8, 2024}}</ref> (The latter term appears mainly in an educational context, in opposition to [[elementary algebra]], which is concerned with the older way of manipulating formulas.)

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:<ref name=MSC/>
* [[group theory]];
* [[field (mathematics)|field theory]];
* [[vector space]]s, whose study is essentially the same as [[linear algebra]];
* [[ring theory]];
* [[commutative algebra]], which is the study of [[commutative ring]]s, includes the study of [[polynomial]]s, and is a foundational part of [[algebraic geometry]];
* [[homological algebra]];
* [[Lie algebra]] and [[Lie group]] theory;
* [[Boolean algebra]], which is widely used for the study of the logical structure of [[computer]]s.

The study of types of algebraic structures as [[mathematical object]]s is the purpose of [[universal algebra]] and [[category theory]].<ref>{{cite book |last=Riche |first=Jacques |editor1-last=Beziau |editor1-first=J. Y. |editor2-last=Costa-Leite |editor2-first=Alexandre |year=2007 |chapter=From Universal Algebra to Universal Logic |pages=3–39 |title=Perspectives on Universal Logic |publisher=Polimetrica International Scientific Publisher |publication-place=Milano, Italy |isbn=978-88-7699-077-9 |oclc=647049731 |url={{GBurl|id=ZoRN9T1GUVwC|p=3}} |access-date=February 8, 2024}}</ref> The latter applies to every [[mathematical structure]] (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as [[topological space]]s; this particular area of application is called [[algebraic topology]].<ref>{{cite book |last=Krömer |first=Ralph |year=2007 |title=Tool and Object: A History and Philosophy of Category Theory |pages=xxi–xxv, 1–91 |series=Science Networks - Historical Studies |volume=32 |publisher=[[Springer Science & Business Media]] |publication-place=Germany |isbn=978-3-7643-7523-2 |lccn=2007920230 |oclc=85242858 |url={{GBurl|id=41bHxtHxjUAC|pg=PR20}} |access-date=February 8, 2024}}</ref>

=== Calculus and analysis ===
{{Main|Calculus|Mathematical analysis}}
[[File:Cauchy sequence illustration.svg|thumb|A [[Cauchy sequence]] consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses (from left to right).]]

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians [[Isaac Newton|Newton]] and [[Leibniz]].<ref>{{cite book |last=Guicciardini |first=Niccolo |author-link=Niccolò Guicciardini |editor1-last=Schliesser |editor1-first=Eric |editor2-last=Smeenk |editor2-first=Chris |year=2017 |chapter=The Newton–Leibniz Calculus Controversy, 1708–1730 |title=The Oxford Handbook of Newton |series=Oxford Handbooks |publisher=[[Oxford University Press]] |doi=10.1093/oxfordhb/9780199930418.013.9 |isbn=978-0-19-993041-8 |oclc=975829354 |chapter-url=https://core.ac.uk/download/pdf/187993169.pdf |url-status=live |archive-url=https://web.archive.org/web/20221109163253/https://core.ac.uk/download/pdf/187993169.pdf |archive-date=November 9, 2022 |access-date=February 9, 2024}}</ref> It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by [[Euler]] with the introduction of the concept of a [[function (mathematics)|function]] and many other results.<ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |date=September 1998 |title=Leonhard Euler |website=MacTutor |publisher=[[University of St Andrews]] |publication-place=Scotland, UK |url=https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ |url-status=live |archive-url=https://web.archive.org/web/20221109164921/https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ |archive-date=November 9, 2022 |access-date=February 9, 2024}}</ref> 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]]s, and [[complex analysis]], where variables represent [[complex number]]s. Analysis includes many subareas shared by other areas of mathematics which include:<ref name=MSC/>
* [[Multivariable calculus]]
* [[Functional analysis]], where variables represent varying functions;
* [[Integration (mathematics)|Integration]], [[measure theory]] and [[potential theory]], all strongly related with [[probability theory]] on a [[Continuum (set theory)|continuum]];
* [[Ordinary differential equation]]s;
* [[Partial differential equation]]s;
* [[Numerical analysis]], mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

=== Discrete mathematics ===
{{Main|Discrete mathematics}}
[[File:Markovkate_01.svg|right|thumb|A diagram representing a two-state [[Markov chain]]. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.]]
Discrete mathematics, broadly speaking, is the study of individual, [[Countable set|countable]] mathematical objects. An example is the set of all integers.<ref>{{cite journal |last=Franklin |first=James |author-link=James Franklin (philosopher) |date=July 2017 |title=Discrete and Continuous: A Fundamental Dichotomy in Mathematics |journal=Journal of Humanistic Mathematics |volume=7 |issue=2 |pages=355–378 |url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1334&context=jhm |doi=10.5642/jhummath.201702.18 |doi-access=free |issn=2159-8118 |lccn=2011202231 |oclc=700943261 |s2cid=6945363 |access-date=February 9, 2024}}</ref> Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.{{efn|However, some advanced methods of analysis are sometimes used; for example, methods of [[complex analysis]] applied to [[generating series]].}} [[Algorithm]]s{{emdash}}especially their [[implementation]] and [[computational complexity]]{{emdash}}play a major role in discrete mathematics.<ref>{{cite book |last=Maurer |first=Stephen B. |editor1-last=Rosenstein |editor1-first=Joseph G. |editor2-last=Franzblau |editor2-first=Deborah S. |editor3-last=Roberts |editor3-first=Fred S. |editor3-link=Fred S. Roberts |year=1997 |chapter=What is Discrete Mathematics? The Many Answers |pages=121–124 |title=Discrete Mathematics in the Schools |series=DIMACS: Series in Discrete Mathematics and Theoretical Computer Science |volume=36 |publisher=[[American Mathematical Society]] |doi=10.1090/dimacs/036/13 |isbn=0-8218-0448-0 |issn=1052-1798 |lccn=97023277 |oclc=37141146 |s2cid=67358543 |chapter-url={{GBurl|id=EvuQdO3h-DQC|p=121}} |access-date=February 9, 2024}}</ref>

The [[four color theorem]] and [[Kepler conjecture|optimal sphere packing]] were two major problems of discrete mathematics solved in the second half of the 20th century.<ref>{{cite book |last=Hales |first=Thomas C. |title=Turing's Legacy |author-link=Thomas Callister Hales |editor-last=Downey |editor-first=Rod |editor-link=Rod Downey |year=2014 |pages=260–261 |chapter=Turing's Legacy: Developments from Turing's Ideas in Logic |publisher=[[Cambridge University Press]] |series=Lecture Notes in Logic |volume=42 |doi=10.1017/CBO9781107338579.001 |isbn=978-1-107-04348-0 |lccn=2014000240 |oclc=867717052 |s2cid=19315498 |chapter-url={{GBurl|id=fYgaBQAAQBAJ|p=260}} |access-date=February 9, 2024}}</ref> The [[P versus NP problem]], which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of [[Computationally expensive|computationally difficult]] problems.<ref>{{cite conference |last=Sipser |first=Michael |author-link=Michael Sipser |date=July 1992 |title=The History and Status of the P versus NP Question |conference=STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing |pages=603–618 |doi=10.1145/129712.129771 |s2cid=11678884}}</ref>

Discrete mathematics includes:<ref name=MSC/><!-- Scope of [[Discrete Mathematics (journal)]] [https://www.journals.elsevier.com/discrete-mathematics]The research areas covered by Discrete Mathematics include graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, discrete probability, and parts of cryptography.

Discrete Mathematics generally does not include research on dynamical systems, differential equations, or discrete Laplacian operators within its scope. It also does not publish articles that are principally focused on linear algebra, abstract algebraic structures, or fuzzy sets unless they are highly related to one of the main areas of interest. Also, papers focused primarily on applied problems or experimental results fall outside our scope.

In [[Discrete Mathematics and Computer Science]] [https://dmtcs.episciences.org/page/policies]
General

Analysis of algorithms

Automata, logics and semantics

Combinatorics

Discrete algorithms

Distributed Computing and networking

Graph Theory
 -->
* [[Combinatorics]], the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or [[subset]]s of a given [[set (mathematics)|set]]; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of [[geometric shape]]s
* [[Graph theory]] and [[hypergraph]]s
* [[Coding theory]], including [[error correcting code]]s and a part of [[cryptography]]
* [[Matroid]] theory
* [[Discrete geometry]]
* [[Discrete probability distribution]]s
* [[Game theory]] (although [[continuous game]]s are also studied, most common games, such as [[chess]] and [[poker]] are discrete)
* [[Discrete optimization]], including [[combinatorial optimization]], [[integer programming]], [[constraint programming]]

=== Mathematical logic and set theory ===
{{Main|Mathematical logic|Set theory}}
[[File:Venn A intersect B.svg|thumb|The [[Venn diagram]] is a commonly used method to illustrate the relations between sets.]]

The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.<ref name=Ewald_2018>{{cite web
 | first=William
 | last=Ewald
 | date=November 17, 2018
| title=The Emergence of First-Order Logic
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/settheory-early/
 | access-date=November 2, 2022
 | archive-date=May 12, 2021
| archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/
 | url-status=live
 }}</ref><ref name="Ferreirós_2020">{{cite web
 | first=José
 | last=Ferreirós
 | date=June 18, 2020
| title=The Early Development of Set Theory
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/settheory-early/
 | access-date=November 2, 2022
 | archive-date=May 12, 2021
| archive-url=https://web.archive.org/web/20210512135148/https://plato.stanford.edu/entries/settheory-early/
 | url-status=live
 }}</ref> Before this period, sets were not considered to be mathematical objects, and [[logic]], although used for mathematical proofs, belonged to [[philosophy]] and was not specifically studied by mathematicians.<ref>{{Cite journal
 | title=The Road to Modern Logic—An Interpretation
 | last=Ferreirós
 | first=José
 | journal=Bulletin of Symbolic Logic
 | volume=7
 | issue=4
 | pages=441–484
 | date=2001
 | doi=10.2307/2687794
 | jstor=2687794
 | hdl=11441/38373
 | s2cid=43258676
 | url=https://idus.us.es/xmlui/bitstream/11441/38373/1/The%20road%20to%20modern%20logic.pdf
 | access-date=November 11, 2022
 | archive-url=https://web.archive.org/web/20230202133703/https://idus.us.es/bitstream/handle/11441/38373/The%20road%20to%20modern%20logic.pdf?sequence=1
 | archive-date=February 2, 2023
 | url-status=live
 }}</ref>

Before [[Georg Cantor|Cantor]]'s study of [[infinite set]]s, mathematicians were reluctant to consider [[actual infinite|actually infinite]] collections, and considered [[infinity]] to be the result of endless [[enumeration]]. Cantor's work offended many mathematicians not only by considering actually infinite sets<ref>{{cite web | first=Natalie | last=Wolchover | author-link=Natalie Wolchover | date=December 3, 2013 | title=Dispute over Infinity Divides Mathematicians | website=[[Scientific American]] | url=https://www.scientificamerican.com/article/infinity-logic-law/ | access-date=November 1, 2022 | archive-date=November 2, 2022 | archive-url=https://web.archive.org/web/20221102011848/https://www.scientificamerican.com/article/infinity-logic-law/ | url-status=live }}</ref> but by showing that this implies different sizes of infinity, per [[Cantor's diagonal argument]]. This led to the [[controversy over Cantor's theory|controversy over Cantor's set theory]].<ref>{{cite web
 | title=Wittgenstein's analysis on Cantor's diagonal argument
 | last=Zhuang | first=C. | website=[[PhilArchive]]
 | url=https://philarchive.org/archive/ZHUWAO
 | access-date=November 18, 2022 }}</ref>

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring [[mathematical rigour]]. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics.<ref>{{cite web
 | title="Clarifying the nature of the infinite": the development of metamathematics and proof theory
 | first1=Jeremy
 | last1=Avigad
 | author1-link=Jeremy Avigad
 | first2=Erich H.
 | last2=Reck
 | date=December 11, 2001
| work=Carnegie Mellon Technical Report CMU-PHIL-120
 | url=https://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf
 | access-date=November 12, 2022
 | archive-date=October 9, 2022
| archive-url=https://web.archive.org/web/20221009074025/https://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf
 | url-status=live
 }}</ref> It was eventually solved in mainstream mathematics by systematizing the axiomatic method 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.<ref name=Warner_2013/> For example, in [[Peano arithmetic]], the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.<ref>{{cite book
 | title=Numbers, Sets and Axioms: The Apparatus of Mathematics
 | first=Alan G.
 | last=Hamilton
 | pages=3–4
 | year=1982
 | isbn=978-0-521-28761-6
 | publisher=Cambridge University Press
 | url={{GBurl|id=OXfmTHXvRXMC|p=3}}
 | access-date=November 12, 2022
}}</ref> This [[abstraction (mathematics)|mathematical abstraction]] from reality is embodied in the modern philosophy of [[Formalism (philosophy of mathematics)|formalism]], as founded by [[David Hilbert]] around 1910.<ref name="Snapper">{{Cite journal |doi=10.2307/2689412 |title=The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism |journal=Mathematics Magazine |date=September 1979 |first=Ernst |last=Snapper |author-link=Ernst Snapper |volume=52 |issue=4 |pages=207–216 |jstor=2689412 }}</ref>

The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion{{emdash}}sometimes called "intuition"{{emdash}}to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, [[Gödel's incompleteness theorems]] assert, roughly speaking that, in every [[Consistency|consistent]] [[formal system]] that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.<ref name=Raatikainen_2005>{{cite journal | title=On the Philosophical Relevance of Gödel's Incompleteness Theorems | first=Panu | last=Raatikainen | journal=Revue Internationale de Philosophie | volume=59 | issue=4 | date=October 2005 | pages=513–534 | doi=10.3917/rip.234.0513 | url=https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm | jstor=23955909 | s2cid=52083793 | access-date=November 12, 2022 | archive-date=November 12, 2022 | archive-url=https://web.archive.org/web/20221112212555/https://www.cairn.info/revue-internationale-de-philosophie-2005-4-page-513.htm | url-status=live }}</ref> This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by [[L. E. J. Brouwer|Brouwer]], who promoted [[intuitionistic logic]], which explicitly lacks the [[law of excluded middle]].<ref>{{cite web
 | title=Intuitionistic Logic
 | date=September 4, 2018
| first=Joan
 | last=Moschovakis
 | author-link=Joan Moschovakis
 | website=Stanford Encyclopedia of Philosophy
 | url=https://plato.stanford.edu/entries/logic-intuitionistic/
 | access-date=November 12, 2022
 | archive-date=December 16, 2022
| archive-url=https://web.archive.org/web/20221216154821/https://plato.stanford.edu/entries/logic-intuitionistic/
 | url-status=live
 }}</ref><ref>{{cite journal
 | title=At the Heart of Analysis: Intuitionism and Philosophy
 | first=Charles | last=McCarty
 | journal=Philosophia Scientiæ, Cahier spécial 6
 | year=2006 | pages=81–94 | doi=10.4000/philosophiascientiae.411
| doi-access=free}}</ref>

These problems and debates led to a wide expansion of mathematical logic, with subareas such as [[model theory]] (modeling some logical theories inside other theories), [[proof theory]], [[type theory]], [[computability theory]] and [[computational complexity theory]].<ref name=MSC/> Although these aspects of mathematical logic were introduced before the rise of [[computer]]s, their use in [[compiler]] design, [[computer program|program certification]], [[proof assistant]]s and other aspects of [[computer science]], contributed in turn to the expansion of these logical theories.<ref>{{cite web
 | last1=Halpern | first1=Joseph | author1-link=Joseph Halpern
 | last2=Harper | first2=Robert | author2-link=Robert Harper (computer scientist)
 | last3=Immerman | first3=Neil | author3-link=Neil Immerman
 | last4=Kolaitis | first4=Phokion | author4-link=Phokion Kolaitis
 | last5=Vardi | first5=Moshe | author5-link=Moshe Vardi
 | last6=Vianu | first6=Victor | author6-link=Victor Vianu
 | title=On the Unusual Effectiveness of Logic in Computer Science
 | url=https://www.cs.cmu.edu/~rwh/papers/unreasonable/basl.pdf
 | access-date=January 15, 2021 | date=2001 | archive-date=March 3, 2021
| archive-url=https://web.archive.org/web/20210303115643/https://www.cs.cmu.edu/~rwh/papers/unreasonable/basl.pdf
 | url-status=live }}</ref>

=== Statistics and other decision sciences ===
{{Main|Statistics|Probability theory}}
[[File:IllustrationCentralTheorem.png|upright=1.5|thumb|right|Whatever the form of a random population [[Probability distribution|distribution]] (μ), the sampling [[mean]] (x̄) tends to a [[Gaussian]] distribution and its [[variance]] (σ) is given by the [[central limit theorem]] of probability theory.<ref>{{cite book |last=Rouaud |first=Mathieu |date=April 2017 |orig-date=First published July 2013 |title=Probability, Statistics and Estimation |page=10 |url=http://www.incertitudes.fr/book.pdf |url-status=live |archive-url=https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf |archive-date=October 9, 2022 |access-date=February 13, 2024}}</ref>]]

The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially [[probability theory]]. Statisticians generate data with [[random sampling]] or randomized [[design of experiments|experiments]].<ref>{{cite book |last=Rao |first=C. Radhakrishna |author-link=C. R. Rao |year=1997 |orig-date=1989 |title=Statistics and Truth: Putting Chance to Work |edition=2nd |pages=3–17, 63–70 |publisher=World Scientific |isbn=981-02-3111-3 |lccn=97010349 |mr=1474730 |oclc=36597731}}</ref> The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from [[observational study|observational studies]] is done using [[statistical model]]s and the theory of [[statistical inference|inference]], using [[model selection]] and [[estimation theory|estimation]]. The models and consequential [[Scientific method#Predictions from the hypothesis|predictions]] should then be [[statistical hypothesis testing|tested]] against [[Scientific method#Evaluation and improvement|new data]].{{efn|Like other mathematical sciences such as [[physics]] and [[computer science]], statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.}}

[[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.<ref name="RaoOpt">{{cite book |last=Rao |first=C. Radhakrishna |author-link=C.R. Rao |editor1-last=Arthanari |editor1-first=T.S. |editor2-last=Dodge |editor2-first=Yadolah |editor2-link=Yadolah Dodge |chapter=Foreword |title=Mathematical programming in statistics |series=Wiley Series in Probability and Mathematical Statistics |publisher=Wiley |location=New York |year=1981 |pages=vii–viii |isbn=978-0-471-08073-2 |lccn=80021637 |mr=607328 |oclc=6707805}}</ref> Because of its use of [[mathematical optimization|optimization]], the mathematical theory of statistics overlaps with other [[decision science]]s, such as [[operations research]], [[control theory]], and [[mathematical economics]].{{sfn|Whittle|1994|pp=10–11, 14–18}}

=== Computational mathematics ===
{{Main|Computational mathematics}}
Computational mathematics is the study of [[mathematical problem]]s that are typically too large for human, numerical capacity.<ref>{{cite web
 | title=G I Marchuk's plenary: ICM 1970
 | first=Gurii Ivanovich
 | last=Marchuk
 | website=MacTutor
 | date=April 2020
 | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
 | url=https://mathshistory.st-andrews.ac.uk/Extras/Computational_mathematics/
 | access-date=November 13, 2022
 | archive-date=November 13, 2022
| archive-url=https://web.archive.org/web/20221113155409/https://mathshistory.st-andrews.ac.uk/Extras/Computational_mathematics/
 | url-status=live
 }}</ref><ref>{{cite conference | title=Grand Challenges, High Performance Computing, and Computational Science | last1=Johnson | first1=Gary M. | last2=Cavallini | first2=John S. | conference=Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage | date=September 1991 | page=28 |lccn=91018998 |publisher=World Scientific | editor1-first=Kang Hoh | editor1-last=Phua | editor2-first=Kia Fock | editor2-last=Loe | url={{GBurl|id=jYNIDwAAQBAJ|p=28}} | access-date=November 13, 2022 }}</ref> [[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]] with special focus on [[rounding error]]s.<ref>{{cite book |last=Trefethen |first=Lloyd N. |author-link=Lloyd N. Trefethen |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor2-link=June Barrow-Green |editor3-last=Leader |editor3-first=Imre |editor3-link=Imre Leader |year=2008 |chapter=Numerical Analysis |pages=604–615 |title=The Princeton Companion to Mathematics |publisher=[[Princeton University Press]] |isbn=978-0-691-11880-2 |lccn=2008020450 |mr=2467561 |oclc=227205932 |url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |url-status=live |archive-url=https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |archive-date=March 7, 2023 |access-date=February 15, 2024}}</ref> 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]].
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