Mathematics: Difference between revisions - Wikipedia

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'''Mathematics''' is a field of study that discovers and organizes ~~abstract objects,~~ 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>

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== Areas of mathematics ==

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

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

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

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| url-status=live

}}</ref><ref>{{cite conference | title=Grand Challenges, High Performance Computing, and Computational Science | last1=Johnson | first1=Gary M. | last2=Cavallini | first2=John S. | conference=Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage | date=September 1991 | page=28 |lccn=91018998 |publisher=World Scientific | editor1-first=Kang Hoh | editor1-last=Phua | editor2-first=Kia Fock | editor2-last=Loe | url={{GBurl|id=jYNIDwAAQBAJ|p=28}} | access-date=November 13, 2022 }}</ref> [[Numerical analysis]] studies methods for problems in [[analysis (mathematics)|analysis]] using [[functional analysis]] and [[approximation theory]]; numerical analysis broadly includes the study of [[approximation]] and [[discretization]] with special focus on [[rounding error]]s.<ref>{{cite book |last=Trefethen |first=Lloyd N. |author-link=Lloyd N. Trefethen |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor2-link=June Barrow-Green |editor3-last=Leader |editor3-first=Imre |editor3-link=Imre Leader |year=2008 |chapter=Numerical Analysis |pages=604–615 |title=The Princeton Companion to Mathematics |publisher=[[Princeton University Press]] |isbn=978-0-691-11880-2 |lccn=2008020450 |mr=2467561 |oclc=227205932 |url=http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |url-status=live |archive-url=https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf |archive-date=March 7, 2023 |access-date=February 15, 2024}}</ref> Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-[[numerical linear algebra|matrix]]-and-[[graph theory]]. Other areas of computational mathematics include [[computer algebra]] and [[symbolic computation]].

== History ==

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=== Ancient ===

[[File:Plimpton 322.jpg|thumb|The Babylonian mathematical tablet ''[[Plimpton 322]]'', dated to 1800 BC]]

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> 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|~~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.<ref>[https://www.ms.uky.edu/~dhje223/CrestOfThePeacockCh4-pages-2-21.pdf/ Mesopotamia] pg 10. Retrieved June 1, 2024</ref> 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 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}}

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| 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 [[Gödel's incompleteness theorems|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}}

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{{Main|Mathematical notation|Language of mathematics|Glossary of mathematics}}

[[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) [[summation]] notation|class=skin-invert-image]]

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 |date=October 2000 |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/ |url-status=live |archive-url=https://web.archive.org/web/20221116150905/https://www.stephenwolfram.com/publications/mathematical-notation-past-future/ |archive-date=November 16, 2022 |access-date=February 3, 2024}}</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 |last1=Douglas |first1=Heather |last2=Headley |first2=Marcia Gail |last3=Hadden |first3=Stephanie |last4=LeFevre |first4=Jo-Anne |author4-link=Jo-Anne LeFevre |date=December 3, 2020 |title=Knowledge of Mathematical Symbols Goes Beyond Numbers |journal=Journal of Numerical Cognition |volume=6 |issue=3 |pages=322–354 |doi=10.5964/jnc.v6i3.293 |doi-access=free |eissn=2363-8761 |s2cid=228085700}}</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 |last1=Letourneau |first1=Mary |last2=Wright Sharp |first2=Jennifer |date=October 2017 |title=AMS Style Guide |page=75 |publisher=[[American Mathematical Society]] |url=https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf |url-status=live |archive-url=https://web.archive.org/web/20221208063650/https://www.ams.org//publications/authors/AMS-StyleGuide-online.pdf |archive-date=December 8, 2022 |access-date=February 3, 2024}}</ref> All these symbols are generally grouped according to specific rules to form expressions and formulas.<ref>{{cite journal |last1=Jansen |first1=Anthony R. |last2=Marriott |first2=Kim |last3=Yelland |first3=Greg W. |year=2000 |title=Constituent Structure in Mathematical Expressions |journal=Proceedings of the Annual Meeting of the Cognitive Science Society |volume=22 |publisher=[[University of California Merced]] |eissn=1069-7977 |oclc=68713073 |url=https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |url-status=live |archive-url=https://web.archive.org/web/20221116152222/https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |archive-date=November 16, 2022 |access-date=February 3, 2024}}</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 |last=Rossi |first=Richard J. |year=2006 |title=Theorems, Corollaries, Lemmas, and Methods of Proof |series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts |publisher=[[John Wiley & Sons]] |pages=1–14, 47–48 |isbn=978-0-470-04295-3 |lccn=2006041609 |oclc=64085024}}</ref>

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Mathematization of the social sciences is not without risk. 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.<ref>{{Cite web|url=https://www.factcheck.org/2023/01/bidens-misleading-unemployment-statistic/|title=Biden's Misleading Unemployment Statistic – FactCheck.org}}</ref><ref>{{Cite web|url=https://www.minneapolisfed.org/article/2010/modern-macroeconomic-models-as-tools-for-economic-policy|title=Modern Macroeconomic Models as Tools for Economic Policy | Federal Reserve Bank of Minneapolis|website=www.minneapolisfed.org}}</ref>

~~== 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> As of 2023, astrology is no longer considered a science, but [[pseudoscience]].<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 ==

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{{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 ===

[[File:FieldsMedalFront.jpg|thumb|The front side of the [[Fields Medal]] with an illustration of the Greek [[polymath]] [[Archimedes]]]]

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* [[Law (mathematics)]]

* [[List of mathematical jargon]]

* [[Lists of mathematicians]]

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=== Sources ===

* {{cite book |last=Bouleau |first=Nicolas|author-link=Nicolas Bouleau |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}}