Editing Mathematics (section)

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'''Mathematics''' is a field of study that discovers and organizes methods, [[Mathematical theory|theories]] and [[theorem]]s that are developed and [[Mathematical proof|proved]] for the needs of [[empirical sciences]] and mathematics itself. There are many areas of mathematics, which include [[number theory]] (the study of numbers), [[algebra]] (the study of formulas and related structures), [[geometry]] (the study of shapes and spaces that contain them), [[Mathematical analysis|analysis]] (the study of continuous changes), and [[set theory]] (presently used as a foundation for all mathematics).

Mathematics involves the description and manipulation of [[mathematical object|abstract objects]] that consist of either [[abstraction (mathematics)|abstraction]]s from nature or{{emdash}}in modern mathematics{{emdash}}purely abstract entities that are stipulated to have certain properties, called [[axiom]]s.  Mathematics uses pure [[reason]] to [[proof (mathematics)|prove]] properties of objects, a ''proof'' consisting 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 book |last=Hipólito |first=Inês Viegas |editor1-last=Kanzian |editor1-first=Christian |editor2-last=Mitterer |editor2-first=Josef |editor2-link=Josef Mitterer |editor3-last=Neges |editor3-first=Katharina |date=August 9–15, 2015 |chapter=Abstract Cognition and the Nature of Mathematical Proof |pages=132–134 |title=Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums |trans-title=Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium |volume=23 |language=de, en |publisher=Austrian Ludwig Wittgenstein Society |location=Kirchberg am Wechsel, Austria |issn=1022-3398 |oclc=236026294 |url=https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133 |url-status=live |archive-url=https://web.archive.org/web/20221107221937/https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133 |archive-date=November 7, 2022 |access-date=January 17, 2024}} ([https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof at ResearchGate] {{open access}} {{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 of 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|1988|page=12}}<ref name=wigner1960 />

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 |last=Wise |first=David |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|website=[[The University of Georgia]] |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=January 18, 2024}}</ref> Since its beginning, mathematics was primarily divided into geometry and [[arithmetic]] (the manipulation of [[natural number]]s and [[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 |last=Alexander |first=Amir |author-link=Amir Alexander |date=September 2011 |title=The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics? |journal=Isis |volume=102 |number=3 |pages=475–480 |doi=10.1086/661620 |issn=0021-1753 |mr=2884913 |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>{{cite journal |last=Kleiner |first=Israel |author-link=Israel Kleiner (mathematician) |date=December 1991 |title=Rigor and Proof in Mathematics: A Historical Perspective |journal=Mathematics Magazine |publisher=Taylor & Francis, Ltd. |volume=64 |issue=5 |pages=291–314 |doi=10.1080/0025570X.1991.11977625 |jstor=2690647 |issn=0025-570X |eissn=1930-0980 |lccn=47003192 |mr=1141557 |oclc=1756877 |s2cid=7787171}}</ref> which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary [[Mathematics Subject Classification]] lists more than sixty first-level areas of mathematics.

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