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'''Mathematics''' is an area of [[knowledge]] that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.<!-- Please, do not link articles to words that are used in their non-technical meaning, such as "numbers", "quantity", "formula", "structure", "shape", "space". Linking such articles would provide too much technical information about those abstractions. --> These topics are represented in modern mathematics with the major subdisciplines of [[number theory]],<ref name="OED">{{cite web
| url=http://oed.com/view/Entry/114974
| title=mathematics, ''n.''
| publisher=Oxford University Press
| website=Oxford English Dictionary
| year=2012 | access-date=June 16, 2012
| quote=The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
| archive-url=https://web.archive.org/web/20191116075558/https://www.oed.com/view/Entry/114974
| archive-date=November 16, 2019 | url-status=live
}}</ref> [[algebra]],<ref name="Kneebone">{{cite book
| title=Mathematical Logic and the Foundations of Mathematics: An Introductory Survey
| publisher=Dover
| last=Kneebone | first=G. T. | year=1963 | page=4
| url={{GBurl|id=tCXxf4vbXCcC|p=4}}
| isbn=978-0-486-41712-7
| quote=Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
| access-date=June 20, 2015
}}</ref> [[geometry]],<ref name=OED /> and [[mathematical analysis|analysis]],<ref name="LaTorre">{{cite book
| title=Calculus Concepts: An Informal Approach to the Mathematics of Change
| publisher=Cengage Learning
| first1=Donald R. | last1=LaTorre | first2=John W. | last2=Kenelly
| first3=Sherry S. | last3=Biggers | first4=Laurel R. | last4=Carpenter
| first5=Iris B. | last5=Reed | first6=Cynthia R. | last6=Harris
| year=2011 | page=2 | isbn=978-1-4390-4957-0
| url={{GBurl|id=1Ebu2Tij4QsC|p=2}}
| quote=Calculus is the study of change—how things change, and how quickly they change.
| access-date=June 20, 2015
}}</ref><ref name="Ramana">{{cite book
| title=Applied Mathematics
| publisher=Tata McGraw–Hill Education
| first=B. V. |last=Ramana
| year=2007 | page=2.10
| url={{GBurl|id=XCRC6BeKhIIC|pg=SA2–PA10}}
| isbn=978-0-07-066753-2
| quote=The mathematical study of change, motion, growth or decay is calculus.
| access-date=July 30, 2022
}}</ref> respectively.<!--<<< Please do NOT change the opening sentence without discussion; much time and discussion have been invested in its current form.--> There is no general consensus among mathematicians about a common definition for their [[academic discipline]].
Most mathematical activity involves the discovery of properties of [[mathematical object|abstract objects]] and the use of pure [[reason]] to [[proof (mathematics)|prove]] them. These objects consist of either [[abstraction (mathematics)|abstraction]]s from nature or{{emdash}}in modern mathematics{{emdash}}entities that are stipulated to have certain properties, called [[axiom]]s. A ''proof'' consists of a succession of applications of [[inference rule|deductive rules]] to already established results. These results include previously proved [[theorem]]s, axioms, and{{emdash}}in case of abstraction from nature{{emdash}}some basic properties that are considered true starting points of the theory under consideration.<ref>{{cite conference
| title=Abstract Cognition and the Nature of Mathematical Proof
| book-title=Realism – Relativism – Constructivism
| conference=38th International Wittgenstein Symposium, August 9–15, 2015. Kirchberg am Wechsel, Austria
| first=Inês Viegas
| last=Hipólito
| year=2015
| pages=132–134
| publisher=Austrian Ludwig Wittgenstein Society
| editor1-first=Christian
| editor1-last=Kanzian
| editor2-first=Josef
| editor2-last=Mitterer
| editor2-link=Josef Mitterer
| editor3-first=Katharina
| editor3-last=Neges
| url=https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133
| access-date=November 5, 2022
| archive-date=November 7, 2022
| archive-url=https://web.archive.org/web/20221107221937/https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133
| url-status=live
}} ([https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof at ResearchGate] {{Webarchive|url=https://web.archive.org/web/20221105145638/https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof |date=November 5, 2022 }})</ref><!-- Commenting out the following pending discussion on talk: Contrary to [[physical law]]s, the validity of a theorem (its truth) does not rely on any [[experimentation]] but on the correctness of its reasoning (though experimentation is often useful for discovering new theorems of interest). -->
Mathematics is essential in the [[natural science]]s, [[engineering]], [[medicine]], [[finance]], [[computer science]] and the [[social sciences]]. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as [[statistics]] and [[game theory]], are developed in close correlation with their applications and are often grouped under [[applied mathematics]]. Other areas are developed independently from any application (and are therefore called [[pure mathematics]]), but often later find practical applications.{{sfn|Peterson|2001|p=12}}<ref name=wigner1960 /> The problem of [[integer factorization]], for example, which goes back to [[Euclid]] in 300 BC, had no practical application before its use in the [[RSA cryptosystem]], now widely used for the security of [[computer network]]s.
Historically, the concept of a proof and its associated [[mathematical rigour]] first appeared in [[Greek mathematics]], most notably in Euclid's ''[[Euclid's Elements|Elements]]''.<ref>{{Cite web|url=http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Wise/essay7/essay7.htm|title=Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion|last=Wise|first=David|website=jwilson.coe.uga.edu|url-status=live|archive-url=https://web.archive.org/web/20190601004355/http://jwilson.coe.uga.edu/emt668/EMAT6680.F99/Wise/essay7/essay7.htm|archive-date=June 1, 2019|access-date=October 26, 2019}}</ref> Since its beginning, mathematics was primarily divided into geometry and [[arithmetic]] (the manipulation of [[natural number]]s and [[fraction (mathematics)|fractions]]), until the 16th and 17th centuries, when algebra{{efn|Here, ''algebra'' is taken in its modern sense, which is, roughly speaking, the art of manipulating [[formula]]s.}} and [[infinitesimal calculus]] were introduced as new fields. Since then, the interaction between mathematical innovations and [[timeline of scientific discoveries|scientific discoveries]] has led to a correlated increase in the development of both.<ref>{{cite journal
| title=The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?
| first=Amir | last=Alexander | author-link=Amir Alexander
| journal=Isis | year=2011 | volume=102 | number=3
| pages=475–480 | doi=10.1086/661620 | pmid=22073771 | s2cid=21629993 }}</ref> At the end of the 19th century, the [[foundational crisis of mathematics]] led to the systematization of the [[axiomatic method]],<ref name=Kleiner_1991/> which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary [[Mathematics Subject Classification]] lists more than 60 first-level areas of mathematics.
{{TOC limit|3}}
== Etymology ==
The word ''mathematics'' comes from [[Ancient Greek]] ''máthēma'' (''{{Lang-grc|{{wikt-lang|en|μάθημα}}|label=none}}''), meaning "that which is learnt",<ref name=EOD_n>{{cite encyclopedia
| title=mathematic (n.)
| first=Douglas
| last=Harper
| dictionary=[[Online Etymology Dictionary]]
| url=https://www.etymonline.com/word/mathematic
| access-date=November 26, 2022
| archive-date=March 7, 2013
| archive-url=https://web.archive.org/web/20130307093926/http://etymonline.com/index.php?term=mathematic&allowed_in_frame=0
| url-status=live
}}</ref> "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in [[Classical antiquity|Classical times]].<ref>Both meanings can be found in Plato, the narrower in {{cite book|title=Republic|title-link=Republic (Plato)|at=[https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 510c]}} {{Cite web |url=http://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 |title=Plato, Republic, Book 6, section 510c |access-date=June 19, 2019 |archive-date=February 24, 2021 |archive-url=https://web.archive.org/web/20210224152747/http://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 |url-status=live |postscript=,}} but Plato did not use a ''math-'' word; Aristotle did, commenting on it. {{LSJ|maqhmatiko/s|μαθηματική|ref}}. {{cite book|title=OED Online|at=Mathematics}}</ref> Its [[adjective]] is ''mathēmatikós'' ({{lang|grc|μαθηματικός}}), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".<ref>{{cite encyclopedia
| title=mathematical (adj.)
| first=Douglas
| last=Harper
| dictionary=Online Etymology Dictionary
| url=https://www.etymonline.com/word/mathematical
| access-date=November 26, 2022
| archive-date=November 26, 2022
| archive-url=https://web.archive.org/web/20221126170916/https://www.etymonline.com/word/mathematical
| url-status=live
}}</ref> In particular, ''mathēmatikḗ tékhnē'' ({{lang|grc|μαθηματικὴ τέχνη}}; {{lang-la|ars mathematica}}) meant "the mathematical art".<ref name=EOD_n/>
Similarly, one of the two main schools of thought in [[Pythagoreanism]] was known as the ''mathēmatikoi'' (μαθηματικοί){{emdash}}which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of [[arithmetic]] and geometry. By the time of [[Aristotle]] (384–322 BC) this meaning was fully established.<ref>{{cite journal
| title=The Etymology of Mathematical Terms
| first=Margaret W. | last=Perisho
| journal=Pi Mu Epsilon Journal
| volume=4 | issue=2 | date=Spring 1965 | pages=62–66
| jstor=24338341 }}</ref>
In Latin, and in English until around 1700, the term ''mathematics'' more commonly meant "[[astrology]]" (or sometimes "[[astronomy]]") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, [[Saint Augustine]]'s warning that Christians should beware of ''mathematici'', meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.<ref name="Boas">{{cite book | title=Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas, Jr | publisher=Cambridge University Press | last=Boas | first=Ralph | author-link=Ralph P. Boas Jr. | year=1995 | orig-date=1991 | page=257 | chapter-url={{GBurl|id=f-EWj5WtQHgC|p=257}} | chapter=What Augustine Didn't Say About Mathematicians | isbn=978-0-88385-323-8 | access-date=January 17, 2018 }}</ref>
The apparent [[plural]] form in English goes back to the Latin [[Neuter (grammar)|neuter]] plural {{lang|la|mathematica}} ([[Cicero]]), based on the Greek plural ''ta mathēmatiká'' ({{lang|el|τὰ μαθηματικά}}) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective ''mathematic(al)'' and formed the noun ''mathematics'' anew, after the pattern of ''[[physics]]'' and ''[[metaphysics]]'', inherited from Greek.<ref>''[[The Oxford Dictionary of English Etymology]]'', ''[[Oxford English Dictionary]]'', ''sub'' "mathematics", "mathematic", "mathematics".</ref> In English, the noun ''mathematics'' takes a singular verb. It is often shortened to ''maths'' or, in North America, ''math''.<ref name=maths>[http://oed.com/view/Entry/114982 "maths, ''n.''"] and [http://oed.com/view/Entry/114962 "math, ''n.3''"] {{Webarchive|url=https://web.archive.org/web/20200404201407/http://oed.com/view/Entry/114982 |date=April 4, 2020 }}. ''Oxford English Dictionary'', on-line version (2012).</ref>
== 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 | author-link=Eric Temple Bell | first=E. T. | last=Bell | title=The Development of Mathematics | series=Dover Books on Mathematics | page=3 | publisher=Courier Corporation | isbn=978-0-486-15228-8 | year=2012 | edition=reprint, revised | url={{GBurl|id=4lTCAgAAQBAJ|p=3}} | access-date=November 11, 2022 }}</ref> Some types of [[pseudoscience]], such as [[numerology]] and astrology, were not then clearly distinguished from mathematics.<ref>{{cite book | title=Mathematics in History, Culture, Philosophy, and Science | first=Sarju | last=Tiwari | year=1992 | page=27 | publisher=Mittal Publications | isbn=978-81-7099-404-6 | url={{GBurl|id=-yyDJUD0DNwC|p=27}} | access-date=March 19, 2023 }}</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 | title=Mathematics in Society and History | first=S. | last=Restivo | author-link=Sal Restivo | date=December 2013 | pages=14–15 | publisher=Springer Netherlands | isbn=978-94-011-2944-2 | url={{GBurl|id=V0RuCQAAQBAJ|p=14}} | access-date=March 19, 2023 }}</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
| title=Leonhard Euler and the Foundations of Celestial Mechanics
| first=Dora
| last=Musielak
| author-link=Dora Musielak
| pages=1–183
| year=2022
| isbn=978-3-031-12322-1
| publisher=Springer International Publishing
| url={{GBurl|id=ZY2ZEAAAQBAJ|p=2}}
| access-date=March 19, 2023
}}</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
| title=The roots of combinatorics
| first=N. L. | last=Biggs
| journal=Historia Mathematica
| volume=6 | issue=2 | date=May 1979 | pages=109–136
| doi=10.1016/0315-0860(79)90074-0
| doi-access=free}}</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 | title=Splash Talk: The Foundational Crisis of Mathematics | year=2013 | first=Evan | last=Warner | publisher=Columbia University | pages=1–17 | url=https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf | access-date=November 4, 2022 | archive-date=April 14, 2021 | archive-url=https://web.archive.org/web/20210414191543/https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf | url-status=live }}</ref><ref name=Kleiner_1991/> The 2020 [[Mathematics Subject Classification]] contains no less than {{em|sixty-three}} first-level areas.<ref>{{cite journal | title=Mathematics Subject Classification 2020 | first1=Edward | last1=Dunne | first2=Klaus | last2=Hulek | author2-link=Klaus Hulek | journal=Notices of the American Mathematical Society | volume=67 | issue=3 | date=March 2020 | url=https://www.ams.org/journals/notices/202003/rnoti-p410.pdf?trk=2052&cat=collection | access-date=November 4, 2022 | archive-date=November 20, 2022 | archive-url=https://web.archive.org/web/20221120014912/https://www.ams.org/journals/notices/202003/rnoti-p410.pdf?trk=2052&cat=collection | url-status=live }}</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/>
=== 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
| title=Fundamentals of Number Theory
| first=William J.
| last=LeVeque
| author-link=William J. LeVeque
| date=January 5, 2014
| pages=1–30
| isbn=978-0-486-14150-3
| publisher=Dover Publications
| url={{GBurl|id=0y3DAgAAQBAJ|p=1}}
| access-date=March 19, 2023
}}</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 | title=The Queen of Mathematics: A Historically Motivated Guide to Number Theory | first=Jay | last=Goldman | year=1997 | pages=1–3 | publisher=CRC Press | isbn=978-1-4398-6462-3 | url={{GBurl|id=A0FZDwAAQBAJ|pg=PP1}} | access-date=November 11, 2022 }}</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
| title=Number Theory, An Approach Through History From Hammurapi to Legendre
| first=André
| last=Weil
| author-link=André Weil
| year=2007
| isbn=978-0-8176-4571-7
| pages=1–3
| publisher=Birkhäuser Boston
| url={{GBurl|id=Ar7gBwAAQBAJ|p=1}}
| access-date=March 19, 2023
}}</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
| title=From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem
| first=Israel | last=Kleiner | author-link=Israel Kleiner (mathematician)
| journal=Elemente der Mathematik
| volume=55 | pages=19–37 | date=February 2000
| doi=10.1007/PL00000079
| s2cid=53319514 | doi-access=free}}</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 | title=The Goldbach Conjecture | pages=1–18 | first=Yuan | last=Wang | author-link=Wang Yuan (mathematician) | publisher=[[World Scientific]] | year=2002 | edition=revised | isbn=978-981-277-660-0 | volume=4 | series=Series in pure mathematics | url={{GBurl|id=VAY9nTreXkcC|p=1}} | access-date=November 11, 2022 }}</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>{{cite web
| title=MSC2020-Mathematics Subject Classification System
| website=zbMath
| publisher=Associate Editors of Mathematical Reviews and zbMATH
| url=https://zbmath.org/static/msc2020.pdf
| access-date=November 26, 2022
| archive-date=July 31, 2020
| archive-url=https://web.archive.org/web/20200731133802/https://zbmath.org/static/msc2020.pdf
| url-status=live
}}</ref>
=== 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 journal
| title=A Survey of the Development of Geometry up to 1870
| last=Straume | first=Eldar
| journal=ePrint | date=September 2014
| arxiv=1409.1140 | bibcode=2014arXiv1409.1140S }}</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 | title=The Foundations of Geometry | first=David | last=Hilbert | author-link=David Hilbert | year=1962 | page=1 | publisher=Open Court Publishing Company | url={{GBurl|id=0AA8AQAAMAAJ}} | access-date=March 22, 2023 }}</ref><ref>{{cite book
| title=Geometry: Euclid and Beyond
| first=Robin
| last=Hartshorne
| author-link=Robin Hartshorne
| pages=9–13
| isbn=978-0-387-22676-7
| publisher=Springer New York
| date=November 11, 2013
| url={{GBurl|id=C5fSBwAAQBAJ|p=9}}
| access-date=March 19, 2023
| archive-date=<!-- March 23, 2023 -->
}}</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 | title=History of Analytic Geometry | first=Carl B. | last=Boyer | author-link=Carl B. Boyer | pages=74–102 | isbn=978-0-486-15451-0 | publisher=Dover Publications | date=June 28, 2012 | url={{GBurl|id=2T4i5fXZbOYC|p=74}} | access-date=March 19, 2023 }}</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 | title=Reconstructing the Unity of Mathematics circa 1900 | first=David J. | last=Stump | journal=Perspectives on Science | year=1997 | volume=5 | issue=3 | page=383 | doi=10.1162/posc_a_00532 | s2cid=117709681 | url=https://repository.usfca.edu/phil/10 | access-date=November 6, 2022 | archive-date=November 6, 2022 | archive-url=https://web.archive.org/web/20221106142816/https://repository.usfca.edu/phil/10/ | url-status=live }}</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
| title=Non-Euclidean geometry
| first1=J. J.
| last1=O'Connor
| first2=E. F.
| last2=Robertson
| website=MacTuror
| date=February 1996
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/
| access-date=November 6, 2022
| archive-date=November 6, 2022
| archive-url=https://web.archive.org/web/20221106142807/https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/
| url-status=live
}}</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=D. | year=2008
| title=Adventures in group theory: Rubik's Cube, Merlin's machine, and other mathematical toys
| pages=219–232 | edition=2nd | isbn=978-0-8018-9013-0
| publisher=Johns Hopkins University Press
}}</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
| title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria
| first1=Jean | last1=Christianidis | first2=Jeffrey | last2=Oaks
| journal=Historia Mathematica
| volume=40 | issue=2 | date=May 2013 | pages=127–163
| doi=10.1016/j.hm.2012.09.001
| doi-access=free }}</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
| first=Lisa
| last=Lim
| date=December 21, 2018
| title=Where algebra got its x from, and Xmas its X
| url=https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas
| access-date=August 9, 2022
| website=South China Morning Post
| language=en
| archive-date=August 9, 2022
| archive-url=https://web.archive.org/web/20220809001307/https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas
| url-status=live
}}</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 | title=François Viète's revolution in algebra | last=Oaks | first=J. A. | journal=Archive for History of Exact Sciences | volume=72 | pages=245–302 | year=2018 | issue=3 | url=https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf | access-date=November 8, 2022 | doi=10.1007/s00407-018-0208-0 | s2cid=125704699 | archive-date=November 8, 2022 | archive-url=https://web.archive.org/web/20221108162134/https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf | url-status=live }}</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 | title=Modern Algebra and the Rise of Mathematical Structures | first=Leo | last=Corry | author-link=Leo Corry | date=December 6, 2012 | pages=247–252 | isbn=978-3-0348-7917-0 | publisher=Birkhäuser Basel | url={{GBurl|id=8G0FCAAAQBAJ|p=247}} | access-date=March 19, 2023 }}</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
| chapter=From Universal Algebra to Universal Logic
| first=Jacques
| last=Riche
| title=Perspective on Universal Logic
| publisher=Polimetrica International Scientific
| publication-place=Italy
| pages=3–39
| year=2007
| isbn=978-88-7699-077-9
| editor1-first=J. Y.
| editor1-last=Beziau
| editor2-first=Alexandre
| editor2-last=Costa-Leite
| url={{GBurl|id=ZoRN9T1GUVwC|p=3}}
| access-date=November 25, 2022
| archive-date=<!-- March 7, 2023 -->
}}</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
| title=Tool and Object: A History and Philosophy of Category Theory
| first=Ralph
| last=Krömer
| publisher=Springer Science & Business Media
| series=Science Networks. Historical Studies
| volume=32
| isbn=978-3-7643-7524-9
| year=2007
| pages=xxi–xxv, 1–91
| url={{GBurl|id=41bHxtHxjUAC|pg=PR20}}
| access-date=November 25, 2022
}}</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
| chapter=The Newton–Leibniz Calculus Controversy, 1708–1730
| first=Niccolo
| last=Guicciardini
| author-link=Niccolò Guicciardini
| title=The Oxford Handbook of Newton
| series=Oxford handbooks
| publisher=Oxford University Press
| editor1-first=Eric
| editor1-last=Schliesser
| editor2-first=Chris
| editor2-last=Smeenk
| isbn=978-0-19-993041-8
| year=2017
| chapter-url=https://core.ac.uk/download/pdf/187993169.pdf
| access-date=November 9, 2022
| doi=10.1093/oxfordhb/9780199930418.013.9
| archive-date=November 9, 2022
| archive-url=https://web.archive.org/web/20221109163253/https://core.ac.uk/download/pdf/187993169.pdf
| url-status=live
}}</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 | title=Leonhard Euler | website=MacTutor | date=September 1998 | first1=J. J. | last1=O'Connor | first2=E. F. | last2=Robertson | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland | url=https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ | access-date=November 9, 2022 | archive-date=November 9, 2022 | archive-url=https://web.archive.org/web/20221109164921/https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ | url-status=live }}</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 | title=Discrete and continuous: a fundamental dichotomy in mathematics | last1=Franklin | first1=James | author-link=James Franklin (philosopher) | journal=Journal of Humanistic Mathematics | year=2017 | volume=7 | issue=2 | pages=355–378 | url=https://scholarship.claremont.edu/jhm/vol7/iss2/18/ | doi=10.5642/jhummath.201702.18 | s2cid=6945363 | access-date=November 10, 2022 | doi-access=free | archive-date=November 10, 2022 | archive-url=https://web.archive.org/web/20221110155912/https://scholarship.claremont.edu/jhm/vol7/iss2/18/ | url-status=live }}</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
| chapter=What is Discrete Mathematics? The Many Answers
| first=Stephen B.
| last=Maurer
| title=Discrete Mathematics in the Schools
| publisher=American Mathematical Society
| editor1-first=Joseph G.
| editor1-last=Rosenstein
| editor2-first=Deborah S.
| editor2-last=Franzblau
| editor3-first=Fred S.
| editor3-last=Roberts
| isbn=978-0-8218-8578-9
| pages=121–124
| volume=36
| year=1997
| series=DIMACS: Series in Discrete Mathematics and Theoretical Computer Science
| url={{GBurl|id=EvuQdO3h-DQC|p=121}}
| access-date=November 10, 2022
}}</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
| title=Turing's Legacy: Developments from Turing's Ideas in Logic
| first=Rod
| last=Downey
| author-link=Rod Downey
| pages=260–261
| publisher=Cambridge University Press
| year=2014
| series=Issue 42 of Lecture Notes in Logic
| isbn=978-1-107-04348-0
| url={{GBurl|id=fYgaBQAAQBAJ|p=260}}
| access-date=November 10, 2022
}}</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
| title=The history and status of the P versus NP question
| first=Michael | last=Sipser | author-link=Michael Sipser
| conference=STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing
| date=July 1992 | pages=603–618
| doi=10.1145/129712.129771 }}</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"/>
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 |title=Probability, Statistics and Estimation|year=2013 |page=10 |url=http://www.incertitudes.fr/book.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf |archive-date=October 9, 2022 |url-status=live }}</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 | author-link=C. R. Rao | last=Rao | first=C.R. | year=1997 | title=Statistics and Truth: Putting Chance to Work | publisher=World Scientific | isbn=978-981-02-3111-8 }}</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 |editor1-last=Arthanari |editor1-first=T.S. |editor2-last=Dodge |editor2-first=Yadolah |editor2-link=Yadolah Dodge |last=Rao |first=C.R. |author-link=C.R. Rao |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 |mr=607328 }}</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 | 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
| chapter=Numerical Analysis
| title= The Princeton Companion to Mathematics
| editor1-first=Timothy
| editor1-last=Gowers
| editor2-first=June
| editor2-last=Barrow-Green
| editor3-first=Imre
| editor3-last=Leader
| publisher=Princeton University Press
| year=2008
| isbn=978-0-691-11880-2
| pages=604–615
| url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf
| access-date=November 13, 2022
| archive-date=March 7, 2023
| archive-url=https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf
| url-status=live
}}</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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== History ==
{{Main|History of mathematics}}
=== Ancient ===
The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,<ref>{{cite journal |title=Abstract representations of numbers in the animal and human brain |journal=Trends in Neurosciences |volume=21 |issue=8 |date=Aug 1998 |pages=355–361 |doi=10.1016/S0166-2236(98)01263-6 |pmid=9720604 |last1=Dehaene |first1=Stanislas | author1-link=Stanislas Dehaene |last2=Dehaene-Lambertz |first2=Ghislaine |author2-link=Ghislaine Dehaene-Lambertz | last3=Cohen |first3=Laurent|s2cid=17414557 }}</ref> was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are {{em|two}} of them. As evidenced by [[tally sticks|tallies]] found on bone, in addition to recognizing how to [[counting|count]] physical objects, [[prehistoric]] peoples may have also known how to count abstract quantities, like time{{emdash}}days, seasons, or years.<ref>See, for example, {{cite book | first=Raymond L. | last=Wilder|author-link=Raymond L. Wilder|title=Evolution of Mathematical Concepts; an Elementary Study|at=passim}}</ref><ref>{{Cite book|last=Zaslavsky|first=Claudia|author-link=Claudia Zaslavsky|title=Africa Counts: Number and Pattern in African Culture.|date=1999|publisher=Chicago Review Press|isbn=978-1-61374-115-3|oclc=843204342}}</ref>
[[File:Plimpton 322.jpg|thumb|The Babylonian mathematical tablet ''[[Plimpton 322]]'', dated to 1800 BC]]
Evidence for more complex mathematics does not appear until around 3000 {{Abbr|BC|Before Christ}}, when the [[Babylonia]]ns and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.{{sfn|Kline|1990|loc=Chapter 1}} The oldest mathematical texts from [[Mesopotamia]] and [[Ancient Egypt|Egypt]] are from 2000 to 1800 BC. Many early texts mention [[Pythagorean triple]]s 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.{{sfn|Boyer|1991|loc="Mesopotamia" pp. 24–27}}
In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some [[Ancient Greece|Ancient Greeks]] such as the [[Pythagoreans]] appeared to have considered it a subject in its own right.<ref>{{cite book | last=Heath | first=Thomas Little | author-link=Thomas Heath (classicist) |url=https://archive.org/details/historyofgreekma0002heat/page/n14 |url-access=registration |page=1 |title=A History of Greek Mathematics: From Thales to Euclid |location=New York |publisher=Dover Publications |date=1981 |orig-date=1921 |isbn=978-0-486-24073-2}}</ref> Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.<ref>{{Cite journal |last=Mueller |first=I. |date=1969 |title=Euclid's Elements and the Axiomatic Method |journal=The British Journal for the Philosophy of Science |volume=20 |issue=4 |pages=289–309 |doi=10.1093/bjps/20.4.289 |jstor=686258 |issn=0007-0882}}</ref> His book, ''[[Euclid's Elements|Elements]]'', is widely considered the most successful and influential textbook of all time.{{sfn|Boyer|1991|loc="Euclid of Alexandria" p. 119}} The greatest mathematician of antiquity is often held to be [[Archimedes]] ({{Circa|287|212 BC}}) of [[Syracuse, Italy|Syracuse]].{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 120}} 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.{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 130}} Other notable achievements of Greek mathematics are [[conic sections]] ([[Apollonius of Perga]], 3rd century BC),{{sfn|Boyer|1991|loc="Apollonius of Perga" p. 145}} [[trigonometry]] ([[Hipparchus of Nicaea]], 2nd century BC),{{sfn|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 162}} and the beginnings of algebra (Diophantus, 3rd century AD).{{sfn|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 180}}
[[File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BC and the 2nd 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]].<ref>{{cite book
| title=Number Theory and Its History
| first=Øystein
| last=Ore
| author-link=Øystein Ore
| publisher=Courier Corporation
| pages=19–24
| year=1988
| isbn=978-0-486-65620-5
| url={{GBurl|id=Sl_6BPp7S0AC|pg=IA19}}
| access-date=November 14, 2022
}}</ref> Other notable developments of Indian mathematics include the modern definition and approximation of [[sine]] and [[cosine]], and an early form of [[infinite series]].<ref>{{cite journal
| title=On the Use of Series in Hindu Mathematics
| first=A. N. | last=Singh | journal=Osiris
| volume=1 | date=January 1936 | pages=606–628
| doi=10.1086/368443 | jstor=301627
| s2cid=144760421 }}</ref><ref>{{cite book
| chapter=Use of series in India
| last1=Kolachana | first1=A. | last2=Mahesh | first2=K.
| last3=Ramasubramanian | first3=K.
| title=Studies in Indian Mathematics and Astronomy
| series=Sources and Studies in the History of Mathematics and Physical Sciences
| pages=438–461 | publisher=Springer | publication-place=Singapore
| isbn=978-981-13-7325-1 | year=2019
| doi=10.1007/978-981-13-7326-8_20 | s2cid=190176726 }}</ref>
=== Medieval and later ===
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's ''Algebra'']]
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. Other achievements of the Islamic period include advances in [[spherical trigonometry]] and the addition of the [[decimal point]] to the Arabic numeral system.<ref>{{Cite book | last=Saliba | first=George | author-link=George Saliba | title=A history of Arabic astronomy: planetary theories during the golden age of Islam | date=1994 | publisher=New York University Press | isbn=978-0-8147-7962-0 | oclc=28723059 }}</ref> Many notable mathematicians from this period were Persian, such as Al-Khwarismi, [[Omar Khayyam]] and [[Sharaf al-Dīn al-Ṭūsī]].<ref>{{cite journal
| title=Contributions of Islamic scholars to the scientific enterprise
| first=Yasmeen M.
| last=Faruqi
| journal=International Education Journal
| year=2006
| volume=7
| issue=4
| pages=391–399
| publisher=Shannon Research Press
| url=https://eric.ed.gov/?id=EJ854295
| access-date=November 14, 2022
| archive-date=November 14, 2022
| archive-url=https://web.archive.org/web/20221114165547/https://eric.ed.gov/?id=EJ854295
| url-status=live
}}</ref> The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.<ref>{{cite journal | title=Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages | first=Richard | last=Lorch | journal=Science in Context | volume=14 | issue=1–2 | date=June 2001 | pages=313–331 | publisher=Cambridge University Press | doi=10.1017/S0269889701000114 | s2cid=146539132 | url=https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | access-date=December 5, 2022 | archive-date=December 17, 2022 | archive-url=https://web.archive.org/web/20221217160922/https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | url-status=live }}</ref>
During the [[early modern period]], mathematics began to develop at an accelerating pace in [[Western Europe]], with innovations that revolutionized mathematics, such as the introduction of variables and [[#Symbolic notation and terminology|symbolic notation]] by François Viète (1540–1603), the introduction of [[History of logarithms|logarithms]] by [[John Napier]] in 1614, which greatly simplified numerical calculations, especially for [[astronomy]] and [[marine navigation]], the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and [[Gottfried Wilhelm Leibniz|Gottfried Leibniz]] (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.
[[File:Carl Friedrich Gauss 1840 by Jensen.jpg|thumb|left|[[Carl Friedrich Gauss]]]]
Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, [[differential geometry and topology|differential geometry]], [[matrix theory]], number theory, and [[statistics]].<ref>{{cite journal
| title=History of Mathematics After the Sixteenth Century
| first=Raymond Clare | last=Archibald | author-link=Raymond Clare Archibald
| journal=The American Mathematical Monthly
| series=Part 2: Outline of the History of Mathematics
| volume=56 | issue=1 | date=January 1949 | pages=35–56
| doi=10.2307/2304570 | jstor=2304570
}}</ref> In the early 20th century, [[Kurt Gödel]] transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system{{emdash}}if powerful enough to describe arithmetic{{emdash}}will contain true propositions that cannot be proved.<ref name=Raatikainen_2005/>
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and [[science]], to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the ''[[Bulletin of the American Mathematical Society]]'', "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 theorems and their proofs."{{sfn|Sevryuk|2006|pp=101–109}}
== Symbolic notation and terminology ==
{{main|Mathematical notation|Language of mathematics|Glossary of mathematics}}
[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) [[summation]] notation]]
Mathematical notation is widely used in science and [[engineering]] for representing complex [[concept]]s and [[property (philosophy)|properties]] in a concise, unambiguous, and accurate way. This notation consists of [[glossary of mathematical symbols|symbols]] used for representing [[operation (mathematics)|operation]]s, unspecified numbers, [[relation (mathematics)|relation]]s and any other mathematical objects, and then assembling them into [[expression (mathematics)|expression]]s and formulas.<ref>{{cite conference
| last=Wolfram
| first=Stephan
| author-link=Stephen Wolfram
| title=Mathematical notation: Past and future
| conference=MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA.
| url=https://www.stephenwolfram.com/publications/mathematical-notation-past-future/
| date=October 2000
| access-date=November 16, 2022
| archive-date=November 16, 2022
| archive-url=https://web.archive.org/web/20221116150905/https://www.stephenwolfram.com/publications/mathematical-notation-past-future/
| url-status=live
}}</ref> More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally [[Latin alphabet|Latin]] or [[Greek alphabet|Greek]] letters, and often include [[subscript]]s. Operation and relations are generally represented by specific [[Glossary of mathematical symbols|symbols]] or [[glyph]]s,<ref>{{cite journal
| title=Knowledge of Mathematical Symbols Goes Beyond Numbers
| first1=Heather | last1=Douglas | first2=Marcia Gail | last2=Headley
| first3=Stephanie | last3=Hadden | first4=Jo-Anne | last4=LeFevre|author4-link=Jo-Anne LeFevre
| journal=Journal of Numerical Cognition
| date=December 3, 2020 | volume=6 | issue=3 | pages=322–354
| doi=10.5964/jnc.v6i3.293
| s2cid=228085700 | doi-access=free }}</ref> such as {{math|+}} ([[plus sign|plus]]), {{math|×}} ([[multiplication sign|multiplication]]), <math display =inline>\int</math> ([[integral sign|integral]]), {{math|1==}} ([[equals sign|equal]]), and {{math|<}} ([[less-than sign|less than]]).<ref name=AMS>{{cite web | title=AMS style guide | year=2017 | page=75 | first1=Mary | last1=Letourneau | first2=Jennifer | last2=Wright Sharp | publisher=[[American Mathematical Society]] | url=https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf | access-date=November 16, 2022 | archive-date=December 8, 2022 | archive-url=https://web.archive.org/web/20221208063650/https://www.ams.org//publications/authors/AMS-StyleGuide-online.pdf | url-status=live }}</ref> All these symbols are generally grouped according to specific rules to form expressions and formulas.<ref>{{cite conference
| title=Constituent structure in mathematical expressions
| last1=Jansen
| first1=A. R.
| last2=Marriott
| first2=K.
| last3=Yelland
| first3=G. W.
| conference=Proceedings of the 22th Annual Meeting of the Cognitive Science Society
| volume=22
| page=238
| year=2000
| url=https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf
| access-date=November 16, 2022
| archive-date=November 16, 2022
| archive-url=https://web.archive.org/web/20221116152222/https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf
| url-status=live
}}</ref> Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of [[noun phrase]]s and formulas play the role of [[clause]]s.
Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous [[Technical definition|definitions]] that provide a standard foundation for communication. An axiom or [[postulate]] is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a [[conjecture]]. Through a series of rigorous arguments employing [[deductive reasoning]], a statement that is [[formal proof|proven]] to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a [[Lemma (mathematics)|lemma]]. A proven instance that forms part of a more general finding is termed a [[corollary]].<ref>{{cite book
| title=Theorems, Corollaries, Lemmas, and Methods of Proof
| first=Richard J.
| last=Rossi
| pages=1–14, 47–48
| year=2011
| publisher=John Wiley & Sons
| isbn=978-1-118-03057-8
| series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
| url={{GBurl|id=kSwVGbBtel8C|pg=PR7}}
| access-date=November 17, 2022
}}</ref>
Numerous technical terms used in mathematics are [[neologism]]s, such as ''[[polynomial]]'' and ''[[homeomorphism]]''.<ref>{{cite web
| title=Earliest Uses of Some Words of Mathematics
| website=MacTutor
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/
| access-date=November 19, 2022
| archive-date=September 29, 2022
| archive-url=https://web.archive.org/web/20220929032236/https://mathshistory.st-andrews.ac.uk/Miller/mathword/
| url-status=live
}}</ref> Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "[[logical disjunction|or]]" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "[[exclusive or]]"). Finally, many mathematical terms are common words that are used with a completely different meaning.<ref>{{cite journal
| title=The New Language of Mathematics
| first=Daniel S.
| last=Silver
| journal=The American Scientist
| year=2017
| volume=105
| issue=6
| pages=364–371
| doi=10.1511/2017.105.6.364
| url=https://www.americanscientist.org/article/the-new-language-of-mathematics
| access-date=November 19, 2022
| archive-date=October 22, 2022
| archive-url=https://web.archive.org/web/20221022091706/https://www.americanscientist.org/article/the-new-language-of-mathematics
| url-status=live
}}</ref> This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every [[free module]] is [[flat module|flat]]" and "a [[field (mathematics)|field]] is always a [[ring (mathematics)|ring]]".
== Relationship with sciences ==
Mathematics is used in most [[science]]s for [[Mathematical model|modeling]] phenomena, which then allows predictions to be made from experimental laws.<ref>{{cite book | title=Modelling Mathematical Methods and Scientific Computation | first1=Nicola | last1=Bellomo | first2=Luigi | last2=Preziosi | publisher=CRC Press | date=December 22, 1994 | page=1 | isbn=978-0-8493-8331-1 | series=Mathematical Modeling | volume=1 | url={{GBurl|id=pJAvWaRYo3UC}} | access-date=November 16, 2022 }}</ref> The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.<ref>{{cite journal
| title=Mathematical Models and Reality: A Constructivist Perspective
| first=Christian | last=Hennig
| journal=Foundations of Science
| volume=15 | pages=29–48 | year=2010
| doi=10.1007/s10699-009-9167-x
| s2cid=6229200 | url=https://www.researchgate.net/publication/225691477
| access-date=November 17, 2022
}}</ref> Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.<ref>{{cite journal | title=Models in Science | date=February 4, 2020 | first1=Roman | last1=Frigg | author-link=Roman Frigg | first2=Stephan | last2=Hartmann | author2-link=Stephan Hartmann | website=Stanford Encyclopedia of Philosophy | url=https://seop.illc.uva.nl/entries/models-science/ | access-date=November 17, 2022 | archive-date=November 17, 2022 | archive-url=https://web.archive.org/web/20221117162412/https://seop.illc.uva.nl/entries/models-science/ | url-status=live }}</ref> For example, the [[perihelion precession of Mercury]] could only be explained after the emergence of [[Einstein]]'s [[general relativity]], which replaced [[Newton's law of gravitation]] as a better mathematical model.<ref>{{cite book | last=Stewart | first=Ian | author-link=Ian Stewart (mathematician) | chapter=Mathematics, Maps, and Models | title=The Map and the Territory: Exploring the Foundations of Science, Thought and Reality | pages=345–356 | publisher=Springer | year=2018 | editor1-first=Shyam | editor1-last=Wuppuluri | editor2-first=Francisco Antonio | editor2-last=Doria | isbn=978-3-319-72478-2 | series=The Frontiers Collection | chapter-url={{GBurl|id=mRBMDwAAQBAJ|p=345}} | doi=10.1007/978-3-319-72478-2_18 | access-date=November 17, 2022 }}</ref>
There is still a [[philosophy of mathematics|philosophical]] debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is [[falsifiable]], which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a [[counterexample]]. Similarly as in science, [[mathematical theory|theories]] and results (theorems) are often obtained from [[experimentation]].<ref>{{Cite web|url=https://undsci.berkeley.edu/article/mathematics|title=The science checklist applied: Mathematics|website=undsci.berkeley.edu|access-date=October 27, 2019|archive-url=https://web.archive.org/web/20191027021023/https://undsci.berkeley.edu/article/mathematics|archive-date=October 27, 2019|url-status=live}}</ref> In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).<ref>{{cite book | last=Mackay | first=A. L. | year=1991 | title=Dictionary of Scientific Quotations | location=London | page=100 | isbn=978-0-7503-0106-0 | publisher=Taylor & Francis | url={{GBurl|id=KwESE88CGa8C|q=durch planmässiges Tattonieren}} | access-date=March 19, 2023 }}</ref> However, some authors emphasize that mathematics differs from the modern notion of science by not {{em|relying}} on empirical evidence.<ref name="Bishop1991">{{cite book | last1 = Bishop | first1 = Alan | year = 1991 | chapter = Environmental activities and mathematical culture | title = Mathematical Enculturation: A Cultural Perspective on Mathematics Education | chapter-url = {{GBurl|id=9AgrBgAAQBAJ|p=54}} | pages = 20–59 | location = Norwell, Massachusetts | publisher = Kluwer Academic Publishers | isbn = 978-0-7923-1270-3 | access-date = April 5, 2020 }}</ref><ref>{{cite book | title=Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists | last1=Shasha | first1=Dennis Elliot | author1-link=Dennis Elliot Shasha | last2=Lazere | first2=Cathy A. | publisher=Springer | year=1998 | page=228 | isbn=978-0-387-98269-4 }}</ref><ref name="Nickles2013">{{cite book | last=Nickles | first=Thomas | year=2013 | chapter=The Problem of Demarcation | title=Philosophy of Pseudoscience: Reconsidering the Demarcation Problem | page=104 | location=Chicago | publisher=The University of Chicago Press | isbn=978-0-226-05182-6 }}</ref><ref name="Pigliucci2014">{{Cite magazine | year=2014| last=Pigliucci| first=Massimo | author-link=Massimo Pigliucci | title=Are There 'Other' Ways of Knowing? | magazine=[[Philosophy Now]]| url=https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | access-date=April 6, 2020| archive-date=May 13, 2020 | archive-url=https://web.archive.org/web/20200513190522/https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | url-status=live}}</ref>
<!-- What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections. -->
=== Pure and applied mathematics ===
{{main|Applied mathematics|Pure mathematics}}
{{multiple image
| footer = Isaac Newton (left) and [[Gottfried Wilhelm Leibniz]] developed infinitesimal calculus.
| total_width = 330
| width1 = 407
| height1 = 559
| image1 = GodfreyKneller-IsaacNewton-1689.jpg
| alt1 = Isaac Newton
| width2 = 320
| height2 = 390
| image2 = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
| alt2 = Gottfried Wilhelm von Leibniz
}}
Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of [[technology]] and science, and there was no clear distinction between pure and applied mathematics.<ref name=Ferreirós_2007>{{cite book
| title=The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
| last=Ferreirós | first=J.
| chapter=Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss
| pages=235–268 | year=2007 | isbn=978-3-540-34720-0
| editor1-first=Catherine | editor1-last=Goldstein | editor1-link=Catherine Goldstein
| editor2-first=Norbert | editor2-last=Schappacher
| editor3-first=Joachim | editor3-last=Schwermer
| publisher=Springer Science & Business Media
| chapter-url={{GBurl|id=IUFTcOsMTysC|p=235}}
}}</ref> For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the [[planet]]s with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.<ref>{{cite journal
| title=Mathematical vs. Experimental Traditions in the Development of Physical Science
| first=Thomas S. | last=Kuhn | author-link=Thomas Kuhn
| journal=The Journal of Interdisciplinary History
| year=1976 | volume=7 | issue=1 | pages=1–31 | publisher=The MIT Press
| jstor=202372 | doi=10.2307/202372
}}</ref> However, a notable exception occurred with the tradition of [[pure mathematics in Ancient Greece]].<ref>{{cite book
| chapter=The two cultures of mathematics in ancient Greece
| first=Markus
| last=Asper
| year=2009
| title=The Oxford Handbook of the History of Mathematics
| editor1-first=Eleanor
| editor1-last=Robson
| editor2-first=Jacqueline
| editor2-last=Stedall
| pages=107–132
| isbn=978-0-19-921312-2
| publisher=OUP Oxford
| series=Oxford Handbooks in Mathematics
| chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=107}}
| access-date=November 18, 2022
}}</ref>
In the 19th century, mathematicians such as [[Karl Weierstrass]] and [[Richard Dedekind]] increasingly focused their research on internal problems, that is, ''pure mathematics''.<ref name=Ferreirós_2007/><ref>{{cite journal
| title=How applied mathematics became pure
| last=Maddy | first=P. | author-link=Penelope Maddy
| journal=The Review of Symbolic Logic
| year=2008
| volume=1
| issue=1
| pages=16–41
| doi=10.1017/S1755020308080027
| s2cid=18122406
| url=http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
| access-date=November 19, 2022
| archive-date=August 12, 2017
| archive-url=https://web.archive.org/web/20170812012210/http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
| url-status=live
}}</ref> This led to split mathematics into ''pure mathematics'' and ''applied mathematics'', the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.<ref>{{cite book
| title=The Best Writing on Mathematics, 2016
| chapter=In Defense of Pure Mathematics
| first=Daniel S.
| last=Silver
| pages=17–26
| isbn=978-0-691-17529-4
| year=2017
| editor1-first=Mircea
| editor1-last=Pitici
| publisher=Princeton University Press
| chapter-url={{GBurl|id=RXGYDwAAQBAJ|p=17}}
| access-date=November 19, 2022
}}</ref>
The aftermath of [[World War II]] led to a surge in the development of applied mathematics in the US and elsewhere.<ref>{{cite journal | title=The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account | first=Karen Hunger | last=Parshall | author-link=Karen Hunger Parshall | journal=Bulletin of the American Mathematical Society | volume=59 | year=2022 | issue=3 | pages=405–427 | doi=10.1090/bull/1754 | s2cid=249561106 | url=https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | access-date=November 20, 2022 | doi-access=free | archive-date=November 20, 2022 | archive-url=https://web.archive.org/web/20221120151259/https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | url-status=live }}</ref><ref>{{cite journal
| title=The History Of Applied Mathematics And The History Of Society
| first=Michael | last=Stolz
| journal=Synthese
| volume=133 | pages=43–57 | year=2002
| doi=10.1023/A:1020823608217
| s2cid=34271623 | url=https://www.researchgate.net/publication/226795930
| access-date=November 20, 2022
}}</ref> Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".<ref>{{cite journal
| title=On the role of applied mathematics
| journal=[[Advances in Mathematics]] | first=C. C . | last=Lin
| volume=19 | issue=3 | date=March 1976 | pages=267–288
| doi=10.1016/0001-8708(76)90024-4 | doi-access=free
}}</ref><ref>{{cite conference
| title=Applying Pure Mathematics
| first=Anthony
| last=Peressini
| conference=Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers
| volume=66
| date=September 1999
| pages=S1–S13
| jstor=188757
| access-date=November 30, 2022
| url=https://www.academia.edu/download/32799272/ApplyingMathPSA.pdf
| archive-url=https://web.archive.org/web/20240102210931/https://d1wqtxts1xzle7.cloudfront.net/32799272/ApplyingMathPSA-libre.pdf?1391205742=&response-content-disposition=inline%3B+filename%3DApplying_Pure_Mathematics.pdf&Expires=1704233371&Signature=BvNJyYufdj9BiKFe94w6gdXLpAfr7T5JIv~RU74R2uT0O9Ngj6i4cdBtYYOSB6D4V-MgButb6lKNhIGGQogw0e0sHVFkJUy5TRsoCiQ-MLabpZOf74E5SGLMFIExhGVAw7SKrSFaQsFGhfbaRMxbMP~u-wRdJAz6ve6kbWr6oq-doQeEOlRfO4EByNCUYx-KAk3~cBsH1Q2WNZ5QiVObMI1ufQ7zkQM1bqzOumLu6g07F~pt~Cds~lftuQufHomoTH-V9H9iKQgUyc3-4bEB1y1Jdngs7WWg76LcSGn65bPK8dxvsZzKaLDGfoK5jamZkA8z3-xxiMIPL8c6YETjZA__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=January 2, 2024
| url-status=live
}}</ref>
An example of the first case is the [[theory of distributions]], introduced by [[Laurent Schwartz]] for validating computations done in [[quantum mechanics]], which became immediately an important tool of (pure) mathematical analysis.<ref>{{cite conference
| title=Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century
| last=Lützen
| first=J.
| year=2011
| editor1-last=Schlote
| editor1-first=K. H.
| editor2-last=Schneider
| editor2-first=M.
| publisher=Verlag Harri Deutsch
| publication-place=Frankfurt am Main
| chapter=Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century
| chapter-url=https://slub.qucosa.de/api/qucosa%3A16267/zip/
| access-date=November 19, 2022
| archive-date=March 23, 2023
| archive-url=https://web.archive.org/web/20230323164143/https://slub.qucosa.de/api/qucosa%3A16267/zip/
| url-status=live
}}</ref> An example of the second case is the [[decidability of the first-order theory of the real numbers]], a problem of pure mathematics that was proved true by [[Alfred Tarski]], with an algorithm that is impossible to [[implementation (computer science)|implement]] because of a computational complexity that is much too high.<ref>{{cite journal
| title=Model theory and exponentiation
| last=Marker
| first=Dave
| journal=Notices of the American Mathematical Society
| volume=43
| issue=7
| date=July 1996
| pages=753–759
| url=https://www.ams.org/notices/199607/
| access-date=November 19, 2022
| archive-date=March 13, 2014
| archive-url=https://web.archive.org/web/20140313004011/http://www.ams.org/notices/199607/
| url-status=live
}}</ref> For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, [[George E. Collins|George Collins]] introduced the [[cylindrical algebraic decomposition]] that became a fundamental tool in [[real algebraic geometry]].<ref>{{cite conference
| title=Cylindrical Algebraic Decomposition in the RegularChains Library
| first1=Changbo | last1=Chen | first2=Marc Moreno | last2=Maza
| date=August 2014 | volume=8592
| publisher=Springer | publication-place=Berlin
| conference=International Congress on Mathematical Software 2014
| series=Lecture Notes in Computer Science
| url=https://www.researchgate.net/publication/268067322
| access-date=November 19, 2022 | doi=10.1007/978-3-662-44199-2_65 }}</ref>
In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.<ref>{{cite journal
| title=Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy
| first1=José Antonio | last1=Pérez-Escobar | first2=Deniz | last2=Sarikaya
| journal=European Journal for Philosophy of Science
| volume=12 | issue=1 | pages=1–22 | year=2021
| doi=10.1007/s13194-021-00435-9 | s2cid=245465895
| doi-access=free }}</ref><ref>{{cite book
| chapter=Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity
| last=Takase
| first=M.
| title=A Mathematical Approach to Research Problems of Science and Technology
| series=Mathematics for Industry
| volume=5
| year=2014
| pages=393–399
| publisher=Springer
| publication-place=Tokyo
| chapter-url={{GBurl|id=UeElBAAAQBAJ|p=393}}
| doi=10.1007/978-4-431-55060-0_29
| isbn=978-4-431-55059-4
| access-date=November 20, 2022
}}</ref> The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".<ref name=MSC/> However, these terms are still used in names of some [[university]] departments, such as at the [[Faculty of Mathematics, University of Cambridge|Faculty of Mathematics]] at the [[University of Cambridge]].
=== Unreasonable effectiveness ===
The [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|unreasonable effectiveness of mathematics]] is a phenomenon that was named and first made explicit by physicist [[Eugene Wigner]].<ref name=wigner1960>{{cite journal
| title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences
| last=Wigner | first=Eugene | author-link=Eugene Wigner
| journal=[[Communications on Pure and Applied Mathematics]]
| volume=13 | issue=1 | pages=1–14 | year=1960
| doi=10.1002/cpa.3160130102 | bibcode=1960CPAM...13....1W
| s2cid=6112252 | url=https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
| url-status=live | archive-url=https://web.archive.org/web/20110228152633/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
| archive-date=February 28, 2011 | df=mdy-all
}}</ref> It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.<ref>{{cite journal
| title=Revisiting the 'unreasonable effectiveness' of mathematics
| first=Sundar | last=Sarukkai
| journal=Current Science
| volume=88 | issue=3 | date=February 10, 2005 | pages=415–423
| jstor=24110208
}}</ref> Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the [[prime factorization]] of natural numbers that was discovered more than 2,000 years before its common use for secure [[internet]] communications through the [[RSA cryptosystem]].<ref>{{cite book
| chapter=History of Integer Factoring
| pages=41–77
| first=Samuel S.
| last=Wagstaff, Jr.
| title=Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL
| editor1-first=Joppe W.
| editor1-last=Bos
| editor2-first=Martijn
| editor2-last=Stam
| series=London Mathematical Society Lecture Notes Series 469
| publisher=Cambridge University Press
| year=2021
| chapter-url=https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
| access-date=November 20, 2022
| archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120155733/https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
| url-status=live
}}</ref> A second historical example is the theory of [[ellipse]]s. They were studied by the [[Greek mathematics|ancient Greek mathematicians]] as [[conic section]]s (that is, intersections of [[cone]]s with planes). It is almost 2,000 years later that [[Johannes Kepler]] discovered that the [[trajectories]] of the planets are ellipses.<ref>{{cite web
| title=Curves: Ellipse
| website=MacTutor
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
| access-date=November 20, 2022
| archive-date=October 14, 2022
| archive-url=https://web.archive.org/web/20221014051943/https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
| url-status=live
}}</ref>
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and [[manifold]]s. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, [[Albert Einstein]] developed the [[theory of relativity]] that uses fundamentally these concepts. In particular, [[spacetime]] of [[special relativity]] is a non-Euclidean space of dimension four, and spacetime of [[general relativity]] is a (curved) manifold of dimension four.<ref>{{cite web
| title=Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry
| first=Vasudevan
| last=Mukunth
| website=The Wire
| date=September 10, 2015
| url=https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
| access-date=November 20, 2022
| archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120191206/https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
| url-status=live
}}</ref><ref>{{cite journal
| title=The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics
| first1=Edwin B. | last1=Wilson | first2=Gilbert N. | last2=Lewis
| journal=Proceedings of the American Academy of Arts and Sciences
| volume=48 | issue=11 | date=November 1912 | pages=389–507
| doi=10.2307/20022840 | jstor=20022840 }}</ref>
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the [[positron]] and the [[omega baryon|baryon]] <math>\Omega^{-}.</math> In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown [[particle]], and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.<ref name=borel /><ref>{{cite journal
| title=Discovering the Positron (I)
| first=Norwood Russell | last=Hanson | author-link=Norwood Russell Hanson
| journal=The British Journal for the Philosophy of Science
| volume=12 | issue=47 | date=November 1961 | pages=194–214
| publisher=The University of Chicago Press
| jstor=685207 | doi=10.1093/bjps/xiii.49.54
}}</ref><ref>{{cite journal
| title=Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω<sup>–</sup> particle
| first=Michele | last=Ginammi
| journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
| volume=53 | date=February 2016 | pages=20–27
| doi=10.1016/j.shpsb.2015.12.001
| bibcode=2016SHPMP..53...20G }}</ref>
=== Specific sciences ===
{{Essay-like|date=December 2022|subsection}}
==== Physics ====
{{Main|Relationship between mathematics and physics}}
[[File:Pendule schema.gif|thumb|Diagram of a pendulum]]
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,<ref>{{Cite book |last1=Wagh |first1=Sanjay Moreshwar |url={{GBurl|id=-DmfVjBUPksC|p=3}} |title=Essentials of Physics |last2=Deshpande |first2=Dilip Abasaheb |date=September 27, 2012 |publisher=PHI Learning Pvt. Ltd. |isbn=978-81-203-4642-0 |page=3 |language=en |access-date=January 3, 2023 }}</ref> and is also the motivation of major mathematical developments.<ref>{{Cite conference |last=Atiyah |first=Michael |author-link=Michael Atiyah |date=1990 |title=On the Work of Edward Witten |url=http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |conference=Proceedings of the International Congress of Mathematicians |page=31 |archive-url=https://web.archive.org/web/20130928095313/http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |archive-date=September 28, 2013 |access-date=December 29, 2022}}</ref>
==== Computing ====
{{Further|Theoretical computer science|Computational mathematics}}
The rise of technology in the 20th century opened the way to a new science: [[computing]].{{Efn|[[Ada Lovelace]], in the 1840s, is known for having written the first computer program ever in collaboration with [[Charles Babbage]]}} This field is closely related to mathematics in several ways. [[Theoretical computer science]] is essentially mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in [[cryptography]] and [[coding theory]]. [[Discrete mathematics]] is useful in many areas of computer science, such as [[Computational complexity theory|complexity theory]], [[information theory]], [[graph theory]], and so on.{{Citation needed|date=December 2022}}
In return, computing has also become essential for obtaining new results. This is a group of techniques known as [[experimental mathematics]], which is the use of ''experimentation'' to discover mathematical insights.<ref>{{Cite web |last1=Borwein |first1=J. |last2=Borwein |first2=P. |last3=Girgensohn |first3=R. |last4=Parnes |first4=S. |date=1996 |title=Conclusion |url=http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |url-status=dead |archive-url=https://web.archive.org/web/20080121081424/http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |archive-date=January 21, 2008 |website=oldweb.cecm.sfu.ca}}</ref> The most well-known example is the [[Four color theorem|four-color theorem]], which was proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where the rule was that the mathematician should verify each part of the proof. In 1998, the [[Kepler conjecture]] on [[sphere packing]] seemed to also be partially proven by computer. An international team had since worked on writing a formal proof; it was finished (and verified) in 2015.<ref>{{cite journal |last1=Hales |first1=Thomas |last2=Adams |first2=Mark |last3=Bauer |first3=Gertrud |last4=Dang |first4=Tat Dat |last5=Harrison |first5=John |last6=Hoang |first6=Le Truong |last7=Kaliszyk |first7=Cezary |last8=Magron |first8=Victor |last9=Mclaughlin |first9=Sean |last10=Nguyen |first10=Tat Thang |last11=Nguyen |first11=Quang Truong |last12=Nipkow |first12=Tobias |last13=Obua |first13=Steven |last14=Pleso |first14=Joseph |last15=Rute |first15=Jason |last16=Solovyev |first16=Alexey |last17=Ta |first17=Thi Hoai An |last18=Tran |first18=Nam Trung |last19=Trieu |first19=Thi Diep |last20=Urban |first20=Josef |last21=Vu |first21=Ky |last22=Zumkeller |first22=Roland |title=A Formal Proof of the Kepler Conjecture |journal=Forum of Mathematics, Pi |date=2017 |volume=5 |page=e2 |doi=10.1017/fmp.2017.1 |s2cid=216912822 |url=https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |language=en |issn=2050-5086 |access-date=February 25, 2023 |archive-date=December 4, 2020 |archive-url=https://web.archive.org/web/20201204053232/https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |url-status=live |hdl=2066/176365 |hdl-access=free }}</ref>
Once written formally, a proof can be verified using a program called a [[proof assistant]].<ref name=":1">{{Cite journal |last=Geuvers |first=H. |date=February 2009 |title=Proof assistants: History, ideas and future |url=https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |journal=Sādhanā |volume=34 |pages=3–4 |doi=10.1007/s12046-009-0001-5 |s2cid=14827467 |doi-access=free |access-date=December 29, 2022 |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204107/https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |url-status=live |hdl=2066/75958 |hdl-access=free }}</ref> These programs are useful in situations where one is uncertain about a proof's correctness.<ref name=":1" />
A major open problem in theoretical computer science is [[P versus NP problem|P versus NP]]. It is one of the seven [[Millennium Prize Problems]].<ref>{{Cite web |title=P versus NP problem {{!}} mathematics |url=https://www.britannica.com/science/P-versus-NP-problem |access-date=December 29, 2022 |website=Britannica |language=en |archive-date=December 6, 2022 |archive-url=https://web.archive.org/web/20221206044556/https://www.britannica.com/science/P-versus-NP-problem |url-status=live }}</ref>
==== Biology and chemistry ====
{{Main|Mathematical and theoretical biology|Mathematical chemistry}}
[[File:Giant Pufferfish skin pattern detail.jpg|thumb|The skin of this [[giant pufferfish]] exhibits a [[Turing pattern]], which can be modeled by [[reaction–diffusion system]]s.]]
[[Biology]] uses probability extensively – for example, in ecology or [[neurobiology]].<ref name=":2">{{Cite book |last=Millstein |first=Roberta |author-link=Roberta Millstein |title=The Oxford Handbook of Probability and Philosophy |date=September 8, 2016 |editor-last=Hájek |editor-first=Alan |pages=601–622 |chapter=Probability in Biology: The Case of Fitness |doi=10.1093/oxfordhb/9780199607617.013.27 |editor-last2=Hitchcock |editor-first2=Christopher |chapter-url=http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054456/http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |url-status=live }}</ref> Most of the discussion of probability in biology, however, centers on the concept of [[evolutionary fitness]].<ref name=":2" />
Ecology heavily uses modeling to simulate [[population dynamics]],<ref name=":2" /><ref>See for example Anne Laurent, Roland Gamet, Jérôme Pantel, ''Tendances nouvelles en modélisation pour l'environnement, actes du congrès «Programme environnement, vie et sociétés»'' 15-17 janvier 1996, CNRS</ref> study ecosystems such as the predator-prey model, measure pollution diffusion,{{Sfn|Bouleau|1999|pp=282–283}} or to assess climate change.{{Sfn|Bouleau|1999|p=285}} The dynamics of a population can be modeled by coupled differential equations, such as the [[Lotka–Volterra equations]].<ref>{{Cite web |date=January 5, 2022 |title=1.4: The Lotka-Volterra Predator-Prey Model |url=https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01%3A_Population_Dynamics/1.04%3A_The_Lotka-Volterra_Predator-Prey_Model |access-date=December 29, 2022 |website=Mathematics LibreTexts |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204111/https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01:_Population_Dynamics/1.04:_The_Lotka-Volterra_Predator-Prey_Model |url-status=live }}</ref> However, there is the problem of [[model validation]]. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.{{Sfn|Bouleau|1999|p=287}}
Genotype evolution can be modeled with the [[Hardy-Weinberg principle]].{{Citation needed|date=December 2022}}
[[Phylogeography]] uses probabilistic models.{{Citation needed|date=December 2022}}
Medicine uses [[statistical hypothesis testing]], run on data from [[clinical trial]]s, to determine whether a new treatment works.{{Citation needed|date=December 2022}}
Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. It turns out that the form of [[macromolecules]] in biology is variable and determines the action. Such modeling uses Euclidean geometry; neighboring atoms form a [[polyhedron]] whose distances and angles are fixed by the laws of interaction.{{Citation needed|date=December 2022}}
==== Earth sciences ====
{{Main|Geomathematics}}
[[Structural geology]] and climatology use probabilistic models to predict the risk of natural catastrophes.{{Citation needed|date=December 2022}} Similarly, [[meteorology]], [[oceanography]], and [[planetology]] also use mathematics due to their heavy use of models.{{Citation needed|date=December 2022}}
==== Social sciences ====
{{Further|Mathematical economics|Historical dynamics}}
Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, [[economics]], [[sociology]],<ref>{{Cite journal |last=Edling |first=Christofer R. |date=2002 |title=Mathematics in Sociology |url=https://www.annualreviews.org/doi/10.1146/annurev.soc.28.110601.140942 |journal=Annual Review of Sociology |language=en |volume=28 |issue=1 |pages=197–220 |doi=10.1146/annurev.soc.28.110601.140942 |issn=0360-0572}}</ref> and [[psychology]].<ref>{{Citation |last=Batchelder |first=William H. |title=Mathematical Psychology: History |date=2015-01-01 |url=https://www.sciencedirect.com/science/article/pii/B978008097086843059X |encyclopedia=International Encyclopedia of the Social & Behavioral Sciences (Second Edition) |pages=808–815 |editor-last=Wright |editor-first=James D. |access-date=2023-09-30 |place=Oxford |publisher=Elsevier |isbn=978-0-08-097087-5}}</ref>
[[File:Supply-demand-equilibrium.svg|thumb|[[Supply and demand|Supply and demand curves]], like this one, are a staple of mathematical economics.]]
The fundamental postulate of mathematical economics is that of the rational individual actor – ''[[Homo economicus]]'' ({{Literal translation|economic man}}).<ref name=":3">{{Cite book |last=Zak |first=Paul J. |url={{GBurl|id=6QrvmNo2qD4C|p=158}} |title=Moral Markets: The Critical Role of Values in the Economy |date=2010 |page=158 |publisher=Princeton University Press |isbn=978-1-4008-3736-6 |language=en |access-date=January 3, 2023 }}</ref> In this model, the individual seeks to maximize their [[rational choice theory|self-interest]],<ref name=":3" /> and always makes optimal choices using [[perfect information]].<ref name=":4">{{Cite web |last=Kim |first=Oliver W. |date=May 29, 2014 |title=Meet Homo Economicus |url=https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |access-date=December 29, 2022 |website=The Harvard Crimson |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204106/https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |url-status=live }}</ref>{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual [[calculations]] are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be difficult to discover by a "literary" analysis.{{Citation needed|date=December 2022}} For example, explanations of [[economic cycles]] are not trivial. Without mathematical modeling, it is hard to go beyond statistical observations or unproven speculation.{{Citation needed|date=December 2022}}
However, many people have rejected or criticized the concept of ''Homo economicus''.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}}
At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, [[Nikolai Kondratiev]] discerned the ~50-year-long [[Kondratiev cycle]], which explains phases of economic growth or crisis.<ref>{{Cite web |title=Kondratiev, Nikolai Dmitrievich {{!}} Encyclopedia.com |url=https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/kondratiev-nikolai-dmitrievich |access-date=December 29, 2022 |website=www.encyclopedia.com |archive-date=July 1, 2016 |archive-url=https://web.archive.org/web/20160701224009/http://www.encyclopedia.com/doc/1G2-3404100667.html |url-status=live }}</ref> Towards the end of the 19th century, {{Ill|Nicolas-Remi Brück|fr}} and {{Ill|Charles Henri Lagrange|fr}} extended their analysis into [[geopolitics]].<ref>{{Cite web|url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=##+Math%C3%A9matique+de+l'histoire,org%E3%80%91|title=Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. | The Online Books Page|website=onlinebooks.library.upenn.edu}}</ref> [[Peter Turchin]] has worked on developing [[cliodynamics]] since the 1990s.<ref>{{Cite web |title=Cliodynamics: a science for predicting the future |url=https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |access-date=December 29, 2022 |website=ZDNET |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204104/https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |url-status=live }}</ref>
Even so, mathematization of the social sciences is not without danger. In the controversial book ''[[Fashionable Nonsense]]'' (1997), [[Alan Sokal|Sokal]] and [[Jean Bricmont|Bricmont]] denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.<ref>{{cite book|last=Sokal|first=Alan|url=https://archive.org/details/fashionablenonse00soka|title=Fashionable Nonsense|author2=Jean Bricmont|publisher=Picador|year=1998|isbn=978-0-312-19545-8|location=New York|oclc=39605994|author-link=Alan Sokal|author2-link=Jean Bricmont}}</ref> The study of [[complex systems]] (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.{{Citation needed|date=December 2022}}
== Relationship with astrology and esotericism ==
Some renowned mathematicians have also been considered to be renowned astrologists; for example, [[Ptolemy]], Arab astronomers, [[Regiomantus]], [[Gerolamo Cardano|Cardano]], [[Kepler]], or [[John Dee]]. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, [[Theodor Zwinger]] wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions".<ref>{{Cite book |last=Beaujouan |first=Guy |url={{GBurl|id=92n7ZE8Iww8C|p=130}} |title=Comprendre et maîtriser la nature au Moyen Age: mélanges d'histoire des sciences offerts à Guy Beaujouan |date=1994 |publisher=Librairie Droz |isbn=978-2-600-00040-6 |page=130 |language=fr |access-date=January 3, 2023 }}</ref>
Astrology is no longer considered a science.<ref>{{Cite web |title=L'astrologie à l'épreuve : ça ne marche pas, ça n'a jamais marché ! / Afis Science – Association française pour l'information scientifique |url=https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |access-date=December 28, 2022 |website=Afis Science – Association française pour l’information scientifique |language=fr |archive-date=January 29, 2023 |archive-url=https://web.archive.org/web/20230129204349/https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |url-status=live }}</ref>
==Philosophy==
{{Main|Philosophy of mathematics}}
===Reality===
The connection between mathematics and material reality has led to philosophical debates since at least the time of [[Pythagoras]]. The ancient philosopher [[Plato]] argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as [[Mathematical Platonism|Platonism]]. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.<ref name=SEP-Platonism>{{cite encyclopedia |title=Platonism in Metaphysics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Balaguer |first=Mark |editor-last=Zalta |editor-first=Edward N. |year=2016 |edition=Spring 2016 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/spr2016/entries/platonism |access-date=April 2, 2022 |archive-date=January 30, 2022 |archive-url=https://web.archive.org/web/20220130174043/https://plato.stanford.edu/archives/spr2016/entries/platonism/ |url-status=live }}</ref>
[[Armand Borel]] summarized this view of mathematics reality as follows, and provided quotations of [[G. H. Hardy]], [[Charles Hermite]], [[Henri Poincaré]] and Albert Einstein that support his views.<ref name=borel />
{{blockquote| Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.<ref>See {{cite journal
| first=L. | last=White | year=1947
| title=The locus of mathematical reality: An anthropological footnote
| journal=[[Philosophy of Science (journal)|Philosophy of Science]]
| volume=14|issue=4 | pages=289–303
| doi=10.1086/286957 | s2cid=119887253
| id=189303 | postscript=;
}} also in {{cite book
| first=J. R. | last=Newman | year=1956
| title=The World of Mathematics
| publisher=Simon and Schuster | location=New York
| volume=4 | pages=2348–2364
}}</ref> Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a {{em|feeling}} of an objective existence, of a reality of mathematics ...}}
Nevertheless, Platonism and the concurrent views on abstraction do not explain the [[#Unreasonable effectiveness|unreasonable effectiveness]] of mathematics.<ref>{{cite book
| title=The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature
| first=Mauro
| last=Dorato
| year=2005
| chapter=Why are laws mathematical?
| pages=31–66
| isbn=978-0-7546-3994-7
| publisher=Ashgate
| chapter-url=https://www.academia.edu/download/52076815/2ch.pdf
| access-date=December 5, 2022
| archive-url=https://web.archive.org/web/20230817111932/https://d1wqtxts1xzle7.cloudfront.net/52076815/2ch-libre.pdf?1488997736=&response-content-disposition=inline%3B+filename%3DChapter_2_of_the_book_the_software_of_th.pdf&Expires=1692274771&Signature=PXpNLBsmWMkz9YUs6~LUOfXNkmkCAmDfxQUoWOkGJKP4YqPGQUFMuP1I0xFycLZkL0dyfGwdGQ7mPk44nvmpM3YpKBSeVCZRXtDMiwgqs1JhEWrJovAhrchPLM1mGn3pw5P6LPo0sDZsl7uaPoZHMyCyJpayHvFtpyj1oUMIdmGuYM5P3euy1R87g6xlKyNAp~~BR5I4gVpopzLoeZn7d3oEnOOua0GjsqsZ6H9mEgcZMpH-qF8w9iFa9aSXFpqxagQwcVVkg7DXkOjVV5jyzctBUKQtOQQ~-9EN1y-c9pFV-Xu-NNuoN3Ij6K4SwvjYv0a8DMs8T5SVj1Kz9i4CEQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=August 17, 2023
| url-status=live
}}</ref>
=== Proposed definitions ===
{{Main|Definitions of mathematics}}
There is no general consensus about a definition of mathematics or its [[epistemology|epistemological status]]{{emdash}}that is, its place among other human activities.<ref name="Mura">{{cite journal
| title=Images of Mathematics Held by University Teachers of Mathematical Sciences
| last=Mura | first=Roberta | date=Dec 1993
| journal=Educational Studies in Mathematics
| volume=25 | issue=4 | pages=375–85
| doi=10.1007/BF01273907 | jstor=3482762 | s2cid=122351146
}}</ref><ref name="Runge">{{cite book
| title=Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry
| last1=Tobies
| first1=Renate
| author1-link=Renate Tobies
| first2=Helmut
| last2=Neunzert
| publisher=Springer
| year=2012
| isbn=978-3-0348-0229-1
| page=9
| url={{GBurl|id=EDm0eQqFUQ4C|p=9}}
| quote=[I]t is first necessary to ask what is meant by ''mathematics'' in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
| access-date=June 20, 2015
}}</ref> A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.<ref name="Mura" /> There is not even consensus on whether mathematics is an art or a science.<ref name="Runge" /> Some just say, "mathematics is what mathematicians do".<ref name="Mura" /> This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.<ref>{{cite conference
| title="What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it
| first1=Günter M. | last1=Ziegler | author1-link=Günter M. Ziegler
| first2=Andreas | last2=Loos | editor-last=Kaiser | editor-first=G.
| conference=Proceedings of the 13th International Congress on Mathematical Education
| series=ICME-13 Monographs
| date=November 2, 2017 | pages=63–77 | publisher=Springer
| doi=10.1007/978-3-319-62597-3_5
| isbn=978-3-319-62596-6 }}</ref>
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.<ref name="Franklin">{{Cite book | last=Franklin | first=James | author-link=James Franklin (philosopher) | title=Philosophy of Mathematics | date= 2009 | isbn=978-0-08-093058-9 | pages=104–106 | publisher=Elsevier | url={{GBurl|id=mbn35b2ghgkC|p=104}} | access-date=June 20, 2015 }}</ref> In the 19th century, when mathematicians began to address topics{{mdash}}such as infinite sets{{mdash}}which have no clear-cut relation to physical reality, a variety of new definitions were given.<ref name="Cajori">{{cite book
| title=A History of Mathematics
| last=Cajori
| first=Florian
| author-link=Florian Cajori
| publisher=American Mathematical Society (1991 reprint)
| year=1893
| isbn=978-0-8218-2102-2
| pages=285–286
| url={{GBurl|id=mGJRjIC9fZgC|p=285}}
| access-date=June 20, 2015
}}</ref> With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.
Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems{{emdash}}assertions whose validity relies on a proof, that is, a purely-logical deduction.<ref>{{cite journal | title=The Methodology of Mathematics | first1=Ronald | last1=Brown | author1-link=Ronald Brown (mathematician) | first2=Timothy | last2=Porter | journal=The Mathematical Gazette | volume=79 | issue=485 | pages=321–334 | date=January 2000 | doi=10.2307/3618304 | jstor=3618304 | s2cid=178923299 | url=https://cds.cern.ch/record/280311 | access-date=November 25, 2022 | archive-date=March 23, 2023 | archive-url=https://web.archive.org/web/20230323164159/https://cds.cern.ch/record/280311 | url-status=live }}</ref> Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.<ref>{{cite journal
| last=Strauss | first=Danie | year=2011
| title=Defining mathematics
| journal=Acta Academica
| volume=43 | issue=4 | pages=1–28
| url=https://www.researchgate.net/publication/290955899
| access-date=November 25, 2022
}}</ref>
=== Rigor ===
{{See also|Logic}}
Mathematical reasoning requires [[Mathematical rigor|rigor]]. This means that the definitions must be absolutely unambiguous and the [[proof (mathematics)|proof]]s must be reducible to a succession of applications of [[inference rule]]s,{{efn|This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without [[computer]]s and [[proof assistant]]s. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.}} without any use of empirical evidence and [[intuition]].{{efn|This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.}}<ref>{{cite journal | title=Mathematical Rigor and Proof | first=Yacin | last=Hamami | journal=The Review of Symbolic Logic | volume=15 | issue=2 | date=June 2022 | pages=409–449 | url=https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | access-date=November 21, 2022 | doi=10.1017/S1755020319000443 | s2cid=209980693 | archive-date=December 5, 2022 | archive-url=https://web.archive.org/web/20221205114343/https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | url-status=live }}</ref> Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' [[concision]], rigorous proofs can require hundreds of pages to express. The emergence of [[computer-assisted proof]]s has allowed proof lengths to further expand,{{efn|For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software}}<ref>{{cite book
| last=Peterson | first=Ivars | author-link=Ivars Peterson
| title=The Mathematical Tourist
| publisher=Freeman | isbn=978-0-7167-1953-3
| year=1988 | page=4
| quote=A few complain that the computer program can't be verified properly | postscript=none}}, (in reference to the Haken–Apple proof of the Four Color Theorem).
</ref> such as the 255-page [[Feit–Thompson theorem]].{{efn|The book containing the complete proof has more than 1,000 pages.}} The result of this trend is a philosophy of the [[Quasi-empiricism in mathematics|quasi-empiricist]] proof that can not be considered infallible, but has a probability attached to it.<ref name=Kleiner_1991/>
The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.<ref name=Kleiner_1991>{{cite journal
| title=Rigor and Proof in Mathematics: A Historical Perspective
| first=Israel | last=Kleiner | author-link=Israel Kleiner (mathematician)
| journal=Mathematics Magazine
| volume=64 | issue=5 | date=December 1991 | pages=291–314
| publisher=Taylor & Francis, Ltd.
| jstor=2690647 | doi=10.1080/0025570X.1991.11977625 }}</ref>
At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and [[Weierstrass function]]) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the [[Apodicticity|apodictic]] inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.<ref name=Kleiner_1991/> It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a [[pleonasm]]. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.<ref>{{cite journal
| title=On the Reliability of Mathematical Proofs
| first=V. Ya. | last=Perminov
| journal=Philosophy of Mathematics
| volume=42 | issue=167 (4) | year=1988 | pages=500–508
| publisher=Revue Internationale de Philosophie
}}</ref>
Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.<ref>{{cite journal
| title=Teachers' perceptions of the official curriculum: Problem solving and rigor
| first1=Jon D. | last1=Davis | first2=Amy Roth | last2=McDuffie
| first3=Corey | last3=Drake | first4=Amanda L. | last4=Seiwell
| journal=International Journal of Educational Research
| volume=93 | year=2019 | pages=91–100
| doi=10.1016/j.ijer.2018.10.002 | s2cid=149753721 }}</ref>
== Training and practice ==
=== Education ===
{{main|Mathematics education}}
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a [[human activity]], the practice of mathematics has a social side, which includes [[Mathematics education|education]], [[Mathematician|careers]], [[List of mathematics awards|recognition]], [[Popular mathematics|popularization]], and so on. In education, mathematics is a core part of the curriculum and forms an important element of the [[STEM]] academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, [[statistician]], [[actuary]], [[financial analyst]], [[economist]], [[accountant]], [[commodity trader]], or [[Information technology consulting|computer consultant]].<ref>{{cite book
| title=Mathematicians and Statisticians: A Practical Career Guide
| first=Kezia
| last=Endsley
| year=2021
| series=Practical Career Guides
| isbn=978-1-5381-4517-3
| publisher=Rowman & Littlefield
| pages=1–3
| url={{GBurl|id=1cEYEAAAQBAJ|p=3}}
| access-date=November 29, 2022
}}</ref>
Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.<ref>{{cite book
| title=The Oxford Handbook of the History of Mathematics
| first=Eleanor | last=Robson | author-link=Eleanor Robson
| year=2009
| chapter=Mathematics education in an Old Babylonian scribal school
| editor1-first=Eleanor | editor1-last=Robson
| editor2-first=Jacqueline | editor2-last=Stedall | editor2-link=Jackie Stedall
| publisher=OUP Oxford
| isbn=978-0-19-921312-2
| chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=199}}
| access-date=November 24, 2022
}}</ref> Comparable evidence has been unearthed for scribal mathematics training in the [[ancient Near East]] and then for the [[Greco-Roman world]] starting around 300 BCE.<ref>{{cite book
| chapter=Mathematics Education in Antiquity
| first1=Alain | last1=Bernard
| first2=Christine | last2=Proust | author2-link=Christine Proust
| first3=Micah | last3=Ross
| title=Handbook on the History of Mathematics Education
| editor1-last=Karp | editor1-first=A.
| editor2-last=Schubring | editor2-first=G.
| year=2014 | pages=27–53 | isbn=978-1-4614-9154-5
| publisher=Springer | publication-place=New York
| doi=10.1007/978-1-4614-9155-2_3
}}</ref> The oldest known mathematics textbook is the [[Rhind papyrus]], dated from {{Circa|1650 BCE}} in Egypt.<ref>{{cite journal
| title=The World's First Mathematics Textbook
| first=Underwood | last=Dudley
| journal=Math Horizons
| volume=9 | issue=4 | date=April 2002 | pages=8–11
| publisher=Taylor & Francis, Ltd.
| doi=10.1080/10724117.2002.11975154 | jstor=25678363
| s2cid=126067145 }}</ref> Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized [[oral tradition]] since the [[Vedic period]] ({{c.|1500|500 BCE}}).<ref>{{cite conference
| title=Indian pedagogy and problem solving in ancient Thamizhakam
| last=Subramarian
| first=F.
| conference=History and Pedagogy of Mathematics conference, July 16–20, 2012
| url=http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
| access-date=November 29, 2022
| archive-date=November 28, 2022
| archive-url=https://web.archive.org/web/20221128082654/http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
| url-status=live
}}</ref> In [[Imperial China]] during the [[Tang dynasty]] (618–907 CE), a mathematics curriculum was adopted for the [[Imperial examination|civil service exam]] to join the state bureaucracy.<ref>{{cite book
| chapter=Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?
| first=Man Keung | last=Siu
| series=Series on Mathematics Education
| title=How Chinese Learn Mathematics
| pages=157–185 | year=2004 | volume=1 | isbn=978-981-256-014-8
| doi=10.1142/9789812562241_0006
| url=https://scholar.archive.org/work/3fb5lb2qsfg35gf2cv6viaydny/access/wayback/http://hkumath.hku.hk:80/~mks/Chapter%206-Siu.pdf
| access-date=November 26, 2022 }}</ref>
Following the [[Dark Age]]s, mathematics education in Europe was provided by religious schools as part of the [[Quadrivium]]. Formal instruction in [[pedagogy]] began with [[Jesuit]] schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was ''[[L'Enseignement Mathématique]]'', which began publication in 1899.<ref>{{cite journal
| title=The History of Mathematical Education
| journal=The American Mathematical Monthly
| volume=74 | issue=1 | pages=38–55
| publisher=Taylor & Francis, Ltd.
| doi=10.2307/2314867 | jstor=2314867 | last1=Jones
| first1=Phillip S.
| year=1967
}}</ref> The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component{{emdash}}initially for its military applications.<ref>{{cite journal
| title=Introduction: the history of mathematics teaching. Indicators for modernization processes in societies
| first1=Gert | last1=Schubring | first2=Fulvia | last2=Furinghetti
| first3=Man Keung | last3=Siu
| journal=ZDM Mathematics Education
| volume=44 | pages=457–459 | date=August 2012
| issue=4 | doi=10.1007/s11858-012-0445-7
| s2cid=145507519 | doi-access=free }}</ref> While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.<ref>{{Cite book | chapter=Examining eTIMSS Country Differences Between eTIMSS Data and Bridge Data: A Look at Country-Level Mode of Administration Effects | title=TIMSS 2019 International Results in Mathematics and Science | first1=Matthias | last1=von Davier | first2=Pierre | last2=Foy | first3=Michael O. | last3=Martin | first4=Ina V.S. | last4=Mullis | publisher=[[TIMSS]] & [[PIRLS]] International Study Center, [[Lynch School of Education and Human Development]] and [[International Association for the Evaluation of Educational Achievement]] | isbn=978-1-889938-54-7 | page=13.1 | language=en-US | year=2020 | url=https://files.eric.ed.gov/fulltext/ED610099.pdf | access-date=November 29, 2022 | archive-date=November 29, 2022 | archive-url=https://web.archive.org/web/20221129163908/https://files.eric.ed.gov/fulltext/ED610099.pdf | url-status=live }}</ref>
During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.<ref>{{cite journal
| title=Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career
| first1=Heather T.
| last1=Rowan-Kenyon
| first2=Amy K.
| last2=Swan
| first3=Marie F.
| last3=Creager
| journal=The Career Development Quarterly
| volume=60
| issue=1
| date=March 2012
| pages=2–15
| doi=10.1002/j.2161-0045.2012.00001.x
| url=https://www.academia.edu/download/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl.pdf
| access-date=November 29, 2022
| archive-url=https://web.archive.org/web/20231122212933/https://d1wqtxts1xzle7.cloudfront.net/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl-libre.pdf?1464293840=&response-content-disposition=inline%3B+filename%3DSocial_Cognitive_Factors_Support_and_Eng.pdf&Expires=1700692172&Signature=cs9KfTPxoPh859wY~ExtJyAl9NpYb3X-2P4rDel1Z3z7DwehsHLRggoZtgi1pMsamxYobu9dVK4G7OsqfvNxcuwz3uKh1pnCMZQEz~ahVtPb4kvN-4dmwExJplzoxWu31o3SJOfuBt0GGE-0Hl8eLfPBg5agmtkjSwAWQwlqGrjp3YgYZGjbNxOEAM4t1l4qvoWXidWvSHHcEUNvlKYwCDvG0~QhGTmA6ldxmfS1ovf0adog-qqvjGxxJuSjtP6O8zCTwkPXYwi2e8giI0H6b5fNarHc-2q~-NRnVVtYKhvSBcwC22kNZoA7s8sp8ix9KIdM3uxiUIBRBRC-4aaVoQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=November 22, 2023
| url-status=live
}}</ref> Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as [[math anxiety]] or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.<ref>{{cite journal
| title=Spotlight on math anxiety
| first1=Silke | last1=Luttenberger
| first2=Sigrid | last2=Wimmer
| first3=Manuela | last3=Paechter
| journal=Psychology Research and Behavior Management
| year=2018 | volume=11 | pages=311–322
| doi=10.2147/PRBM.S141421 | pmid=30123014
| pmc=6087017 | doi-access=free }}</ref>
=== Psychology (aesthetic, creativity and intuition) ===
The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a [[computer program]]. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.<ref>{{cite journal
| title=The Outlook of the Mathematicians' Creative Processes
| first=Narges | last=Yaftian
| journal=Procedia - Social and Behavioral Sciences
| volume=191 | date=June 2, 2015 | pages=2519–2525
| doi=10.1016/j.sbspro.2015.04.617
| doi-access=free}}</ref><ref>{{cite journal
| title=The Frontage of Creativity and Mathematical Creativity
| first1=Mehdi | last1=Nadjafikhah | first2=Narges | last2=Yaftian
| journal=Procedia - Social and Behavioral Sciences
| volume=90 | date=October 10, 2013 | pages=344–350
| doi=10.1016/j.sbspro.2013.07.101
| doi-access=free}}</ref> An extreme example is [[Apery's theorem]]: [[Roger Apery]] provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.<ref>{{cite journal
| title=A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)
| first=A.
| last=van der Poorten
| journal=[[The Mathematical Intelligencer]]
| volume=1
| issue=4
| year=1979
| pages=195–203
| doi=10.1007/BF03028234
| s2cid=121589323
| url=http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
| access-date=November 22, 2022
| archive-date=September 6, 2015
| archive-url=https://web.archive.org/web/20150906015716/http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
| url-status=live
}}</ref>
Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving [[puzzle]]s.<ref>{{cite book
| title=Famous Puzzles of Great Mathematicians
| first=Miodrag
| last=Petkovi
| date=September 2, 2009
| publisher=American Mathematical Society
| pages=xiii–xiv
| isbn=978-0-8218-4814-2
| url={{GBurl|id=AZlwAAAAQBAJ|pg=PR13}}
| access-date=November 25, 2022
}}</ref> This aspect of mathematical activity is emphasized in [[recreational mathematics]].
Mathematicians can find an [[aesthetic]] value to mathematics. Like [[beauty]], it is hard to define, it is commonly related to ''elegance'', which involves qualities like [[simplicity]], [[symmetry]], completeness, and generality. G. H. Hardy in ''[[A Mathematician's Apology]]'' expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.<ref>{{cite book
| title=A Mathematician's Apology
| last=Hardy | first=G. H. | author-link=G. H. Hardy
| publisher=Cambridge University Press | year=1940
| url=https://archive.org/details/hardy_annotated/
| isbn=978-0-521-42706-7 | access-date=November 22, 2022
}} See also ''[[A Mathematician's Apology]]''.</ref> [[Paul Erdős]] expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book ''[[Proofs from THE BOOK]]'', inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the [[fast Fourier transform]] for [[harmonic analysis]].<ref>{{cite journal
| title=Reflections on Paul Erdős on His Birth Centenary, Part II
| first1=Noga | last1=Alon | first2=Dan | last2=Goldston
| first3=András | last3=Sárközy | first4=József | last4=Szabados
| first5=Gérald | last5=Tenenbaum | first6=Stephan Ramon | last6=Garcia
| first7=Amy L. | last7=Shoemaker
| journal=Notices of the American Mathematical Society
| date=March 2015 | volume=62 | issue=3 | pages=226–247
| editor1-first=Krishnaswami | editor1-last=Alladi
| editor2-first=Steven G. | editor2-last=Krantz
| doi=10.1090/noti1223
| doi-access=free }}</ref>
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional [[liberal arts]].<ref>See, for example [[Bertrand Russell]]'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his {{cite book | title=History of Western Philosophy | year=1919 | page=60 }}</ref> One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are ''created'' (as in art) or ''discovered'' (as in science).<ref name=borel>{{Cite journal
| last=Borel | first=Armand | author-link=Armand Borel
| title=Mathematics: Art and Science
| journal=The Mathematical Intelligencer
| volume=5 | issue=4 | pages=9–17 | year=1983
| publisher=Springer | issn=1027-488X
| doi=10.4171/news/103/8| doi-access=free
}}</ref> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as [[intuitionism]], which primarily identifies mathematics with certain creative processes in the mind.<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> Intuitionism is in turn one flavor of a stance known as [[Constructivism (mathematics)|constructivism]], which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like [[existential proof]]s based on the law of excluded middle.<ref name=SEP-Intuitionism>{{cite encyclopedia |title=Intuitionism in the Philosophy of Mathematics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Iemhoff |first=Rosalie|author-link=Rosalie Iemhoff |editor-last=Zalta |editor-first=Edward N. |year=2020 |edition=Fall 2020 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/fall2020/entries/intuitionism |access-date=April 2, 2022 |archive-date=April 21, 2022 |archive-url=https://web.archive.org/web/20220421162527/https://plato.stanford.edu/archives/fall2020/entries/intuitionism/ |url-status=live }}</ref>
In the end, neither constructivism nor intuitionism displaced [[classical mathematics]] or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.<ref name=SEP-Intuitionism />
== Cultural impact ==
{{Globalize|date=December 2022|1=section|2=West}}
=== Artistic expression ===
{{Main|Mathematics and art}}
Notes that sound well together to a Western ear are sounds whose fundamental [[frequencies]] of vibration are in simple ratios. For example, an octave doubles the frequency and a [[perfect fifth]] multiplies it by <math>\frac{3}{2}</math>.<ref>{{cite journal | last = Cazden | first = Norman | date = October 1959 | doi = 10.1177/002242945900700205 | issue = 2 | journal = Journal of Research in Music Education | jstor = 3344215 | pages = 197–220 | title = Musical intervals and simple number ratios | volume = 7| s2cid = 220636812 }}</ref><ref>{{cite journal | last = Budden | first = F. J. | date = October 1967 | doi = 10.2307/3613237 | issue = 377 | journal = The Mathematical Gazette | jstor = 3613237 | pages = 204–215 | publisher = Cambridge University Press ({CUP}) | title = Modern mathematics and music | volume = 51| s2cid = 126119711 }}</ref>
[[File:Julia set (highres 01).jpg|thumb|[[Fractal]] with a scaling symmetry and a central symmetry]]
Humans, as well as some other animals, find symmetric patterns to be more beautiful.<ref>{{Cite journal |last1=Enquist |first1=Magnus |last2=Arak |first2=Anthony |date=November 1994 |title=Symmetry, beauty and evolution |url=https://www.nature.com/articles/372169a0 |journal=Nature |language=en |volume=372 |issue=6502 |pages=169–172 |doi=10.1038/372169a0 |pmid=7969448 |bibcode=1994Natur.372..169E |s2cid=4310147 |issn=1476-4687 |access-date=December 29, 2022 |archive-date=December 28, 2022 |archive-url=https://web.archive.org/web/20221228052049/https://www.nature.com/articles/372169a0 |url-status=live }}</ref> Mathematically, the symmetries of an object form a group known as the [[symmetry group]].<ref>{{Cite web |last=Hestenes |first=David |date=1999 |title=Symmetry Groups |url=http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |access-date=December 29, 2022 |website=geocalc.clas.asu.edu |archive-date=January 1, 2023 |archive-url=https://web.archive.org/web/20230101210124/http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |url-status=live }}</ref>
For example, the group underlying mirror symmetry is the [[cyclic group]] of two elements, <math>\mathbb{Z}/2\mathbb{Z}</math>. A [[Rorschach test]] is a figure invariant by this symmetry,<ref>{{cite encyclopedia | last = Bender | first = Sara | editor1-last = Carducci | editor1-first = Bernardo J. | editor2-last = Nave | editor2-first = Christopher S. | editor3-last = Mio | editor3-first = Jeffrey S. | editor4-last = Riggio | editor4-first = Ronald E. | title = The Rorschach Test | date = September 2020 | doi = 10.1002/9781119547167.ch131 | pages = 367–376 | publisher = Wiley | encyclopedia = The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment| isbn = 9781119057512 }}</ref> as are [[butterfly]] and animal bodies more generally (at least on the surface).<ref>{{cite book|title=Symmetry|volume=47|series=Princeton Science Library|first=Hermann|last=Weyl|author-link=Hermann Weyl|publisher=Princeton University Press|year=2015|isbn=9781400874347|page=[https://books.google.com/books?hl=en&lr=&id=GG1FCQAAQBAJ&pg=PA4 4]}}</ref> Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.{{Citation needed|date=December 2022}} Furthermore, [[fractals]] possess (usually approximate{{Citation needed|date=December 2022}}) [[self-similarity]].<ref>{{Cite web |last=Bradley |first=Larry |date=2010 |title=Fractals – Chaos & Fractals |url=https://www.stsci.edu/~lbradley/seminar/fractals.html |access-date=December 29, 2022 |website=www.stsci.edu |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054609/https://www.stsci.edu/~lbradley/seminar/fractals.html |url-status=live }}</ref><ref>{{Cite web |title=Self-similarity |url=https://math.bu.edu/DYSYS/chaos-game/node5.html |access-date=December 29, 2022 |website=math.bu.edu |archive-date=March 2, 2023 |archive-url=https://web.archive.org/web/20230302132911/http://math.bu.edu/DYSYS/chaos-game/node5.html |url-status=live }}</ref>
=== Popularization ===
{{Main|Popular mathematics}}Popular mathematics is the act of presenting mathematics without technical terms.<ref>{{Cite conference |last=Kissane |first=Barry |date=July 2009 |title=Popular mathematics |url=https://researchrepository.murdoch.edu.au/id/eprint/6242/ |conference=22nd Biennial Conference of The Australian Association of Mathematics Teachers |location=Fremantle, Western Australia |publisher=Australian Association of Mathematics Teachers |pages=125–126 |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054610/https://researchrepository.murdoch.edu.au/id/eprint/6242/ |url-status=live }}</ref> Presenting mathematics may be hard since the general public suffers from [[mathematical anxiety]] and mathematical objects are highly abstract.<ref>{{Cite book |last=Steen |first=L. A. |url={{GBurl|id=-d3TBwAAQBAJ|dq="popular mathematics" analogies|p=2}} |title=Mathematics Today Twelve Informal Essays |date=2012|publisher=Springer Science & Business Media |isbn=978-1-4613-9435-8 |page=2 |language=en |access-date=January 3, 2023 }}</ref> However, popular mathematics writing can overcome this by using applications or cultural links.<ref>{{Cite book |last=Pitici |first=Mircea |url={{GBurl|id=9nGQDQAAQBAJ|dq="popular mathematics" analogies|p=331}} |title=The Best Writing on Mathematics 2016 |date=2017|publisher=Princeton University Press |isbn=978-1-4008-8560-2 |language=en |access-date=January 3, 2023 }}</ref> Despite this, mathematics is rarely the topic of popularization in printed or televised media.
== Awards and prize problems ==
{{Main category|Mathematics awards}}
[[File:FieldsMedalFront.jpg|thumb|The front side of the [[Fields Medal]] with an illustration of the Greek [[polymath]] [[Archimedes]]]]
The most prestigious award in mathematics is the [[Fields Medal]],{{sfn|Monastyrsky|2001|p=1|ps=: "The Fields Medal is now indisputably the best known and most influential award in mathematics."}}{{sfn|Riehm|2002|pp=778–782}} established in 1936 and awarded every four years (except around [[World War II in Yugoslavia|World War II]]) to up to four individuals.<ref>{{Cite web |title=Fields Medal {{!}} International Mathematical Union (IMU) |url=https://www.mathunion.org/imu-awards/fields-medal |access-date=February 21, 2022 |website=www.mathunion.org |archive-date=December 26, 2018 |archive-url=https://web.archive.org/web/20181226015744/https://www.mathunion.org/imu-awards/fields-medal |url-status=live }}</ref><ref name="StAndrews-Fields">{{Cite web |title=Fields Medal |url=https://mathshistory.st-andrews.ac.uk/Honours/FieldsMedal/ |access-date=February 21, 2022 |website=Maths History |language=en |archive-date=March 22, 2019 |archive-url=https://web.archive.org/web/20190322134417/http://www-history.mcs.st-andrews.ac.uk/Honours/FieldsMedal.html |url-status=live }}</ref> It is considered the mathematical equivalent of the [[Nobel Prize]].<ref name="StAndrews-Fields" />
Other prestigious mathematics awards include:<ref>{{cite web
| title=Honours/Prizes Index
| website=MacTutor History of Mathematics Archive
| url=https://mathshistory.st-andrews.ac.uk/Honours/
| access-date=February 20, 2023
| archive-date=December 17, 2021
| archive-url=https://web.archive.org/web/20211217235828/https://mathshistory.st-andrews.ac.uk/Honours/
| url-status=live
}}</ref>
* The [[Abel Prize]], instituted in 2002<ref>{{Cite web|title=About the Abel Prize|publisher=The Abel Prize|url=https://abelprize.no/page/about-abel-prize|access-date=January 23, 2022|archive-date=April 14, 2022|archive-url=https://web.archive.org/web/20220414060442/https://abelprize.no/page/about-abel-prize|url-status=live}}</ref> and first awarded in 2003<ref>{{Cite encyclopedia|title=Abel Prize {{!}} mathematics award|encyclopedia=Encyclopedia Britannica|url=https://www.britannica.com/science/Abel-Prize|access-date=January 23, 2022|language=en|archive-date=January 26, 2020|archive-url=https://web.archive.org/web/20200126120202/https://www.britannica.com/science/Abel-Prize|url-status=live}}</ref>
* The [[Chern Medal]] for lifetime achievement, introduced in 2009<ref>{{Cite web |date=June 1, 2009 |title=Chern Medal Award|url=https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |url-status=live |archive-url=https://web.archive.org/web/20090617012953/https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |archive-date=June 17, 2009 |access-date=February 21, 2022 |website=www.mathunion.org}}</ref> and first awarded in 2010<ref>{{Cite web |title=Chern Medal Award|publisher=International Mathematical Union (IMU)|url=https://www.mathunion.org/imu-awards/chern-medal-award |access-date=January 23, 2022|archive-date=August 25, 2010 |archive-url=https://web.archive.org/web/20100825071850/http://www.mathunion.org/general/prizes/chern/details |url-status=live }}</ref>
* The [[American Mathematical Society|AMS]] [[Leroy P. Steele Prize]], awarded since 1970<ref>{{cite web
| title=The Leroy P Steele Prize of the AMS
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
| access-date=November 17, 2022
| archive-date=November 17, 2022
| archive-url=https://web.archive.org/web/20221117201134/https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
| url-status=live
}}</ref>
* The [[Wolf Prize in Mathematics]], also for lifetime achievement,<ref>{{Cite book |last1=Chern |first1=S. S. |last2=Hirzebruch |first2=F. |date=September 2000 |title=Wolf Prize in Mathematics |url=https://www.worldscientific.com/worldscibooks/10.1142/4149 |language=en |doi=10.1142/4149 |isbn=978-981-02-3945-9 |access-date=February 21, 2022 |archive-date=February 21, 2022 |archive-url=https://web.archive.org/web/20220221171351/https://www.worldscientific.com/worldscibooks/10.1142/4149 |url-status=live }}</ref> instituted in 1978<ref>{{Cite web|title=The Wolf Prize|url=https://wolffund.org.il/the-wolf-prize/|url-status=live|archive-url=https://web.archive.org/web/20200112205029/https://wolffund.org.il/the-wolf-prize/|archive-date=January 12, 2020|access-date=January 23, 2022|website=Wolf Foundation|language=en-US}}</ref>
A famous list of 23 [[open problem]]s, called "[[Hilbert's problems]]", was compiled in 1900 by German mathematician David Hilbert.<ref name=":0">{{Cite web|date=May 6, 2020|title=Hilbert's Problems: 23 and Math|url=https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|access-date=January 23, 2022|website=Simons Foundation|language=en-US|archive-date=January 23, 2022|archive-url=https://web.archive.org/web/20220123011430/https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|url-status=live}}</ref> This list has achieved great celebrity among mathematicians,<ref>{{cite book
| chapter=Deciding the undecidable: Wrestling with Hilbert's problems
| first=Solomon
| last=Feferman
| author-link=Solomon Feferman
| title=In the Light of Logic
| year=1998
| publisher=Oxford University Press
| isbn=978-0-19-508030-8
| pages=3–27
| series=Logic and Computation in Philosophy series
| chapter-url=https://math.stanford.edu/~feferman/papers/deciding.pdf
| url={{GBurl|id=1rjnCwAAQBAJ}}
| access-date=November 29, 2022
}}</ref> and, {{as of|2022|lc=yes}}, at least thirteen of the problems (depending how some are interpreted) have been solved.<ref name=":0"/><!-- Namely: problems 1, 3, 4; 5, 7, 10; 13, 14, 17; 18, 19, 20; 21 have been solved. (The semicolons are to make counting easier). ~Duckmather -->
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.<ref>{{Cite web|title=The Millennium Prize Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems/millennium-prize-problems|access-date=January 23, 2022|archive-date=July 3, 2015|archive-url=https://web.archive.org/web/20150703184941/http://www.claymath.org/millennium-problems/millennium-prize-problems|url-status=live}}</ref> To date, only one of these problems, the [[Poincaré conjecture]], has been solved.<ref>{{Cite web|title=Millennium Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems|access-date=January 23, 2022|archive-date=December 20, 2018|archive-url=https://web.archive.org/web/20181220122925/http://www.claymath.org/millennium-problems|url-status=live}}</ref><!-- NOTE that this website describes the answer to each problem as "unknown" EXCEPT for the Poincaré conjecture, where it mentions "Perelman's proof". ~Duckmather -->
== See also ==
{{Portal|Mathematics}}
{{div col|colwidth=22em}}
* [[List of mathematical jargon]]
* [[Lists of mathematicians]]
* [[Lists of mathematics topics]]
* [[Mathematical constant]]
* [[Mathematical sciences]]
* [[Mathematics and art]]
* [[Mathematics education]]
* [[Outline of mathematics]]
* [[Philosophy of mathematics]]
* [[Relationship between mathematics and physics]]
* [[Science, technology, engineering, and mathematics]]
{{div col end}}
== References ==
=== Notes ===
{{notelist}}
===Citations===
{{Reflist|30em
<!--
|refs=
<ref name=future>[[Lynn Steen|Steen, L.A.]] (April 29, 1988). ''The Science of Patterns'' [[Science (journal)|Science]], 240: 611–16. And summarized at [http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx Association for Supervision and Curriculum Development] {{webarchive|url=https://web.archive.org/web/20101028101034/http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx |date=October 28, 2010 }}, www.ascd.org.</ref>
<ref name=devlin>[[Keith Devlin|Devlin, Keith]], ''Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe'' (Scientific American Paperback Library) 1996, {{isbn|978-0716750475}}</ref>
-->
}}
=== Bibliography ===
{{refbegin|30em}}
* {{cite book |last=Bouleau |first=Nicolas |title=Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur |publisher=L'Harmattan |year=1999 |isbn=978-2-7384-8125-2}}
* {{cite book |last1=Boyer |first1=Carl Benjamin |author1-link=Carl Benjamin Boyer |title=A History of Mathematics |date=1991 |publisher=[[Wiley (publisher)|Wiley]] |location=New York |isbn=978-0-471-54397-8 |edition=2nd |url=https://archive.org/details/historyofmathema00boye/page/n3/mode/2up |chapter= |url-access=registration }}
* {{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=An Introduction to the History of Mathematics |edition=6th |publisher=Saunders |year=1990 |isbn=978-0-03-029558-4 |ref=none}}
* {{cite book |last=Kleiner |first=Izraïl' |author-link=Israel Kleiner (mathematician) |title=A History of Abstract Algebra |publisher=Springer Science & Business Media |isbn=978-0-8176-4684-4 |year=2007 |url={{GBurl|id=RTLRBK-wj6wC|pg=PR1}} |access-date=November 11, 2022 }}
* {{cite book |last=Kline |first=Morris |author-link=Morris Kline |title=Mathematical Thought from Ancient to Modern Times |url=https://archive.org/details/mathematicalthou00klin |url-access=registration |location=New York |publisher=Oxford University Press |edition=|year=1990 |isbn=978-0-19-506135-2 }}
* {{cite journal |last=Monastyrsky |first=Michael |url=http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf |year=2001 |title=Some Trends in Modern Mathematics and the Fields Medal |journal=CMS – Notes – de la SMC |volume=33 |issue=2–3 |publisher=Canadian Mathematical Society |access-date=July 28, 2006 |archive-date=August 13, 2006 |archive-url=https://web.archive.org/web/20060813224844/http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf |url-status=live }}
* {{cite book |last=Oakley |first=Barbara |author-link=Barbara Oakley |title=A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) |date=2014 |publisher=Penguin Random House |location=New York |url=https://archive.org/details/isbn_9780399165245 |url-access=registration |quote=A Mind for Numbers. |isbn=978-0-399-16524-5 |ref=none }}
* {{cite journal |last=Peirce |first=Benjamin |author-link=Benjamin Peirce |pages=97–229 |title=Linear associative algebra |editor-link=Charles Sanders Peirce |editor-first=Charles Sanders |editor-last=Peirce |edition=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph |journal=American Journal of Mathematics |volume=4 |year=1881 |url={{GBurl|id=De0GAAAAYAAJ|q=Peirce Benjamin Linear Associative Algebra|p=1}} |id=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. ''Google'' [{{GBurl|id=LQgPAAAAIAAJ|p=221}} Eprint] and as an extract, D. Van Nostrand, 1882, ''Google'' [https://archive.org/details/bub_gb_De0GAAAAYAAJ Eprint] |issue=1–4 |doi=10.2307/2369153 |jstor=2369153 |hdl=2027/hvd.32044030622997 |hdl-access=free |access-date=November 17, 2020 |ref=none }}.
* {{cite book |last=Peterson |first=Ivars |title=Mathematical Tourist, New and Updated Snapshots of Modern Mathematics |publisher=Owl Books |year=2001 |isbn=978-0-8050-7159-7}}
* {{cite book |last=Popper |first=Karl R. |author-link=Karl Popper |title=In Search of a Better World: Lectures and Essays from Thirty Years |location=New York |publisher=Routledge |chapter=On knowledge |year=1995 |isbn=978-0-415-13548-1 |bibcode=1992sbwl.book.....P |url-access=registration |url=https://archive.org/details/insearchofbetter00karl |ref=none }}
* {{cite journal |last=Riehm |first=Carl |title=The Early History of the Fields Medal |journal=Notices of the AMS |volume=49 |issue=7 |pages=778–782 |date=August 2002 |url=https://www.ams.org/notices/200207/comm-riehm.pdf |access-date=October 2, 2006 |archive-date=October 26, 2006 |archive-url=https://web.archive.org/web/20061026000014/http://www.ams.org/notices/200207/comm-riehm.pdf |url-status=live }}
* {{cite journal |last=Sevryuk |first=Mikhail B. |date=January 2006 |title=Book Reviews |journal=[[Bulletin of the American Mathematical Society]] |volume=43 |issue=1 |pages=101–109 |url=https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf |access-date=June 24, 2006 |doi=10.1090/S0273-0979-05-01069-4 |archive-date=July 23, 2006 |archive-url=https://web.archive.org/web/20060723082901/http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf |url-status=live |doi-access=free }}
* {{cite book |last=Whittle |first=Peter |author-link=Peter Whittle (mathematician) |chapter=Almost home |editor-link=Frank Kelly (mathematician) |editor-first=F.P. |editor-last=Kelly |year=1994 |title=Probability, statistics and optimisation: A Tribute to Peter Whittle |location=Chichester |publisher=John Wiley |isbn=978-0-471-94829-2 |pages=1–28 |chapter-url=http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair |edition=previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" |url-status=live |archive-url=https://web.archive.org/web/20131219080017/http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair |archive-date=December 19, 2013 }}
{{refend}}
== Further reading ==
{{Library resources box |by=no |onlinebooks=yes |others=yes |about=yes |label=Mathematics}}
{{refbegin}}
* {{cite book |last=Benson |first=Donald C. |title=The Moment of Proof: Mathematical Epiphanies |publisher=Oxford University Press |year=1999 |url=https://archive.org/details/momentofproofmat00bens/page/n5/mode/2up |url-access=registration |isbn=978-0-19-513919-8 |ref=none}}
* {{cite book |last1=Davis |first1=Philip J. |author1-link=Philip J. Davis |last2=Hersh |first2=Reuben |author2-link=Reuben Hersh |title=The Mathematical Experience |publisher=Mariner Books |location=Boston; New York |edition=Reprint |year=1999 |isbn=978-0-395-92968-1 |title-link=The Mathematical Experience |ref=none}} Available [https://archive.org/details/mathematicalexpe0000davi/page/n5/mode/2up online] (registration required).
* {{cite book |last1=Courant |first1=Richard |author-link1=Richard Courant |last2=Robbins |first2=Herbert |author-link2=Herbert Robbins |title=What Is Mathematics?: An Elementary Approach to Ideas and Methods |location=New York |publisher=Oxford University Press |edition=2nd |year=1996 |isbn=978-0-19-510519-3 |url=https://archive.org/details/whatismathematic0000cour/page/n5/mode/2up |url-access=registration |ref=none}}
* {{cite book |last=Gullberg |first=Jan |author-link=Jan Gullberg |title=Mathematics: From the Birth of Numbers |url=https://archive.org/details/mathematicsfromb1997gull/page/n5/mode/2up |url-access=registration |publisher=W.W. Norton & Company |edition= |year=1997 |isbn=978-0-393-04002-9 |ref=none}}
* {{cite book |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |title=Encyclopaedia of Mathematics |publisher=Kluwer Academic Publishers |year=2000 |title-link=Encyclopaedia of Mathematics |ref=none}} – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and [https://encyclopediaofmath.org/wiki/Special:AllPages online]. {{Webarchive|url=https://archive.today/20121220135247/http://www.encyclopediaofmath.org/ |date=December 20, 2012 |ref=none }}.
* {{cite book |last=Hodgkin |first=Luke Howard |title=A History of Mathematics: From Mesopotamia to Modernity |publisher=Oxford University Press |year=2005 |isbn=978-0-19-152383-0}}
* {{cite book |last=Jourdain |first=Philip E. B. |author-link=Philip Jourdain |chapter=The Nature of Mathematics |title=The World of Mathematics |editor=James R. Newman |publisher=Dover Publications |year=2003 |isbn=978-0-486-43268-7 |ref=none}}
<!-- * {{cite book |last=Maier |first=Annaliese |author-link=Anneliese Maier |title=At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy |editor=Steven Sargent |location=Philadelphia |publisher=University of Pennsylvania Press |year=1982 |ref=none}}
-->
* {{cite book |last1=Pappas |first1=Theoni |author1-link=Theoni Pappas |title=The Joy Of Mathematics |date=1986 |publisher=Wide World Publishing |location=San Carlos, California |isbn=978-0-933174-65-8 |url=https://archive.org/details/joyofmathematics0000papp_t0z1/page/n3/mode/2up |url-access=registration |ref=none}}
* {{cite book |last=Waltershausen |first=Wolfgang Sartorius von |author-link=Wolfgang Sartorius von Waltershausen |title=Gauss zum Gedächtniss |year=1965 |orig-date=first published 1856 |publisher=Sändig Reprint Verlag H. R. Wohlwend |isbn=978-3-253-01702-5 |ref=no}}
{{refend}}
{{Areas of mathematics}}
{{Authority control}}
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| portal2 = Arithmetic
| portal3 = History of science
| portal4 = Science
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| n = y
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[[Category:Mathematics| ]]
[[Category:Formal sciences]]
[[Category:Main topic articles]]' |
New page wikitext, after the edit (new_wikitext) | '{{Short description|Area of knowledge}}
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{{Use American English|date=August 2022}}
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{{Math topics TOC}}
'''Mathematics''' is an area of [[knowledge]] that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.<!-- Please, do not link articles to words that are used in their non-technical meaning, such as "numbers", "quantity", "formula", "structure", "shape", "space". Linking such articles would provide too much technical information about those abstractions. --> These topics are represented in modern mathematics with the major subdisciplines of [[number theory]],<ref name="OED">{{cite web
| url=http://oed.com/view/Entry/114974
| title=mathematics, ''n.''
| publisher=Oxford University Press
| website=Oxford English Dictionary
| year=2012 | access-date=June 16, 2012
| quote=The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.
| archive-url=https://web.archive.org/web/20191116075558/https://www.oed.com/view/Entry/114974
| archive-date=November 16, 2019 | url-status=live
}}</ref> [[algebra]],<ref name="Kneebone">{{cite book
| title=Mathematical Logic and the Foundations of Mathematics: An Introductory Survey
| publisher=Dover
| last=Kneebone | first=G. T. | year=1963 | page=4
| url={{GBurl|id=tCXxf4vbXCcC|p=4}}
| isbn=978-0-486-41712-7
| quote=Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.
| access-date=June 20, 2015
}}</ref> [[geometry]],<ref name=OED /> and [[mathematical analysis|analysis]],<ref name="LaTorre">{{cite book
| title=Calculus Concepts: An Informal Approach to the Mathematics of Change
| publisher=Cengage Learning
| first1=Donald R. | last1=LaTorre | first2=John W. | last2=Kenelly
| first3=Sherry S. | last3=Biggers | first4=Laurel R. | last4=Carpenter
| first5=Iris B. | last5=Reed | first6=Cynthia R. | last6=Harris
| year=2011 | page=2 | isbn=978-1-4390-4957-0
| url={{GBurl|id=1Ebu2Tij4QsC|p=2}}
| quote=Calculus is the study of change—how things change, and how quickly they change.
| access-date=June 20, 2015
}}</ref><ref name="Ramana">{{cite book
| title=Applied Mathematics
| publisher=Tata McGraw–Hill Education
| first=B. V. |last=Ramana
| year=2007 | page=2.10
| url={{GBurl|id=XCRC6BeKhIIC|pg=SA2–PA10}}
| isbn=978-0-07-066753-2
| quote=The mathematical study of change, motion, growth or decay is calculus.
| access-date=July 30, 2022
}}</ref> respectively.<!--<<< Please do NOT change the opening sentence without discussion; much time and discussion have been invested in its current form.--> There is no general consensus among mathematicians about a common definition for their [[academic discipline]].
Most mathematical activity involves the discovery of properties of [[mathematical object|abstract objects]] and the use of pure [[reason]] to [[proof (mathematics)|prove]] them. These objects consist of either [[abstraction (mathematics)|abstraction]]s from nature or{{emdash}}in modern mathematics{{emdash}}entities that are stipulated to have certain properties, called [[axiom]]s. A ''proof'' consists of a succession of applications of [[inference rule|deductive rules]] to already established results. These results include previously proved [[theorem]]s, axioms, and{{emdash}}in case of abstraction from nature{{emdash}}some basic properties that are considered true starting points of the theory under consideration.<ref>{{cite conference
| title=Abstract Cognition and the Nature of Mathematical Proof
| book-title=Realism – Relativism – Constructivism
| conference=38th International Wittgenstein Symposium, August 9–15, 2015. Kirchberg am Wechsel, Austria
| first=Inês Viegas
| last=Hipólito
| year=2015
| pages=132–134
| publisher=Austrian Ludwig Wittgenstein Society
| editor1-first=Christian
| editor1-last=Kanzian
| editor2-first=Josef
| editor2-last=Mitterer
| editor2-link=Josef Mitterer
| editor3-first=Katharina
| editor3-last=Neges
| url=https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133
| access-date=November 5, 2022
| archive-date=November 7, 2022
| archive-url=https://web.archive.org/web/20221107221937/https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133
| url-status=live
}} ([https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof at ResearchGate] {{Webarchive|url=https://web.archive.org/web/20221105145638/https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof |date=November 5, 2022 }})</ref><!-- Commenting out the following pending discussion on talk: Contrary to [[physical law]]s, the validity of a theorem (its truth) does not rely on any [[experimentation]] but on the correctness of its reasoning (though experimentation is often useful for discovering new theorems of interest). -->
Mathematics is essential in the [[natural science]]s, [[engineering]], [[medicine]], [[finance]], [[computer science]] and the [[social sciences]]. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as [[statistics]] and [[game theory]], are developed in close correlation with their applications and are often grouped under [[applied mathematics]]. Other areas are developed independently from any application (and are therefore called [[pure mathematics]]), but often later find practical applications.{{sfn|Peterson|2001|p=12}}<ref name=wigner1960 /> The problem of [[integer factorization]], for example, which goes back to [[Euclid]] in 300 BC, had no practical application before its use in the [[RSA cryptosystem]], now widely used for the security of [[computer network]]s.
Historically, the concept of a proof and its associated [[mathematical rigour]] first appeared in [[Greek mathematics]], most notably in Euclid's ''[[Euclid's Elements|Elements]]''.<ref>{{Cite web|url=http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Wise/essay7/essay7.htm|title=Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion|last=Wise|first=David|website=jwilson.coe.uga.edu|url-status=live|archive-url=https://web.archive.org/web/20190601004355/http://jwilson.coe.uga.edu/emt668/EMAT6680.F99/Wise/essay7/essay7.htm|archive-date=June 1, 2019|access-date=October 26, 2019}}</ref> Since its beginning, mathematics was primarily divided into geometry and [[arithmetic]] (the manipulation of [[natural number]]s and [[fraction (mathematics)|fractions]]), until the 16th and 17th centuries, when algebra{{efn|Here, ''algebra'' is taken in its modern sense, which is, roughly speaking, the art of manipulating [[formula]]s.}} and [[infinitesimal calculus]] were introduced as new fields. Since then, the interaction between mathematical innovations and [[timeline of scientific discoveries|scientific discoveries]] has led to a correlated increase in the development of both.<ref>{{cite journal
| title=The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?
| first=Amir | last=Alexander | author-link=Amir Alexander
| journal=Isis | year=2011 | volume=102 | number=3
| pages=475–480 | doi=10.1086/661620 | pmid=22073771 | s2cid=21629993 }}</ref> At the end of the 19th century, the [[foundational crisis of mathematics]] led to the systematization of the [[axiomatic method]],<ref name=Kleiner_1991/> which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary [[Mathematics Subject Classification]] lists more than 60 first-level areas of mathematics.
{{TOC limit|3}}
== Etymology ==
The word ''mathematics'' comes from [[Ancient Greek]] ''máthēma'' (''{{Lang-grc|{{wikt-lang|en|μάθημα}}|label=none}}''), meaning "that which is learnt",<ref name=EOD_n>{{cite encyclopedia
| title=mathematic (n.)
| first=Douglas
| last=Harper
| dictionary=[[Online Etymology Dictionary]]
| url=https://www.etymonline.com/word/mathematic
| access-date=November 26, 2022
| archive-date=March 7, 2013
| archive-url=https://web.archive.org/web/20130307093926/http://etymonline.com/index.php?term=mathematic&allowed_in_frame=0
| url-status=live
}}</ref> "what one gets to know", hence also "study" and "science". The word came to have the narrower and more technical meaning of "mathematical study" even in [[Classical antiquity|Classical times]].<ref>Both meanings can be found in Plato, the narrower in {{cite book|title=Republic|title-link=Republic (Plato)|at=[https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 510c]}} {{Cite web |url=http://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 |title=Plato, Republic, Book 6, section 510c |access-date=June 19, 2019 |archive-date=February 24, 2021 |archive-url=https://web.archive.org/web/20210224152747/http://www.perseus.tufts.edu/hopper/text?doc=Plat.+Rep.+6.510c&fromdoc=Perseus%3Atext%3A1999.01.0168 |url-status=live |postscript=,}} but Plato did not use a ''math-'' word; Aristotle did, commenting on it. {{LSJ|maqhmatiko/s|μαθηματική|ref}}. {{cite book|title=OED Online|at=Mathematics}}</ref> Its [[adjective]] is ''mathēmatikós'' ({{lang|grc|μαθηματικός}}), meaning "related to learning" or "studious", which likewise further came to mean "mathematical".<ref>{{cite encyclopedia
| title=mathematical (adj.)
| first=Douglas
| last=Harper
| dictionary=Online Etymology Dictionary
| url=https://www.etymonline.com/word/mathematical
| access-date=November 26, 2022
| archive-date=November 26, 2022
| archive-url=https://web.archive.org/web/20221126170916/https://www.etymonline.com/word/mathematical
| url-status=live
}}</ref> In particular, ''mathēmatikḗ tékhnē'' ({{lang|grc|μαθηματικὴ τέχνη}}; {{lang-la|ars mathematica}}) meant "the mathematical art".<ref name=EOD_n/>
Similarly, one of the two main schools of thought in [[Pythagoreanism]] was known as the ''mathēmatikoi'' (μαθηματικοί){{emdash}}which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of [[arithmetic]] and geometry. By the time of [[Aristotle]] (384–322 BC) this meaning was fully established.<ref>{{cite journal
| title=The Etymology of Mathematical Terms
| first=Margaret W. | last=Perisho
| journal=Pi Mu Epsilon Journal
| volume=4 | issue=2 | date=Spring 1965 | pages=62–66
| jstor=24338341 }}</ref>
In Latin, and in English until around 1700, the term ''mathematics'' more commonly meant "[[astrology]]" (or sometimes "[[astronomy]]") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, [[Saint Augustine]]'s warning that Christians should beware of ''mathematici'', meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.<ref name="Boas">{{cite book | title=Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas, Jr | publisher=Cambridge University Press | last=Boas | first=Ralph | author-link=Ralph P. Boas Jr. | year=1995 | orig-date=1991 | page=257 | chapter-url={{GBurl|id=f-EWj5WtQHgC|p=257}} | chapter=What Augustine Didn't Say About Mathematicians | isbn=978-0-88385-323-8 | access-date=January 17, 2018 }}</ref>
The apparent [[plural]] form in English goes back to the Latin [[Neuter (grammar)|neuter]] plural {{lang|la|mathematica}} ([[Cicero]]), based on the Greek plural ''ta mathēmatiká'' ({{lang|el|τὰ μαθηματικά}}) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective ''mathematic(al)'' and formed the noun ''mathematics'' anew, after the pattern of ''[[physics]]'' and ''[[metaphysics]]'', inherited from Greek.<ref>''[[The Oxford Dictionary of English Etymology]]'', ''[[Oxford English Dictionary]]'', ''sub'' "mathematics", "mathematic", "mathematics".</ref> In English, the noun ''mathematics'' takes a singular verb. It is often shortened to ''maths'' or, in North America, ''math''.<ref name=maths>[http://oed.com/view/Entry/114982 "maths, ''n.''"] and [http://oed.com/view/Entry/114962 "math, ''n.3''"] {{Webarchive|url=https://web.archive.org/web/20200404201407/http://oed.com/view/Entry/114982 |date=April 4, 2020 }}. ''Oxford English Dictionary'', on-line version (2012).</ref>
== 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 | author-link=Eric Temple Bell | first=E. T. | last=Bell | title=The Development of Mathematics | series=Dover Books on Mathematics | page=3 | publisher=Courier Corporation | isbn=978-0-486-15228-8 | year=2012 | edition=reprint, revised | url={{GBurl|id=4lTCAgAAQBAJ|p=3}} | access-date=November 11, 2022 }}</ref> Some types of [[pseudoscience]], such as [[numerology]] and astrology, were not then clearly distinguished from mathematics.<ref>{{cite book | title=Mathematics in History, Culture, Philosophy, and Science | first=Sarju | last=Tiwari | year=1992 | page=27 | publisher=Mittal Publications | isbn=978-81-7099-404-6 | url={{GBurl|id=-yyDJUD0DNwC|p=27}} | access-date=March 19, 2023 }}</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 | title=Mathematics in Society and History | first=S. | last=Restivo | author-link=Sal Restivo | date=December 2013 | pages=14–15 | publisher=Springer Netherlands | isbn=978-94-011-2944-2 | url={{GBurl|id=V0RuCQAAQBAJ|p=14}} | access-date=March 19, 2023 }}</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
| title=Leonhard Euler and the Foundations of Celestial Mechanics
| first=Dora
| last=Musielak
| author-link=Dora Musielak
| pages=1–183
| year=2022
| isbn=978-3-031-12322-1
| publisher=Springer International Publishing
| url={{GBurl|id=ZY2ZEAAAQBAJ|p=2}}
| access-date=March 19, 2023
}}</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
| title=The roots of combinatorics
| first=N. L. | last=Biggs
| journal=Historia Mathematica
| volume=6 | issue=2 | date=May 1979 | pages=109–136
| doi=10.1016/0315-0860(79)90074-0
| doi-access=free}}</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 | title=Splash Talk: The Foundational Crisis of Mathematics | year=2013 | first=Evan | last=Warner | publisher=Columbia University | pages=1–17 | url=https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf | access-date=November 4, 2022 | archive-date=April 14, 2021 | archive-url=https://web.archive.org/web/20210414191543/https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf | url-status=live }}</ref><ref name=Kleiner_1991/> The 2020 [[Mathematics Subject Classification]] contains no less than {{em|sixty-three}} first-level areas.<ref>{{cite journal | title=Mathematics Subject Classification 2020 | first1=Edward | last1=Dunne | first2=Klaus | last2=Hulek | author2-link=Klaus Hulek | journal=Notices of the American Mathematical Society | volume=67 | issue=3 | date=March 2020 | url=https://www.ams.org/journals/notices/202003/rnoti-p410.pdf?trk=2052&cat=collection | access-date=November 4, 2022 | archive-date=November 20, 2022 | archive-url=https://web.archive.org/web/20221120014912/https://www.ams.org/journals/notices/202003/rnoti-p410.pdf?trk=2052&cat=collection | url-status=live }}</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/>
=== 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 is the study of the integers (e.g. whole numbers) and related object.Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients.
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
| title=Fundamentals of Number Theory
| first=William J.
| last=LeVeque
| author-link=William J. LeVeque
| date=January 5, 2014
| pages=1–30
| isbn=978-0-486-14150-3
| publisher=Dover Publications
| url={{GBurl|id=0y3DAgAAQBAJ|p=1}}
| access-date=March 19, 2023
}}</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 | title=The Queen of Mathematics: A Historically Motivated Guide to Number Theory | first=Jay | last=Goldman | year=1997 | pages=1–3 | publisher=CRC Press | isbn=978-1-4398-6462-3 | url={{GBurl|id=A0FZDwAAQBAJ|pg=PP1}} | access-date=November 11, 2022 }}</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
| title=Number Theory, An Approach Through History From Hammurapi to Legendre
| first=André
| last=Weil
| author-link=André Weil
| year=2007
| isbn=978-0-8176-4571-7
| pages=1–3
| publisher=Birkhäuser Boston
| url={{GBurl|id=Ar7gBwAAQBAJ|p=1}}
| access-date=March 19, 2023
}}</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
| title=From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem
| first=Israel | last=Kleiner | author-link=Israel Kleiner (mathematician)
| journal=Elemente der Mathematik
| volume=55 | pages=19–37 | date=February 2000
| doi=10.1007/PL00000079
| s2cid=53319514 | doi-access=free}}</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 | title=The Goldbach Conjecture | pages=1–18 | first=Yuan | last=Wang | author-link=Wang Yuan (mathematician) | publisher=[[World Scientific]] | year=2002 | edition=revised | isbn=978-981-277-660-0 | volume=4 | series=Series in pure mathematics | url={{GBurl|id=VAY9nTreXkcC|p=1}} | access-date=November 11, 2022 }}</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>{{cite web
| title=MSC2020-Mathematics Subject Classification System
| website=zbMath
| publisher=Associate Editors of Mathematical Reviews and zbMATH
| url=https://zbmath.org/static/msc2020.pdf
| access-date=November 26, 2022
| archive-date=July 31, 2020
| archive-url=https://web.archive.org/web/20200731133802/https://zbmath.org/static/msc2020.pdf
| url-status=live
}}</ref>
=== 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 journal
| title=A Survey of the Development of Geometry up to 1870
| last=Straume | first=Eldar
| journal=ePrint | date=September 2014
| arxiv=1409.1140 | bibcode=2014arXiv1409.1140S }}</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 | title=The Foundations of Geometry | first=David | last=Hilbert | author-link=David Hilbert | year=1962 | page=1 | publisher=Open Court Publishing Company | url={{GBurl|id=0AA8AQAAMAAJ}} | access-date=March 22, 2023 }}</ref><ref>{{cite book
| title=Geometry: Euclid and Beyond
| first=Robin
| last=Hartshorne
| author-link=Robin Hartshorne
| pages=9–13
| isbn=978-0-387-22676-7
| publisher=Springer New York
| date=November 11, 2013
| url={{GBurl|id=C5fSBwAAQBAJ|p=9}}
| access-date=March 19, 2023
| archive-date=<!-- March 23, 2023 -->
}}</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 | title=History of Analytic Geometry | first=Carl B. | last=Boyer | author-link=Carl B. Boyer | pages=74–102 | isbn=978-0-486-15451-0 | publisher=Dover Publications | date=June 28, 2012 | url={{GBurl|id=2T4i5fXZbOYC|p=74}} | access-date=March 19, 2023 }}</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 | title=Reconstructing the Unity of Mathematics circa 1900 | first=David J. | last=Stump | journal=Perspectives on Science | year=1997 | volume=5 | issue=3 | page=383 | doi=10.1162/posc_a_00532 | s2cid=117709681 | url=https://repository.usfca.edu/phil/10 | access-date=November 6, 2022 | archive-date=November 6, 2022 | archive-url=https://web.archive.org/web/20221106142816/https://repository.usfca.edu/phil/10/ | url-status=live }}</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
| title=Non-Euclidean geometry
| first1=J. J.
| last1=O'Connor
| first2=E. F.
| last2=Robertson
| website=MacTuror
| date=February 1996
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/
| access-date=November 6, 2022
| archive-date=November 6, 2022
| archive-url=https://web.archive.org/web/20221106142807/https://mathshistory.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry/
| url-status=live
}}</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=D. | year=2008
| title=Adventures in group theory: Rubik's Cube, Merlin's machine, and other mathematical toys
| pages=219–232 | edition=2nd | isbn=978-0-8018-9013-0
| publisher=Johns Hopkins University Press
}}</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
| title=Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria
| first1=Jean | last1=Christianidis | first2=Jeffrey | last2=Oaks
| journal=Historia Mathematica
| volume=40 | issue=2 | date=May 2013 | pages=127–163
| doi=10.1016/j.hm.2012.09.001
| doi-access=free }}</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
| first=Lisa
| last=Lim
| date=December 21, 2018
| title=Where algebra got its x from, and Xmas its X
| url=https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas
| access-date=August 9, 2022
| website=South China Morning Post
| language=en
| archive-date=August 9, 2022
| archive-url=https://web.archive.org/web/20220809001307/https://www.scmp.com/magazines/post-magazine/short-reads/article/2178856/where-x-we-use-algebra-came-and-x-xmas
| url-status=live
}}</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 | title=François Viète's revolution in algebra | last=Oaks | first=J. A. | journal=Archive for History of Exact Sciences | volume=72 | pages=245–302 | year=2018 | issue=3 | url=https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf | access-date=November 8, 2022 | doi=10.1007/s00407-018-0208-0 | s2cid=125704699 | archive-date=November 8, 2022 | archive-url=https://web.archive.org/web/20221108162134/https://researchoutreach.org/wp-content/uploads/2019/02/Jeffrey-Oaks.pdf | url-status=live }}</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 | title=Modern Algebra and the Rise of Mathematical Structures | first=Leo | last=Corry | author-link=Leo Corry | date=December 6, 2012 | pages=247–252 | isbn=978-3-0348-7917-0 | publisher=Birkhäuser Basel | url={{GBurl|id=8G0FCAAAQBAJ|p=247}} | access-date=March 19, 2023 }}</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
| chapter=From Universal Algebra to Universal Logic
| first=Jacques
| last=Riche
| title=Perspective on Universal Logic
| publisher=Polimetrica International Scientific
| publication-place=Italy
| pages=3–39
| year=2007
| isbn=978-88-7699-077-9
| editor1-first=J. Y.
| editor1-last=Beziau
| editor2-first=Alexandre
| editor2-last=Costa-Leite
| url={{GBurl|id=ZoRN9T1GUVwC|p=3}}
| access-date=November 25, 2022
| archive-date=<!-- March 7, 2023 -->
}}</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
| title=Tool and Object: A History and Philosophy of Category Theory
| first=Ralph
| last=Krömer
| publisher=Springer Science & Business Media
| series=Science Networks. Historical Studies
| volume=32
| isbn=978-3-7643-7524-9
| year=2007
| pages=xxi–xxv, 1–91
| url={{GBurl|id=41bHxtHxjUAC|pg=PR20}}
| access-date=November 25, 2022
}}</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
| chapter=The Newton–Leibniz Calculus Controversy, 1708–1730
| first=Niccolo
| last=Guicciardini
| author-link=Niccolò Guicciardini
| title=The Oxford Handbook of Newton
| series=Oxford handbooks
| publisher=Oxford University Press
| editor1-first=Eric
| editor1-last=Schliesser
| editor2-first=Chris
| editor2-last=Smeenk
| isbn=978-0-19-993041-8
| year=2017
| chapter-url=https://core.ac.uk/download/pdf/187993169.pdf
| access-date=November 9, 2022
| doi=10.1093/oxfordhb/9780199930418.013.9
| archive-date=November 9, 2022
| archive-url=https://web.archive.org/web/20221109163253/https://core.ac.uk/download/pdf/187993169.pdf
| url-status=live
}}</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 | title=Leonhard Euler | website=MacTutor | date=September 1998 | first1=J. J. | last1=O'Connor | first2=E. F. | last2=Robertson | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland | url=https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ | access-date=November 9, 2022 | archive-date=November 9, 2022 | archive-url=https://web.archive.org/web/20221109164921/https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ | url-status=live }}</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 | title=Discrete and continuous: a fundamental dichotomy in mathematics | last1=Franklin | first1=James | author-link=James Franklin (philosopher) | journal=Journal of Humanistic Mathematics | year=2017 | volume=7 | issue=2 | pages=355–378 | url=https://scholarship.claremont.edu/jhm/vol7/iss2/18/ | doi=10.5642/jhummath.201702.18 | s2cid=6945363 | access-date=November 10, 2022 | doi-access=free | archive-date=November 10, 2022 | archive-url=https://web.archive.org/web/20221110155912/https://scholarship.claremont.edu/jhm/vol7/iss2/18/ | url-status=live }}</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
| chapter=What is Discrete Mathematics? The Many Answers
| first=Stephen B.
| last=Maurer
| title=Discrete Mathematics in the Schools
| publisher=American Mathematical Society
| editor1-first=Joseph G.
| editor1-last=Rosenstein
| editor2-first=Deborah S.
| editor2-last=Franzblau
| editor3-first=Fred S.
| editor3-last=Roberts
| isbn=978-0-8218-8578-9
| pages=121–124
| volume=36
| year=1997
| series=DIMACS: Series in Discrete Mathematics and Theoretical Computer Science
| url={{GBurl|id=EvuQdO3h-DQC|p=121}}
| access-date=November 10, 2022
}}</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
| title=Turing's Legacy: Developments from Turing's Ideas in Logic
| first=Rod
| last=Downey
| author-link=Rod Downey
| pages=260–261
| publisher=Cambridge University Press
| year=2014
| series=Issue 42 of Lecture Notes in Logic
| isbn=978-1-107-04348-0
| url={{GBurl|id=fYgaBQAAQBAJ|p=260}}
| access-date=November 10, 2022
}}</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
| title=The history and status of the P versus NP question
| first=Michael | last=Sipser | author-link=Michael Sipser
| conference=STOC '92: Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing
| date=July 1992 | pages=603–618
| doi=10.1145/129712.129771 }}</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"/>
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 |title=Probability, Statistics and Estimation|year=2013 |page=10 |url=http://www.incertitudes.fr/book.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf |archive-date=October 9, 2022 |url-status=live }}</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 | author-link=C. R. Rao | last=Rao | first=C.R. | year=1997 | title=Statistics and Truth: Putting Chance to Work | publisher=World Scientific | isbn=978-981-02-3111-8 }}</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 |editor1-last=Arthanari |editor1-first=T.S. |editor2-last=Dodge |editor2-first=Yadolah |editor2-link=Yadolah Dodge |last=Rao |first=C.R. |author-link=C.R. Rao |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 |mr=607328 }}</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 | 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
| chapter=Numerical Analysis
| title= The Princeton Companion to Mathematics
| editor1-first=Timothy
| editor1-last=Gowers
| editor2-first=June
| editor2-last=Barrow-Green
| editor3-first=Imre
| editor3-last=Leader
| publisher=Princeton University Press
| year=2008
| isbn=978-0-691-11880-2
| pages=604–615
| url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf
| access-date=November 13, 2022
| archive-date=March 7, 2023
| archive-url=https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf
| url-status=live
}}</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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| [[Fluid dynamics]] || [[Numerical analysis]] || [[Mathematical optimization|Optimization]] || [[Statistics]] || [[Cryptography]]
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== History ==
{{Main|History of mathematics}}
=== Ancient ===
The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,<ref>{{cite journal |title=Abstract representations of numbers in the animal and human brain |journal=Trends in Neurosciences |volume=21 |issue=8 |date=Aug 1998 |pages=355–361 |doi=10.1016/S0166-2236(98)01263-6 |pmid=9720604 |last1=Dehaene |first1=Stanislas | author1-link=Stanislas Dehaene |last2=Dehaene-Lambertz |first2=Ghislaine |author2-link=Ghislaine Dehaene-Lambertz | last3=Cohen |first3=Laurent|s2cid=17414557 }}</ref> was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are {{em|two}} of them. As evidenced by [[tally sticks|tallies]] found on bone, in addition to recognizing how to [[counting|count]] physical objects, [[prehistoric]] peoples may have also known how to count abstract quantities, like time{{emdash}}days, seasons, or years.<ref>See, for example, {{cite book | first=Raymond L. | last=Wilder|author-link=Raymond L. Wilder|title=Evolution of Mathematical Concepts; an Elementary Study|at=passim}}</ref><ref>{{Cite book|last=Zaslavsky|first=Claudia|author-link=Claudia Zaslavsky|title=Africa Counts: Number and Pattern in African Culture.|date=1999|publisher=Chicago Review Press|isbn=978-1-61374-115-3|oclc=843204342}}</ref>
[[File:Plimpton 322.jpg|thumb|The Babylonian mathematical tablet ''[[Plimpton 322]]'', dated to 1800 BC]]
Evidence for more complex mathematics does not appear until around 3000 {{Abbr|BC|Before Christ}}, when the [[Babylonia]]ns and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.{{sfn|Kline|1990|loc=Chapter 1}} The oldest mathematical texts from [[Mesopotamia]] and [[Ancient Egypt|Egypt]] are from 2000 to 1800 BC. Many early texts mention [[Pythagorean triple]]s 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.{{sfn|Boyer|1991|loc="Mesopotamia" pp. 24–27}}
In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some [[Ancient Greece|Ancient Greeks]] such as the [[Pythagoreans]] appeared to have considered it a subject in its own right.<ref>{{cite book | last=Heath | first=Thomas Little | author-link=Thomas Heath (classicist) |url=https://archive.org/details/historyofgreekma0002heat/page/n14 |url-access=registration |page=1 |title=A History of Greek Mathematics: From Thales to Euclid |location=New York |publisher=Dover Publications |date=1981 |orig-date=1921 |isbn=978-0-486-24073-2}}</ref> Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.<ref>{{Cite journal |last=Mueller |first=I. |date=1969 |title=Euclid's Elements and the Axiomatic Method |journal=The British Journal for the Philosophy of Science |volume=20 |issue=4 |pages=289–309 |doi=10.1093/bjps/20.4.289 |jstor=686258 |issn=0007-0882}}</ref> His book, ''[[Euclid's Elements|Elements]]'', is widely considered the most successful and influential textbook of all time.{{sfn|Boyer|1991|loc="Euclid of Alexandria" p. 119}} The greatest mathematician of antiquity is often held to be [[Archimedes]] ({{Circa|287|212 BC}}) of [[Syracuse, Italy|Syracuse]].{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 120}} 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.{{sfn|Boyer|1991|loc="Archimedes of Syracuse" p. 130}} Other notable achievements of Greek mathematics are [[conic sections]] ([[Apollonius of Perga]], 3rd century BC),{{sfn|Boyer|1991|loc="Apollonius of Perga" p. 145}} [[trigonometry]] ([[Hipparchus of Nicaea]], 2nd century BC),{{sfn|Boyer|1991|loc="Greek Trigonometry and Mensuration" p. 162}} and the beginnings of algebra (Diophantus, 3rd century AD).{{sfn|Boyer|1991|loc="Revival and Decline of Greek Mathematics" p. 180}}
[[File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BC and the 2nd 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]].<ref>{{cite book
| title=Number Theory and Its History
| first=Øystein
| last=Ore
| author-link=Øystein Ore
| publisher=Courier Corporation
| pages=19–24
| year=1988
| isbn=978-0-486-65620-5
| url={{GBurl|id=Sl_6BPp7S0AC|pg=IA19}}
| access-date=November 14, 2022
}}</ref> Other notable developments of Indian mathematics include the modern definition and approximation of [[sine]] and [[cosine]], and an early form of [[infinite series]].<ref>{{cite journal
| title=On the Use of Series in Hindu Mathematics
| first=A. N. | last=Singh | journal=Osiris
| volume=1 | date=January 1936 | pages=606–628
| doi=10.1086/368443 | jstor=301627
| s2cid=144760421 }}</ref><ref>{{cite book
| chapter=Use of series in India
| last1=Kolachana | first1=A. | last2=Mahesh | first2=K.
| last3=Ramasubramanian | first3=K.
| title=Studies in Indian Mathematics and Astronomy
| series=Sources and Studies in the History of Mathematics and Physical Sciences
| pages=438–461 | publisher=Springer | publication-place=Singapore
| isbn=978-981-13-7325-1 | year=2019
| doi=10.1007/978-981-13-7326-8_20 | s2cid=190176726 }}</ref>
=== Medieval and later ===
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|A page from al-Khwārizmī's ''Algebra'']]
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. Other achievements of the Islamic period include advances in [[spherical trigonometry]] and the addition of the [[decimal point]] to the Arabic numeral system.<ref>{{Cite book | last=Saliba | first=George | author-link=George Saliba | title=A history of Arabic astronomy: planetary theories during the golden age of Islam | date=1994 | publisher=New York University Press | isbn=978-0-8147-7962-0 | oclc=28723059 }}</ref> Many notable mathematicians from this period were Persian, such as Al-Khwarismi, [[Omar Khayyam]] and [[Sharaf al-Dīn al-Ṭūsī]].<ref>{{cite journal
| title=Contributions of Islamic scholars to the scientific enterprise
| first=Yasmeen M.
| last=Faruqi
| journal=International Education Journal
| year=2006
| volume=7
| issue=4
| pages=391–399
| publisher=Shannon Research Press
| url=https://eric.ed.gov/?id=EJ854295
| access-date=November 14, 2022
| archive-date=November 14, 2022
| archive-url=https://web.archive.org/web/20221114165547/https://eric.ed.gov/?id=EJ854295
| url-status=live
}}</ref> The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.<ref>{{cite journal | title=Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages | first=Richard | last=Lorch | journal=Science in Context | volume=14 | issue=1–2 | date=June 2001 | pages=313–331 | publisher=Cambridge University Press | doi=10.1017/S0269889701000114 | s2cid=146539132 | url=https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | access-date=December 5, 2022 | archive-date=December 17, 2022 | archive-url=https://web.archive.org/web/20221217160922/https://epub.ub.uni-muenchen.de/15929/1/greek-arabic-latin.pdf | url-status=live }}</ref>
During the [[early modern period]], mathematics began to develop at an accelerating pace in [[Western Europe]], with innovations that revolutionized mathematics, such as the introduction of variables and [[#Symbolic notation and terminology|symbolic notation]] by François Viète (1540–1603), the introduction of [[History of logarithms|logarithms]] by [[John Napier]] in 1614, which greatly simplified numerical calculations, especially for [[astronomy]] and [[marine navigation]], the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and [[Gottfried Wilhelm Leibniz|Gottfried Leibniz]] (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems.
[[File:Carl Friedrich Gauss 1840 by Jensen.jpg|thumb|left|[[Carl Friedrich Gauss]]]]
Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, [[differential geometry and topology|differential geometry]], [[matrix theory]], number theory, and [[statistics]].<ref>{{cite journal
| title=History of Mathematics After the Sixteenth Century
| first=Raymond Clare | last=Archibald | author-link=Raymond Clare Archibald
| journal=The American Mathematical Monthly
| series=Part 2: Outline of the History of Mathematics
| volume=56 | issue=1 | date=January 1949 | pages=35–56
| doi=10.2307/2304570 | jstor=2304570
}}</ref> In the early 20th century, [[Kurt Gödel]] transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system{{emdash}}if powerful enough to describe arithmetic{{emdash}}will contain true propositions that cannot be proved.<ref name=Raatikainen_2005/>
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and [[science]], to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the ''[[Bulletin of the American Mathematical Society]]'', "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 theorems and their proofs."{{sfn|Sevryuk|2006|pp=101–109}}
== Symbolic notation and terminology ==
{{main|Mathematical notation|Language of mathematics|Glossary of mathematics}}
[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) [[summation]] notation]]
Mathematical notation is widely used in science and [[engineering]] for representing complex [[concept]]s and [[property (philosophy)|properties]] in a concise, unambiguous, and accurate way. This notation consists of [[glossary of mathematical symbols|symbols]] used for representing [[operation (mathematics)|operation]]s, unspecified numbers, [[relation (mathematics)|relation]]s and any other mathematical objects, and then assembling them into [[expression (mathematics)|expression]]s and formulas.<ref>{{cite conference
| last=Wolfram
| first=Stephan
| author-link=Stephen Wolfram
| title=Mathematical notation: Past and future
| conference=MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA.
| url=https://www.stephenwolfram.com/publications/mathematical-notation-past-future/
| date=October 2000
| access-date=November 16, 2022
| archive-date=November 16, 2022
| archive-url=https://web.archive.org/web/20221116150905/https://www.stephenwolfram.com/publications/mathematical-notation-past-future/
| url-status=live
}}</ref> More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally [[Latin alphabet|Latin]] or [[Greek alphabet|Greek]] letters, and often include [[subscript]]s. Operation and relations are generally represented by specific [[Glossary of mathematical symbols|symbols]] or [[glyph]]s,<ref>{{cite journal
| title=Knowledge of Mathematical Symbols Goes Beyond Numbers
| first1=Heather | last1=Douglas | first2=Marcia Gail | last2=Headley
| first3=Stephanie | last3=Hadden | first4=Jo-Anne | last4=LeFevre|author4-link=Jo-Anne LeFevre
| journal=Journal of Numerical Cognition
| date=December 3, 2020 | volume=6 | issue=3 | pages=322–354
| doi=10.5964/jnc.v6i3.293
| s2cid=228085700 | doi-access=free }}</ref> such as {{math|+}} ([[plus sign|plus]]), {{math|×}} ([[multiplication sign|multiplication]]), <math display =inline>\int</math> ([[integral sign|integral]]), {{math|1==}} ([[equals sign|equal]]), and {{math|<}} ([[less-than sign|less than]]).<ref name=AMS>{{cite web | title=AMS style guide | year=2017 | page=75 | first1=Mary | last1=Letourneau | first2=Jennifer | last2=Wright Sharp | publisher=[[American Mathematical Society]] | url=https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf | access-date=November 16, 2022 | archive-date=December 8, 2022 | archive-url=https://web.archive.org/web/20221208063650/https://www.ams.org//publications/authors/AMS-StyleGuide-online.pdf | url-status=live }}</ref> All these symbols are generally grouped according to specific rules to form expressions and formulas.<ref>{{cite conference
| title=Constituent structure in mathematical expressions
| last1=Jansen
| first1=A. R.
| last2=Marriott
| first2=K.
| last3=Yelland
| first3=G. W.
| conference=Proceedings of the 22th Annual Meeting of the Cognitive Science Society
| volume=22
| page=238
| year=2000
| url=https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf
| access-date=November 16, 2022
| archive-date=November 16, 2022
| archive-url=https://web.archive.org/web/20221116152222/https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf
| url-status=live
}}</ref> Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of [[noun phrase]]s and formulas play the role of [[clause]]s.
Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous [[Technical definition|definitions]] that provide a standard foundation for communication. An axiom or [[postulate]] is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a [[conjecture]]. Through a series of rigorous arguments employing [[deductive reasoning]], a statement that is [[formal proof|proven]] to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a [[Lemma (mathematics)|lemma]]. A proven instance that forms part of a more general finding is termed a [[corollary]].<ref>{{cite book
| title=Theorems, Corollaries, Lemmas, and Methods of Proof
| first=Richard J.
| last=Rossi
| pages=1–14, 47–48
| year=2011
| publisher=John Wiley & Sons
| isbn=978-1-118-03057-8
| series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
| url={{GBurl|id=kSwVGbBtel8C|pg=PR7}}
| access-date=November 17, 2022
}}</ref>
Numerous technical terms used in mathematics are [[neologism]]s, such as ''[[polynomial]]'' and ''[[homeomorphism]]''.<ref>{{cite web
| title=Earliest Uses of Some Words of Mathematics
| website=MacTutor
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/
| access-date=November 19, 2022
| archive-date=September 29, 2022
| archive-url=https://web.archive.org/web/20220929032236/https://mathshistory.st-andrews.ac.uk/Miller/mathword/
| url-status=live
}}</ref> Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "[[logical disjunction|or]]" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "[[exclusive or]]"). Finally, many mathematical terms are common words that are used with a completely different meaning.<ref>{{cite journal
| title=The New Language of Mathematics
| first=Daniel S.
| last=Silver
| journal=The American Scientist
| year=2017
| volume=105
| issue=6
| pages=364–371
| doi=10.1511/2017.105.6.364
| url=https://www.americanscientist.org/article/the-new-language-of-mathematics
| access-date=November 19, 2022
| archive-date=October 22, 2022
| archive-url=https://web.archive.org/web/20221022091706/https://www.americanscientist.org/article/the-new-language-of-mathematics
| url-status=live
}}</ref> This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every [[free module]] is [[flat module|flat]]" and "a [[field (mathematics)|field]] is always a [[ring (mathematics)|ring]]".
== Relationship with sciences ==
Mathematics is used in most [[science]]s for [[Mathematical model|modeling]] phenomena, which then allows predictions to be made from experimental laws.<ref>{{cite book | title=Modelling Mathematical Methods and Scientific Computation | first1=Nicola | last1=Bellomo | first2=Luigi | last2=Preziosi | publisher=CRC Press | date=December 22, 1994 | page=1 | isbn=978-0-8493-8331-1 | series=Mathematical Modeling | volume=1 | url={{GBurl|id=pJAvWaRYo3UC}} | access-date=November 16, 2022 }}</ref> The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.<ref>{{cite journal
| title=Mathematical Models and Reality: A Constructivist Perspective
| first=Christian | last=Hennig
| journal=Foundations of Science
| volume=15 | pages=29–48 | year=2010
| doi=10.1007/s10699-009-9167-x
| s2cid=6229200 | url=https://www.researchgate.net/publication/225691477
| access-date=November 17, 2022
}}</ref> Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.<ref>{{cite journal | title=Models in Science | date=February 4, 2020 | first1=Roman | last1=Frigg | author-link=Roman Frigg | first2=Stephan | last2=Hartmann | author2-link=Stephan Hartmann | website=Stanford Encyclopedia of Philosophy | url=https://seop.illc.uva.nl/entries/models-science/ | access-date=November 17, 2022 | archive-date=November 17, 2022 | archive-url=https://web.archive.org/web/20221117162412/https://seop.illc.uva.nl/entries/models-science/ | url-status=live }}</ref> For example, the [[perihelion precession of Mercury]] could only be explained after the emergence of [[Einstein]]'s [[general relativity]], which replaced [[Newton's law of gravitation]] as a better mathematical model.<ref>{{cite book | last=Stewart | first=Ian | author-link=Ian Stewart (mathematician) | chapter=Mathematics, Maps, and Models | title=The Map and the Territory: Exploring the Foundations of Science, Thought and Reality | pages=345–356 | publisher=Springer | year=2018 | editor1-first=Shyam | editor1-last=Wuppuluri | editor2-first=Francisco Antonio | editor2-last=Doria | isbn=978-3-319-72478-2 | series=The Frontiers Collection | chapter-url={{GBurl|id=mRBMDwAAQBAJ|p=345}} | doi=10.1007/978-3-319-72478-2_18 | access-date=November 17, 2022 }}</ref>
There is still a [[philosophy of mathematics|philosophical]] debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is [[falsifiable]], which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a [[counterexample]]. Similarly as in science, [[mathematical theory|theories]] and results (theorems) are often obtained from [[experimentation]].<ref>{{Cite web|url=https://undsci.berkeley.edu/article/mathematics|title=The science checklist applied: Mathematics|website=undsci.berkeley.edu|access-date=October 27, 2019|archive-url=https://web.archive.org/web/20191027021023/https://undsci.berkeley.edu/article/mathematics|archive-date=October 27, 2019|url-status=live}}</ref> In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).<ref>{{cite book | last=Mackay | first=A. L. | year=1991 | title=Dictionary of Scientific Quotations | location=London | page=100 | isbn=978-0-7503-0106-0 | publisher=Taylor & Francis | url={{GBurl|id=KwESE88CGa8C|q=durch planmässiges Tattonieren}} | access-date=March 19, 2023 }}</ref> However, some authors emphasize that mathematics differs from the modern notion of science by not {{em|relying}} on empirical evidence.<ref name="Bishop1991">{{cite book | last1 = Bishop | first1 = Alan | year = 1991 | chapter = Environmental activities and mathematical culture | title = Mathematical Enculturation: A Cultural Perspective on Mathematics Education | chapter-url = {{GBurl|id=9AgrBgAAQBAJ|p=54}} | pages = 20–59 | location = Norwell, Massachusetts | publisher = Kluwer Academic Publishers | isbn = 978-0-7923-1270-3 | access-date = April 5, 2020 }}</ref><ref>{{cite book | title=Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists | last1=Shasha | first1=Dennis Elliot | author1-link=Dennis Elliot Shasha | last2=Lazere | first2=Cathy A. | publisher=Springer | year=1998 | page=228 | isbn=978-0-387-98269-4 }}</ref><ref name="Nickles2013">{{cite book | last=Nickles | first=Thomas | year=2013 | chapter=The Problem of Demarcation | title=Philosophy of Pseudoscience: Reconsidering the Demarcation Problem | page=104 | location=Chicago | publisher=The University of Chicago Press | isbn=978-0-226-05182-6 }}</ref><ref name="Pigliucci2014">{{Cite magazine | year=2014| last=Pigliucci| first=Massimo | author-link=Massimo Pigliucci | title=Are There 'Other' Ways of Knowing? | magazine=[[Philosophy Now]]| url=https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | access-date=April 6, 2020| archive-date=May 13, 2020 | archive-url=https://web.archive.org/web/20200513190522/https://philosophynow.org/issues/102/Are_There_Other_Ways_of_Knowing | url-status=live}}</ref>
<!-- What precedes is only one aspect of the relationship between mathematics and other sciences. Other aspects are considered in the next subsections. -->
=== Pure and applied mathematics ===
{{main|Applied mathematics|Pure mathematics}}
{{multiple image
| footer = Isaac Newton (left) and [[Gottfried Wilhelm Leibniz]] developed infinitesimal calculus.
| total_width = 330
| width1 = 407
| height1 = 559
| image1 = GodfreyKneller-IsaacNewton-1689.jpg
| alt1 = Isaac Newton
| width2 = 320
| height2 = 390
| image2 = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
| alt2 = Gottfried Wilhelm von Leibniz
}}
Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of [[technology]] and science, and there was no clear distinction between pure and applied mathematics.<ref name=Ferreirós_2007>{{cite book
| title=The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae
| last=Ferreirós | first=J.
| chapter=Ό Θεὸς Άριθμητίζει: The Rise of Pure Mathematics as Arithmetic with Gauss
| pages=235–268 | year=2007 | isbn=978-3-540-34720-0
| editor1-first=Catherine | editor1-last=Goldstein | editor1-link=Catherine Goldstein
| editor2-first=Norbert | editor2-last=Schappacher
| editor3-first=Joachim | editor3-last=Schwermer
| publisher=Springer Science & Business Media
| chapter-url={{GBurl|id=IUFTcOsMTysC|p=235}}
}}</ref> For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the [[planet]]s with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.<ref>{{cite journal
| title=Mathematical vs. Experimental Traditions in the Development of Physical Science
| first=Thomas S. | last=Kuhn | author-link=Thomas Kuhn
| journal=The Journal of Interdisciplinary History
| year=1976 | volume=7 | issue=1 | pages=1–31 | publisher=The MIT Press
| jstor=202372 | doi=10.2307/202372
}}</ref> However, a notable exception occurred with the tradition of [[pure mathematics in Ancient Greece]].<ref>{{cite book
| chapter=The two cultures of mathematics in ancient Greece
| first=Markus
| last=Asper
| year=2009
| title=The Oxford Handbook of the History of Mathematics
| editor1-first=Eleanor
| editor1-last=Robson
| editor2-first=Jacqueline
| editor2-last=Stedall
| pages=107–132
| isbn=978-0-19-921312-2
| publisher=OUP Oxford
| series=Oxford Handbooks in Mathematics
| chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=107}}
| access-date=November 18, 2022
}}</ref>
In the 19th century, mathematicians such as [[Karl Weierstrass]] and [[Richard Dedekind]] increasingly focused their research on internal problems, that is, ''pure mathematics''.<ref name=Ferreirós_2007/><ref>{{cite journal
| title=How applied mathematics became pure
| last=Maddy | first=P. | author-link=Penelope Maddy
| journal=The Review of Symbolic Logic
| year=2008
| volume=1
| issue=1
| pages=16–41
| doi=10.1017/S1755020308080027
| s2cid=18122406
| url=http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
| access-date=November 19, 2022
| archive-date=August 12, 2017
| archive-url=https://web.archive.org/web/20170812012210/http://pgrim.org/philosophersannual/pa28articles/maddyhowapplied.pdf
| url-status=live
}}</ref> This led to split mathematics into ''pure mathematics'' and ''applied mathematics'', the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred.<ref>{{cite book
| title=The Best Writing on Mathematics, 2016
| chapter=In Defense of Pure Mathematics
| first=Daniel S.
| last=Silver
| pages=17–26
| isbn=978-0-691-17529-4
| year=2017
| editor1-first=Mircea
| editor1-last=Pitici
| publisher=Princeton University Press
| chapter-url={{GBurl|id=RXGYDwAAQBAJ|p=17}}
| access-date=November 19, 2022
}}</ref>
The aftermath of [[World War II]] led to a surge in the development of applied mathematics in the US and elsewhere.<ref>{{cite journal | title=The American Mathematical Society and Applied Mathematics from the 1920s to the 1950s: A Revisionist Account | first=Karen Hunger | last=Parshall | author-link=Karen Hunger Parshall | journal=Bulletin of the American Mathematical Society | volume=59 | year=2022 | issue=3 | pages=405–427 | doi=10.1090/bull/1754 | s2cid=249561106 | url=https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | access-date=November 20, 2022 | doi-access=free | archive-date=November 20, 2022 | archive-url=https://web.archive.org/web/20221120151259/https://www.ams.org/journals/bull/2022-59-03/S0273-0979-2022-01754-5/home.html | url-status=live }}</ref><ref>{{cite journal
| title=The History Of Applied Mathematics And The History Of Society
| first=Michael | last=Stolz
| journal=Synthese
| volume=133 | pages=43–57 | year=2002
| doi=10.1023/A:1020823608217
| s2cid=34271623 | url=https://www.researchgate.net/publication/226795930
| access-date=November 20, 2022
}}</ref> Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".<ref>{{cite journal
| title=On the role of applied mathematics
| journal=[[Advances in Mathematics]] | first=C. C . | last=Lin
| volume=19 | issue=3 | date=March 1976 | pages=267–288
| doi=10.1016/0001-8708(76)90024-4 | doi-access=free
}}</ref><ref>{{cite conference
| title=Applying Pure Mathematics
| first=Anthony
| last=Peressini
| conference=Philosophy of Science. Proceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributed Papers
| volume=66
| date=September 1999
| pages=S1–S13
| jstor=188757
| access-date=November 30, 2022
| url=https://www.academia.edu/download/32799272/ApplyingMathPSA.pdf
| archive-url=https://web.archive.org/web/20240102210931/https://d1wqtxts1xzle7.cloudfront.net/32799272/ApplyingMathPSA-libre.pdf?1391205742=&response-content-disposition=inline%3B+filename%3DApplying_Pure_Mathematics.pdf&Expires=1704233371&Signature=BvNJyYufdj9BiKFe94w6gdXLpAfr7T5JIv~RU74R2uT0O9Ngj6i4cdBtYYOSB6D4V-MgButb6lKNhIGGQogw0e0sHVFkJUy5TRsoCiQ-MLabpZOf74E5SGLMFIExhGVAw7SKrSFaQsFGhfbaRMxbMP~u-wRdJAz6ve6kbWr6oq-doQeEOlRfO4EByNCUYx-KAk3~cBsH1Q2WNZ5QiVObMI1ufQ7zkQM1bqzOumLu6g07F~pt~Cds~lftuQufHomoTH-V9H9iKQgUyc3-4bEB1y1Jdngs7WWg76LcSGn65bPK8dxvsZzKaLDGfoK5jamZkA8z3-xxiMIPL8c6YETjZA__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=January 2, 2024
| url-status=live
}}</ref>
An example of the first case is the [[theory of distributions]], introduced by [[Laurent Schwartz]] for validating computations done in [[quantum mechanics]], which became immediately an important tool of (pure) mathematical analysis.<ref>{{cite conference
| title=Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century
| last=Lützen
| first=J.
| year=2011
| editor1-last=Schlote
| editor1-first=K. H.
| editor2-last=Schneider
| editor2-first=M.
| publisher=Verlag Harri Deutsch
| publication-place=Frankfurt am Main
| chapter=Examples and reflections on the interplay between mathematics and physics in the 19th and 20th century
| chapter-url=https://slub.qucosa.de/api/qucosa%3A16267/zip/
| access-date=November 19, 2022
| archive-date=March 23, 2023
| archive-url=https://web.archive.org/web/20230323164143/https://slub.qucosa.de/api/qucosa%3A16267/zip/
| url-status=live
}}</ref> An example of the second case is the [[decidability of the first-order theory of the real numbers]], a problem of pure mathematics that was proved true by [[Alfred Tarski]], with an algorithm that is impossible to [[implementation (computer science)|implement]] because of a computational complexity that is much too high.<ref>{{cite journal
| title=Model theory and exponentiation
| last=Marker
| first=Dave
| journal=Notices of the American Mathematical Society
| volume=43
| issue=7
| date=July 1996
| pages=753–759
| url=https://www.ams.org/notices/199607/
| access-date=November 19, 2022
| archive-date=March 13, 2014
| archive-url=https://web.archive.org/web/20140313004011/http://www.ams.org/notices/199607/
| url-status=live
}}</ref> For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, [[George E. Collins|George Collins]] introduced the [[cylindrical algebraic decomposition]] that became a fundamental tool in [[real algebraic geometry]].<ref>{{cite conference
| title=Cylindrical Algebraic Decomposition in the RegularChains Library
| first1=Changbo | last1=Chen | first2=Marc Moreno | last2=Maza
| date=August 2014 | volume=8592
| publisher=Springer | publication-place=Berlin
| conference=International Congress on Mathematical Software 2014
| series=Lecture Notes in Computer Science
| url=https://www.researchgate.net/publication/268067322
| access-date=November 19, 2022 | doi=10.1007/978-3-662-44199-2_65 }}</ref>
In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas.<ref>{{cite journal
| title=Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy
| first1=José Antonio | last1=Pérez-Escobar | first2=Deniz | last2=Sarikaya
| journal=European Journal for Philosophy of Science
| volume=12 | issue=1 | pages=1–22 | year=2021
| doi=10.1007/s13194-021-00435-9 | s2cid=245465895
| doi-access=free }}</ref><ref>{{cite book
| chapter=Pure Mathematics and Applied Mathematics are Inseparably Intertwined: Observation of the Early Analysis of the Infinity
| last=Takase
| first=M.
| title=A Mathematical Approach to Research Problems of Science and Technology
| series=Mathematics for Industry
| volume=5
| year=2014
| pages=393–399
| publisher=Springer
| publication-place=Tokyo
| chapter-url={{GBurl|id=UeElBAAAQBAJ|p=393}}
| doi=10.1007/978-4-431-55060-0_29
| isbn=978-4-431-55059-4
| access-date=November 20, 2022
}}</ref> The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".<ref name=MSC/> However, these terms are still used in names of some [[university]] departments, such as at the [[Faculty of Mathematics, University of Cambridge|Faculty of Mathematics]] at the [[University of Cambridge]].
=== Unreasonable effectiveness ===
The [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|unreasonable effectiveness of mathematics]] is a phenomenon that was named and first made explicit by physicist [[Eugene Wigner]].<ref name=wigner1960>{{cite journal
| title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences
| last=Wigner | first=Eugene | author-link=Eugene Wigner
| journal=[[Communications on Pure and Applied Mathematics]]
| volume=13 | issue=1 | pages=1–14 | year=1960
| doi=10.1002/cpa.3160130102 | bibcode=1960CPAM...13....1W
| s2cid=6112252 | url=https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
| url-status=live | archive-url=https://web.archive.org/web/20110228152633/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
| archive-date=February 28, 2011 | df=mdy-all
}}</ref> It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.<ref>{{cite journal
| title=Revisiting the 'unreasonable effectiveness' of mathematics
| first=Sundar | last=Sarukkai
| journal=Current Science
| volume=88 | issue=3 | date=February 10, 2005 | pages=415–423
| jstor=24110208
}}</ref> Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the [[prime factorization]] of natural numbers that was discovered more than 2,000 years before its common use for secure [[internet]] communications through the [[RSA cryptosystem]].<ref>{{cite book
| chapter=History of Integer Factoring
| pages=41–77
| first=Samuel S.
| last=Wagstaff, Jr.
| title=Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL
| editor1-first=Joppe W.
| editor1-last=Bos
| editor2-first=Martijn
| editor2-last=Stam
| series=London Mathematical Society Lecture Notes Series 469
| publisher=Cambridge University Press
| year=2021
| chapter-url=https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
| access-date=November 20, 2022
| archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120155733/https://www.cs.purdue.edu/homes/ssw/chapter3.pdf
| url-status=live
}}</ref> A second historical example is the theory of [[ellipse]]s. They were studied by the [[Greek mathematics|ancient Greek mathematicians]] as [[conic section]]s (that is, intersections of [[cone]]s with planes). It is almost 2,000 years later that [[Johannes Kepler]] discovered that the [[trajectories]] of the planets are ellipses.<ref>{{cite web
| title=Curves: Ellipse
| website=MacTutor
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
| access-date=November 20, 2022
| archive-date=October 14, 2022
| archive-url=https://web.archive.org/web/20221014051943/https://mathshistory.st-andrews.ac.uk/Curves/Ellipse/
| url-status=live
}}</ref>
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and [[manifold]]s. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, [[Albert Einstein]] developed the [[theory of relativity]] that uses fundamentally these concepts. In particular, [[spacetime]] of [[special relativity]] is a non-Euclidean space of dimension four, and spacetime of [[general relativity]] is a (curved) manifold of dimension four.<ref>{{cite web
| title=Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry
| first=Vasudevan
| last=Mukunth
| website=The Wire
| date=September 10, 2015
| url=https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
| access-date=November 20, 2022
| archive-date=November 20, 2022
| archive-url=https://web.archive.org/web/20221120191206/https://thewire.in/science/beyond-the-surface-of-einsteins-relativity-lay-a-chimerical-geometry
| url-status=live
}}</ref><ref>{{cite journal
| title=The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics
| first1=Edwin B. | last1=Wilson | first2=Gilbert N. | last2=Lewis
| journal=Proceedings of the American Academy of Arts and Sciences
| volume=48 | issue=11 | date=November 1912 | pages=389–507
| doi=10.2307/20022840 | jstor=20022840 }}</ref>
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the [[positron]] and the [[omega baryon|baryon]] <math>\Omega^{-}.</math> In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown [[particle]], and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.<ref name=borel /><ref>{{cite journal
| title=Discovering the Positron (I)
| first=Norwood Russell | last=Hanson | author-link=Norwood Russell Hanson
| journal=The British Journal for the Philosophy of Science
| volume=12 | issue=47 | date=November 1961 | pages=194–214
| publisher=The University of Chicago Press
| jstor=685207 | doi=10.1093/bjps/xiii.49.54
}}</ref><ref>{{cite journal
| title=Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω<sup>–</sup> particle
| first=Michele | last=Ginammi
| journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
| volume=53 | date=February 2016 | pages=20–27
| doi=10.1016/j.shpsb.2015.12.001
| bibcode=2016SHPMP..53...20G }}</ref>
=== Specific sciences ===
{{Essay-like|date=December 2022|subsection}}
==== Physics ====
{{Main|Relationship between mathematics and physics}}
[[File:Pendule schema.gif|thumb|Diagram of a pendulum]]
Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,<ref>{{Cite book |last1=Wagh |first1=Sanjay Moreshwar |url={{GBurl|id=-DmfVjBUPksC|p=3}} |title=Essentials of Physics |last2=Deshpande |first2=Dilip Abasaheb |date=September 27, 2012 |publisher=PHI Learning Pvt. Ltd. |isbn=978-81-203-4642-0 |page=3 |language=en |access-date=January 3, 2023 }}</ref> and is also the motivation of major mathematical developments.<ref>{{Cite conference |last=Atiyah |first=Michael |author-link=Michael Atiyah |date=1990 |title=On the Work of Edward Witten |url=http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |conference=Proceedings of the International Congress of Mathematicians |page=31 |archive-url=https://web.archive.org/web/20130928095313/http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf |archive-date=September 28, 2013 |access-date=December 29, 2022}}</ref>
==== Computing ====
{{Further|Theoretical computer science|Computational mathematics}}
The rise of technology in the 20th century opened the way to a new science: [[computing]].{{Efn|[[Ada Lovelace]], in the 1840s, is known for having written the first computer program ever in collaboration with [[Charles Babbage]]}} This field is closely related to mathematics in several ways. [[Theoretical computer science]] is essentially mathematical in nature. Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in [[cryptography]] and [[coding theory]]. [[Discrete mathematics]] is useful in many areas of computer science, such as [[Computational complexity theory|complexity theory]], [[information theory]], [[graph theory]], and so on.{{Citation needed|date=December 2022}}
In return, computing has also become essential for obtaining new results. This is a group of techniques known as [[experimental mathematics]], which is the use of ''experimentation'' to discover mathematical insights.<ref>{{Cite web |last1=Borwein |first1=J. |last2=Borwein |first2=P. |last3=Girgensohn |first3=R. |last4=Parnes |first4=S. |date=1996 |title=Conclusion |url=http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |url-status=dead |archive-url=https://web.archive.org/web/20080121081424/http://oldweb.cecm.sfu.ca/organics/vault/expmath/expmath/html/node16.html |archive-date=January 21, 2008 |website=oldweb.cecm.sfu.ca}}</ref> The most well-known example is the [[Four color theorem|four-color theorem]], which was proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where the rule was that the mathematician should verify each part of the proof. In 1998, the [[Kepler conjecture]] on [[sphere packing]] seemed to also be partially proven by computer. An international team had since worked on writing a formal proof; it was finished (and verified) in 2015.<ref>{{cite journal |last1=Hales |first1=Thomas |last2=Adams |first2=Mark |last3=Bauer |first3=Gertrud |last4=Dang |first4=Tat Dat |last5=Harrison |first5=John |last6=Hoang |first6=Le Truong |last7=Kaliszyk |first7=Cezary |last8=Magron |first8=Victor |last9=Mclaughlin |first9=Sean |last10=Nguyen |first10=Tat Thang |last11=Nguyen |first11=Quang Truong |last12=Nipkow |first12=Tobias |last13=Obua |first13=Steven |last14=Pleso |first14=Joseph |last15=Rute |first15=Jason |last16=Solovyev |first16=Alexey |last17=Ta |first17=Thi Hoai An |last18=Tran |first18=Nam Trung |last19=Trieu |first19=Thi Diep |last20=Urban |first20=Josef |last21=Vu |first21=Ky |last22=Zumkeller |first22=Roland |title=A Formal Proof of the Kepler Conjecture |journal=Forum of Mathematics, Pi |date=2017 |volume=5 |page=e2 |doi=10.1017/fmp.2017.1 |s2cid=216912822 |url=https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |language=en |issn=2050-5086 |access-date=February 25, 2023 |archive-date=December 4, 2020 |archive-url=https://web.archive.org/web/20201204053232/https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC |url-status=live |hdl=2066/176365 |hdl-access=free }}</ref>
Once written formally, a proof can be verified using a program called a [[proof assistant]].<ref name=":1">{{Cite journal |last=Geuvers |first=H. |date=February 2009 |title=Proof assistants: History, ideas and future |url=https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |journal=Sādhanā |volume=34 |pages=3–4 |doi=10.1007/s12046-009-0001-5 |s2cid=14827467 |doi-access=free |access-date=December 29, 2022 |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204107/https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |url-status=live |hdl=2066/75958 |hdl-access=free }}</ref> These programs are useful in situations where one is uncertain about a proof's correctness.<ref name=":1" />
A major open problem in theoretical computer science is [[P versus NP problem|P versus NP]]. It is one of the seven [[Millennium Prize Problems]].<ref>{{Cite web |title=P versus NP problem {{!}} mathematics |url=https://www.britannica.com/science/P-versus-NP-problem |access-date=December 29, 2022 |website=Britannica |language=en |archive-date=December 6, 2022 |archive-url=https://web.archive.org/web/20221206044556/https://www.britannica.com/science/P-versus-NP-problem |url-status=live }}</ref>
==== Biology and chemistry ====
{{Main|Mathematical and theoretical biology|Mathematical chemistry}}
[[File:Giant Pufferfish skin pattern detail.jpg|thumb|The skin of this [[giant pufferfish]] exhibits a [[Turing pattern]], which can be modeled by [[reaction–diffusion system]]s.]]
[[Biology]] uses probability extensively – for example, in ecology or [[neurobiology]].<ref name=":2">{{Cite book |last=Millstein |first=Roberta |author-link=Roberta Millstein |title=The Oxford Handbook of Probability and Philosophy |date=September 8, 2016 |editor-last=Hájek |editor-first=Alan |pages=601–622 |chapter=Probability in Biology: The Case of Fitness |doi=10.1093/oxfordhb/9780199607617.013.27 |editor-last2=Hitchcock |editor-first2=Christopher |chapter-url=http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054456/http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf |url-status=live }}</ref> Most of the discussion of probability in biology, however, centers on the concept of [[evolutionary fitness]].<ref name=":2" />
Ecology heavily uses modeling to simulate [[population dynamics]],<ref name=":2" /><ref>See for example Anne Laurent, Roland Gamet, Jérôme Pantel, ''Tendances nouvelles en modélisation pour l'environnement, actes du congrès «Programme environnement, vie et sociétés»'' 15-17 janvier 1996, CNRS</ref> study ecosystems such as the predator-prey model, measure pollution diffusion,{{Sfn|Bouleau|1999|pp=282–283}} or to assess climate change.{{Sfn|Bouleau|1999|p=285}} The dynamics of a population can be modeled by coupled differential equations, such as the [[Lotka–Volterra equations]].<ref>{{Cite web |date=January 5, 2022 |title=1.4: The Lotka-Volterra Predator-Prey Model |url=https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01%3A_Population_Dynamics/1.04%3A_The_Lotka-Volterra_Predator-Prey_Model |access-date=December 29, 2022 |website=Mathematics LibreTexts |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204111/https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01:_Population_Dynamics/1.04:_The_Lotka-Volterra_Predator-Prey_Model |url-status=live }}</ref> However, there is the problem of [[model validation]]. This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.{{Sfn|Bouleau|1999|p=287}}
Genotype evolution can be modeled with the [[Hardy-Weinberg principle]].{{Citation needed|date=December 2022}}
[[Phylogeography]] uses probabilistic models.{{Citation needed|date=December 2022}}
Medicine uses [[statistical hypothesis testing]], run on data from [[clinical trial]]s, to determine whether a new treatment works.{{Citation needed|date=December 2022}}
Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions. It turns out that the form of [[macromolecules]] in biology is variable and determines the action. Such modeling uses Euclidean geometry; neighboring atoms form a [[polyhedron]] whose distances and angles are fixed by the laws of interaction.{{Citation needed|date=December 2022}}
==== Earth sciences ====
{{Main|Geomathematics}}
[[Structural geology]] and climatology use probabilistic models to predict the risk of natural catastrophes.{{Citation needed|date=December 2022}} Similarly, [[meteorology]], [[oceanography]], and [[planetology]] also use mathematics due to their heavy use of models.{{Citation needed|date=December 2022}}
==== Social sciences ====
{{Further|Mathematical economics|Historical dynamics}}
Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, [[economics]], [[sociology]],<ref>{{Cite journal |last=Edling |first=Christofer R. |date=2002 |title=Mathematics in Sociology |url=https://www.annualreviews.org/doi/10.1146/annurev.soc.28.110601.140942 |journal=Annual Review of Sociology |language=en |volume=28 |issue=1 |pages=197–220 |doi=10.1146/annurev.soc.28.110601.140942 |issn=0360-0572}}</ref> and [[psychology]].<ref>{{Citation |last=Batchelder |first=William H. |title=Mathematical Psychology: History |date=2015-01-01 |url=https://www.sciencedirect.com/science/article/pii/B978008097086843059X |encyclopedia=International Encyclopedia of the Social & Behavioral Sciences (Second Edition) |pages=808–815 |editor-last=Wright |editor-first=James D. |access-date=2023-09-30 |place=Oxford |publisher=Elsevier |isbn=978-0-08-097087-5}}</ref>
[[File:Supply-demand-equilibrium.svg|thumb|[[Supply and demand|Supply and demand curves]], like this one, are a staple of mathematical economics.]]
The fundamental postulate of mathematical economics is that of the rational individual actor – ''[[Homo economicus]]'' ({{Literal translation|economic man}}).<ref name=":3">{{Cite book |last=Zak |first=Paul J. |url={{GBurl|id=6QrvmNo2qD4C|p=158}} |title=Moral Markets: The Critical Role of Values in the Economy |date=2010 |page=158 |publisher=Princeton University Press |isbn=978-1-4008-3736-6 |language=en |access-date=January 3, 2023 }}</ref> In this model, the individual seeks to maximize their [[rational choice theory|self-interest]],<ref name=":3" /> and always makes optimal choices using [[perfect information]].<ref name=":4">{{Cite web |last=Kim |first=Oliver W. |date=May 29, 2014 |title=Meet Homo Economicus |url=https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |access-date=December 29, 2022 |website=The Harvard Crimson |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204106/https://www.thecrimson.com/column/homo-economicus/article/2014/9/19/Harvard-homo-economicus-fiction/ |url-status=live }}</ref>{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual [[calculations]] are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be difficult to discover by a "literary" analysis.{{Citation needed|date=December 2022}} For example, explanations of [[economic cycles]] are not trivial. Without mathematical modeling, it is hard to go beyond statistical observations or unproven speculation.{{Citation needed|date=December 2022}}
However, many people have rejected or criticized the concept of ''Homo economicus''.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}} Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.<ref name=":4" />{{Better source needed|reason=this is an opinion essay, not a scholarly work|date=December 2022}}
At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, [[Nikolai Kondratiev]] discerned the ~50-year-long [[Kondratiev cycle]], which explains phases of economic growth or crisis.<ref>{{Cite web |title=Kondratiev, Nikolai Dmitrievich {{!}} Encyclopedia.com |url=https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/kondratiev-nikolai-dmitrievich |access-date=December 29, 2022 |website=www.encyclopedia.com |archive-date=July 1, 2016 |archive-url=https://web.archive.org/web/20160701224009/http://www.encyclopedia.com/doc/1G2-3404100667.html |url-status=live }}</ref> Towards the end of the 19th century, {{Ill|Nicolas-Remi Brück|fr}} and {{Ill|Charles Henri Lagrange|fr}} extended their analysis into [[geopolitics]].<ref>{{Cite web|url=https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=##+Math%C3%A9matique+de+l'histoire,org%E3%80%91|title=Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. | The Online Books Page|website=onlinebooks.library.upenn.edu}}</ref> [[Peter Turchin]] has worked on developing [[cliodynamics]] since the 1990s.<ref>{{Cite web |title=Cliodynamics: a science for predicting the future |url=https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |access-date=December 29, 2022 |website=ZDNET |language=en |archive-date=December 29, 2022 |archive-url=https://web.archive.org/web/20221229204104/https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/ |url-status=live }}</ref>
Even so, mathematization of the social sciences is not without danger. In the controversial book ''[[Fashionable Nonsense]]'' (1997), [[Alan Sokal|Sokal]] and [[Jean Bricmont|Bricmont]] denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.<ref>{{cite book|last=Sokal|first=Alan|url=https://archive.org/details/fashionablenonse00soka|title=Fashionable Nonsense|author2=Jean Bricmont|publisher=Picador|year=1998|isbn=978-0-312-19545-8|location=New York|oclc=39605994|author-link=Alan Sokal|author2-link=Jean Bricmont}}</ref> The study of [[complex systems]] (evolution of unemployment, business capital, demographic evolution of a population, etc.) uses mathematical knowledge. However, the choice of counting criteria, particularly for unemployment, or of models, can be subject to controversy.{{Citation needed|date=December 2022}}
== Relationship with astrology and esotericism ==
Some renowned mathematicians have also been considered to be renowned astrologists; for example, [[Ptolemy]], Arab astronomers, [[Regiomantus]], [[Gerolamo Cardano|Cardano]], [[Kepler]], or [[John Dee]]. In the Middle Ages, astrology was considered a science that included mathematics. In his encyclopedia, [[Theodor Zwinger]] wrote that astrology was a mathematical science that studied the "active movement of bodies as they act on other bodies". He reserved to mathematics the need to "calculate with probability the influences [of stars]" to foresee their "conjunctions and oppositions".<ref>{{Cite book |last=Beaujouan |first=Guy |url={{GBurl|id=92n7ZE8Iww8C|p=130}} |title=Comprendre et maîtriser la nature au Moyen Age: mélanges d'histoire des sciences offerts à Guy Beaujouan |date=1994 |publisher=Librairie Droz |isbn=978-2-600-00040-6 |page=130 |language=fr |access-date=January 3, 2023 }}</ref>
Astrology is no longer considered a science.<ref>{{Cite web |title=L'astrologie à l'épreuve : ça ne marche pas, ça n'a jamais marché ! / Afis Science – Association française pour l'information scientifique |url=https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |access-date=December 28, 2022 |website=Afis Science – Association française pour l’information scientifique |language=fr |archive-date=January 29, 2023 |archive-url=https://web.archive.org/web/20230129204349/https://www.afis.org/L-astrologie-a-l-epreuve-ca-ne-marche-pas-ca-n-a-jamais-marche |url-status=live }}</ref>
==Philosophy==
{{Main|Philosophy of mathematics}}
===Reality===
The connection between mathematics and material reality has led to philosophical debates since at least the time of [[Pythagoras]]. The ancient philosopher [[Plato]] argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as [[Mathematical Platonism|Platonism]]. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.<ref name=SEP-Platonism>{{cite encyclopedia |title=Platonism in Metaphysics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Balaguer |first=Mark |editor-last=Zalta |editor-first=Edward N. |year=2016 |edition=Spring 2016 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/spr2016/entries/platonism |access-date=April 2, 2022 |archive-date=January 30, 2022 |archive-url=https://web.archive.org/web/20220130174043/https://plato.stanford.edu/archives/spr2016/entries/platonism/ |url-status=live }}</ref>
[[Armand Borel]] summarized this view of mathematics reality as follows, and provided quotations of [[G. H. Hardy]], [[Charles Hermite]], [[Henri Poincaré]] and Albert Einstein that support his views.<ref name=borel />
{{blockquote| Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.<ref>See {{cite journal
| first=L. | last=White | year=1947
| title=The locus of mathematical reality: An anthropological footnote
| journal=[[Philosophy of Science (journal)|Philosophy of Science]]
| volume=14|issue=4 | pages=289–303
| doi=10.1086/286957 | s2cid=119887253
| id=189303 | postscript=;
}} also in {{cite book
| first=J. R. | last=Newman | year=1956
| title=The World of Mathematics
| publisher=Simon and Schuster | location=New York
| volume=4 | pages=2348–2364
}}</ref> Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a {{em|feeling}} of an objective existence, of a reality of mathematics ...}}
Nevertheless, Platonism and the concurrent views on abstraction do not explain the [[#Unreasonable effectiveness|unreasonable effectiveness]] of mathematics.<ref>{{cite book
| title=The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature
| first=Mauro
| last=Dorato
| year=2005
| chapter=Why are laws mathematical?
| pages=31–66
| isbn=978-0-7546-3994-7
| publisher=Ashgate
| chapter-url=https://www.academia.edu/download/52076815/2ch.pdf
| access-date=December 5, 2022
| archive-url=https://web.archive.org/web/20230817111932/https://d1wqtxts1xzle7.cloudfront.net/52076815/2ch-libre.pdf?1488997736=&response-content-disposition=inline%3B+filename%3DChapter_2_of_the_book_the_software_of_th.pdf&Expires=1692274771&Signature=PXpNLBsmWMkz9YUs6~LUOfXNkmkCAmDfxQUoWOkGJKP4YqPGQUFMuP1I0xFycLZkL0dyfGwdGQ7mPk44nvmpM3YpKBSeVCZRXtDMiwgqs1JhEWrJovAhrchPLM1mGn3pw5P6LPo0sDZsl7uaPoZHMyCyJpayHvFtpyj1oUMIdmGuYM5P3euy1R87g6xlKyNAp~~BR5I4gVpopzLoeZn7d3oEnOOua0GjsqsZ6H9mEgcZMpH-qF8w9iFa9aSXFpqxagQwcVVkg7DXkOjVV5jyzctBUKQtOQQ~-9EN1y-c9pFV-Xu-NNuoN3Ij6K4SwvjYv0a8DMs8T5SVj1Kz9i4CEQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=August 17, 2023
| url-status=live
}}</ref>
=== Proposed definitions ===
{{Main|Definitions of mathematics}}
There is no general consensus about a definition of mathematics or its [[epistemology|epistemological status]]{{emdash}}that is, its place among other human activities.<ref name="Mura">{{cite journal
| title=Images of Mathematics Held by University Teachers of Mathematical Sciences
| last=Mura | first=Roberta | date=Dec 1993
| journal=Educational Studies in Mathematics
| volume=25 | issue=4 | pages=375–85
| doi=10.1007/BF01273907 | jstor=3482762 | s2cid=122351146
}}</ref><ref name="Runge">{{cite book
| title=Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry
| last1=Tobies
| first1=Renate
| author1-link=Renate Tobies
| first2=Helmut
| last2=Neunzert
| publisher=Springer
| year=2012
| isbn=978-3-0348-0229-1
| page=9
| url={{GBurl|id=EDm0eQqFUQ4C|p=9}}
| quote=[I]t is first necessary to ask what is meant by ''mathematics'' in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.
| access-date=June 20, 2015
}}</ref> A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.<ref name="Mura" /> There is not even consensus on whether mathematics is an art or a science.<ref name="Runge" /> Some just say, "mathematics is what mathematicians do".<ref name="Mura" /> This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.<ref>{{cite conference
| title="What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it
| first1=Günter M. | last1=Ziegler | author1-link=Günter M. Ziegler
| first2=Andreas | last2=Loos | editor-last=Kaiser | editor-first=G.
| conference=Proceedings of the 13th International Congress on Mathematical Education
| series=ICME-13 Monographs
| date=November 2, 2017 | pages=63–77 | publisher=Springer
| doi=10.1007/978-3-319-62597-3_5
| isbn=978-3-319-62596-6 }}</ref>
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.<ref name="Franklin">{{Cite book | last=Franklin | first=James | author-link=James Franklin (philosopher) | title=Philosophy of Mathematics | date= 2009 | isbn=978-0-08-093058-9 | pages=104–106 | publisher=Elsevier | url={{GBurl|id=mbn35b2ghgkC|p=104}} | access-date=June 20, 2015 }}</ref> In the 19th century, when mathematicians began to address topics{{mdash}}such as infinite sets{{mdash}}which have no clear-cut relation to physical reality, a variety of new definitions were given.<ref name="Cajori">{{cite book
| title=A History of Mathematics
| last=Cajori
| first=Florian
| author-link=Florian Cajori
| publisher=American Mathematical Society (1991 reprint)
| year=1893
| isbn=978-0-8218-2102-2
| pages=285–286
| url={{GBurl|id=mGJRjIC9fZgC|p=285}}
| access-date=June 20, 2015
}}</ref> With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.
Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems{{emdash}}assertions whose validity relies on a proof, that is, a purely-logical deduction.<ref>{{cite journal | title=The Methodology of Mathematics | first1=Ronald | last1=Brown | author1-link=Ronald Brown (mathematician) | first2=Timothy | last2=Porter | journal=The Mathematical Gazette | volume=79 | issue=485 | pages=321–334 | date=January 2000 | doi=10.2307/3618304 | jstor=3618304 | s2cid=178923299 | url=https://cds.cern.ch/record/280311 | access-date=November 25, 2022 | archive-date=March 23, 2023 | archive-url=https://web.archive.org/web/20230323164159/https://cds.cern.ch/record/280311 | url-status=live }}</ref> Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.<ref>{{cite journal
| last=Strauss | first=Danie | year=2011
| title=Defining mathematics
| journal=Acta Academica
| volume=43 | issue=4 | pages=1–28
| url=https://www.researchgate.net/publication/290955899
| access-date=November 25, 2022
}}</ref>
=== Rigor ===
{{See also|Logic}}
Mathematical reasoning requires [[Mathematical rigor|rigor]]. This means that the definitions must be absolutely unambiguous and the [[proof (mathematics)|proof]]s must be reducible to a succession of applications of [[inference rule]]s,{{efn|This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without [[computer]]s and [[proof assistant]]s. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.}} without any use of empirical evidence and [[intuition]].{{efn|This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.}}<ref>{{cite journal | title=Mathematical Rigor and Proof | first=Yacin | last=Hamami | journal=The Review of Symbolic Logic | volume=15 | issue=2 | date=June 2022 | pages=409–449 | url=https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | access-date=November 21, 2022 | doi=10.1017/S1755020319000443 | s2cid=209980693 | archive-date=December 5, 2022 | archive-url=https://web.archive.org/web/20221205114343/https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf | url-status=live }}</ref> Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' [[concision]], rigorous proofs can require hundreds of pages to express. The emergence of [[computer-assisted proof]]s has allowed proof lengths to further expand,{{efn|For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software}}<ref>{{cite book
| last=Peterson | first=Ivars | author-link=Ivars Peterson
| title=The Mathematical Tourist
| publisher=Freeman | isbn=978-0-7167-1953-3
| year=1988 | page=4
| quote=A few complain that the computer program can't be verified properly | postscript=none}}, (in reference to the Haken–Apple proof of the Four Color Theorem).
</ref> such as the 255-page [[Feit–Thompson theorem]].{{efn|The book containing the complete proof has more than 1,000 pages.}} The result of this trend is a philosophy of the [[Quasi-empiricism in mathematics|quasi-empiricist]] proof that can not be considered infallible, but has a probability attached to it.<ref name=Kleiner_1991/>
The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged logical, deductive reasoning. However, this rigorous approach would tend to discourage exploration of new approaches, such as irrational numbers and concepts of infinity. The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rigorous approaches, while transitioning from geometric methods to algebraic and then arithmetic proofs.<ref name=Kleiner_1991>{{cite journal
| title=Rigor and Proof in Mathematics: A Historical Perspective
| first=Israel | last=Kleiner | author-link=Israel Kleiner (mathematician)
| journal=Mathematics Magazine
| volume=64 | issue=5 | date=December 1991 | pages=291–314
| publisher=Taylor & Francis, Ltd.
| jstor=2690647 | doi=10.1080/0025570X.1991.11977625 }}</ref>
At the end of the 19th century, it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and [[Weierstrass function]]) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the [[Apodicticity|apodictic]] inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.<ref name=Kleiner_1991/> It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a [[pleonasm]]. Where a special concept of rigor comes into play is in the socialized aspects of a proof, wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted for many years or even decades, it can then be considered as reliable.<ref>{{cite journal
| title=On the Reliability of Mathematical Proofs
| first=V. Ya. | last=Perminov
| journal=Philosophy of Mathematics
| volume=42 | issue=167 (4) | year=1988 | pages=500–508
| publisher=Revue Internationale de Philosophie
}}</ref>
Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.<ref>{{cite journal
| title=Teachers' perceptions of the official curriculum: Problem solving and rigor
| first1=Jon D. | last1=Davis | first2=Amy Roth | last2=McDuffie
| first3=Corey | last3=Drake | first4=Amanda L. | last4=Seiwell
| journal=International Journal of Educational Research
| volume=93 | year=2019 | pages=91–100
| doi=10.1016/j.ijer.2018.10.002 | s2cid=149753721 }}</ref>
== Training and practice ==
=== Education ===
{{main|Mathematics education}}
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a [[human activity]], the practice of mathematics has a social side, which includes [[Mathematics education|education]], [[Mathematician|careers]], [[List of mathematics awards|recognition]], [[Popular mathematics|popularization]], and so on. In education, mathematics is a core part of the curriculum and forms an important element of the [[STEM]] academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, [[statistician]], [[actuary]], [[financial analyst]], [[economist]], [[accountant]], [[commodity trader]], or [[Information technology consulting|computer consultant]].<ref>{{cite book
| title=Mathematicians and Statisticians: A Practical Career Guide
| first=Kezia
| last=Endsley
| year=2021
| series=Practical Career Guides
| isbn=978-1-5381-4517-3
| publisher=Rowman & Littlefield
| pages=1–3
| url={{GBurl|id=1cEYEAAAQBAJ|p=3}}
| access-date=November 29, 2022
}}</ref>
Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.<ref>{{cite book
| title=The Oxford Handbook of the History of Mathematics
| first=Eleanor | last=Robson | author-link=Eleanor Robson
| year=2009
| chapter=Mathematics education in an Old Babylonian scribal school
| editor1-first=Eleanor | editor1-last=Robson
| editor2-first=Jacqueline | editor2-last=Stedall | editor2-link=Jackie Stedall
| publisher=OUP Oxford
| isbn=978-0-19-921312-2
| chapter-url={{GBurl|id=xZMSDAAAQBAJ|p=199}}
| access-date=November 24, 2022
}}</ref> Comparable evidence has been unearthed for scribal mathematics training in the [[ancient Near East]] and then for the [[Greco-Roman world]] starting around 300 BCE.<ref>{{cite book
| chapter=Mathematics Education in Antiquity
| first1=Alain | last1=Bernard
| first2=Christine | last2=Proust | author2-link=Christine Proust
| first3=Micah | last3=Ross
| title=Handbook on the History of Mathematics Education
| editor1-last=Karp | editor1-first=A.
| editor2-last=Schubring | editor2-first=G.
| year=2014 | pages=27–53 | isbn=978-1-4614-9154-5
| publisher=Springer | publication-place=New York
| doi=10.1007/978-1-4614-9155-2_3
}}</ref> The oldest known mathematics textbook is the [[Rhind papyrus]], dated from {{Circa|1650 BCE}} in Egypt.<ref>{{cite journal
| title=The World's First Mathematics Textbook
| first=Underwood | last=Dudley
| journal=Math Horizons
| volume=9 | issue=4 | date=April 2002 | pages=8–11
| publisher=Taylor & Francis, Ltd.
| doi=10.1080/10724117.2002.11975154 | jstor=25678363
| s2cid=126067145 }}</ref> Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized [[oral tradition]] since the [[Vedic period]] ({{c.|1500|500 BCE}}).<ref>{{cite conference
| title=Indian pedagogy and problem solving in ancient Thamizhakam
| last=Subramarian
| first=F.
| conference=History and Pedagogy of Mathematics conference, July 16–20, 2012
| url=http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
| access-date=November 29, 2022
| archive-date=November 28, 2022
| archive-url=https://web.archive.org/web/20221128082654/http://hpm2012.onpcs.com/Proceeding/OT2/T2-10.pdf
| url-status=live
}}</ref> In [[Imperial China]] during the [[Tang dynasty]] (618–907 CE), a mathematics curriculum was adopted for the [[Imperial examination|civil service exam]] to join the state bureaucracy.<ref>{{cite book
| chapter=Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?
| first=Man Keung | last=Siu
| series=Series on Mathematics Education
| title=How Chinese Learn Mathematics
| pages=157–185 | year=2004 | volume=1 | isbn=978-981-256-014-8
| doi=10.1142/9789812562241_0006
| url=https://scholar.archive.org/work/3fb5lb2qsfg35gf2cv6viaydny/access/wayback/http://hkumath.hku.hk:80/~mks/Chapter%206-Siu.pdf
| access-date=November 26, 2022 }}</ref>
Following the [[Dark Age]]s, mathematics education in Europe was provided by religious schools as part of the [[Quadrivium]]. Formal instruction in [[pedagogy]] began with [[Jesuit]] schools in the 16th and 17th century. Most mathematical curriculum remained at a basic and practical level until the nineteenth century, when it began to flourish in France and Germany. The oldest journal addressing instruction in mathematics was ''[[L'Enseignement Mathématique]]'', which began publication in 1899.<ref>{{cite journal
| title=The History of Mathematical Education
| journal=The American Mathematical Monthly
| volume=74 | issue=1 | pages=38–55
| publisher=Taylor & Francis, Ltd.
| doi=10.2307/2314867 | jstor=2314867 | last1=Jones
| first1=Phillip S.
| year=1967
}}</ref> The Western advancements in science and technology led to the establishment of centralized education systems in many nation-states, with mathematics as a core component{{emdash}}initially for its military applications.<ref>{{cite journal
| title=Introduction: the history of mathematics teaching. Indicators for modernization processes in societies
| first1=Gert | last1=Schubring | first2=Fulvia | last2=Furinghetti
| first3=Man Keung | last3=Siu
| journal=ZDM Mathematics Education
| volume=44 | pages=457–459 | date=August 2012
| issue=4 | doi=10.1007/s11858-012-0445-7
| s2cid=145507519 | doi-access=free }}</ref> While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.<ref>{{Cite book | chapter=Examining eTIMSS Country Differences Between eTIMSS Data and Bridge Data: A Look at Country-Level Mode of Administration Effects | title=TIMSS 2019 International Results in Mathematics and Science | first1=Matthias | last1=von Davier | first2=Pierre | last2=Foy | first3=Michael O. | last3=Martin | first4=Ina V.S. | last4=Mullis | publisher=[[TIMSS]] & [[PIRLS]] International Study Center, [[Lynch School of Education and Human Development]] and [[International Association for the Evaluation of Educational Achievement]] | isbn=978-1-889938-54-7 | page=13.1 | language=en-US | year=2020 | url=https://files.eric.ed.gov/fulltext/ED610099.pdf | access-date=November 29, 2022 | archive-date=November 29, 2022 | archive-url=https://web.archive.org/web/20221129163908/https://files.eric.ed.gov/fulltext/ED610099.pdf | url-status=live }}</ref>
During school, mathematical capabilities and positive expectations have a strong association with career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and peer groups can influence the level of interest in mathematics.<ref>{{cite journal
| title=Social Cognitive Factors, Support, and Engagement: Early Adolescents' Math Interests as Precursors to Choice of Career
| first1=Heather T.
| last1=Rowan-Kenyon
| first2=Amy K.
| last2=Swan
| first3=Marie F.
| last3=Creager
| journal=The Career Development Quarterly
| volume=60
| issue=1
| date=March 2012
| pages=2–15
| doi=10.1002/j.2161-0045.2012.00001.x
| url=https://www.academia.edu/download/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl.pdf
| access-date=November 29, 2022
| archive-url=https://web.archive.org/web/20231122212933/https://d1wqtxts1xzle7.cloudfront.net/45974312/j.2161-0045.2012.00001.x20160526-3995-67kydl-libre.pdf?1464293840=&response-content-disposition=inline%3B+filename%3DSocial_Cognitive_Factors_Support_and_Eng.pdf&Expires=1700692172&Signature=cs9KfTPxoPh859wY~ExtJyAl9NpYb3X-2P4rDel1Z3z7DwehsHLRggoZtgi1pMsamxYobu9dVK4G7OsqfvNxcuwz3uKh1pnCMZQEz~ahVtPb4kvN-4dmwExJplzoxWu31o3SJOfuBt0GGE-0Hl8eLfPBg5agmtkjSwAWQwlqGrjp3YgYZGjbNxOEAM4t1l4qvoWXidWvSHHcEUNvlKYwCDvG0~QhGTmA6ldxmfS1ovf0adog-qqvjGxxJuSjtP6O8zCTwkPXYwi2e8giI0H6b5fNarHc-2q~-NRnVVtYKhvSBcwC22kNZoA7s8sp8ix9KIdM3uxiUIBRBRC-4aaVoQ__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
| archive-date=November 22, 2023
| url-status=live
}}</ref> Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as [[math anxiety]] or math phobia, and is considered the most prominent of the disorders impacting academic performance. Math anxiety can develop due to various factors such as parental and teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come from changes in instructional approaches, by interactions with parents and teachers, and by tailored treatments for the individual.<ref>{{cite journal
| title=Spotlight on math anxiety
| first1=Silke | last1=Luttenberger
| first2=Sigrid | last2=Wimmer
| first3=Manuela | last3=Paechter
| journal=Psychology Research and Behavior Management
| year=2018 | volume=11 | pages=311–322
| doi=10.2147/PRBM.S141421 | pmid=30123014
| pmc=6087017 | doi-access=free }}</ref>
=== Psychology (aesthetic, creativity and intuition) ===
The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a [[computer program]]. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process.<ref>{{cite journal
| title=The Outlook of the Mathematicians' Creative Processes
| first=Narges | last=Yaftian
| journal=Procedia - Social and Behavioral Sciences
| volume=191 | date=June 2, 2015 | pages=2519–2525
| doi=10.1016/j.sbspro.2015.04.617
| doi-access=free}}</ref><ref>{{cite journal
| title=The Frontage of Creativity and Mathematical Creativity
| first1=Mehdi | last1=Nadjafikhah | first2=Narges | last2=Yaftian
| journal=Procedia - Social and Behavioral Sciences
| volume=90 | date=October 10, 2013 | pages=344–350
| doi=10.1016/j.sbspro.2013.07.101
| doi-access=free}}</ref> An extreme example is [[Apery's theorem]]: [[Roger Apery]] provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.<ref>{{cite journal
| title=A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)
| first=A.
| last=van der Poorten
| journal=[[The Mathematical Intelligencer]]
| volume=1
| issue=4
| year=1979
| pages=195–203
| doi=10.1007/BF03028234
| s2cid=121589323
| url=http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
| access-date=November 22, 2022
| archive-date=September 6, 2015
| archive-url=https://web.archive.org/web/20150906015716/http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf
| url-status=live
}}</ref>
Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some mathematicians can see their activity as a game, more specifically as solving [[puzzle]]s.<ref>{{cite book
| title=Famous Puzzles of Great Mathematicians
| first=Miodrag
| last=Petkovi
| date=September 2, 2009
| publisher=American Mathematical Society
| pages=xiii–xiv
| isbn=978-0-8218-4814-2
| url={{GBurl|id=AZlwAAAAQBAJ|pg=PR13}}
| access-date=November 25, 2022
}}</ref> This aspect of mathematical activity is emphasized in [[recreational mathematics]].
Mathematicians can find an [[aesthetic]] value to mathematics. Like [[beauty]], it is hard to define, it is commonly related to ''elegance'', which involves qualities like [[simplicity]], [[symmetry]], completeness, and generality. G. H. Hardy in ''[[A Mathematician's Apology]]'' expressed the belief that the aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He also identified other criteria such as significance, unexpectedness, and inevitability, which contribute to mathematical aesthetic.<ref>{{cite book
| title=A Mathematician's Apology
| last=Hardy | first=G. H. | author-link=G. H. Hardy
| publisher=Cambridge University Press | year=1940
| url=https://archive.org/details/hardy_annotated/
| isbn=978-0-521-42706-7 | access-date=November 22, 2022
}} See also ''[[A Mathematician's Apology]]''.</ref> [[Paul Erdős]] expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book ''[[Proofs from THE BOOK]]'', inspired by Erdős, is a collection of particularly succinct and revelatory mathematical arguments. Some examples of particularly elegant results included are Euclid's proof that there are infinitely many prime numbers and the [[fast Fourier transform]] for [[harmonic analysis]].<ref>{{cite journal
| title=Reflections on Paul Erdős on His Birth Centenary, Part II
| first1=Noga | last1=Alon | first2=Dan | last2=Goldston
| first3=András | last3=Sárközy | first4=József | last4=Szabados
| first5=Gérald | last5=Tenenbaum | first6=Stephan Ramon | last6=Garcia
| first7=Amy L. | last7=Shoemaker
| journal=Notices of the American Mathematical Society
| date=March 2015 | volume=62 | issue=3 | pages=226–247
| editor1-first=Krishnaswami | editor1-last=Alladi
| editor2-first=Steven G. | editor2-last=Krantz
| doi=10.1090/noti1223
| doi-access=free }}</ref>
Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional [[liberal arts]].<ref>See, for example [[Bertrand Russell]]'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his {{cite book | title=History of Western Philosophy | year=1919 | page=60 }}</ref> One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are ''created'' (as in art) or ''discovered'' (as in science).<ref name=borel>{{Cite journal
| last=Borel | first=Armand | author-link=Armand Borel
| title=Mathematics: Art and Science
| journal=The Mathematical Intelligencer
| volume=5 | issue=4 | pages=9–17 | year=1983
| publisher=Springer | issn=1027-488X
| doi=10.4171/news/103/8| doi-access=free
}}</ref> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as [[intuitionism]], which primarily identifies mathematics with certain creative processes in the mind.<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> Intuitionism is in turn one flavor of a stance known as [[Constructivism (mathematics)|constructivism]], which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like [[existential proof]]s based on the law of excluded middle.<ref name=SEP-Intuitionism>{{cite encyclopedia |title=Intuitionism in the Philosophy of Mathematics |encyclopedia=The Stanford Encyclopedia of Philosophy |last=Iemhoff |first=Rosalie|author-link=Rosalie Iemhoff |editor-last=Zalta |editor-first=Edward N. |year=2020 |edition=Fall 2020 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/archives/fall2020/entries/intuitionism |access-date=April 2, 2022 |archive-date=April 21, 2022 |archive-url=https://web.archive.org/web/20220421162527/https://plato.stanford.edu/archives/fall2020/entries/intuitionism/ |url-status=live }}</ref>
In the end, neither constructivism nor intuitionism displaced [[classical mathematics]] or achieved mainstream acceptance. However, these programs have motivated specific developments, such as intuitionistic logic and other foundational insights, which are appreciated in their own right.<ref name=SEP-Intuitionism />
== Cultural impact ==
{{Globalize|date=December 2022|1=section|2=West}}
=== Artistic expression ===
{{Main|Mathematics and art}}
Notes that sound well together to a Western ear are sounds whose fundamental [[frequencies]] of vibration are in simple ratios. For example, an octave doubles the frequency and a [[perfect fifth]] multiplies it by <math>\frac{3}{2}</math>.<ref>{{cite journal | last = Cazden | first = Norman | date = October 1959 | doi = 10.1177/002242945900700205 | issue = 2 | journal = Journal of Research in Music Education | jstor = 3344215 | pages = 197–220 | title = Musical intervals and simple number ratios | volume = 7| s2cid = 220636812 }}</ref><ref>{{cite journal | last = Budden | first = F. J. | date = October 1967 | doi = 10.2307/3613237 | issue = 377 | journal = The Mathematical Gazette | jstor = 3613237 | pages = 204–215 | publisher = Cambridge University Press ({CUP}) | title = Modern mathematics and music | volume = 51| s2cid = 126119711 }}</ref>
[[File:Julia set (highres 01).jpg|thumb|[[Fractal]] with a scaling symmetry and a central symmetry]]
Humans, as well as some other animals, find symmetric patterns to be more beautiful.<ref>{{Cite journal |last1=Enquist |first1=Magnus |last2=Arak |first2=Anthony |date=November 1994 |title=Symmetry, beauty and evolution |url=https://www.nature.com/articles/372169a0 |journal=Nature |language=en |volume=372 |issue=6502 |pages=169–172 |doi=10.1038/372169a0 |pmid=7969448 |bibcode=1994Natur.372..169E |s2cid=4310147 |issn=1476-4687 |access-date=December 29, 2022 |archive-date=December 28, 2022 |archive-url=https://web.archive.org/web/20221228052049/https://www.nature.com/articles/372169a0 |url-status=live }}</ref> Mathematically, the symmetries of an object form a group known as the [[symmetry group]].<ref>{{Cite web |last=Hestenes |first=David |date=1999 |title=Symmetry Groups |url=http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |access-date=December 29, 2022 |website=geocalc.clas.asu.edu |archive-date=January 1, 2023 |archive-url=https://web.archive.org/web/20230101210124/http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |url-status=live }}</ref>
For example, the group underlying mirror symmetry is the [[cyclic group]] of two elements, <math>\mathbb{Z}/2\mathbb{Z}</math>. A [[Rorschach test]] is a figure invariant by this symmetry,<ref>{{cite encyclopedia | last = Bender | first = Sara | editor1-last = Carducci | editor1-first = Bernardo J. | editor2-last = Nave | editor2-first = Christopher S. | editor3-last = Mio | editor3-first = Jeffrey S. | editor4-last = Riggio | editor4-first = Ronald E. | title = The Rorschach Test | date = September 2020 | doi = 10.1002/9781119547167.ch131 | pages = 367–376 | publisher = Wiley | encyclopedia = The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment| isbn = 9781119057512 }}</ref> as are [[butterfly]] and animal bodies more generally (at least on the surface).<ref>{{cite book|title=Symmetry|volume=47|series=Princeton Science Library|first=Hermann|last=Weyl|author-link=Hermann Weyl|publisher=Princeton University Press|year=2015|isbn=9781400874347|page=[https://books.google.com/books?hl=en&lr=&id=GG1FCQAAQBAJ&pg=PA4 4]}}</ref> Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.{{Citation needed|date=December 2022}} Furthermore, [[fractals]] possess (usually approximate{{Citation needed|date=December 2022}}) [[self-similarity]].<ref>{{Cite web |last=Bradley |first=Larry |date=2010 |title=Fractals – Chaos & Fractals |url=https://www.stsci.edu/~lbradley/seminar/fractals.html |access-date=December 29, 2022 |website=www.stsci.edu |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054609/https://www.stsci.edu/~lbradley/seminar/fractals.html |url-status=live }}</ref><ref>{{Cite web |title=Self-similarity |url=https://math.bu.edu/DYSYS/chaos-game/node5.html |access-date=December 29, 2022 |website=math.bu.edu |archive-date=March 2, 2023 |archive-url=https://web.archive.org/web/20230302132911/http://math.bu.edu/DYSYS/chaos-game/node5.html |url-status=live }}</ref>
=== Popularization ===
{{Main|Popular mathematics}}Popular mathematics is the act of presenting mathematics without technical terms.<ref>{{Cite conference |last=Kissane |first=Barry |date=July 2009 |title=Popular mathematics |url=https://researchrepository.murdoch.edu.au/id/eprint/6242/ |conference=22nd Biennial Conference of The Australian Association of Mathematics Teachers |location=Fremantle, Western Australia |publisher=Australian Association of Mathematics Teachers |pages=125–126 |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054610/https://researchrepository.murdoch.edu.au/id/eprint/6242/ |url-status=live }}</ref> Presenting mathematics may be hard since the general public suffers from [[mathematical anxiety]] and mathematical objects are highly abstract.<ref>{{Cite book |last=Steen |first=L. A. |url={{GBurl|id=-d3TBwAAQBAJ|dq="popular mathematics" analogies|p=2}} |title=Mathematics Today Twelve Informal Essays |date=2012|publisher=Springer Science & Business Media |isbn=978-1-4613-9435-8 |page=2 |language=en |access-date=January 3, 2023 }}</ref> However, popular mathematics writing can overcome this by using applications or cultural links.<ref>{{Cite book |last=Pitici |first=Mircea |url={{GBurl|id=9nGQDQAAQBAJ|dq="popular mathematics" analogies|p=331}} |title=The Best Writing on Mathematics 2016 |date=2017|publisher=Princeton University Press |isbn=978-1-4008-8560-2 |language=en |access-date=January 3, 2023 }}</ref> Despite this, mathematics is rarely the topic of popularization in printed or televised media.
== Awards and prize problems ==
{{Main category|Mathematics awards}}
[[File:FieldsMedalFront.jpg|thumb|The front side of the [[Fields Medal]] with an illustration of the Greek [[polymath]] [[Archimedes]]]]
The most prestigious award in mathematics is the [[Fields Medal]],{{sfn|Monastyrsky|2001|p=1|ps=: "The Fields Medal is now indisputably the best known and most influential award in mathematics."}}{{sfn|Riehm|2002|pp=778–782}} established in 1936 and awarded every four years (except around [[World War II in Yugoslavia|World War II]]) to up to four individuals.<ref>{{Cite web |title=Fields Medal {{!}} International Mathematical Union (IMU) |url=https://www.mathunion.org/imu-awards/fields-medal |access-date=February 21, 2022 |website=www.mathunion.org |archive-date=December 26, 2018 |archive-url=https://web.archive.org/web/20181226015744/https://www.mathunion.org/imu-awards/fields-medal |url-status=live }}</ref><ref name="StAndrews-Fields">{{Cite web |title=Fields Medal |url=https://mathshistory.st-andrews.ac.uk/Honours/FieldsMedal/ |access-date=February 21, 2022 |website=Maths History |language=en |archive-date=March 22, 2019 |archive-url=https://web.archive.org/web/20190322134417/http://www-history.mcs.st-andrews.ac.uk/Honours/FieldsMedal.html |url-status=live }}</ref> It is considered the mathematical equivalent of the [[Nobel Prize]].<ref name="StAndrews-Fields" />
Other prestigious mathematics awards include:<ref>{{cite web
| title=Honours/Prizes Index
| website=MacTutor History of Mathematics Archive
| url=https://mathshistory.st-andrews.ac.uk/Honours/
| access-date=February 20, 2023
| archive-date=December 17, 2021
| archive-url=https://web.archive.org/web/20211217235828/https://mathshistory.st-andrews.ac.uk/Honours/
| url-status=live
}}</ref>
* The [[Abel Prize]], instituted in 2002<ref>{{Cite web|title=About the Abel Prize|publisher=The Abel Prize|url=https://abelprize.no/page/about-abel-prize|access-date=January 23, 2022|archive-date=April 14, 2022|archive-url=https://web.archive.org/web/20220414060442/https://abelprize.no/page/about-abel-prize|url-status=live}}</ref> and first awarded in 2003<ref>{{Cite encyclopedia|title=Abel Prize {{!}} mathematics award|encyclopedia=Encyclopedia Britannica|url=https://www.britannica.com/science/Abel-Prize|access-date=January 23, 2022|language=en|archive-date=January 26, 2020|archive-url=https://web.archive.org/web/20200126120202/https://www.britannica.com/science/Abel-Prize|url-status=live}}</ref>
* The [[Chern Medal]] for lifetime achievement, introduced in 2009<ref>{{Cite web |date=June 1, 2009 |title=Chern Medal Award|url=https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |url-status=live |archive-url=https://web.archive.org/web/20090617012953/https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |archive-date=June 17, 2009 |access-date=February 21, 2022 |website=www.mathunion.org}}</ref> and first awarded in 2010<ref>{{Cite web |title=Chern Medal Award|publisher=International Mathematical Union (IMU)|url=https://www.mathunion.org/imu-awards/chern-medal-award |access-date=January 23, 2022|archive-date=August 25, 2010 |archive-url=https://web.archive.org/web/20100825071850/http://www.mathunion.org/general/prizes/chern/details |url-status=live }}</ref>
* The [[American Mathematical Society|AMS]] [[Leroy P. Steele Prize]], awarded since 1970<ref>{{cite web
| title=The Leroy P Steele Prize of the AMS
| publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
| url=https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
| access-date=November 17, 2022
| archive-date=November 17, 2022
| archive-url=https://web.archive.org/web/20221117201134/https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
| url-status=live
}}</ref>
* The [[Wolf Prize in Mathematics]], also for lifetime achievement,<ref>{{Cite book |last1=Chern |first1=S. S. |last2=Hirzebruch |first2=F. |date=September 2000 |title=Wolf Prize in Mathematics |url=https://www.worldscientific.com/worldscibooks/10.1142/4149 |language=en |doi=10.1142/4149 |isbn=978-981-02-3945-9 |access-date=February 21, 2022 |archive-date=February 21, 2022 |archive-url=https://web.archive.org/web/20220221171351/https://www.worldscientific.com/worldscibooks/10.1142/4149 |url-status=live }}</ref> instituted in 1978<ref>{{Cite web|title=The Wolf Prize|url=https://wolffund.org.il/the-wolf-prize/|url-status=live|archive-url=https://web.archive.org/web/20200112205029/https://wolffund.org.il/the-wolf-prize/|archive-date=January 12, 2020|access-date=January 23, 2022|website=Wolf Foundation|language=en-US}}</ref>
A famous list of 23 [[open problem]]s, called "[[Hilbert's problems]]", was compiled in 1900 by German mathematician David Hilbert.<ref name=":0">{{Cite web|date=May 6, 2020|title=Hilbert's Problems: 23 and Math|url=https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|access-date=January 23, 2022|website=Simons Foundation|language=en-US|archive-date=January 23, 2022|archive-url=https://web.archive.org/web/20220123011430/https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|url-status=live}}</ref> This list has achieved great celebrity among mathematicians,<ref>{{cite book
| chapter=Deciding the undecidable: Wrestling with Hilbert's problems
| first=Solomon
| last=Feferman
| author-link=Solomon Feferman
| title=In the Light of Logic
| year=1998
| publisher=Oxford University Press
| isbn=978-0-19-508030-8
| pages=3–27
| series=Logic and Computation in Philosophy series
| chapter-url=https://math.stanford.edu/~feferman/papers/deciding.pdf
| url={{GBurl|id=1rjnCwAAQBAJ}}
| access-date=November 29, 2022
}}</ref> and, {{as of|2022|lc=yes}}, at least thirteen of the problems (depending how some are interpreted) have been solved.<ref name=":0"/><!-- Namely: problems 1, 3, 4; 5, 7, 10; 13, 14, 17; 18, 19, 20; 21 have been solved. (The semicolons are to make counting easier). ~Duckmather -->
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.<ref>{{Cite web|title=The Millennium Prize Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems/millennium-prize-problems|access-date=January 23, 2022|archive-date=July 3, 2015|archive-url=https://web.archive.org/web/20150703184941/http://www.claymath.org/millennium-problems/millennium-prize-problems|url-status=live}}</ref> To date, only one of these problems, the [[Poincaré conjecture]], has been solved.<ref>{{Cite web|title=Millennium Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems|access-date=January 23, 2022|archive-date=December 20, 2018|archive-url=https://web.archive.org/web/20181220122925/http://www.claymath.org/millennium-problems|url-status=live}}</ref><!-- NOTE that this website describes the answer to each problem as "unknown" EXCEPT for the Poincaré conjecture, where it mentions "Perelman's proof". ~Duckmather -->
== See also ==
{{Portal|Mathematics}}
{{div col|colwidth=22em}}
* [[List of mathematical jargon]]
* [[Lists of mathematicians]]
* [[Lists of mathematics topics]]
* [[Mathematical constant]]
* [[Mathematical sciences]]
* [[Mathematics and art]]
* [[Mathematics education]]
* [[Outline of mathematics]]
* [[Philosophy of mathematics]]
* [[Relationship between mathematics and physics]]
* [[Science, technology, engineering, and mathematics]]
{{div col end}}
== References ==
=== Notes ===
{{notelist}}
===Citations===
{{Reflist|30em
<!--
|refs=
<ref name=future>[[Lynn Steen|Steen, L.A.]] (April 29, 1988). ''The Science of Patterns'' [[Science (journal)|Science]], 240: 611–16. And summarized at [http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx Association for Supervision and Curriculum Development] {{webarchive|url=https://web.archive.org/web/20101028101034/http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx |date=October 28, 2010 }}, www.ascd.org.</ref>
<ref name=devlin>[[Keith Devlin|Devlin, Keith]], ''Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe'' (Scientific American Paperback Library) 1996, {{isbn|978-0716750475}}</ref>
-->
}}
=== Bibliography ===
{{refbegin|30em}}
* {{cite book |last=Bouleau |first=Nicolas |title=Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur |publisher=L'Harmattan |year=1999 |isbn=978-2-7384-8125-2}}
* {{cite book |last1=Boyer |first1=Carl Benjamin |author1-link=Carl Benjamin Boyer |title=A History of Mathematics |date=1991 |publisher=[[Wiley (publisher)|Wiley]] |location=New York |isbn=978-0-471-54397-8 |edition=2nd |url=https://archive.org/details/historyofmathema00boye/page/n3/mode/2up |chapter= |url-access=registration }}
* {{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=An Introduction to the History of Mathematics |edition=6th |publisher=Saunders |year=1990 |isbn=978-0-03-029558-4 |ref=none}}
* {{cite book |last=Kleiner |first=Izraïl' |author-link=Israel Kleiner (mathematician) |title=A History of Abstract Algebra |publisher=Springer Science & Business Media |isbn=978-0-8176-4684-4 |year=2007 |url={{GBurl|id=RTLRBK-wj6wC|pg=PR1}} |access-date=November 11, 2022 }}
* {{cite book |last=Kline |first=Morris |author-link=Morris Kline |title=Mathematical Thought from Ancient to Modern Times |url=https://archive.org/details/mathematicalthou00klin |url-access=registration |location=New York |publisher=Oxford University Press |edition=|year=1990 |isbn=978-0-19-506135-2 }}
* {{cite journal |last=Monastyrsky |first=Michael |url=http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf |year=2001 |title=Some Trends in Modern Mathematics and the Fields Medal |journal=CMS – Notes – de la SMC |volume=33 |issue=2–3 |publisher=Canadian Mathematical Society |access-date=July 28, 2006 |archive-date=August 13, 2006 |archive-url=https://web.archive.org/web/20060813224844/http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf |url-status=live }}
* {{cite book |last=Oakley |first=Barbara |author-link=Barbara Oakley |title=A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) |date=2014 |publisher=Penguin Random House |location=New York |url=https://archive.org/details/isbn_9780399165245 |url-access=registration |quote=A Mind for Numbers. |isbn=978-0-399-16524-5 |ref=none }}
* {{cite journal |last=Peirce |first=Benjamin |author-link=Benjamin Peirce |pages=97–229 |title=Linear associative algebra |editor-link=Charles Sanders Peirce |editor-first=Charles Sanders |editor-last=Peirce |edition=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph |journal=American Journal of Mathematics |volume=4 |year=1881 |url={{GBurl|id=De0GAAAAYAAJ|q=Peirce Benjamin Linear Associative Algebra|p=1}} |id=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. ''Google'' [{{GBurl|id=LQgPAAAAIAAJ|p=221}} Eprint] and as an extract, D. Van Nostrand, 1882, ''Google'' [https://archive.org/details/bub_gb_De0GAAAAYAAJ Eprint] |issue=1–4 |doi=10.2307/2369153 |jstor=2369153 |hdl=2027/hvd.32044030622997 |hdl-access=free |access-date=November 17, 2020 |ref=none }}.
* {{cite book |last=Peterson |first=Ivars |title=Mathematical Tourist, New and Updated Snapshots of Modern Mathematics |publisher=Owl Books |year=2001 |isbn=978-0-8050-7159-7}}
* {{cite book |last=Popper |first=Karl R. |author-link=Karl Popper |title=In Search of a Better World: Lectures and Essays from Thirty Years |location=New York |publisher=Routledge |chapter=On knowledge |year=1995 |isbn=978-0-415-13548-1 |bibcode=1992sbwl.book.....P |url-access=registration |url=https://archive.org/details/insearchofbetter00karl |ref=none }}
* {{cite journal |last=Riehm |first=Carl |title=The Early History of the Fields Medal |journal=Notices of the AMS |volume=49 |issue=7 |pages=778–782 |date=August 2002 |url=https://www.ams.org/notices/200207/comm-riehm.pdf |access-date=October 2, 2006 |archive-date=October 26, 2006 |archive-url=https://web.archive.org/web/20061026000014/http://www.ams.org/notices/200207/comm-riehm.pdf |url-status=live }}
* {{cite journal |last=Sevryuk |first=Mikhail B. |date=January 2006 |title=Book Reviews |journal=[[Bulletin of the American Mathematical Society]] |volume=43 |issue=1 |pages=101–109 |url=https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf |access-date=June 24, 2006 |doi=10.1090/S0273-0979-05-01069-4 |archive-date=July 23, 2006 |archive-url=https://web.archive.org/web/20060723082901/http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf |url-status=live |doi-access=free }}
* {{cite book |last=Whittle |first=Peter |author-link=Peter Whittle (mathematician) |chapter=Almost home |editor-link=Frank Kelly (mathematician) |editor-first=F.P. |editor-last=Kelly |year=1994 |title=Probability, statistics and optimisation: A Tribute to Peter Whittle |location=Chichester |publisher=John Wiley |isbn=978-0-471-94829-2 |pages=1–28 |chapter-url=http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair |edition=previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" |url-status=live |archive-url=https://web.archive.org/web/20131219080017/http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair |archive-date=December 19, 2013 }}
{{refend}}
== Further reading ==
{{Library resources box |by=no |onlinebooks=yes |others=yes |about=yes |label=Mathematics}}
{{refbegin}}
* {{cite book |last=Benson |first=Donald C. |title=The Moment of Proof: Mathematical Epiphanies |publisher=Oxford University Press |year=1999 |url=https://archive.org/details/momentofproofmat00bens/page/n5/mode/2up |url-access=registration |isbn=978-0-19-513919-8 |ref=none}}
* {{cite book |last1=Davis |first1=Philip J. |author1-link=Philip J. Davis |last2=Hersh |first2=Reuben |author2-link=Reuben Hersh |title=The Mathematical Experience |publisher=Mariner Books |location=Boston; New York |edition=Reprint |year=1999 |isbn=978-0-395-92968-1 |title-link=The Mathematical Experience |ref=none}} Available [https://archive.org/details/mathematicalexpe0000davi/page/n5/mode/2up online] (registration required).
* {{cite book |last1=Courant |first1=Richard |author-link1=Richard Courant |last2=Robbins |first2=Herbert |author-link2=Herbert Robbins |title=What Is Mathematics?: An Elementary Approach to Ideas and Methods |location=New York |publisher=Oxford University Press |edition=2nd |year=1996 |isbn=978-0-19-510519-3 |url=https://archive.org/details/whatismathematic0000cour/page/n5/mode/2up |url-access=registration |ref=none}}
* {{cite book |last=Gullberg |first=Jan |author-link=Jan Gullberg |title=Mathematics: From the Birth of Numbers |url=https://archive.org/details/mathematicsfromb1997gull/page/n5/mode/2up |url-access=registration |publisher=W.W. Norton & Company |edition= |year=1997 |isbn=978-0-393-04002-9 |ref=none}}
* {{cite book |editor-last=Hazewinkel |editor-first=Michiel |editor-link=Michiel Hazewinkel |title=Encyclopaedia of Mathematics |publisher=Kluwer Academic Publishers |year=2000 |title-link=Encyclopaedia of Mathematics |ref=none}} – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and [https://encyclopediaofmath.org/wiki/Special:AllPages online]. {{Webarchive|url=https://archive.today/20121220135247/http://www.encyclopediaofmath.org/ |date=December 20, 2012 |ref=none }}.
* {{cite book |last=Hodgkin |first=Luke Howard |title=A History of Mathematics: From Mesopotamia to Modernity |publisher=Oxford University Press |year=2005 |isbn=978-0-19-152383-0}}
* {{cite book |last=Jourdain |first=Philip E. B. |author-link=Philip Jourdain |chapter=The Nature of Mathematics |title=The World of Mathematics |editor=James R. Newman |publisher=Dover Publications |year=2003 |isbn=978-0-486-43268-7 |ref=none}}
<!-- * {{cite book |last=Maier |first=Annaliese |author-link=Anneliese Maier |title=At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy |editor=Steven Sargent |location=Philadelphia |publisher=University of Pennsylvania Press |year=1982 |ref=none}}
-->
* {{cite book |last1=Pappas |first1=Theoni |author1-link=Theoni Pappas |title=The Joy Of Mathematics |date=1986 |publisher=Wide World Publishing |location=San Carlos, California |isbn=978-0-933174-65-8 |url=https://archive.org/details/joyofmathematics0000papp_t0z1/page/n3/mode/2up |url-access=registration |ref=none}}
* {{cite book |last=Waltershausen |first=Wolfgang Sartorius von |author-link=Wolfgang Sartorius von Waltershausen |title=Gauss zum Gedächtniss |year=1965 |orig-date=first published 1856 |publisher=Sändig Reprint Verlag H. R. Wohlwend |isbn=978-3-253-01702-5 |ref=no}}
{{refend}}
{{Areas of mathematics}}
{{Authority control}}
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[[Category:Mathematics| ]]
[[Category:Formal sciences]]
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