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1.5 Discrete mathematics </div> </a> <ul id="toc-Discrete_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_logic_and_set_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_logic_and_set_theory"> <div class="vector-toc-text"> 1.6 Mathematical logic and set theory </div> </a> <ul id="toc-Mathematical_logic_and_set_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistics_and_other_decision_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistics_and_other_decision_sciences"> <div class="vector-toc-text"> 1.7 Statistics and other decision sciences </div> </a> <ul id="toc-Statistics_and_other_decision_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_mathematics"> <div class="vector-toc-text"> 1.8 Computational mathematics </div> </a> <ul id="toc-Computational_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> 2.1 Etymology </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ancient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient"> <div class="vector-toc-text"> 2.2 Ancient </div> </a> <ul id="toc-Ancient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medieval_and_later" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medieval_and_later"> <div class="vector-toc-text"> 2.3 Medieval and later </div> </a> <ul id="toc-Medieval_and_later-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symbolic_notation_and_terminology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Symbolic_notation_and_terminology"> <div class="vector-toc-text"> 3 Symbolic notation and terminology </div> </a> <ul id="toc-Symbolic_notation_and_terminology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_with_sciences" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_with_sciences"> <div class="vector-toc-text"> 4 Relationship with sciences </div> </a> <button aria-controls="toc-Relationship_with_sciences-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relationship with sciences subsection </button> <ul id="toc-Relationship_with_sciences-sublist" class="vector-toc-list"> <li id="toc-Pure_and_applied_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_and_applied_mathematics"> <div class="vector-toc-text"> 4.1 Pure and applied mathematics </div> </a> <ul id="toc-Pure_and_applied_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unreasonable_effectiveness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unreasonable_effectiveness"> <div class="vector-toc-text"> 4.2 Unreasonable effectiveness </div> </a> <ul id="toc-Unreasonable_effectiveness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_sciences"> <div class="vector-toc-text"> 4.3 Specific sciences </div> </a> <ul id="toc-Specific_sciences-sublist" class="vector-toc-list"> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> 4.3.1 Physics </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Computing"> <div class="vector-toc-text"> 4.3.2 Computing </div> </a> <ul id="toc-Computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biology_and_chemistry" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Biology_and_chemistry"> <div class="vector-toc-text"> 4.3.3 Biology and chemistry </div> </a> <ul id="toc-Biology_and_chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Earth_sciences" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Earth_sciences"> <div class="vector-toc-text"> 4.3.4 Earth sciences </div> </a> <ul id="toc-Earth_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Social_sciences" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Social_sciences"> <div class="vector-toc-text"> 4.3.5 Social sciences </div> </a> <ul id="toc-Social_sciences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Philosophy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Philosophy"> <div class="vector-toc-text"> 5 Philosophy </div> </a> <button aria-controls="toc-Philosophy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Philosophy subsection </button> <ul id="toc-Philosophy-sublist" class="vector-toc-list"> <li id="toc-Reality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reality"> <div class="vector-toc-text"> 5.1 Reality </div> </a> <ul id="toc-Reality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proposed_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proposed_definitions"> <div class="vector-toc-text"> 5.2 Proposed definitions </div> </a> <ul id="toc-Proposed_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigor"> <div class="vector-toc-text"> 5.3 Rigor </div> </a> <ul id="toc-Rigor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Training_and_practice" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Training_and_practice"> <div class="vector-toc-text"> 6 Training and practice </div> </a> <button aria-controls="toc-Training_and_practice-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Training and practice subsection </button> <ul id="toc-Training_and_practice-sublist" class="vector-toc-list"> <li id="toc-Education" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Education"> <div class="vector-toc-text"> 6.1 Education </div> </a> <ul id="toc-Education-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Psychology_(aesthetic,_creativity_and_intuition)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Psychology_(aesthetic,_creativity_and_intuition)"> <div class="vector-toc-text"> 6.2 Psychology (aesthetic, creativity and intuition) </div> </a> <ul id="toc-Psychology_(aesthetic,_creativity_and_intuition)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cultural_impact" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cultural_impact"> <div class="vector-toc-text"> 7 Cultural impact </div> </a> <button aria-controls="toc-Cultural_impact-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Cultural impact subsection </button> <ul id="toc-Cultural_impact-sublist" class="vector-toc-list"> <li id="toc-Artistic_expression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Artistic_expression"> <div class="vector-toc-text"> 7.1 Artistic expression </div> </a> <ul id="toc-Artistic_expression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Popularization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Popularization"> <div class="vector-toc-text"> 7.2 Popularization </div> </a> <ul id="toc-Popularization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Awards_and_prize_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Awards_and_prize_problems"> <div class="vector-toc-text"> 7.3 Awards and prize problems </div> </a> <ul id="toc-Awards_and_prize_problems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9.1 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 9.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 9.3 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 10 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Mathematics</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 254 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-254" aria-hidden="true" > 254 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Afrikaans" lang="af" hreflang="af" data-title="Wiskunde" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mathematik" title="Mathematik – Alemannic" lang="gsw" hreflang="gsw" data-title="Mathematik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%B5%E1%88%9D%E1%88%85%E1%88%AD%E1%89%B0_%E1%88%82%E1%88%B3%E1%89%A5" title="ትምህርተ ሂሳብ – Amharic" lang="am" hreflang="am" data-title="ትምህርተ ሂሳብ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Angika" lang="anp" hreflang="anp" data-title="गणित" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/R%C4%ABmcr%C3%A6ft" title="Rīmcræft – Old English" lang="ang" hreflang="ang" data-title="Rīmcræft" data-language-autonym="Ænglisc" data-language-local-name="Old English" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-ar badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="رياضيات – Arabic" lang="ar" hreflang="ar" data-title="رياضيات" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Matematicas" title="Matematicas – Aragonese" lang="an" hreflang="an" data-title="Matematicas" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-roa-rup mw-list-item"><a href="https://roa-rup.wikipedia.org/wiki/Mathematic%C3%A2" title="Mathematicâ – Aromanian" lang="rup" hreflang="rup" data-title="Mathematicâ" data-language-autonym="Armãneashti" data-language-local-name="Aromanian" class="interlanguage-link-target">Armãneashti</a></li><li class="interlanguage-link interwiki-frp mw-list-item"><a href="https://frp.wikipedia.org/wiki/Mat%C3%A8matiques" title="Matèmatiques – Arpitan" lang="frp" hreflang="frp" data-title="Matèmatiques" data-language-autonym="Arpetan" data-language-local-name="Arpitan" class="interlanguage-link-target">Arpetan</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – Assamese" lang="as" hreflang="as" data-title="গণিত" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Matem%C3%A1tiques" title="Matemátiques – Asturian" lang="ast" hreflang="ast" data-title="Matemátiques" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-awa mw-list-item"><a href="https://awa.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Awadhi" lang="awa" hreflang="awa" data-title="गणित" data-language-autonym="अवधी" data-language-local-name="Awadhi" class="interlanguage-link-target">अवधी</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Papapykuaa" title="Papapykuaa – Guarani" lang="gn" hreflang="gn" data-title="Papapykuaa" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-ay mw-list-item"><a href="https://ay.wikipedia.org/wiki/Jakhu" title="Jakhu – Aymara" lang="ay" hreflang="ay" data-title="Jakhu" data-language-autonym="Aymar aru" data-language-local-name="Aymara" class="interlanguage-link-target">Aymar aru</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyaziyyat" title="Riyaziyyat – Azerbaijani" lang="az" hreflang="az" data-title="Riyaziyyat" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="ریاضیات – South Azerbaijani" lang="azb" hreflang="azb" data-title="ریاضیات" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bm mw-list-item"><a href="https://bm.wikipedia.org/wiki/Matematiki" title="Matematiki – Bambara" lang="bm" hreflang="bm" data-title="Matematiki" data-language-autonym="Bamanankan" data-language-local-name="Bambara" class="interlanguage-link-target">Bamanankan</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – Bangla" lang="bn" hreflang="bn" data-title="গণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Matamatika" title="Matamatika – Banjar" lang="bjn" hreflang="bjn" data-title="Matamatika" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target">Banjar</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-ha%CC%8Dk" title="Sò͘-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-map-bms mw-list-item"><a href="https://map-bms.wikipedia.org/wiki/Matematika" title="Matematika – Banyumasan" lang="jv-x-bms" hreflang="jv-x-bms" data-title="Matematika" data-language-autonym="Basa Banyumasan" data-language-local-name="Banyumasan" class="interlanguage-link-target">Basa Banyumasan</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Bashkir" lang="ba" hreflang="ba" data-title="Математика" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – Belarusian" lang="be" hreflang="be" data-title="Матэматыка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Матэматыка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматыка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Bhojpuri" lang="bh" hreflang="bh" data-title="गणित" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target">भोजपुरी</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Matematika" title="Matematika – Central Bikol" lang="bcl" hreflang="bcl" data-title="Matematika" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bi mw-list-item"><a href="https://bi.wikipedia.org/wiki/Matematikis" title="Matematikis – Bislama" lang="bi" hreflang="bi" data-title="Matematikis" data-language-autonym="Bislama" data-language-local-name="Bislama" class="interlanguage-link-target">Bislama</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Bulgarian" lang="bg" hreflang="bg" data-title="Математика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Mathematik" title="Mathematik – Bavarian" lang="bar" hreflang="bar" data-title="Mathematik" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A2%E0%BE%A9%E0%BD%B2%E0%BD%A6%E0%BC%8B%E0%BD%A2%E0%BD%B2%E0%BD%82" title="རྩིས་རིག – Tibetan" lang="bo" hreflang="bo" data-title="རྩིས་རིག" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target">བོད་ཡིག</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matematika" title="Matematika – Bosnian" lang="bs" hreflang="bs" data-title="Matematika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Matematik" title="Matematik – Breton" lang="br" hreflang="br" data-title="Matematik" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D0%B3%D0%BE%D0%B9_%D1%83%D1%85%D0%B0%D0%B0%D0%BD" title="Тоогой ухаан – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Тоогой ухаан" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matem%C3%A0tiques" title="Matemàtiques – Catalan" lang="ca" hreflang="ca" data-title="Matemàtiques" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Chuvash" lang="cv" hreflang="cv" data-title="Математика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Isip" title="Isip – Cebuano" lang="ceb" hreflang="ceb" data-title="Isip" data-language-autonym="Cebuano" data-language-local-name="Cebuano" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematika" title="Matematika – Czech" lang="cs" hreflang="cs" data-title="Matematika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-ch mw-list-item"><a href="https://ch.wikipedia.org/wiki/Matematika" title="Matematika – Chamorro" lang="ch" hreflang="ch" data-title="Matematika" data-language-autonym="Chamoru" data-language-local-name="Chamorro" class="interlanguage-link-target">Chamoru</a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Matematica" title="Matematica – Chavacano" lang="cbk" hreflang="cbk" data-title="Matematica" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target">Chavacano de Zamboanga</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Masvomhu" title="Masvomhu – Shona" lang="sn" hreflang="sn" data-title="Masvomhu" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Matematica" title="Matematica – Corsican" lang="co" hreflang="co" data-title="Matematica" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target">Corsu</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mathemateg" title="Mathemateg – Welsh" lang="cy" hreflang="cy" data-title="Mathemateg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-dag mw-list-item"><a href="https://dag.wikipedia.org/wiki/Laasabu_malibu" title="Laasabu malibu – Dagbani" lang="dag" hreflang="dag" data-title="Laasabu malibu" data-language-autonym="Dagbanli" data-language-local-name="Dagbani" class="interlanguage-link-target">Dagbanli</a></li><li class="interlanguage-link interwiki-da badge-Q17559452 badge-recommendedarticle mw-list-item" title="recommended article"><a href="https://da.wikipedia.org/wiki/Matematik" title="Matematik – Danish" lang="da" hreflang="da" data-title="Matematik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D9%85%D8%A7%D8%B7" title="لماط – Moroccan Arabic" lang="ary" hreflang="ary" data-title="لماط" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Mathematik" title="Mathematik – German" lang="de" hreflang="de" data-title="Mathematik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dv mw-list-item"><a href="https://dv.wikipedia.org/wiki/%DE%83%DE%A8%DE%94%DE%A7%DE%9F%DE%A8%DE%87%DE%B0%DE%94%DE%A7%DE%8C%DE%AA" title="ރިޔާޟިއްޔާތު – Divehi" lang="dv" hreflang="dv" data-title="ރިޔާޟިއްޔާތު" data-language-autonym="ދިވެހިބަސް" data-language-local-name="Divehi" class="interlanguage-link-target">ދިވެހިބަސް</a></li><li class="interlanguage-link interwiki-nv mw-list-item"><a href="https://nv.wikipedia.org/wiki/A%C5%82hii%CA%BCn%C3%ADn%C3%A1%CA%BCiidz%C3%B3%C3%B3h" title="Ałhiiʼnínáʼiidzóóh – Navajo" lang="nv" hreflang="nv" data-title="Ałhiiʼnínáʼiidzóóh" data-language-autonym="Diné bizaad" data-language-local-name="Navajo" class="interlanguage-link-target">Diné bizaad</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Matematika" title="Matematika – Lower Sorbian" lang="dsb" hreflang="dsb" data-title="Matematika" data-language-autonym="Dolnoserbski" data-language-local-name="Lower Sorbian" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatika" title="Matemaatika – Estonian" lang="et" hreflang="et" data-title="Matemaatika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC" title="Μαθηματικά – Greek" lang="el" hreflang="el" data-title="Μαθηματικά" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Matem%C3%A2tica" title="Matemâtica – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Matemâtica" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Erzya" lang="myv" hreflang="myv" data-title="Математика" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Spanish" lang="es" hreflang="es" data-title="Matemáticas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Matematiko" title="Matematiko – Esperanto" lang="eo" hreflang="eo" data-title="Matematiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Extremaduran" lang="ext" hreflang="ext" data-title="Matemáticas" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matematika" title="Matematika – Basque" lang="eu" hreflang="eu" data-title="Matematika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA" title="ریاضیات – Persian" lang="fa" hreflang="fa" data-title="ریاضیات" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematics" title="Mathematics – Fiji Hindi" lang="hif" hreflang="hif" data-title="Mathematics" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/St%C3%B8ddfr%C3%B8%C3%B0i" title="Støddfrøði – Faroese" lang="fo" hreflang="fo" data-title="Støddfrøði" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques" title="Mathématiques – French" lang="fr" hreflang="fr" data-title="Mathématiques" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Western Frisian" lang="fy" hreflang="fy" data-title="Wiskunde" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Matematiche" title="Matematiche – Friulian" lang="fur" hreflang="fur" data-title="Matematiche" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target">Furlan</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Matamaitic" title="Matamaitic – Irish" lang="ga" hreflang="ga" data-title="Matamaitic" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Maddaght" title="Maddaght – Manx" lang="gv" hreflang="gv" data-title="Maddaght" data-language-autonym="Gaelg" data-language-local-name="Manx" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Matamataig" title="Matamataig – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Matamataig" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matem%C3%A1ticas" title="Matemáticas – Galician" lang="gl" hreflang="gl" data-title="Matemáticas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Ingush" lang="inh" hreflang="inh" data-title="Математика" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Gan" lang="gan" hreflang="gan" data-title="數學" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%97%E0%AA%A3%E0%AA%BF%E0%AA%A4" title="ગણિત – Gujarati" lang="gu" hreflang="gu" data-title="ગણિત" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-gom mw-list-item"><a href="https://gom.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4%E0%B3%8D" title="ಗಣಿತ್ – Goan Konkani" lang="gom" hreflang="gom" data-title="ಗಣಿತ್" data-language-autonym="गोंयची कोंकणी / Gõychi Konknni" data-language-local-name="Goan Konkani" class="interlanguage-link-target">गोंयची कोंकणी / Gõychi Konknni</a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Su-ho%CC%8Dk" title="Su-ho̍k – Hakka Chinese" lang="hak" hreflang="hak" data-title="Su-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="Hakka Chinese" class="interlanguage-link-target">客家語 / Hak-kâ-ngî</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Kalmyk" lang="xal" hreflang="xal" data-title="Математика" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99" title="수학 – Korean" lang="ko" hreflang="ko" data-title="수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Lissafi" title="Lissafi – Hausa" lang="ha" hreflang="ha" data-title="Lissafi" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target">Hausa</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/Makemakika" title="Makemakika – Hawaiian" lang="haw" hreflang="haw" data-title="Makemakika" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1" title="Մաթեմատիկա – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Hindi" lang="hi" hreflang="hi" data-title="गणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matematika" title="Matematika – Croatian" lang="hr" hreflang="hr" data-title="Matematika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Matematika" title="Matematika – Gorontalo" lang="gor" hreflang="gor" data-title="Matematika" data-language-autonym="Bahasa Hulontalo" data-language-local-name="Gorontalo" class="interlanguage-link-target">Bahasa Hulontalo</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Matematiko" title="Matematiko – Ido" lang="io" hreflang="io" data-title="Matematiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/%E1%BB%8Cm%C3%BAm%C3%BA-%C3%B3n%C3%BA%E1%BB%8Dg%E1%BB%A5g%E1%BB%A5" title="Ọmúmú-ónúọgụgụ – Igbo" lang="ig" hreflang="ig" data-title="Ọmúmú-ónúọgụgụ" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Matematika" title="Matematika – Iloko" lang="ilo" hreflang="ilo" data-title="Matematika" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-bpy mw-list-item"><a href="https://bpy.wikipedia.org/wiki/%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="গণিত – Bishnupriya" lang="bpy" hreflang="bpy" data-title="গণিত" data-language-autonym="বিষ্ণুপ্রিয়া মণিপুরী" data-language-local-name="Bishnupriya" class="interlanguage-link-target">বিষ্ণুপ্রিয়া মণিপুরী</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matematika" title="Matematika – Indonesian" lang="id" hreflang="id" data-title="Matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ia.wikipedia.org/wiki/Mathematica" title="Mathematica – Interlingua" lang="ia" hreflang="ia" data-title="Mathematica" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Matematica" title="Matematica – Interlingue" lang="ie" hreflang="ie" data-title="Matematica" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C3%A6" title="Математикæ – Ossetic" lang="os" hreflang="os" data-title="Математикæ" data-language-autonym="Ирон" data-language-local-name="Ossetic" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Mathematics" title="I-Mathematics – Xhosa" lang="xh" hreflang="xh" data-title="I-Mathematics" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Umchazazibalo" title="Umchazazibalo – Zulu" lang="zu" hreflang="zu" data-title="Umchazazibalo" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0i" title="Stærðfræði – Icelandic" lang="is" hreflang="is" data-title="Stærðfræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matematica" title="Matematica – Italian" lang="it" hreflang="it" data-title="Matematica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה – Hebrew" lang="he" hreflang="he" data-title="מתמטיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://jv.wikipedia.org/wiki/Mat%C3%A9matika" title="Matématika – Javanese" lang="jv" hreflang="jv" data-title="Matématika" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kl mw-list-item"><a href="https://kl.wikipedia.org/wiki/Matematikki" title="Matematikki – Kalaallisut" lang="kl" hreflang="kl" data-title="Matematikki" data-language-autonym="Kalaallisut" data-language-local-name="Kalaallisut" class="interlanguage-link-target">Kalaallisut</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ – Kannada" lang="kn" hreflang="kn" data-title="ಗಣಿತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-pam mw-list-item"><a href="https://pam.wikipedia.org/wiki/Matematica" title="Matematica – Pampanga" lang="pam" hreflang="pam" data-title="Matematica" data-language-autonym="Kapampangan" data-language-local-name="Pampanga" class="interlanguage-link-target">Kapampangan</a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Karachay-Balkar" lang="krc" hreflang="krc" data-title="Математика" data-language-autonym="Къарачай-малкъар" data-language-local-name="Karachay-Balkar" class="interlanguage-link-target">Къарачай-малкъар</a></li><li class="interlanguage-link interwiki-ka badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკა – Georgian" lang="ka" hreflang="ka" data-title="მათემატიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/%DA%AF%D8%B1%D9%9B%D8%A0%D9%86%D8%AF%D9%8F%D8%AA" title="گرٛؠندُت – Kashmiri" lang="ks" hreflang="ks" data-title="گرٛؠندُت" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="Kashmiri" class="interlanguage-link-target">कॉशुर / کٲشُر</a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Matematika" title="Matematika – Kashubian" lang="csb" hreflang="csb" data-title="Matematika" data-language-autonym="Kaszëbsczi" data-language-local-name="Kashubian" class="interlanguage-link-target">Kaszëbsczi</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Kazakh" lang="kk" hreflang="kk" data-title="Математика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Awgrym" title="Awgrym – Cornish" lang="kw" hreflang="kw" data-title="Awgrym" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Hisabati" title="Hisabati – Swahili" lang="sw" hreflang="sw" data-title="Hisabati" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-avk mw-list-item"><a href="https://avk.wikipedia.org/wiki/Solokseropa" title="Solokseropa – Kotava" lang="avk" hreflang="avk" data-title="Solokseropa" data-language-autonym="Kotava" data-language-local-name="Kotava" class="interlanguage-link-target">Kotava</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Matematik" title="Matematik – Haitian Creole" lang="ht" hreflang="ht" data-title="Matematik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Mat%C3%A9matik" title="Matématik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Matématik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Matemat%C3%AEk" title="Matematîk – Kurdish" lang="ku" hreflang="ku" data-title="Matematîk" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BC%D0%BE/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Википедия:Сомо/Математика – Kyrgyz" lang="ky" hreflang="ky" data-title="Википедия:Сомо/Математика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lld mw-list-item"><a href="https://lld.wikipedia.org/wiki/Matematica" title="Matematica – Ladin" lang="lld" hreflang="lld" data-title="Matematica" data-language-autonym="Ladin" data-language-local-name="Ladin" class="interlanguage-link-target">Ladin</a></li><li class="interlanguage-link interwiki-lad mw-list-item"><a href="https://lad.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – Ladino" lang="lad" hreflang="lad" data-title="Matemátika" data-language-autonym="Ladino" data-language-local-name="Ladino" class="interlanguage-link-target">Ladino</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ຄະນິດສາດ – Lao" lang="lo" hreflang="lo" data-title="ຄະນິດສາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://la.wikipedia.org/wiki/Mathematica" title="Mathematica – Latin" lang="la" hreflang="la" data-title="Mathematica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tika" title="Matemātika – Latvian" lang="lv" hreflang="lv" data-title="Matemātika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mathematik" title="Mathematik – Luxembourgish" lang="lb" hreflang="lb" data-title="Mathematik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Lezghian" lang="lez" hreflang="lez" data-title="Математика" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target">Лезги</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematika" title="Matematika – Lithuanian" lang="lt" hreflang="lt" data-title="Matematika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-nia mw-list-item"><a href="https://nia.wikipedia.org/wiki/Matematika" title="Matematika – Nias" lang="nia" hreflang="nia" data-title="Matematika" data-language-autonym="Li Niha" data-language-local-name="Nias" class="interlanguage-link-target">Li Niha</a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Matematica" title="Matematica – Ligurian" lang="lij" hreflang="lij" data-title="Matematica" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target">Ligure</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Mathematiek" title="Mathematiek – Limburgish" lang="li" hreflang="li" data-title="Mathematiek" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Matematica" title="Matematica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Matematica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-olo mw-list-item"><a href="https://olo.wikipedia.org/wiki/Matematiekku" title="Matematiekku – Livvi-Karelian" lang="olo" hreflang="olo" data-title="Matematiekku" data-language-autonym="Livvinkarjala" data-language-local-name="Livvi-Karelian" class="interlanguage-link-target">Livvinkarjala</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/cmaci" title="cmaci – Lojban" lang="jbo" hreflang="jbo" data-title="cmaci" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Ekibalangulo" title="Ekibalangulo – Ganda" lang="lg" hreflang="lg" data-title="Ekibalangulo" data-language-autonym="Luganda" data-language-local-name="Ganda" class="interlanguage-link-target">Luganda</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matematega" title="Matematega – Lombard" lang="lmo" hreflang="lmo" data-title="Matematega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematika" title="Matematika – Hungarian" lang="hu" hreflang="hu" data-title="Matematika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mad mw-list-item"><a href="https://mad.wikipedia.org/wiki/Matematika" title="Matematika – Madurese" lang="mad" hreflang="mad" data-title="Matematika" data-language-autonym="Madhurâ" data-language-local-name="Madurese" class="interlanguage-link-target">Madhurâ</a></li><li class="interlanguage-link interwiki-mai mw-list-item"><a href="https://mai.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Maithili" lang="mai" hreflang="mai" data-title="गणित" data-language-autonym="मैथिली" data-language-local-name="Maithili" class="interlanguage-link-target">मैथिली</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Macedonian" lang="mk" hreflang="mk" data-title="Математика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Matematika" title="Matematika – Malagasy" lang="mg" hreflang="mg" data-title="Matematika" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ഗണിതം – Malayalam" lang="ml" hreflang="ml" data-title="ഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Matematika" title="Matematika – Maltese" lang="mt" hreflang="mt" data-title="Matematika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Marathi" lang="mr" hreflang="mr" data-title="गणित" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკა – Mingrelian" lang="xmf" hreflang="xmf" data-title="მათემატიკა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA" title="رياضيات – Egyptian Arabic" lang="arz" hreflang="arz" data-title="رياضيات" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-mnw mw-list-item"><a href="https://mnw.wikipedia.org/wiki/%E1%80%9E%E1%81%9A%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သၚ်္ချာ – Mon" lang="mnw" hreflang="mnw" data-title="သၚ်္ချာ" data-language-autonym="ဘာသာမန်" data-language-local-name="Mon" class="interlanguage-link-target">ဘာသာမန်</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%B1%D9%82%D9%85" title="رقم – Mazanderani" lang="mzn" hreflang="mzn" data-title="رقم" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matematik" title="Matematik – Malay" lang="ms" hreflang="ms" data-title="Matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mni mw-list-item"><a href="https://mni.wikipedia.org/wiki/%EA%AF%86%EA%AF%A9%EA%AF%81%EA%AF%A4%EA%AF%A1%EA%AF%82%EA%AF%A3%EA%AF%9F" title="ꯆꯩꯁꯤꯡꯂꯣꯟ – Manipuri" lang="mni" hreflang="mni" data-title="ꯆꯩꯁꯤꯡꯂꯣꯟ" data-language-autonym="ꯃꯤꯇꯩ ꯂꯣꯟ" data-language-local-name="Manipuri" class="interlanguage-link-target">ꯃꯤꯇꯩ ꯂꯣꯟ</a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Matematika" title="Matematika – Minangkabau" lang="min" hreflang="min" data-title="Matematika" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target">Minangkabau</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/S%C3%B3-h%C5%8Fk" title="Só-hŏk – Mindong" lang="cdo" hreflang="cdo" data-title="Só-hŏk" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – Mirandese" lang="mwl" hreflang="mwl" data-title="Matemática" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA" title="Математик – Mongolian" lang="mn" hreflang="mn" data-title="Математик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သင်္ချာ – Burmese" lang="my" hreflang="my" data-title="သင်္ချာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nah mw-list-item"><a href="https://nah.wikipedia.org/wiki/Tlap%C5%8Dhualmatiliztli" title="Tlapōhualmatiliztli – Nahuatl" lang="nah" hreflang="nah" data-title="Tlapōhualmatiliztli" data-language-autonym="Nāhuatl" data-language-local-name="Nahuatl" class="interlanguage-link-target">Nāhuatl</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Fika" title="Fika – Fijian" lang="fj" hreflang="fj" data-title="Fika" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Dutch" lang="nl" hreflang="nl" data-title="Wiskunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nds-nl mw-list-item"><a href="https://nds-nl.wikipedia.org/wiki/Wiskunde" title="Wiskunde – Low Saxon" lang="nds-NL" hreflang="nds-NL" data-title="Wiskunde" data-language-autonym="Nedersaksies" data-language-local-name="Low Saxon" class="interlanguage-link-target">Nedersaksies</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="गणित – Nepali" lang="ne" hreflang="ne" data-title="गणित" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B2%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%83%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="ल्याःज्या – Newari" lang="new" hreflang="new" data-title="ल्याःज्या" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – Japanese" lang="ja" hreflang="ja" data-title="数学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Chechen" lang="ce" hreflang="ce" data-title="Математика" data-language-autonym="Нохчийн" data-language-local-name="Chechen" class="interlanguage-link-target">Нохчийн</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Matematiik" title="Matematiik – Northern Frisian" lang="frr" hreflang="frr" data-title="Matematiik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-pih mw-list-item"><a href="https://pih.wikipedia.org/wiki/Maethamatiks" title="Maethamatiks – Norfuk / Pitkern" lang="pih" hreflang="pih" data-title="Maethamatiks" data-language-autonym="Norfuk / Pitkern" data-language-local-name="Norfuk / Pitkern" class="interlanguage-link-target">Norfuk / Pitkern</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematikk" title="Matematikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematikk" title="Matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Caltchul" title="Caltchul – Norman" lang="nrf" hreflang="nrf" data-title="Caltchul" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target">Nouormand</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Matematike" title="Matematike – Novial" lang="nov" hreflang="nov" data-title="Matematike" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Matematicas" title="Matematicas – Occitan" lang="oc" hreflang="oc" data-title="Matematicas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B5" title="Математике – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Математике" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target">Олык марий</a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%97%E0%AC%A3%E0%AC%BF%E0%AC%A4" title="ଗଣିତ – Odia" lang="or" hreflang="or" data-title="ଗଣିତ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target">ଓଡ଼ିଆ</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Herrega" title="Herrega – Oromo" lang="om" hreflang="om" data-title="Herrega" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematika" title="Matematika – Uzbek" lang="uz" hreflang="uz" data-title="Matematika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4" title="ਗਣਿਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਣਿਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pi mw-list-item"><a href="https://pi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%82" title="गणितं – Pali" lang="pi" hreflang="pi" data-title="गणितं" data-language-autonym="पालि" data-language-local-name="Pali" class="interlanguage-link-target">पालि</a></li><li class="interlanguage-link interwiki-pag mw-list-item"><a href="https://pag.wikipedia.org/wiki/Matematiks" title="Matematiks – Pangasinan" lang="pag" hreflang="pag" data-title="Matematiks" data-language-autonym="Pangasinan" data-language-local-name="Pangasinan" class="interlanguage-link-target">Pangasinan</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ریاضی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-blk mw-list-item"><a href="https://blk.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC" title="သင်္ချာ – Pa'O" lang="blk" hreflang="blk" data-title="သင်္ချာ" data-language-autonym="ပအိုဝ်ႏဘာႏသာႏ" data-language-local-name="Pa'O" class="interlanguage-link-target">ပအိုဝ်ႏဘာႏသာႏ</a></li><li class="interlanguage-link interwiki-pap mw-list-item"><a href="https://pap.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – Papiamento" lang="pap" hreflang="pap" data-title="Matemátika" data-language-autonym="Papiamentu" data-language-local-name="Papiamento" class="interlanguage-link-target">Papiamentu</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B4%D9%85%DB%90%D8%B1%D9%BE%D9%88%D9%87%D9%86%D9%87" title="شمېرپوهنه – Pashto" lang="ps" hreflang="ps" data-title="شمېرپوهنه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatix" title="Matimatix – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Matimatix" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%82%E1%9E%8E%E1%9E%B7%E1%9E%8F%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6" title="គណិតវិទ្យា – Khmer" lang="km" hreflang="km" data-title="គណិតវិទ្យា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/Mat%C3%A9matikes" title="Matématikes – Picard" lang="pcd" hreflang="pcd" data-title="Matématikes" data-language-autonym="Picard" data-language-local-name="Picard" class="interlanguage-link-target">Picard</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – Piedmontese" lang="pms" hreflang="pms" data-title="Matemàtica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-tpi mw-list-item"><a href="https://tpi.wikipedia.org/wiki/Ol_matematik" title="Ol matematik – Tok Pisin" lang="tpi" hreflang="tpi" data-title="Ol matematik" data-language-autonym="Tok Pisin" data-language-local-name="Tok Pisin" class="interlanguage-link-target">Tok Pisin</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Mathematik" title="Mathematik – Low German" lang="nds" hreflang="nds" data-title="Mathematik" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Matematyka" title="Matematyka – Polish" lang="pl" hreflang="pl" data-title="Matematyka" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matem%C3%A1tica" title="Matemática – Portuguese" lang="pt" hreflang="pt" data-title="Matemática" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Matematika" title="Matematika – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Matematika" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target">Qaraqalpaqsha</a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Riyaziyat" title="Riyaziyat – Crimean Tatar" lang="crh" hreflang="crh" data-title="Riyaziyat" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target">Qırımtatarca</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matematic%C4%83" title="Matematică – Romanian" lang="ro" hreflang="ro" data-title="Matematică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Yupay_yachay" title="Yupay yachay – Quechua" lang="qu" hreflang="qu" data-title="Yupay yachay" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D1%96%D0%BA%D0%B0" title="Математіка – Rusyn" lang="rue" hreflang="rue" data-title="Математіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Russian" lang="ru" hreflang="ru" data-title="Математика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Yakut" lang="sah" hreflang="sah" data-title="Математика" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sm mw-list-item"><a href="https://sm.wikipedia.org/wiki/Matematika" title="Matematika – Samoan" lang="sm" hreflang="sm" data-title="Matematika" data-language-autonym="Gagana Samoa" data-language-local-name="Samoan" class="interlanguage-link-target">Gagana Samoa</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%AE%E0%A5%8D" title="गणितम् – Sanskrit" lang="sa" hreflang="sa" data-title="गणितम्" data-language-autonym="संस्कृतम्" data-language-local-name="Sanskrit" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%AE%E1%B1%9E%E1%B1%A0%E1%B1%B7%E1%B1%9F" title="ᱮᱞᱠᱷᱟ – Santali" lang="sat" hreflang="sat" data-title="ᱮᱞᱠᱷᱟ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Saraiki" lang="skr" hreflang="skr" data-title="ریاضی" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target">سرائیکی</a></li><li class="interlanguage-link interwiki-sc badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://sc.wikipedia.org/wiki/Matem%C3%A0tica" title="Matemàtica – Sardinian" lang="sc" hreflang="sc" data-title="Matemàtica" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target">Sardu</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Mathematics" title="Mathematics – Scots" lang="sco" hreflang="sco" data-title="Mathematics" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Mathematik" title="Mathematik – Saterland Frisian" lang="stq" hreflang="stq" data-title="Mathematik" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target">Seeltersk</a></li><li class="interlanguage-link interwiki-tn mw-list-item"><a href="https://tn.wikipedia.org/wiki/Dipalo" title="Dipalo – Tswana" lang="tn" hreflang="tn" data-title="Dipalo" data-language-autonym="Setswana" data-language-local-name="Tswana" class="interlanguage-link-target">Setswana</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matematika" title="Matematika – Albanian" lang="sq" hreflang="sq" data-title="Matematika" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Matim%C3%A0tica" title="Matimàtica – Sicilian" lang="scn" hreflang="scn" data-title="Matimàtica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="ගණිතය – Sinhala" lang="si" hreflang="si" data-title="ගණිතය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematics" title="Mathematics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="رياضي – Sindhi" lang="sd" hreflang="sd" data-title="رياضي" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-ss mw-list-item"><a href="https://ss.wikipedia.org/wiki/Tekubala" title="Tekubala – Swati" lang="ss" hreflang="ss" data-title="Tekubala" data-language-autonym="SiSwati" data-language-local-name="Swati" class="interlanguage-link-target">SiSwati</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematika" title="Matematika – Slovak" lang="sk" hreflang="sk" data-title="Matematika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matematika" title="Matematika – Slovenian" lang="sl" hreflang="sl" data-title="Matematika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-cu mw-list-item"><a href="https://cu.wikipedia.org/wiki/%D0%9C%D0%B0%D1%B3%D0%B8%D0%BC%D0%B0%D1%82%D1%97%D0%BA%D0%B0" title="Маѳиматїка – Church Slavic" lang="cu" hreflang="cu" data-title="Маѳиматїка" data-language-autonym="Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ" data-language-local-name="Church Slavic" class="interlanguage-link-target">Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Matymatyka" title="Matymatyka – Silesian" lang="szl" hreflang="szl" data-title="Matymatyka" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tiromaal" title="Tiromaal – Somali" lang="so" hreflang="so" data-title="Tiromaal" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9" title="ماتماتیک – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ماتماتیک" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-srn mw-list-item"><a href="https://srn.wikipedia.org/wiki/Sabi_fu_Teri" title="Sabi fu Teri – Sranan Tongo" lang="srn" hreflang="srn" data-title="Sabi fu Teri" data-language-autonym="Sranantongo" data-language-local-name="Sranan Tongo" class="interlanguage-link-target">Sranantongo</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Serbian" lang="sr" hreflang="sr" data-title="Математика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matematika" title="Matematika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Matematika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Matematika" title="Matematika – Sundanese" lang="su" hreflang="su" data-title="Matematika" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matematiikka" title="Matematiikka – Finnish" lang="fi" hreflang="fi" data-title="Matematiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematik" title="Matematik – Swedish" lang="sv" hreflang="sv" data-title="Matematik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Matematika" title="Matematika – Tagalog" lang="tl" hreflang="tl" data-title="Matematika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="கணிதம் – Tamil" lang="ta" hreflang="ta" data-title="கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tusnakt" title="Tusnakt – Tachelhit" lang="shi" hreflang="shi" data-title="Tusnakt" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tusnakt" title="Tusnakt – Kabyle" lang="kab" hreflang="kab" data-title="Tusnakt" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Tatar" lang="tt" hreflang="tt" data-title="Математика" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B0%A3%E0%B0%BF%E0%B0%A4%E0%B0%82" title="గణితం – Telugu" lang="te" hreflang="te" data-title="గణితం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-tet mw-list-item"><a href="https://tet.wikipedia.org/wiki/Matem%C3%A1tika" title="Matemátika – Tetum" lang="tet" hreflang="tet" data-title="Matemátika" data-language-autonym="Tetun" data-language-local-name="Tetum" class="interlanguage-link-target">Tetun</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="คณิตศาสตร์ – Thai" lang="th" hreflang="th" data-title="คณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-din mw-list-item"><a href="https://din.wikipedia.org/wiki/Aku%C3%ABnk%C3%A4%C5%8B" title="Akuënkäŋ – Dinka" lang="din" hreflang="din" data-title="Akuënkäŋ" data-language-autonym="Thuɔŋjäŋ" data-language-local-name="Dinka" class="interlanguage-link-target">Thuɔŋjäŋ</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82" title="Риёзиёт – Tajik" lang="tg" hreflang="tg" data-title="Риёзиёт" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tcy mw-list-item"><a href="https://tcy.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4" title="ಗಣಿತ – Tulu" lang="tcy" hreflang="tcy" data-title="ಗಣಿತ" data-language-autonym="ತುಳು" data-language-local-name="Tulu" class="interlanguage-link-target">ತುಳು</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematik" title="Matematik – Turkish" lang="tr" hreflang="tr" data-title="Matematik" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Matematika" title="Matematika – Turkmen" lang="tk" hreflang="tk" data-title="Matematika" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-udm mw-list-item"><a href="https://udm.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Udmurt" lang="udm" hreflang="udm" data-title="Математика" data-language-autonym="Удмурт" data-language-local-name="Udmurt" class="interlanguage-link-target">Удмурт</a></li><li class="interlanguage-link interwiki-bug mw-list-item"><a href="https://bug.wikipedia.org/wiki/%E1%A8%86%E1%A8%88%E1%A8%9B%E1%A8%86%E1%A8%88%E1%A8%97%E1%A8%80" title="ᨆᨈᨛᨆᨈᨗᨀ – Buginese" lang="bug" hreflang="bug" data-title="ᨆᨈᨛᨆᨈᨗᨀ" data-language-autonym="Basa Ugi" data-language-local-name="Buginese" class="interlanguage-link-target">Basa Ugi</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Математика – Ukrainian" lang="uk" hreflang="uk" data-title="Математика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="ریاضی – Urdu" lang="ur" hreflang="ur" data-title="ریاضی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-za mw-list-item"><a href="https://za.wikipedia.org/wiki/Soqyoz" title="Soqyoz – Zhuang" lang="za" hreflang="za" data-title="Soqyoz" data-language-autonym="Vahcuengh" data-language-local-name="Zhuang" class="interlanguage-link-target">Vahcuengh</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Matem%C3%A0tega" title="Matemàtega – Venetian" lang="vec" hreflang="vec" data-title="Matemàtega" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Matematik" title="Matematik – Veps" lang="vep" hreflang="vep" data-title="Matematik" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_h%E1%BB%8Dc" title="Toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://vo.wikipedia.org/wiki/Matemat" title="Matemat – Volapük" lang="vo" hreflang="vo" data-title="Matemat" data-language-autonym="Volapük" data-language-local-name="Volapük" class="interlanguage-link-target">Volapük</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Mat%C3%B5maatiga" title="Matõmaatiga – Võro" lang="vro" hreflang="vro" data-title="Matõmaatiga" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Matematike" title="Matematike – Walloon" lang="wa" hreflang="wa" data-title="Matematike" data-language-autonym="Walon" data-language-local-name="Walloon" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8" title="數學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="數學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Wiskunde" title="Wiskunde – West Flemish" lang="vls" hreflang="vls" data-title="Wiskunde" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Matematika" title="Matematika – Waray" lang="war" hreflang="war" data-title="Matematika" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Xayma" title="Xayma – Wolof" lang="wo" hreflang="wo" data-title="Xayma" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target">Wolof</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6" title="数学 – Wu" lang="wuu" hreflang="wuu" data-title="数学" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Dyondzo-Tinhlayo" title="Dyondzo-Tinhlayo – Tsonga" lang="ts" hreflang="ts" data-title="Dyondzo-Tinhlayo" data-language-autonym="Xitsonga" data-language-local-name="Tsonga" class="interlanguage-link-target">Xitsonga</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7" title="מאטעמאטיק – Yiddish" lang="yi" hreflang="yi" data-title="מאטעמאטיק" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a class="mw-selflink selflink">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;"> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup"> <table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="/wiki/Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Number_theory" title="Number theory">Number theory</a></li> <li><a href="/wiki/Geometry" title="Geometry">Geometry</a></li> <li><a href="/wiki/Algebra" title="Algebra">Algebra</a></li> <li><a href="/wiki/Calculus" title="Calculus">Calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></li> <li><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Logic</a> and <a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a> and <a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Physics</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li> <li><a href="/wiki/Geomathematics" title="Geomathematics">Geosciences</a></li> <li><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computation</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li> <li><a href="/wiki/Computational_linguistics" title="Computational linguistics">Linguistics</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Economics</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td> </tr></tbody></table></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/20px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/30px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/40px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics Portal</a></th></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Math_topics_sidebar" title="Template:Math topics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Math_topics_sidebar" title="Template talk:Math topics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Math_topics_sidebar" title="Special:EditPage/Template:Math topics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Mathematics is a field of study that discovers and organizes methods, <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">theories</a> and <a href="/wiki/Theorem" title="Theorem">theorems</a> that are developed and <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a> for the needs of <a href="/wiki/Empirical_sciences" class="mw-redirect" title="Empirical sciences">empirical sciences</a> and mathematics itself. There are many areas of mathematics, which include <a href="/wiki/Number_theory" title="Number theory">number theory</a> (the study of numbers), <a href="/wiki/Algebra" title="Algebra">algebra</a> (the study of formulas and related structures), <a href="/wiki/Geometry" title="Geometry">geometry</a> (the study of shapes and spaces that contain them), <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a> (the study of continuous changes), and <a href="/wiki/Set_theory" title="Set theory">set theory</a> (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of <a href="/wiki/Mathematical_object" title="Mathematical object">abstract objects</a> that consist of either <a href="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">abstractions</a> from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called <a href="/wiki/Axiom" title="Axiom">axioms</a>. Mathematics uses pure <a href="/wiki/Reason" title="Reason">reason</a> to <a href="/wiki/Proof_(mathematics)" class="mw-redirect" title="Proof (mathematics)">prove</a> properties of objects, a proof consisting of a succession of applications of <a href="/wiki/Inference_rule" class="mw-redirect" title="Inference rule">deductive rules</a> to already established results. These results include previously proved <a href="/wiki/Theorem" title="Theorem">theorems</a>, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.<a href="#cite_note-1">[1]</a> Mathematics is essential in the <a href="/wiki/Natural_science" title="Natural science">natural sciences</a>, <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Medicine" title="Medicine">medicine</a>, <a href="/wiki/Finance" title="Finance">finance</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, and the <a href="/wiki/Social_sciences" class="mw-redirect" title="Social sciences">social sciences</a>. 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 <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Game_theory" title="Game theory">game theory</a>, are developed in close correlation with their applications and are often grouped under <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>. Other areas are developed independently from any application (and are therefore called <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a>) but often later find practical applications.<a href="#cite_note-FOOTNOTEPeterson198812-2">[2]</a><a href="#cite_note-wigner1960-3">[3]</a> Historically, the concept of a proof and its associated <a href="/wiki/Mathematical_rigour" class="mw-redirect" title="Mathematical rigour">mathematical rigour</a> first appeared in <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a>, most notably in <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>.<a href="#cite_note-4">[4]</a> Since its beginning, mathematics was primarily divided into geometry and <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> (the manipulation of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> and <a href="/wiki/Fractions" class="mw-redirect" title="Fractions">fractions</a>), until the 16th and 17th centuries, when algebra<a href="#cite_note-5">[a]</a> and <a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">infinitesimal calculus</a> were introduced as new fields. Since then, the interaction between mathematical innovations and <a href="/wiki/Timeline_of_scientific_discoveries" title="Timeline of scientific discoveries">scientific discoveries</a> has led to a correlated increase in the development of both.<a href="#cite_note-6">[5]</a> At the end of the 19th century, the <a href="/wiki/Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a> led to the systematization of the <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic method</a>,<a href="#cite_note-Kleiner_1991-7">[6]</a> which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary <a href="/wiki/Mathematics_Subject_Classification" title="Mathematics Subject Classification">Mathematics Subject Classification</a> lists more than sixty first-level areas of mathematics. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Areas_of_mathematics">Areas of mathematics</h2></div> Before the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>, mathematics was divided into two main areas: <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, regarding the manipulation of numbers, and <a href="/wiki/Geometry" title="Geometry">geometry</a>, regarding the study of shapes.<a href="#cite_note-8">[7]</a> Some types of <a href="/wiki/Pseudoscience" title="Pseudoscience">pseudoscience</a>, such as <a href="/wiki/Numerology" title="Numerology">numerology</a> and <a href="/wiki/Astrology" title="Astrology">astrology</a>, were not then clearly distinguished from mathematics.<a href="#cite_note-9">[8]</a> During the Renaissance, two more areas appeared. <a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a> led to <a href="/wiki/Algebra" title="Algebra">algebra</a> which, roughly speaking, consists of the study and the manipulation of <a href="/wiki/Formula" title="Formula">formulas</a>. <a href="/wiki/Calculus" title="Calculus">Calculus</a>, consisting of the two subfields <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> and <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral calculus</a>, is the study of <a href="/wiki/Continuous_functions" class="mw-redirect" title="Continuous functions">continuous functions</a>, which model the typically <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear relationships</a> between varying quantities, as represented by <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>. This division into four main areas—arithmetic, geometry, algebra, and calculus<a href="#cite_note-10">[9]</a>—endured until the end of the 19th century. Areas such as <a href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</a> and <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a> were then studied by mathematicians, but now are considered as belonging to physics.<a href="#cite_note-11">[10]</a> The subject of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> has been studied for much of recorded history, yet did not become a separate branch of mathematics until the seventeenth century.<a href="#cite_note-12">[11]</a> At the end of the 19th century, the <a href="/wiki/Foundational_crisis_in_mathematics" class="mw-redirect" title="Foundational crisis in mathematics">foundational crisis in mathematics</a> and the resulting systematization of the <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic method</a> led to an explosion of new areas of mathematics.<a href="#cite_note-Warner_2013-13">[12]</a><a href="#cite_note-Kleiner_1991-7">[6]</a> The 2020 <a href="/wiki/Mathematics_Subject_Classification" title="Mathematics Subject Classification">Mathematics Subject Classification</a> contains no less than sixty-three first-level areas.<a href="#cite_note-14">[13]</a> Some of these areas correspond to the older division, as is true regarding <a href="/wiki/Number_theory" title="Number theory">number theory</a> (the modern name for <a href="/wiki/Higher_arithmetic" class="mw-redirect" title="Higher arithmetic">higher arithmetic</a>) 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 <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations</a>.<a href="#cite_note-MSC-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Number_theory" title="Number theory">Number theory</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spirale_Ulam_150.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/220px-Spirale_Ulam_150.jpg" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/330px-Spirale_Ulam_150.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Spirale_Ulam_150.jpg/440px-Spirale_Ulam_150.jpg 2x" data-file-width="750" data-file-height="752" /></a><figcaption>This is the <a href="/wiki/Ulam_spiral" title="Ulam spiral">Ulam spiral</a>, which illustrates the distribution of <a href="/wiki/Prime_numbers" class="mw-redirect" title="Prime numbers">prime numbers</a>. The dark diagonal lines in the spiral hint at the hypothesized approximate <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independence</a> between being prime and being a value of a quadratic polynomial, a conjecture now known as <a href="/wiki/Ulam_spiral#Hardy_and_Littlewood's_Conjecture_F" title="Ulam spiral">Hardy and Littlewood's Conjecture F</a>.</figcaption></figure> Number theory began with the manipulation of <a href="/wiki/Number" title="Number">numbers</a>, that is, <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea55fe9f7155b2404c5e56b357334e6c303740b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ),}"> and later expanded to <a href="/wiki/Integer" title="Integer">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7c394740649bbc95dc603df09ceebaac387c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} )}"> and <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe972c32bdfc7274838beb71596b2001c2bd6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.264ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ).}"> Number theory was once called arithmetic, but nowadays this term is mostly used for <a href="/wiki/Numerical_calculation" class="mw-redirect" title="Numerical calculation">numerical calculations</a>.<a href="#cite_note-16">[15]</a> Number theory dates back to ancient <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Babylon</a> and probably <a href="/wiki/Ancient_China" class="mw-redirect" title="Ancient China">China</a>. Two prominent early number theorists were <a href="/wiki/Euclid" title="Euclid">Euclid</a> of ancient Greece and <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> of Alexandria.<a href="#cite_note-17">[16]</a> The modern study of number theory in its abstract form is largely attributed to <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>. The field came to full fruition with the contributions of <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> and <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>.<a href="#cite_note-18">[17]</a> Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. This conjecture was stated in 1637 by Pierre de Fermat, but it <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">was proved</a> only in 1994 by <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>, who used tools including <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a> from <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/wiki/Category_theory" title="Category theory">category theory</a>, and <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>.<a href="#cite_note-19">[18]</a> Another example is <a href="/wiki/Goldbach%27s_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a>, which asserts that every even integer greater than 2 is the sum of two <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. Stated in 1742 by <a href="/wiki/Christian_Goldbach" title="Christian Goldbach">Christian Goldbach</a>, it remains unproven despite considerable effort.<a href="#cite_note-20">[19]</a> Number theory includes several subareas, including <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>, <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a> (method oriented), <a href="/wiki/Diophantine_equation" title="Diophantine equation">diophantine equations</a>, and <a href="/wiki/Transcendence_theory" class="mw-redirect" title="Transcendence theory">transcendence theory</a> (problem oriented).<a href="#cite_note-MSC-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Geometry">Geometry</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geometry" title="Geometry">Geometry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Triangles_(spherical_geometry).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/220px-Triangles_%28spherical_geometry%29.jpg" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/330px-Triangles_%28spherical_geometry%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Triangles_%28spherical_geometry%29.jpg/440px-Triangles_%28spherical_geometry%29.jpg 2x" data-file-width="2489" data-file-height="2048" /></a><figcaption>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°.</figcaption></figure> Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <a href="/wiki/Angle" title="Angle">angles</a> and <a href="/wiki/Circle" title="Circle">circles</a>, which were developed mainly for the needs of <a href="/wiki/Surveying" title="Surveying">surveying</a> and <a href="/wiki/Architecture" title="Architecture">architecture</a>, but has since blossomed out into many other subfields.<a href="#cite_note-Straume_2014-21">[20]</a> A fundamental innovation was the ancient Greeks' introduction of the concept of <a href="/wiki/Mathematical_proof" title="Mathematical proof">proofs</a>, which require that every assertion must be proved. For example, it is not sufficient to verify by <a href="/wiki/Measurement" title="Measurement">measurement</a> that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (<a href="/wiki/Theorem" title="Theorem">theorems</a>) and a few basic statements. The basic statements are not subject to proof because they are self-evident (<a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulates</a>), or are part of the definition of the subject of study (<a href="/wiki/Axiom" title="Axiom">axioms</a>). This principle, foundational for all mathematics, was first elaborated for geometry, and was systematized by <a href="/wiki/Euclid" title="Euclid">Euclid</a> around 300 BC in his book <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>.<a href="#cite_note-22">[21]</a><a href="#cite_note-23">[22]</a> The resulting <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is the study of shapes and their arrangements <a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">constructed</a> from lines, <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">planes</a> and circles in the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a> (<a href="/wiki/Plane_geometry" class="mw-redirect" title="Plane geometry">plane geometry</a>) and the three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>.<a href="#cite_note-24">[b]</a><a href="#cite_note-Straume_2014-21">[20]</a> Euclidean geometry was developed without change of methods or scope until the 17th century, when <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> introduced what is now called <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>. This constituted a major <a href="/wiki/Paradigm_shift" title="Paradigm shift">change of paradigm</a>: Instead of defining <a href="/wiki/Real_number" title="Real number">real numbers</a> as lengths of <a href="/wiki/Line_segments" class="mw-redirect" title="Line segments">line segments</a> (see <a href="/wiki/Number_line" title="Number line">number line</a>), 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: <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a>, which uses purely geometrical methods, and <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a>, which uses coordinates systemically.<a href="#cite_note-25">[23]</a> Analytic geometry allows the study of <a href="/wiki/Curve" title="Curve">curves</a> unrelated to circles and lines. Such curves can be defined as the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph of functions</a>, the study of which led to <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. They can also be defined as <a href="/wiki/Implicit_equation" class="mw-redirect" title="Implicit equation">implicit equations</a>, often <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a> (which spawned <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions.<a href="#cite_note-Straume_2014-21">[20]</a> In the 19th century, mathematicians discovered <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>, which do not follow the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. By questioning that postulate's truth, this discovery has been viewed as joining <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> in revealing the <a href="/wiki/Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a>. 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.<a href="#cite_note-26">[24]</a><a href="#cite_note-Kleiner_1991-7">[6]</a> In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">do not change</a> under specific transformations of the <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">space</a>.<a href="#cite_note-27">[25]</a> Today's subareas of geometry include:<a href="#cite_note-MSC-15">[14]</a> <ul><li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective geometry</a>, introduced in the 16th century by <a href="/wiki/Girard_Desargues" title="Girard Desargues">Girard Desargues</a>, extends Euclidean geometry by adding <a href="/wiki/Points_at_infinity" class="mw-redirect" title="Points at infinity">points at infinity</a> at which <a href="/wiki/Parallel_lines" class="mw-redirect" title="Parallel lines">parallel lines</a> intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.</li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine geometry</a>, the study of properties relative to <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallelism</a> and independent from the concept of length.</li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a>, the study of curves, surfaces, and their generalizations, which are defined using <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable functions</a>.</li> <li><a href="/wiki/Manifold_theory" class="mw-redirect" title="Manifold theory">Manifold theory</a>, the study of shapes that are not necessarily embedded in a larger space.</li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, the study of distance properties in curved spaces.</li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a>, the study of curves, surfaces, and their generalizations, which are defined using <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>.</li> <li><a href="/wiki/Topology" title="Topology">Topology</a>, the study of properties that are kept under <a href="/wiki/Continuous_deformation" class="mw-redirect" title="Continuous deformation">continuous deformations</a>. <ul><li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic topology</a>, the use in topology of algebraic methods, mainly <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>.</li></ul></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a>, the study of finite configurations in geometry.</li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex geometry</a>, the study of <a href="/wiki/Convex_set" title="Convex set">convex sets</a>, which takes its importance from its applications in <a href="/wiki/Convex_optimization" title="Convex optimization">optimization</a>.</li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex geometry</a>, the geometry obtained by replacing real numbers with <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebra">Algebra</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebra" title="Algebra">Algebra</a></div> <figure class="mw-default-size skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Quadratic_formula.svg" class="mw-file-description"><img alt="refer to caption" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/220px-Quadratic_formula.svg.png" decoding="async" width="220" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/330px-Quadratic_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Quadratic_formula.svg/440px-Quadratic_formula.svg.png 2x" data-file-width="402" data-file-height="124" /></a><figcaption>The <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>, which concisely expresses the solutions of all <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rubik%27s_cube.svg" class="mw-file-description"><img alt="A shuffled 3x3 rubik's cube" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/220px-Rubik%27s_cube.svg.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/330px-Rubik%27s_cube.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/440px-Rubik%27s_cube.svg.png 2x" data-file-width="480" data-file-height="500" /></a><figcaption>The <a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a> is a concrete application of <a href="/wiki/Group_theory" title="Group theory">group theory</a>.<a href="#cite_note-28">[26]</a></figcaption></figure> Algebra is the art of manipulating <a href="/wiki/Equation" title="Equation">equations</a> and formulas. <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a> (3rd century) and <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">al-Khwarizmi</a> (9th century) were the two main precursors of algebra.<a href="#cite_note-29">[27]</a><a href="#cite_note-FOOTNOTEKleiner2007"History_of_Classical_Algebra"_pp._3–5-30">[28]</a> Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution.<a href="#cite_note-31">[29]</a> Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side.<a href="#cite_note-32">[30]</a> The term algebra is derived from the <a href="/wiki/Arabic" title="Arabic">Arabic</a> word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of <a href="/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">his main treatise</a>.<a href="#cite_note-33">[31]</a><a href="#cite_note-34">[32]</a> Algebra became an area in its own right only with <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers.<a href="#cite_note-35">[33]</a> Variables allow mathematicians to describe the operations that have to be done on the numbers represented using <a href="/wiki/Mathematical_formulas" class="mw-redirect" title="Mathematical formulas">mathematical formulas</a>.<a href="#cite_note-36">[34]</a> Until the 19th century, algebra consisted mainly of the study of <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> (presently <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>), and polynomial equations in a single <a href="/wiki/Unknown_(algebra)" class="mw-redirect" title="Unknown (algebra)">unknown</a>, 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 <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular integers</a>, and <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a>), on which generalizations of arithmetic operations are often valid.<a href="#cite_note-FOOTNOTEKleiner2007"History_of_Linear_Algebra"_pp._79–101-37">[35]</a> The concept of <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> addresses this, consisting of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> 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 <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, as established by the influence and works of <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>.<a href="#cite_note-38">[36]</a> 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:<a href="#cite_note-MSC-15">[14]</a> <ul><li><a href="/wiki/Group_theory" title="Group theory">group theory</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field theory</a></li> <li><a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, whose study is essentially the same as <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a></li> <li><a href="/wiki/Ring_theory" title="Ring theory">ring theory</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, which is the study of <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>, includes the study of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, and is a foundational part of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> and <a href="/wiki/Lie_group" title="Lie group">Lie group</a> theory</li> <li><a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, which is widely used for the study of the logical structure of <a href="/wiki/Computer" title="Computer">computers</a></li></ul> The study of types of algebraic structures as <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> is the purpose of <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a> and <a href="/wiki/Category_theory" title="Category theory">category theory</a>.<a href="#cite_note-39">[37]</a> The latter applies to every <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structure</a> (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 <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>; this particular area of application is called <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>.<a href="#cite_note-40">[38]</a> <div class="mw-heading mw-heading3"><h3 id="Calculus_and_analysis">Calculus and analysis</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Calculus" title="Calculus">Calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cauchy_sequence_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/220px-Cauchy_sequence_illustration.svg.png" decoding="async" width="220" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/330px-Cauchy_sequence_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/440px-Cauchy_sequence_illustration.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption>A <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> 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).</figcaption></figure> Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a>.<a href="#cite_note-41">[39]</a> It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a> with the introduction of the concept of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> and many other results.<a href="#cite_note-42">[40]</a> Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.<a href="#cite_note-43">[41]</a> Analysis is further subdivided into <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, where variables represent <a href="/wiki/Real_number" title="Real number">real numbers</a>, and <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, where variables represent <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. Analysis includes many subareas shared by other areas of mathematics which include:<a href="#cite_note-MSC-15">[14]</a> <ul><li><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a>, where variables represent varying functions</li> <li><a href="/wiki/Integration_(mathematics)" class="mw-redirect" title="Integration (mathematics)">Integration</a>, <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> and <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a>, all strongly related with <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> on a <a href="/wiki/Continuum_(set_theory)" title="Continuum (set theory)">continuum</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equations</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a>, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications</li></ul> <div class="mw-heading mw-heading3"><h3 id="Discrete_mathematics">Discrete mathematics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></div> <figure class="mw-default-size mw-halign-right skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Markovkate_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/220px-Markovkate_01.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/330px-Markovkate_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Markovkate_01.svg/440px-Markovkate_01.svg.png 2x" data-file-width="563" data-file-height="563" /></a><figcaption>A diagram representing a two-state <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a>. The states are represented by 'A' and 'E'. The numbers are the probability of flipping the state.</figcaption></figure> Discrete mathematics, broadly speaking, is the study of individual, <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> mathematical objects. An example is the set of all integers.<a href="#cite_note-44">[42]</a> Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.<a href="#cite_note-45">[c]</a> <a href="/wiki/Algorithm" title="Algorithm">Algorithms</a>—especially their <a href="/wiki/Implementation" title="Implementation">implementation</a> and <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a>—play a major role in discrete mathematics.<a href="#cite_note-46">[43]</a> The <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> and <a href="/wiki/Kepler_conjecture" title="Kepler conjecture">optimal sphere packing</a> were two major problems of discrete mathematics solved in the second half of the 20th century.<a href="#cite_note-47">[44]</a> The <a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a>, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of <a href="/wiki/Computationally_expensive" class="mw-redirect" title="Computationally expensive">computationally difficult</a> problems.<a href="#cite_note-48">[45]</a> Discrete mathematics includes:<a href="#cite_note-MSC-15">[14]</a> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a>, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or <a href="/wiki/Subset" title="Subset">subsets</a> of a given <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>; 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 <a href="/wiki/Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shapes</a>.</li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> and <a href="/wiki/Hypergraph" title="Hypergraph">hypergraphs</a></li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a>, including <a href="/wiki/Error_correcting_code" class="mw-redirect" title="Error correcting code">error correcting codes</a> and a part of <a href="/wiki/Cryptography" title="Cryptography">cryptography</a></li> <li><a href="/wiki/Matroid" title="Matroid">Matroid</a> theory</li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a></li> <li><a href="/wiki/Discrete_probability_distribution" class="mw-redirect" title="Discrete probability distribution">Discrete probability distributions</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a> (although <a href="/wiki/Continuous_game" title="Continuous game">continuous games</a> are also studied, most common games, such as <a href="/wiki/Chess" title="Chess">chess</a> and <a href="/wiki/Poker" title="Poker">poker</a> are discrete)</li> <li><a href="/wiki/Discrete_optimization" title="Discrete optimization">Discrete optimization</a>, including <a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a>, <a href="/wiki/Integer_programming" title="Integer programming">integer programming</a>, <a href="/wiki/Constraint_programming" title="Constraint programming">constraint programming</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mathematical_logic_and_set_theory">Mathematical logic and set theory</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a> and <a href="/wiki/Set_theory" title="Set theory">Set theory</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn_A_intersect_B.svg" class="mw-file-description"><img alt="A blue and pink circle and their intersection labeled" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/220px-Venn_A_intersect_B.svg.png" decoding="async" width="220" height="157" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/330px-Venn_A_intersect_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Venn_A_intersect_B.svg/440px-Venn_A_intersect_B.svg.png 2x" data-file-width="350" data-file-height="250" /></a><figcaption>The <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> is a commonly used method to illustrate the relations between sets.</figcaption></figure> The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century.<a href="#cite_note-49">[46]</a><a href="#cite_note-50">[47]</a> Before this period, sets were not considered to be mathematical objects, and <a href="/wiki/Logic" title="Logic">logic</a>, although used for mathematical proofs, belonged to <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> and was not specifically studied by mathematicians.<a href="#cite_note-51">[48]</a> Before <a href="/wiki/Georg_Cantor" title="Georg Cantor">Cantor</a>'s study of <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>, mathematicians were reluctant to consider <a href="/wiki/Actually_infinite" class="mw-redirect" title="Actually infinite">actually infinite</a> collections, and considered <a href="/wiki/Infinity" title="Infinity">infinity</a> to be the result of endless <a href="/wiki/Enumeration" title="Enumeration">enumeration</a>. Cantor's work offended many mathematicians not only by considering actually infinite sets<a href="#cite_note-52">[49]</a> but by showing that this implies different sizes of infinity, per <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">Cantor's diagonal argument</a>. This led to the <a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">controversy over Cantor's set theory</a>.<a href="#cite_note-53">[50]</a> In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring <a href="/wiki/Mathematical_rigour" class="mw-redirect" title="Mathematical rigour">mathematical rigour</a>.<a href="#cite_note-54">[51]</a> This became the foundational crisis of mathematics.<a href="#cite_note-55">[52]</a> It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">formalized set theory</a>. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.<a href="#cite_note-Warner_2013-13">[12]</a> For example, in <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, 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.<a href="#cite_note-56">[53]</a> This <a href="/wiki/Mathematical_abstraction" class="mw-redirect" title="Mathematical abstraction">mathematical abstraction</a> from reality is embodied in the modern philosophy of <a href="/wiki/Formalism_(philosophy_of_mathematics)" title="Formalism (philosophy of mathematics)">formalism</a>, as founded by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> around 1910.<a href="#cite_note-Snapper-57">[54]</a> 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—sometimes called "intuition"—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, <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> assert, roughly speaking that, in every <a href="/wiki/Consistent" class="mw-redirect" title="Consistent">consistent</a> <a href="/wiki/Formal_system" title="Formal system">formal system</a> that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system.<a href="#cite_note-Raatikainen_2005-58">[55]</a> This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by <a href="/wiki/L._E._J._Brouwer" title="L. E. J. Brouwer">Brouwer</a>, who promoted <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, which explicitly lacks the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>.<a href="#cite_note-59">[56]</a><a href="#cite_note-60">[57]</a> These problems and debates led to a wide expansion of mathematical logic, with subareas such as <a href="/wiki/Model_theory" title="Model theory">model theory</a> (modeling some logical theories inside other theories), <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Type_theory" title="Type theory">type theory</a>, <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a> and <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>.<a href="#cite_note-MSC-15">[14]</a> Although these aspects of mathematical logic were introduced before the rise of <a href="/wiki/Computer" title="Computer">computers</a>, their use in <a href="/wiki/Compiler" title="Compiler">compiler</a> design, <a href="/wiki/Formal_verification" title="Formal verification">formal verification</a>, <a href="/wiki/Program_analysis" title="Program analysis">program analysis</a>, <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistants</a> and other aspects of <a href="/wiki/Computer_science" title="Computer science">computer science</a>, contributed in turn to the expansion of these logical theories.<a href="#cite_note-61">[58]</a> <div class="mw-heading mw-heading3"><h3 id="Statistics_and_other_decision_sciences">Statistics and other decision sciences</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Statistics" title="Statistics">Statistics</a> and <a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></div> <figure class="mw-default-size mw-halign-right skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:IllustrationCentralTheorem.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/330px-IllustrationCentralTheorem.png" decoding="async" width="330" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/495px-IllustrationCentralTheorem.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/660px-IllustrationCentralTheorem.png 2x" data-file-width="2771" data-file-height="1166" /></a><figcaption>Whatever the form of a random population <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> (μ), the sampling <a href="/wiki/Mean" title="Mean">mean</a> (x̄) tends to a <a href="/wiki/Gaussian" class="mw-redirect" title="Gaussian">Gaussian</a> distribution and its <a href="/wiki/Variance" title="Variance">variance</a> (σ) is given by the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> of probability theory.<a href="#cite_note-62">[59]</a></figcaption></figure> 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 <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. Statisticians generate data with <a href="/wiki/Random_sampling" class="mw-redirect" title="Random sampling">random sampling</a> or randomized <a href="/wiki/Design_of_experiments" title="Design of experiments">experiments</a>.<a href="#cite_note-63">[60]</a> <a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a> studies <a href="/wiki/Statistical_decision_theory" class="mw-redirect" title="Statistical decision theory">decision problems</a> such as minimizing the <a href="/wiki/Risk" title="Risk">risk</a> (<a href="/wiki/Expected_loss" title="Expected loss">expected loss</a>) of a statistical action, such as using a <a href="/wiki/Statistical_method" class="mw-redirect" title="Statistical method">procedure</a> in, for example, <a href="/wiki/Parameter_estimation" class="mw-redirect" title="Parameter estimation">parameter estimation</a>, <a href="/wiki/Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a>, and <a href="/wiki/Selection_algorithm" title="Selection algorithm">selecting the best</a>. In these traditional areas of <a href="/wiki/Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>, a statistical-decision problem is formulated by minimizing an <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">objective function</a>, like expected loss or <a href="/wiki/Cost" title="Cost">cost</a>, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.<a href="#cite_note-RaoOpt-64">[61]</a> Because of its use of <a href="/wiki/Optimization" class="mw-redirect" title="Optimization">optimization</a>, the mathematical theory of statistics overlaps with other <a href="/wiki/Decision_science" class="mw-redirect" title="Decision science">decision sciences</a>, such as <a href="/wiki/Operations_research" title="Operations research">operations research</a>, <a href="/wiki/Control_theory" title="Control theory">control theory</a>, and <a href="/wiki/Mathematical_economics" title="Mathematical economics">mathematical economics</a>.<a href="#cite_note-FOOTNOTEWhittle199410–11,_14–18-65">[62]</a> <div class="mw-heading mw-heading3"><h3 id="Computational_mathematics">Computational mathematics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></div> Computational mathematics is the study of <a href="/wiki/Mathematical_problem" title="Mathematical problem">mathematical problems</a> that are typically too large for human, numerical capacity.<a href="#cite_note-66">[63]</a><a href="#cite_note-67">[64]</a> <a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> studies methods for problems in <a href="/wiki/Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analysis</a> using <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> and <a href="/wiki/Approximation_theory" title="Approximation theory">approximation theory</a>; numerical analysis broadly includes the study of <a href="/wiki/Approximation" title="Approximation">approximation</a> and <a href="/wiki/Discretization" title="Discretization">discretization</a> with special focus on <a href="/wiki/Rounding_error" class="mw-redirect" title="Rounding error">rounding errors</a>.<a href="#cite_note-68">[65]</a> Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-<a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">matrix</a>-and-<a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>. Other areas of computational mathematics include <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a> and <a href="/wiki/Symbolic_computation" class="mw-redirect" title="Symbolic computation">symbolic computation</a>. <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a></div> <div class="mw-heading mw-heading3"><h3 id="Etymology">Etymology</h3></div> The word mathematics comes from the <a href="/wiki/Ancient_Greek" title="Ancient Greek">Ancient Greek</a> word máthēma (<a href="https://en.wiktionary.org/wiki/%CE%BC%CE%AC%CE%B8%CE%B7%CE%BC%CE%B1#English" class="extiw" title="wikt:μάθημα">μάθημα</a>), meaning 'something learned, knowledge, mathematics', and the derived expression mathēmatikḗ tékhnē (μαθηματικὴ τέχνη), meaning 'mathematical science'. It entered the English language during the <a href="/wiki/Late_Middle_English" class="mw-redirect" title="Late Middle English">Late Middle English</a> period through French and Latin.<a href="#cite_note-69">[66]</a> Similarly, one of the two main schools of thought in <a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreanism</a> was known as the mathēmatikoi (μαθηματικοί)—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 <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a> and geometry. By the time of <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384–322 BC) this meaning was fully established.<a href="#cite_note-70">[67]</a> In Latin and English, until around 1700, the term mathematics more commonly meant "<a href="/wiki/Astrology" title="Astrology">astrology</a>" (or sometimes "<a href="/wiki/Astronomy" title="Astronomy">astronomy</a>") 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, <a href="/wiki/Saint_Augustine" class="mw-redirect" title="Saint Augustine">Saint Augustine</a>'s warning that Christians should beware of mathematici, meaning "astrologers", is sometimes mistranslated as a condemnation of mathematicians.<a href="#cite_note-Boas-71">[68]</a> The apparent <a href="/wiki/Plural" title="Plural">plural</a> form in English goes back to the Latin <a href="/wiki/Neuter_(grammar)" class="mw-redirect" title="Neuter (grammar)">neuter</a> plural mathematica (<a href="/wiki/Cicero" title="Cicero">Cicero</a>), based on the Greek plural ta mathēmatiká (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 <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>, inherited from Greek.<a href="#cite_note-72">[69]</a> In English, the noun mathematics takes a singular verb. It is often shortened to maths<a href="#cite_note-73">[70]</a> or, in North America, math.<a href="#cite_note-74">[71]</a> <div class="mw-heading mw-heading3"><h3 id="Ancient">Ancient</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plimpton_322.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/220px-Plimpton_322.jpg" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/330px-Plimpton_322.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Plimpton_322.jpg/440px-Plimpton_322.jpg 2x" data-file-width="1246" data-file-height="863" /></a><figcaption>The Babylonian mathematical tablet <a href="/wiki/Plimpton_322" title="Plimpton 322">Plimpton 322</a>, dated to 1800 BC</figcaption></figure> In addition to recognizing how to <a href="/wiki/Counting" title="Counting">count</a> physical objects, <a href="/wiki/Prehistoric" class="mw-redirect" title="Prehistoric">prehistoric</a> peoples may have also known how to count abstract quantities, like time—days, seasons, or years.<a href="#cite_note-75">[72]</a><a href="#cite_note-76">[73]</a> Evidence for more complex mathematics does not appear until around 3000 <abbr title="Before Christ">BC</abbr>, when the <a href="/wiki/Babylonia" title="Babylonia">Babylonians</a> and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.<a href="#cite_note-FOOTNOTEKline1990Chapter_1-77">[74]</a> The oldest mathematical texts from <a href="/wiki/Mesopotamia" title="Mesopotamia">Mesopotamia</a> and <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">Egypt</a> are from 2000 to 1800 BC.<a href="#cite_note-78">[75]</a> Many early texts mention <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a> and so, by inference, the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a> (<a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a <a href="/wiki/Sexagesimal" title="Sexagesimal">sexagesimal</a> numeral system which is still in use today for measuring angles and time.<a href="#cite_note-FOOTNOTEBoyer1991"Mesopotamia"_pp._24–27-79">[76]</a> In the 6th century BC, <a href="/wiki/Greek_mathematics" title="Greek mathematics">Greek mathematics</a> began to emerge as a distinct discipline and some <a href="/wiki/Ancient_Greeks" class="mw-redirect" title="Ancient Greeks">Ancient Greeks</a> such as the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a> appeared to have considered it a subject in its own right.<a href="#cite_note-80">[77]</a> Around 300 BC, <a href="/wiki/Euclid" title="Euclid">Euclid</a> 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.<a href="#cite_note-81">[78]</a> His book, <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a>, is widely considered the most successful and influential textbook of all time.<a href="#cite_note-FOOTNOTEBoyer1991"Euclid_of_Alexandria"_p._119-82">[79]</a> The greatest mathematician of antiquity is often held to be <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> (<abbr title="circa">c.</abbr> 287 – c. 212 BC) of <a href="/wiki/Syracuse,_Italy" class="mw-redirect" title="Syracuse, Italy">Syracuse</a>.<a href="#cite_note-FOOTNOTEBoyer1991"Archimedes_of_Syracuse"_p._120-83">[80]</a> He developed formulas for calculating the surface area and volume of <a href="/wiki/Solids_of_revolution" class="mw-redirect" title="Solids of revolution">solids of revolution</a> and used the <a href="/wiki/Method_of_exhaustion" title="Method of exhaustion">method of exhaustion</a> to calculate the <a href="/wiki/Area" title="Area">area</a> under the arc of a <a href="/wiki/Parabola" title="Parabola">parabola</a> with the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">summation of an infinite series</a>, in a manner not too dissimilar from modern calculus.<a href="#cite_note-FOOTNOTEBoyer1991"Archimedes_of_Syracuse"_p._130-84">[81]</a> Other notable achievements of Greek mathematics are <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a> (<a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius of Perga</a>, 3rd century BC),<a href="#cite_note-FOOTNOTEBoyer1991"Apollonius_of_Perga"_p._145-85">[82]</a> <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a> (<a href="/wiki/Hipparchus_of_Nicaea" class="mw-redirect" title="Hipparchus of Nicaea">Hipparchus of Nicaea</a>, 2nd century BC),<a href="#cite_note-FOOTNOTEBoyer1991"Greek_Trigonometry_and_Mensuration"_p._162-86">[83]</a> and the beginnings of algebra (Diophantus, 3rd century AD).<a href="#cite_note-FOOTNOTEBoyer1991"Revival_and_Decline_of_Greek_Mathematics"_p._180-87">[84]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bakhshali_numerals_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/330px-Bakhshali_numerals_2.jpg" decoding="async" width="330" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/495px-Bakhshali_numerals_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Bakhshali_numerals_2.jpg/660px-Bakhshali_numerals_2.jpg 2x" data-file-width="1958" data-file-height="295" /></a><figcaption>The numerals used in the <a href="/wiki/Bakhshali_manuscript" title="Bakhshali manuscript">Bakhshali manuscript</a>, dated between the 2nd century BC and the 2nd century AD</figcaption></figure> The <a href="/wiki/Hindu%E2%80%93Arabic_numeral_system" title="Hindu–Arabic numeral system">Hindu–Arabic numeral system</a> 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 <a href="/wiki/Indian_mathematics" title="Indian mathematics">India</a> and were transmitted to the <a href="/wiki/Western_world" title="Western world">Western world</a> via <a href="/wiki/Islamic_mathematics" class="mw-redirect" title="Islamic mathematics">Islamic mathematics</a>.<a href="#cite_note-88">[85]</a> Other notable developments of Indian mathematics include the modern definition and approximation of <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a>, and an early form of <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>.<a href="#cite_note-89">[86]</a><a href="#cite_note-90">[87]</a> <div class="mw-heading mw-heading3"><h3 id="Medieval_and_later">Medieval and later</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg/150px-Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg" decoding="async" width="150" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg/225px-Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/2/23/Image-Al-Kit%C4%81b_al-mu%E1%B8%ABta%E1%B9%A3ar_f%C4%AB_%E1%B8%A5is%C4%81b_al-%C4%9Fabr_wa-l-muq%C4%81bala.jpg 2x" data-file-width="240" data-file-height="380" /></a><figcaption>A page from <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">al-Khwarizmi</a>'s <a href="/wiki/Al-Jabr" title="Al-Jabr">Al-Jabr</a></figcaption></figure> During the <a href="/wiki/Golden_Age_of_Islam" class="mw-redirect" title="Golden Age of Islam">Golden Age of Islam</a>, 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 <a href="/wiki/Algebra" title="Algebra">algebra</a>. Other achievements of the Islamic period include advances in <a href="/wiki/Spherical_trigonometry" title="Spherical trigonometry">spherical trigonometry</a> and the addition of the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a> to the Arabic numeral system.<a href="#cite_note-91">[88]</a> Many notable mathematicians from this period were Persian, such as <a href="/wiki/Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a>, <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a> and <a href="/wiki/Sharaf_al-D%C4%ABn_al-%E1%B9%AC%C5%ABs%C4%AB" class="mw-redirect" title="Sharaf al-Dīn al-Ṭūsī">Sharaf al-Dīn al-Ṭūsī</a>.<a href="#cite_note-92">[89]</a> The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe.<a href="#cite_note-93">[90]</a> During the <a href="/wiki/Early_modern_period" title="Early modern period">early modern period</a>, mathematics began to develop at an accelerating pace in <a href="/wiki/Western_Europe" title="Western Europe">Western Europe</a>, with innovations that revolutionized mathematics, such as the introduction of variables and <a href="#Symbolic_notation_and_terminology">symbolic notation</a> by <a href="/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (1540–1603), the introduction of <a href="/wiki/History_of_logarithms" title="History of logarithms">logarithms</a> by <a href="/wiki/John_Napier" title="John Napier">John Napier</a> in 1614, which greatly simplified numerical calculations, especially for <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> and <a href="/wiki/Marine_navigation" title="Marine navigation">marine navigation</a>, the introduction of coordinates by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (1596–1650) for reducing geometry to algebra, and the development of calculus by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> (1643–1727) and <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a> (1646–1716). <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> (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.<a href="#cite_note-94">[91]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Carl_Friedrich_Gauss_1840_by_Jensen.jpg/180px-Carl_Friedrich_Gauss_1840_by_Jensen.jpg" decoding="async" width="180" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Carl_Friedrich_Gauss_1840_by_Jensen.jpg/270px-Carl_Friedrich_Gauss_1840_by_Jensen.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Carl_Friedrich_Gauss_1840_by_Jensen.jpg/360px-Carl_Friedrich_Gauss_1840_by_Jensen.jpg 2x" data-file-width="1639" data-file-height="2088" /></a><figcaption><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></figcaption></figure> Perhaps the foremost mathematician of the 19th century was the German mathematician <a href="/wiki/Carl_Gauss" class="mw-redirect" title="Carl Gauss">Carl Gauss</a>, who made numerous contributions to fields such as algebra, analysis, <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, <a href="/wiki/Matrix_theory" class="mw-redirect" title="Matrix theory">matrix theory</a>, number theory, and <a href="/wiki/Statistics" title="Statistics">statistics</a>.<a href="#cite_note-95">[92]</a> In the early 20th century, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> transformed mathematics by publishing <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">his incompleteness theorems</a>, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.<a href="#cite_note-Raatikainen_2005-58">[55]</a> Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and <a href="/wiki/Science" title="Science">science</a>, 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 <a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>, "The number of papers and books included in the <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">Mathematical Reviews</a> (MR) 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."<a href="#cite_note-FOOTNOTESevryuk2006101–109-96">[93]</a> <div class="mw-heading mw-heading2"><h2 id="Symbolic_notation_and_terminology">Symbolic notation and terminology</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a>, <a href="/wiki/Language_of_mathematics" title="Language of mathematics">Language of mathematics</a>, and <a href="/wiki/Glossary_of_mathematics" class="mw-redirect" title="Glossary of mathematics">Glossary of mathematics</a></div> <figure class="mw-default-size skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Sigma_summation_notation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Sigma_summation_notation.svg/220px-Sigma_summation_notation.svg.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Sigma_summation_notation.svg/330px-Sigma_summation_notation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Sigma_summation_notation.svg/440px-Sigma_summation_notation.svg.png 2x" data-file-width="350" data-file-height="240" /></a><figcaption>An explanation of the sigma (Σ) <a href="/wiki/Summation" title="Summation">summation</a> notation</figcaption></figure> Mathematical notation is widely used in science and <a href="/wiki/Engineering" title="Engineering">engineering</a> for representing complex <a href="/wiki/Concept" title="Concept">concepts</a> and <a href="/wiki/Property_(philosophy)" title="Property (philosophy)">properties</a> in a concise, unambiguous, and accurate way. This notation consists of <a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> used for representing <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a>, unspecified numbers, <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a> and any other mathematical objects, and then assembling them into <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> and formulas.<a href="#cite_note-97">[94]</a> More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally <a href="/wiki/Latin_alphabet" title="Latin alphabet">Latin</a> or <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek</a> letters, and often include <a href="/wiki/Subscript" class="mw-redirect" title="Subscript">subscripts</a>. Operation and relations are generally represented by specific <a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> or <a href="/wiki/Glyph" title="Glyph">glyphs</a>,<a href="#cite_note-98">[95]</a> such as + (<a href="/wiki/Plus_sign" class="mw-redirect" title="Plus sign">plus</a>), × (<a href="/wiki/Multiplication_sign" title="Multiplication sign">multiplication</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed07ec6b7c9532ddb859c43b95404125d5b34f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.418ex; height:3.176ex;" alt="{\textstyle \int }"> (<a href="/wiki/Integral_sign" class="mw-redirect" title="Integral sign">integral</a>), = (<a href="/wiki/Equals_sign" title="Equals sign">equal</a>), and < (<a href="/wiki/Less-than_sign" title="Less-than sign">less than</a>).<a href="#cite_note-AMS-99">[96]</a> All these symbols are generally grouped according to specific rules to form expressions and formulas.<a href="#cite_note-100">[97]</a> Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of <a href="/wiki/Noun_phrase" title="Noun phrase">noun phrases</a> and formulas play the role of <a href="/wiki/Clause" title="Clause">clauses</a>. 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 <a href="/wiki/Technical_definition" title="Technical definition">definitions</a> that provide a standard foundation for communication. An axiom or <a href="/wiki/Postulate" class="mw-redirect" title="Postulate">postulate</a> 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 <a href="/wiki/Conjecture" title="Conjecture">conjecture</a>. Through a series of rigorous arguments employing <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a>, a statement that is <a href="/wiki/Formal_proof" title="Formal proof">proven</a> to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemma</a>. A proven instance that forms part of a more general finding is termed a <a href="/wiki/Corollary" title="Corollary">corollary</a>.<a href="#cite_note-101">[98]</a> Numerous technical terms used in mathematics are <a href="/wiki/Neologism" title="Neologism">neologisms</a>, such as <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> and <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>.<a href="#cite_note-102">[99]</a> 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, "<a href="/wiki/Logical_disjunction" title="Logical disjunction">or</a>" 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 "<a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>"). Finally, many mathematical terms are common words that are used with a completely different meaning.<a href="#cite_note-103">[100]</a> 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 <a href="/wiki/Free_module" title="Free module">free module</a> is <a href="/wiki/Flat_module" title="Flat module">flat</a>" and "a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is always a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>". <div class="mw-heading mw-heading2"><h2 id="Relationship_with_sciences">Relationship with sciences</h2></div> Mathematics is used in most <a href="/wiki/Science" title="Science">sciences</a> for <a href="/wiki/Mathematical_model" title="Mathematical model">modeling</a> phenomena, which then allows predictions to be made from experimental laws.<a href="#cite_note-104">[101]</a> The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.<a href="#cite_note-105">[102]</a> Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.<a href="#cite_note-106">[103]</a> For example, the <a href="/wiki/Perihelion_precession_of_Mercury" class="mw-redirect" title="Perihelion precession of Mercury">perihelion precession of Mercury</a> could only be explained after the emergence of <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einstein</a>'s <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, which replaced <a href="/wiki/Newton%27s_law_of_gravitation" class="mw-redirect" title="Newton's law of gravitation">Newton's law of gravitation</a> as a better mathematical model.<a href="#cite_note-107">[104]</a> There is still a <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophical</a> 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 <a href="/wiki/Falsifiable" class="mw-redirect" title="Falsifiable">falsifiable</a>, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a <a href="/wiki/Counterexample" title="Counterexample">counterexample</a>. Similarly as in science, <a href="/wiki/Mathematical_theory" class="mw-redirect" title="Mathematical theory">theories</a> and results (theorems) are often obtained from <a href="/wiki/Experimentation" class="mw-redirect" title="Experimentation">experimentation</a>.<a href="#cite_note-108">[105]</a> 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).<a href="#cite_note-109">[106]</a> However, some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence.<a href="#cite_note-Bishop1991-110">[107]</a><a href="#cite_note-111">[108]</a><a href="#cite_note-Nickles2013-112">[109]</a><a href="#cite_note-Pigliucci2014-113">[110]</a> <div class="mw-heading mw-heading3"><h3 id="Pure_and_applied_mathematics">Pure and applied mathematics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a> and <a href="/wiki/Pure_mathematics" title="Pure mathematics">Pure mathematics</a></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:322px;max-width:322px"><div class="trow"><div class="tsingle" style="width:150px;max-width:150px"><div class="thumbimage" style="height:202px;overflow:hidden"><a href="/wiki/File:GodfreyKneller-IsaacNewton-1689.jpg" class="mw-file-description"><img alt="Isaac Newton" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/148px-GodfreyKneller-IsaacNewton-1689.jpg" decoding="async" width="148" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/222px-GodfreyKneller-IsaacNewton-1689.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/GodfreyKneller-IsaacNewton-1689.jpg/296px-GodfreyKneller-IsaacNewton-1689.jpg 2x" data-file-width="1364" data-file-height="1916" /></a></div></div><div class="tsingle" style="width:168px;max-width:168px"><div class="thumbimage" style="height:202px;overflow:hidden"><a href="/wiki/File:Gottfried_Wilhelm_Leibniz,_Bernhard_Christoph_Francke.jpg" class="mw-file-description"><img alt="Gottfried Wilhelm von Leibniz" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/166px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg" decoding="async" width="166" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/249px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/332px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 2x" data-file-width="4486" data-file-height="5538" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Isaac Newton (left) and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> developed infinitesimal calculus.</div></div></div></div> Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of <a href="/wiki/Technology" title="Technology">technology</a> and science, and there was no clear distinction between pure and applied mathematics.<a href="#cite_note-Ferreirós_2007-114">[111]</a> For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> introduced infinitesimal calculus for explaining the movement of the <a href="/wiki/Planet" title="Planet">planets</a> with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians.<a href="#cite_note-115">[112]</a> However, a notable exception occurred with the tradition of <a href="/wiki/Pure_mathematics_in_Ancient_Greece" class="mw-redirect" title="Pure mathematics in Ancient Greece">pure mathematics in Ancient Greece</a>.<a href="#cite_note-116">[113]</a> The problem of <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a>, for example, which goes back to <a href="/wiki/Euclid" title="Euclid">Euclid</a> in 300 BC, had no practical application before its use in the <a href="/wiki/RSA_cryptosystem" class="mw-redirect" title="RSA cryptosystem">RSA cryptosystem</a>, now widely used for the security of <a href="/wiki/Computer_network" title="Computer network">computer networks</a>.<a href="#cite_note-117">[114]</a> In the 19th century, mathematicians such as <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> increasingly focused their research on internal problems, that is, pure mathematics.<a href="#cite_note-Ferreirós_2007-114">[111]</a><a href="#cite_note-118">[115]</a> 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.<a href="#cite_note-119">[116]</a> The aftermath of <a href="/wiki/World_War_II" title="World War II">World War II</a> led to a surge in the development of applied mathematics in the US and elsewhere.<a href="#cite_note-120">[117]</a><a href="#cite_note-121">[118]</a> 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".<a href="#cite_note-122">[119]</a><a href="#cite_note-123">[120]</a> An example of the first case is the <a href="/wiki/Theory_of_distributions" class="mw-redirect" title="Theory of distributions">theory of distributions</a>, introduced by <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a> for validating computations done in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, which became immediately an important tool of (pure) mathematical analysis.<a href="#cite_note-124">[121]</a> An example of the second case is the <a href="/wiki/Decidability_of_the_first-order_theory_of_the_real_numbers" class="mw-redirect" title="Decidability of the first-order theory of the real numbers">decidability of the first-order theory of the real numbers</a>, a problem of pure mathematics that was proved true by <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, with an algorithm that is impossible to <a href="/wiki/Implementation_(computer_science)" class="mw-redirect" title="Implementation (computer science)">implement</a> because of a computational complexity that is much too high.<a href="#cite_note-125">[122]</a> For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, <a href="/wiki/George_E._Collins" title="George E. Collins">George Collins</a> introduced the <a href="/wiki/Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">cylindrical algebraic decomposition</a> that became a fundamental tool in <a href="/wiki/Real_algebraic_geometry" title="Real algebraic geometry">real algebraic geometry</a>.<a href="#cite_note-126">[123]</a> 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.<a href="#cite_note-127">[124]</a><a href="#cite_note-128">[125]</a> The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics".<a href="#cite_note-MSC-15">[14]</a> However, these terms are still used in names of some <a href="/wiki/University" title="University">university</a> departments, such as at the <a href="/wiki/Faculty_of_Mathematics,_University_of_Cambridge" title="Faculty of Mathematics, University of Cambridge">Faculty of Mathematics</a> at the <a href="/wiki/University_of_Cambridge" title="University of Cambridge">University of Cambridge</a>. <div class="mw-heading mw-heading3"><h3 id="Unreasonable_effectiveness">Unreasonable effectiveness</h3></div> The <a href="/wiki/Unreasonable_effectiveness_of_mathematics" class="mw-redirect" title="Unreasonable effectiveness of mathematics">unreasonable effectiveness of mathematics</a> is a phenomenon that was named and first made explicit by physicist <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a>.<a href="#cite_note-wigner1960-3">[3]</a> 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.<a href="#cite_note-129">[126]</a> Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. A notable example is the <a href="/wiki/Prime_factorization" class="mw-redirect" title="Prime factorization">prime factorization</a> of natural numbers that was discovered more than 2,000 years before its common use for secure <a href="/wiki/Internet" title="Internet">internet</a> communications through the <a href="/wiki/RSA_cryptosystem" class="mw-redirect" title="RSA cryptosystem">RSA cryptosystem</a>.<a href="#cite_note-130">[127]</a> A second historical example is the theory of <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>. They were studied by the <a href="/wiki/Ancient_Greek_mathematicians" class="mw-redirect" title="Ancient Greek mathematicians">ancient Greek mathematicians</a> as <a href="/wiki/Conic_section" title="Conic section">conic sections</a> (that is, intersections of <a href="/wiki/Cone" title="Cone">cones</a> with planes). It was almost 2,000 years later that <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> discovered that the <a href="/wiki/Trajectories" class="mw-redirect" title="Trajectories">trajectories</a> of the planets are ellipses.<a href="#cite_note-131">[128]</a> 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 <a href="/wiki/Manifold" title="Manifold">manifolds</a>. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> developed the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> that uses fundamentally these concepts. In particular, <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> is a non-Euclidean space of dimension four, and spacetime of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> is a (curved) manifold of dimension four.<a href="#cite_note-132">[129]</a><a href="#cite_note-133">[130]</a> 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 <a href="/wiki/Positron" title="Positron">positron</a> and the <a href="/wiki/Omega_baryon" title="Omega baryon">baryon</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{-}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{-}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef44bf9fda1defbf63ab871817d3193ef64683ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.836ex; height:2.509ex;" alt="{\displaystyle \Omega ^{-}.}"> In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown <a href="/wiki/Particle" title="Particle">particle</a>, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.<a href="#cite_note-borel-134">[131]</a><a href="#cite_note-135">[132]</a><a href="#cite_note-136">[133]</a> <div class="mw-heading mw-heading3"><h3 id="Specific_sciences">Specific sciences</h3></div> <div class="mw-heading mw-heading4"><h4 id="Physics">Physics</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relationship_between_mathematics_and_physics" title="Relationship between mathematics and physics">Relationship between mathematics and physics</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pendule_schema.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Pendule_schema.gif/220px-Pendule_schema.gif" decoding="async" width="220" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c5/Pendule_schema.gif 1.5x" data-file-width="291" data-file-height="354" /></a><figcaption>Diagram of a pendulum</figcaption></figure> Mathematics and physics have influenced each other over their modern history. Modern physics uses mathematics abundantly,<a href="#cite_note-137">[134]</a> and is also considered to be the motivation of major mathematical developments.<a href="#cite_note-138">[135]</a> <div class="mw-heading mw-heading4"><h4 id="Computing">Computing</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a> and <a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational mathematics</a></div> Computing is closely related to mathematics in several ways.<a href="#cite_note-139">[136]</a> <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a> is considered to be mathematical in nature.<a href="#cite_note-140">[137]</a> Communication technologies apply branches of mathematics that may be very old (e.g., arithmetic), especially with respect to transmission security, in <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> and <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>. <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a> is useful in many areas of computer science, such as <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity theory</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, and <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>.<a href="#cite_note-141">[138]</a> In 1998, the <a href="/wiki/Kepler_conjecture" title="Kepler conjecture">Kepler conjecture</a> on <a href="/wiki/Sphere_packing" title="Sphere packing">sphere packing</a> seemed to also be partially proven by computer.<a href="#cite_note-142">[139]</a> <div class="mw-heading mw-heading4"><h4 id="Biology_and_chemistry">Biology and chemistry</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical and theoretical biology</a> and <a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Giant_Pufferfish_skin_pattern_detail.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Giant_Pufferfish_skin_pattern_detail.jpg/220px-Giant_Pufferfish_skin_pattern_detail.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/a4/Giant_Pufferfish_skin_pattern_detail.jpg 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>The skin of this <a href="/wiki/Giant_pufferfish" class="mw-redirect" title="Giant pufferfish">giant pufferfish</a> exhibits a <a href="/wiki/Turing_pattern" title="Turing pattern">Turing pattern</a>, which can be modeled by <a href="/wiki/Reaction%E2%80%93diffusion_system" title="Reaction–diffusion system">reaction–diffusion systems</a>.</figcaption></figure> <a href="/wiki/Biology" title="Biology">Biology</a> uses probability extensively in fields such as ecology or <a href="/wiki/Neurobiology" class="mw-redirect" title="Neurobiology">neurobiology</a>.<a href="#cite_note-:2-143">[140]</a> Most discussion of probability centers on the concept of <a href="/wiki/Evolutionary_fitness" class="mw-redirect" title="Evolutionary fitness">evolutionary fitness</a>.<a href="#cite_note-:2-143">[140]</a> Ecology heavily uses modeling to simulate <a href="/wiki/Population_dynamics" title="Population dynamics">population dynamics</a>,<a href="#cite_note-:2-143">[140]</a><a href="#cite_note-144">[141]</a> study ecosystems such as the predator-prey model, measure pollution diffusion,<a href="#cite_note-FOOTNOTEBouleau1999282–283-145">[142]</a> or to assess climate change.<a href="#cite_note-FOOTNOTEBouleau1999285-146">[143]</a> The dynamics of a population can be modeled by coupled differential equations, such as the <a href="/wiki/Lotka%E2%80%93Volterra_equations" title="Lotka–Volterra equations">Lotka–Volterra equations</a>.<a href="#cite_note-147">[144]</a> <a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Statistical hypothesis testing</a>, is run on data from <a href="/wiki/Clinical_trial" title="Clinical trial">clinical trials</a> to determine whether a new treatment works.<a href="#cite_note-148">[145]</a> Since the start of the 20th century, chemistry has used computing to model molecules in three dimensions.<a href="#cite_note-149">[146]</a> <div class="mw-heading mw-heading4"><h4 id="Earth_sciences">Earth sciences</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geomathematics" title="Geomathematics">Geomathematics</a></div> <a href="/wiki/Structural_geology" title="Structural geology">Structural geology</a> and climatology use probabilistic models to predict the risk of natural catastrophes.<a href="#cite_note-150">[147]</a> Similarly, <a href="/wiki/Meteorology" title="Meteorology">meteorology</a>, <a href="/wiki/Oceanography" title="Oceanography">oceanography</a>, and <a href="/wiki/Planetology" class="mw-redirect" title="Planetology">planetology</a> also use mathematics due to their heavy use of models.<a href="#cite_note-151">[148]</a><a href="#cite_note-152">[149]</a><a href="#cite_note-153">[150]</a> <div class="mw-heading mw-heading4"><h4 id="Social_sciences">Social sciences</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a> and <a href="/wiki/Historical_dynamics" title="Historical dynamics">Historical dynamics</a></div> Areas of mathematics used in the social sciences include probability/statistics and differential equations. These are used in linguistics, <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Sociology" title="Sociology">sociology</a>,<a href="#cite_note-154">[151]</a> and <a href="/wiki/Psychology" title="Psychology">psychology</a>.<a href="#cite_note-155">[152]</a> <figure class="mw-default-size skin-invert-image" typeof="mw:File/Thumb"><a href="/wiki/File:Supply-demand-equilibrium.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Supply-demand-equilibrium.svg/220px-Supply-demand-equilibrium.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Supply-demand-equilibrium.svg/330px-Supply-demand-equilibrium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Supply-demand-equilibrium.svg/440px-Supply-demand-equilibrium.svg.png 2x" data-file-width="217" data-file-height="217" /></a><figcaption><a href="/wiki/Supply_and_demand" title="Supply and demand">Supply and demand</a> curves, like this one, are a staple of mathematical economics.</figcaption></figure> Often the fundamental postulate of mathematical economics is that of the rational individual actor – <a href="/wiki/Homo_economicus" title="Homo economicus">Homo economicus</a> (<abbr style="font-size:85%" title="literal translation">lit.</abbr> 'economic man').<a href="#cite_note-:3-156">[153]</a> In this model, the individual seeks to maximize their <a href="/wiki/Rational_choice_theory" class="mw-redirect" title="Rational choice theory">self-interest</a>,<a href="#cite_note-:3-156">[153]</a> and always makes optimal choices using <a href="/wiki/Perfect_information" title="Perfect information">perfect information</a>.<a href="#cite_note-157">[154]</a> This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual <a href="/wiki/Calculations" class="mw-redirect" title="Calculations">calculations</a> are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms. Some reject or criticise the concept of Homo economicus. Economists note that real people have limited information, make poor choices and care about fairness, altruism, not just personal gain.<a href="#cite_note-158">[155]</a> Without mathematical modeling, it is hard to go beyond statistical observations or untestable speculation. Mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions. Models provide clarity and precision, enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real-world data.<a href="#cite_note-159">[156]</a> At the start of the 20th century, there was a development to express historical movements in formulas. In 1922, <a href="/wiki/Nikolai_Kondratiev" title="Nikolai Kondratiev">Nikolai Kondratiev</a> discerned the ~50-year-long <a href="/wiki/Kondratiev_cycle" class="mw-redirect" title="Kondratiev cycle">Kondratiev cycle</a>, which explains phases of economic growth or crisis.<a href="#cite_note-160">[157]</a> Towards the end of the 19th century, mathematicians extended their analysis into <a href="/wiki/Geopolitics" title="Geopolitics">geopolitics</a>.<a href="#cite_note-161">[158]</a> <a href="/wiki/Peter_Turchin" title="Peter Turchin">Peter Turchin</a> developed <a href="/wiki/Cliodynamics" title="Cliodynamics">cliodynamics</a> since the 1990s.<a href="#cite_note-162">[159]</a> Mathematization of the social sciences is not without risk. In the controversial book <a href="/wiki/Fashionable_Nonsense" title="Fashionable Nonsense">Fashionable Nonsense</a> (1997), <a href="/wiki/Alan_Sokal" title="Alan Sokal">Sokal</a> and <a href="/wiki/Jean_Bricmont" title="Jean Bricmont">Bricmont</a> denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences.<a href="#cite_note-163">[160]</a> The study of <a href="/wiki/Complex_systems" class="mw-redirect" title="Complex systems">complex systems</a> (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.<a href="#cite_note-164">[161]</a><a href="#cite_note-165">[162]</a> <div class="mw-heading mw-heading2"><h2 id="Philosophy">Philosophy</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></div> <div class="mw-heading mw-heading3"><h3 id="Reality">Reality</h3></div> The connection between mathematics and material reality has led to philosophical debates since at least the time of <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>. The ancient philosopher <a href="/wiki/Plato" title="Plato">Plato</a> 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 <a href="/wiki/Mathematical_Platonism" class="mw-redirect" title="Mathematical Platonism">Platonism</a>. 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.<a href="#cite_note-SEP-Platonism-166">[163]</a> <a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a> summarized this view of mathematics reality as follows, and provided quotations of <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>, <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> and Albert Einstein that support his views.<a href="#cite_note-borel-134">[131]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"> 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.<a href="#cite_note-167">[164]</a> 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 feeling of an objective existence, of a reality of mathematics ...</blockquote> Nevertheless, Platonism and the concurrent views on abstraction do not explain the <a href="#Unreasonable_effectiveness">unreasonable effectiveness</a> of mathematics.<a href="#cite_note-168">[165]</a> <div class="mw-heading mw-heading3"><h3 id="Proposed_definitions">Proposed definitions</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Definitions_of_mathematics" title="Definitions of mathematics">Definitions of mathematics</a></div> There is no general consensus about the definition of mathematics or its <a href="/wiki/Epistemology" title="Epistemology">epistemological status</a>—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do".<a href="#cite_note-Mura-169">[166]</a><a href="#cite_note-Runge-170">[167]</a> A common approach is to define mathematics by its object of study.<a href="#cite_note-171">[168]</a><a href="#cite_note-FOOTNOTEMura1993379,_381-172">[169]</a><a href="#cite_note-FOOTNOTEBrownPorter1995326-173">[170]</a><a href="#cite_note-174">[171]</a> 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.<a href="#cite_note-Franklin-175">[172]</a> In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.<a href="#cite_note-Cajori-176">[173]</a> With the large number of new areas of mathematics that have appeared since the beginning of the 20th century, defining mathematics by its object of study has become increasingly difficult.<a href="#cite_note-FOOTNOTEDevlin2018[httpsbooksgooglecombooksidgUb7CAAAQBAJpgPA3_3]-177">[174]</a> For example, in lieu of a definition, <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> in <a href="/wiki/Mathematics,_form_and_function" class="mw-redirect" title="Mathematics, form and function">Mathematics, form and function</a> summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:<a href="#cite_note-178">[175]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.</blockquote> Another approach for defining mathematics is to use its methods. For example, an area of study is often qualified as mathematics as soon as one can prove theorems—assertions whose validity relies on a proof, that is, a purely-logical deduction.<a href="#cite_note-179">[d]</a><a href="#cite_note-180">[176]</a>[<a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">failed verification</a>] <div class="mw-heading mw-heading3"><h3 id="Rigor">Rigor</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Logic" title="Logic">Logic</a></div> Mathematical reasoning requires <a href="/wiki/Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">rigor</a>. This means that the definitions must be absolutely unambiguous and the <a href="/wiki/Proof_(mathematics)" class="mw-redirect" title="Proof (mathematics)">proofs</a> must be reducible to a succession of applications of <a href="/wiki/Inference_rule" class="mw-redirect" title="Inference rule">inference rules</a>,<a href="#cite_note-181">[e]</a> without any use of empirical evidence and <a href="/wiki/Intuition" title="Intuition">intuition</a>.<a href="#cite_note-182">[f]</a><a href="#cite_note-183">[177]</a> Rigorous reasoning is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite mathematics' <a href="/wiki/Concision" title="Concision">concision</a>, rigorous proofs can require hundreds of pages to express, such as the 255-page <a href="/wiki/Feit%E2%80%93Thompson_theorem" title="Feit–Thompson theorem">Feit–Thompson theorem</a>.<a href="#cite_note-184">[g]</a> The emergence of <a href="/wiki/Computer-assisted_proof" title="Computer-assisted proof">computer-assisted proofs</a> has allowed proof lengths to further expand.<a href="#cite_note-185">[h]</a><a href="#cite_note-186">[178]</a> The result of this trend is a philosophy of the <a href="/wiki/Quasi-empiricism_in_mathematics" title="Quasi-empiricism in mathematics">quasi-empiricist</a> proof that can not be considered infallible, but has a probability attached to it.<a href="#cite_note-Kleiner_1991-7">[6]</a> 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.<a href="#cite_note-Kleiner_1991-7">[6]</a> 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 <a href="/wiki/Weierstrass_function" title="Weierstrass function">Weierstrass function</a>) and contradictions (Russell's paradox). This was solved by the inclusion of axioms with the <a href="/wiki/Apodictic" class="mw-redirect" title="Apodictic">apodictic</a> inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks.<a href="#cite_note-Kleiner_1991-7">[6]</a> 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 <a href="/wiki/Pleonasm" title="Pleonasm">pleonasm</a>. 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.<a href="#cite_note-187">[179]</a> Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.<a href="#cite_note-188">[180]</a> <div class="mw-heading mw-heading2"><h2 id="Training_and_practice">Training and practice</h2></div> <div class="mw-heading mw-heading3"><h3 id="Education">Education</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></div> Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a <a href="/wiki/Human_activity" class="mw-redirect" title="Human activity">human activity</a>, the practice of mathematics has a social side, which includes <a href="/wiki/Mathematics_education" title="Mathematics education">education</a>, <a href="/wiki/Mathematician" title="Mathematician">careers</a>, <a href="/wiki/List_of_mathematics_awards" title="List of mathematics awards">recognition</a>, <a href="/wiki/Popular_mathematics" title="Popular mathematics">popularization</a>, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the <a href="/wiki/STEM" class="mw-redirect" title="STEM">STEM</a> academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, <a href="/wiki/Statistician" title="Statistician">statistician</a>, <a href="/wiki/Actuary" title="Actuary">actuary</a>, <a href="/wiki/Financial_analyst" title="Financial analyst">financial analyst</a>, <a href="/wiki/Economist" title="Economist">economist</a>, <a href="/wiki/Accountant" title="Accountant">accountant</a>, <a href="/wiki/Commodity_trader" class="mw-redirect" title="Commodity trader">commodity trader</a>, or <a href="/wiki/Information_technology_consulting" title="Information technology consulting">computer consultant</a>.<a href="#cite_note-189">[181]</a> Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia.<a href="#cite_note-190">[182]</a> Comparable evidence has been unearthed for scribal mathematics training in the <a href="/wiki/Ancient_Near_East" title="Ancient Near East">ancient Near East</a> and then for the <a href="/wiki/Greco-Roman_world" title="Greco-Roman world">Greco-Roman world</a> starting around 300 BCE.<a href="#cite_note-191">[183]</a> The oldest known mathematics textbook is the <a href="/wiki/Rhind_papyrus" class="mw-redirect" title="Rhind papyrus">Rhind papyrus</a>, dated from <abbr title="circa">c.</abbr> 1650 BCE in Egypt.<a href="#cite_note-192">[184]</a> Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized <a href="/wiki/Oral_tradition" title="Oral tradition">oral tradition</a> since the <a href="/wiki/Vedic_period" title="Vedic period">Vedic period</a> (<abbr title="circa">c.</abbr> 1500 – c. 500 BCE).<a href="#cite_note-193">[185]</a> In <a href="/wiki/Imperial_China" class="mw-redirect" title="Imperial China">Imperial China</a> during the <a href="/wiki/Tang_dynasty" title="Tang dynasty">Tang dynasty</a> (618–907 CE), a mathematics curriculum was adopted for the <a href="/wiki/Imperial_examination" title="Imperial examination">civil service exam</a> to join the state bureaucracy.<a href="#cite_note-194">[186]</a> Following the <a href="/wiki/Dark_Age" class="mw-redirect" title="Dark Age">Dark Ages</a>, mathematics education in Europe was provided by religious schools as part of the <a href="/wiki/Quadrivium" title="Quadrivium">Quadrivium</a>. Formal instruction in <a href="/wiki/Pedagogy" title="Pedagogy">pedagogy</a> began with <a href="/wiki/Jesuit" class="mw-redirect" title="Jesuit">Jesuit</a> schools in the 16th and 17th century. Most mathematical curricula 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 <a href="/wiki/L%27Enseignement_Math%C3%A9matique" class="mw-redirect" title="L'Enseignement Mathématique">L'Enseignement Mathématique</a>, which began publication in 1899.<a href="#cite_note-195">[187]</a> 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—initially for its military applications.<a href="#cite_note-196">[188]</a> While the content of courses varies, in the present day nearly all countries teach mathematics to students for significant amounts of time.<a href="#cite_note-197">[189]</a> 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.<a href="#cite_note-198">[190]</a> Some students studying math may develop an apprehension or fear about their performance in the subject. This is known as <a href="/wiki/Math_anxiety" class="mw-redirect" title="Math anxiety">math anxiety</a> 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.<a href="#cite_note-199">[191]</a> <div class="mw-heading mw-heading3"><h3 id="Psychology_(aesthetic,_creativity_and_intuition)">Psychology (aesthetic, creativity and intuition)</h3></div> The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a <a href="/wiki/Computer_program" title="Computer program">computer program</a>. 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.<a href="#cite_note-200">[192]</a><a href="#cite_note-201">[193]</a> An extreme example is <a href="/wiki/Apery%27s_theorem" class="mw-redirect" title="Apery's theorem">Apery's theorem</a>: <a href="/wiki/Roger_Apery" class="mw-redirect" title="Roger Apery">Roger Apery</a> provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians.<a href="#cite_note-202">[194]</a> 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 <a href="/wiki/Puzzle" title="Puzzle">puzzles</a>.<a href="#cite_note-203">[195]</a> This aspect of mathematical activity is emphasized in <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>. Mathematicians can find an <a href="/wiki/Aesthetic" class="mw-redirect" title="Aesthetic">aesthetic</a> value to mathematics. Like <a href="/wiki/Beauty" title="Beauty">beauty</a>, it is hard to define, it is commonly related to elegance, which involves qualities like <a href="/wiki/Simplicity" title="Simplicity">simplicity</a>, <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>, completeness, and generality. G. H. Hardy in <a href="/wiki/A_Mathematician%27s_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a> 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 aesthetics.<a href="#cite_note-204">[196]</a> <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> expressed this sentiment more ironically by speaking of "The Book", a supposed divine collection of the most beautiful proofs. The 1998 book <a href="/wiki/Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a>, 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 <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> for <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>.<a href="#cite_note-205">[197]</a> Some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional <a href="/wiki/Liberal_arts" class="mw-redirect" title="Liberal arts">liberal arts</a>.<a href="#cite_note-206">[198]</a> 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).<a href="#cite_note-borel-134">[131]</a> The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. <div class="mw-heading mw-heading2"><h2 id="Cultural_impact">Cultural impact</h2></div> <div class="mw-heading mw-heading3"><h3 id="Artistic_expression">Artistic expression</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></div> Notes that sound well together to a Western ear are sounds whose fundamental <a href="/wiki/Frequencies" class="mw-redirect" title="Frequencies">frequencies</a> of vibration are in simple ratios. For example, an octave doubles the frequency and a <a href="/wiki/Perfect_fifth" title="Perfect fifth">perfect fifth</a> multiplies it by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4d39a9f7beda4ddbeffafaca691c386d3142cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}}">.<a href="#cite_note-207">[199]</a><a href="#cite_note-208">[200]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Julia_set_(highres_01).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Julia_set_%28highres_01%29.jpg/220px-Julia_set_%28highres_01%29.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Julia_set_%28highres_01%29.jpg/330px-Julia_set_%28highres_01%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Julia_set_%28highres_01%29.jpg/440px-Julia_set_%28highres_01%29.jpg 2x" data-file-width="2048" data-file-height="2048" /></a><figcaption><a href="/wiki/Fractal" title="Fractal">Fractal</a> with a scaling symmetry and a central symmetry</figcaption></figure> Humans, as well as some other animals, find symmetric patterns to be more beautiful.<a href="#cite_note-209">[201]</a> Mathematically, the symmetries of an object form a group known as the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>.<a href="#cite_note-210">[202]</a> For example, the group underlying mirror symmetry is the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> of two elements, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">. A <a href="/wiki/Rorschach_test" title="Rorschach test">Rorschach test</a> is a figure invariant by this symmetry,<a href="#cite_note-211">[203]</a> as are <a href="/wiki/Butterfly" title="Butterfly">butterfly</a> and animal bodies more generally (at least on the surface).<a href="#cite_note-212">[204]</a> 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.<a href="#cite_note-213">[205]</a> <a href="/wiki/Fractals" class="mw-redirect" title="Fractals">Fractals</a> possess <a href="/wiki/Self-similarity" title="Self-similarity">self-similarity</a>.<a href="#cite_note-214">[206]</a><a href="#cite_note-215">[207]</a> <div class="mw-heading mw-heading3"><h3 id="Popularization">Popularization</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Popular_mathematics" title="Popular mathematics">Popular mathematics</a></div>Popular mathematics is the act of presenting mathematics without technical terms.<a href="#cite_note-216">[208]</a> Presenting mathematics may be hard since the general public suffers from <a href="/wiki/Mathematical_anxiety" title="Mathematical anxiety">mathematical anxiety</a> and mathematical objects are highly abstract.<a href="#cite_note-217">[209]</a> However, popular mathematics writing can overcome this by using applications or cultural links.<a href="#cite_note-218">[210]</a> Despite this, mathematics is rarely the topic of popularization in printed or televised media. <div class="mw-heading mw-heading3"><h3 id="Awards_and_prize_problems">Awards and prize problems</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main category: <a href="/wiki/Category:Mathematics_awards" title="Category:Mathematics awards">Mathematics awards</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:FieldsMedalFront.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/FieldsMedalFront.jpg/220px-FieldsMedalFront.jpg" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/FieldsMedalFront.jpg/330px-FieldsMedalFront.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/FieldsMedalFront.jpg/440px-FieldsMedalFront.jpg 2x" data-file-width="2493" data-file-height="2398" /></a><figcaption>The front side of the <a href="/wiki/Fields_Medal" title="Fields Medal">Fields Medal</a> with an illustration of the Greek <a href="/wiki/Polymath" title="Polymath">polymath</a> <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></figcaption></figure> The most prestigious award in mathematics is the <a href="/wiki/Fields_Medal" title="Fields Medal">Fields Medal</a>,<a href="#cite_note-FOOTNOTEMonastyrsky20011-219">[211]</a><a href="#cite_note-FOOTNOTERiehm2002778–782-220">[212]</a> established in 1936 and awarded every four years (except around <a href="/wiki/World_War_II_in_Yugoslavia" title="World War II in Yugoslavia">World War II</a>) to up to four individuals.<a href="#cite_note-221">[213]</a><a href="#cite_note-StAndrews-Fields-222">[214]</a> It is considered the mathematical equivalent of the <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a>.<a href="#cite_note-StAndrews-Fields-222">[214]</a> Other prestigious mathematics awards include:<a href="#cite_note-223">[215]</a> <ul><li>The <a href="/wiki/Abel_Prize" title="Abel Prize">Abel Prize</a>, instituted in 2002<a href="#cite_note-224">[216]</a> and first awarded in 2003<a href="#cite_note-225">[217]</a></li> <li>The <a href="/wiki/Chern_Medal" title="Chern Medal">Chern Medal</a> for lifetime achievement, introduced in 2009<a href="#cite_note-226">[218]</a> and first awarded in 2010<a href="#cite_note-227">[219]</a></li> <li>The <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">AMS</a> <a href="/wiki/Leroy_P._Steele_Prize" title="Leroy P. Steele Prize">Leroy P. Steele Prize</a>, awarded since 1970<a href="#cite_note-228">[220]</a></li> <li>The <a href="/wiki/Wolf_Prize_in_Mathematics" title="Wolf Prize in Mathematics">Wolf Prize in Mathematics</a>, also for lifetime achievement,<a href="#cite_note-229">[221]</a> instituted in 1978<a href="#cite_note-230">[222]</a></li></ul> A famous list of 23 <a href="/wiki/Open_problem" title="Open problem">open problems</a>, called "<a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">Hilbert's problems</a>", was compiled in 1900 by German mathematician David Hilbert.<a href="#cite_note-:0-231">[223]</a> This list has achieved great celebrity among mathematicians,<a href="#cite_note-232">[224]</a> and at least thirteen of the problems (depending how some are interpreted) have been solved.<a href="#cite_note-:0-231">[223]</a> A new list of seven important problems, titled the "<a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a>", was published in 2000. Only one of them, the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.<a href="#cite_note-233">[225]</a> To date, only one of these problems, the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, has been solved by the Russian mathematician <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>.<a href="#cite_note-234">[226]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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href="/wiki/Science,_technology,_engineering,_and_mathematics" title="Science, technology, engineering, and mathematics">Science, technology, engineering, and mathematics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><a href="#cite_ref-5">^</a> Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating <a href="/wiki/Formula" title="Formula">formulas</a>. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> This includes <a href="/wiki/Conic_section" title="Conic section">conic sections</a>, which are intersections of <a href="/wiki/Circular_cylinder" class="mw-redirect" title="Circular cylinder">circular cylinders</a> and planes. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> However, some advanced methods of analysis are sometimes used; for example, methods of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> applied to <a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">generating series</a>. </li> <li id="cite_note-179"><a href="#cite_ref-179">^</a> For example, logic belongs to philosophy since <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>. Circa the end of the 19th century, the <a href="/wiki/Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a> implied developments of logic that are specific to mathematics. This allowed eventually the proof of theorems such as <a href="/wiki/G%C3%B6del%27s_theorems" class="mw-redirect" title="Gödel's theorems">Gödel's theorems</a>. Since then, <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> is commonly considered as an area of mathematics. </li> <li id="cite_note-181"><a href="#cite_ref-181">^</a> This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without <a href="/wiki/Computer" title="Computer">computers</a> and <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistants</a>. Even with this modern technology, it may take years of human work for writing down a completely detailed proof. </li> <li id="cite_note-182"><a href="#cite_ref-182">^</a> This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them. </li> <li id="cite_note-184"><a href="#cite_ref-184">^</a> This is the length of the original paper that does not contain the proofs of some previously published auxiliary results. The book devoted to the complete proof has more than 1,000 pages. </li> <li id="cite_note-185"><a href="#cite_ref-185">^</a> For considering as reliable a large computation occurring in a proof, one generally requires two computations using independent software </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em ;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHipólito2015" class="citation book cs1 cs1-prop-foreign-lang-source">Hipólito, Inês Viegas (August 9–15, 2015). "Abstract Cognition and the Nature of Mathematical Proof". In Kanzian, Christian; <a href="/wiki/Josef_Mitterer" title="Josef Mitterer">Mitterer, Josef</a>; Neges, Katharina (eds.). <a rel="nofollow" class="external text" href="https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133">Realismus – Relativismus – Konstruktivismus: Beiträge des 38. Internationalen Wittgenstein Symposiums</a> [Realism – Relativism – Constructivism: Contributions of the 38th International Wittgenstein Symposium] (PDF) (in German and English). Vol. 23. Kirchberg am Wechsel, Austria: Austrian Ludwig Wittgenstein Society. pp. 132–134. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1022-3398">1022-3398</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/236026294">236026294</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221107221937/https://www.alws.at/alws/wp-content/uploads/2018/06/papers-2015.pdf#page=133">Archived</a> (PDF) from the original on November 7, 2022. Retrieved January 17, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abstract+Cognition+and+the+Nature+of+Mathematical+Proof&rft.btitle=Realismus+%E2%80%93+Relativismus+%E2%80%93+Konstruktivismus%3A+Beitr%C3%A4ge+des+38.+Internationalen+Wittgenstein+Symposiums&rft.place=Kirchberg+am+Wechsel%2C+Austria&rft.pages=132-134&rft.pub=Austrian+Ludwig+Wittgenstein+Society&rft.date=2015-08-09%2F2015-08-15&rft_id=info%3Aoclcnum%2F236026294&rft.issn=1022-3398&rft.aulast=Hip%C3%B3lito&rft.aufirst=In%C3%AAs+Viegas&rft_id=https%3A%2F%2Fwww.alws.at%2Falws%2Fwp-content%2Fuploads%2F2018%2F06%2Fpapers-2015.pdf%23page%3D133&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> (<a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof">at ResearchGate</a> <a href="/wiki/Open_access" title="open access publication – free to read"><img alt="Open access icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/9px-Open_Access_logo_PLoS_transparent.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/14px-Open_Access_logo_PLoS_transparent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Open_Access_logo_PLoS_transparent.svg/18px-Open_Access_logo_PLoS_transparent.svg.png 2x" data-file-width="640" data-file-height="1000" /></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221105145638/https://www.researchgate.net/publication/280654540_Abstract_Cognition_and_the_Nature_of_Mathematical_Proof">Archived</a> November 5, 2022, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>) </li> <li id="cite_note-FOOTNOTEPeterson198812-2"><a href="#cite_ref-FOOTNOTEPeterson198812_2-0">^</a> <a href="#CITEREFPeterson1988">Peterson 1988</a>, p. 12. </li> <li id="cite_note-wigner1960-3">^ <a href="#cite_ref-wigner1960_3-0">a</a> <a href="#cite_ref-wigner1960_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWigner1960" class="citation journal cs1"><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner, Eugene</a> (1960). <a rel="nofollow" class="external text" href="https://math.dartmouth.edu/~matc/MathDrama/reading/Wigner.html">"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"</a>. <a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" title="Communications on Pure and Applied Mathematics">Communications on Pure and Applied Mathematics</a>. 13 (1): 1–14. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960CPAM...13....1W">1960CPAM...13....1W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160130102">10.1002/cpa.3160130102</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6112252">6112252</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110228152633/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html">Archived</a> from the original on February 28, 2011.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.atitle=The+Unreasonable+Effectiveness+of+Mathematics+in+the+Natural+Sciences&rft.volume=13&rft.issue=1&rft.pages=1-14&rft.date=1960&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6112252%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160130102&rft_id=info%3Abibcode%2F1960CPAM...13....1W&rft.aulast=Wigner&rft.aufirst=Eugene&rft_id=https%3A%2F%2Fmath.dartmouth.edu%2F~matc%2FMathDrama%2Freading%2FWigner.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWise" class="citation web cs1">Wise, David. <a rel="nofollow" class="external text" href="http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Wise/essay7/essay7.htm">"Eudoxus' Influence on Euclid's Elements with a close look at The Method of Exhaustion"</a>. <a href="/wiki/The_University_of_Georgia" class="mw-redirect" title="The University of Georgia">The University of Georgia</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190601004355/http://jwilson.coe.uga.edu/emt668/EMAT6680.F99/Wise/essay7/essay7.htm">Archived</a> from the original on June 1, 2019. Retrieved January 18, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+University+of+Georgia&rft.atitle=Eudoxus%27+Influence+on+Euclid%27s+Elements+with+a+close+look+at+The+Method+of+Exhaustion&rft.aulast=Wise&rft.aufirst=David&rft_id=http%3A%2F%2Fjwilson.coe.uga.edu%2FEMT668%2FEMAT6680.F99%2FWise%2Fessay7%2Fessay7.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander2011" class="citation journal cs1"><a href="/wiki/Amir_Alexander" title="Amir Alexander">Alexander, Amir</a> (September 2011). "The Skeleton in the Closet: Should Historians of Science Care about the History of Mathematics?". Isis. 102 (3): 475–480. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F661620">10.1086/661620</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-1753">0021-1753</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2884913">2884913</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22073771">22073771</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21629993">21629993</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Isis&rft.atitle=The+Skeleton+in+the+Closet%3A+Should+Historians+of+Science+Care+about+the+History+of+Mathematics%3F&rft.volume=102&rft.issue=3&rft.pages=475-480&rft.date=2011-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21629993%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1086%2F661620&rft.issn=0021-1753&rft_id=info%3Apmid%2F22073771&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2884913%23id-name%3DMR&rft.aulast=Alexander&rft.aufirst=Amir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Kleiner_1991-7">^ <a href="#cite_ref-Kleiner_1991_7-0">a</a> <a href="#cite_ref-Kleiner_1991_7-1">b</a> <a href="#cite_ref-Kleiner_1991_7-2">c</a> <a href="#cite_ref-Kleiner_1991_7-3">d</a> <a href="#cite_ref-Kleiner_1991_7-4">e</a> <a href="#cite_ref-Kleiner_1991_7-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1991" class="citation journal cs1"><a href="/wiki/Israel_Kleiner_(mathematician)" title="Israel Kleiner (mathematician)">Kleiner, Israel</a> (December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine. 64 (5). Taylor & Francis, Ltd.: 291–314. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.1991.11977625">10.1080/0025570X.1991.11977625</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1930-0980">1930-0980</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690647">2690647</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/47003192">47003192</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1141557">1141557</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1756877">1756877</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7787171">7787171</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Rigor+and+Proof+in+Mathematics%3A+A+Historical+Perspective&rft.volume=64&rft.issue=5&rft.pages=291-314&rft.date=1991-12&rft_id=info%3Alccn%2F47003192&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7787171%23id-name%3DS2CID&rft.eissn=1930-0980&rft_id=info%3Adoi%2F10.1080%2F0025570X.1991.11977625&rft_id=info%3Aoclcnum%2F1756877&rft.issn=0025-570X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690647%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1141557%23id-name%3DMR&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1945" class="citation book cs1"><a href="/wiki/Eric_Temple_Bell" title="Eric Temple Bell">Bell, E. T.</a> (1945) [1940]. "General Prospectus". The Development of Mathematics (2nd ed.). Dover Publications. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-27239-9" title="Special:BookSources/978-0-486-27239-9"><bdi>978-0-486-27239-9</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/45010599">45010599</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/523284">523284</a>. <q>... 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.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=General+Prospectus&rft.btitle=The+Development+of+Mathematics&rft.pages=3&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=1945&rft_id=info%3Aoclcnum%2F523284&rft_id=info%3Alccn%2F45010599&rft.isbn=978-0-486-27239-9&rft.aulast=Bell&rft.aufirst=E.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiwari1992" class="citation book cs1">Tiwari, Sarju (1992). "A Mirror of Civilization". Mathematics in History, Culture, Philosophy, and Science (1st ed.). New Delhi, India: Mittal Publications. p. 27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7099-404-6" title="Special:BookSources/978-81-7099-404-6"><bdi>978-81-7099-404-6</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/92909575">92909575</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28115124">28115124</a>. <q>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.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+Mirror+of+Civilization&rft.btitle=Mathematics+in+History%2C+Culture%2C+Philosophy%2C+and+Science&rft.place=New+Delhi%2C+India&rft.pages=27&rft.edition=1st&rft.pub=Mittal+Publications&rft.date=1992&rft_id=info%3Aoclcnum%2F28115124&rft_id=info%3Alccn%2F92909575&rft.isbn=978-81-7099-404-6&rft.aulast=Tiwari&rft.aufirst=Sarju&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRestivo1992" class="citation book cs1"><a href="/wiki/Sal_Restivo" title="Sal Restivo">Restivo, Sal</a> (1992). "Mathematics from the Ground Up". In <a href="/wiki/Mario_Bunge" title="Mario Bunge">Bunge, Mario</a> (ed.). Mathematics in Society and History. Episteme. Vol. 20. <a href="/wiki/Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a>. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-1765-3" title="Special:BookSources/0-7923-1765-3"><bdi>0-7923-1765-3</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/25709270">25709270</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/92013695">92013695</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mathematics+from+the+Ground+Up&rft.btitle=Mathematics+in+Society+and+History&rft.series=Episteme&rft.pages=14&rft.pub=Kluwer+Academic+Publishers&rft.date=1992&rft_id=info%3Aoclcnum%2F92013695&rft_id=info%3Alccn%2F25709270&rft.isbn=0-7923-1765-3&rft.aulast=Restivo&rft.aufirst=Sal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMusielak2022" class="citation book cs1"><a href="/wiki/Dora_Musielak" title="Dora Musielak">Musielak, Dora</a> (2022). Leonhard Euler and the Foundations of Celestial Mechanics. History of Physics. <a href="/wiki/Springer_International_Publishing" class="mw-redirect" title="Springer International Publishing">Springer International Publishing</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-031-12322-1">10.1007/978-3-031-12322-1</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2730-7557">2730-7557</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-031-12321-4" title="Special:BookSources/978-3-031-12321-4"><bdi>978-3-031-12321-4</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2730-7549">2730-7549</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1332780664">1332780664</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:253240718">253240718</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Leonhard+Euler+and+the+Foundations+of+Celestial+Mechanics&rft.series=History+of+Physics&rft.pub=Springer+International+Publishing&rft.date=2022&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A253240718%23id-name%3DS2CID&rft.eissn=2730-7557&rft_id=info%3Adoi%2F10.1007%2F978-3-031-12322-1&rft_id=info%3Aoclcnum%2F1332780664&rft.issn=2730-7549&rft.isbn=978-3-031-12321-4&rft.aulast=Musielak&rft.aufirst=Dora&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBiggs1979" class="citation journal cs1">Biggs, N. L. (May 1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2879%2990074-0">"The roots of combinatorics"</a>. Historia Mathematica. 6 (2): 109–136. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2879%2990074-0">10.1016/0315-0860(79)90074-0</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1090-249X">1090-249X</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0315-0860">0315-0860</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/75642280">75642280</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2240703">2240703</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+roots+of+combinatorics&rft.volume=6&rft.issue=2&rft.pages=109-136&rft.date=1979-05&rft.eissn=1090-249X&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2879%2990074-0&rft_id=info%3Aoclcnum%2F2240703&rft_id=info%3Alccn%2F75642280&rft.issn=0315-0860&rft.aulast=Biggs&rft.aufirst=N.+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0315-0860%252879%252990074-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Warner_2013-13">^ <a href="#cite_ref-Warner_2013_13-0">a</a> <a href="#cite_ref-Warner_2013_13-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner" class="citation web cs1">Warner, Evan. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230322165544/https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf">"Splash Talk: The Foundational Crisis of Mathematics"</a> (PDF). <a href="/wiki/Columbia_University" title="Columbia University">Columbia University</a>. Archived from <a rel="nofollow" class="external text" href="https://www.math.columbia.edu/~warner/notes/SplashTalk.pdf">the original</a> (PDF) on March 22, 2023. Retrieved February 3, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Splash+Talk%3A+The+Foundational+Crisis+of+Mathematics&rft.pub=Columbia+University&rft.aulast=Warner&rft.aufirst=Evan&rft_id=https%3A%2F%2Fwww.math.columbia.edu%2F~warner%2Fnotes%2FSplashTalk.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDunneHulek2020" class="citation journal cs1">Dunne, Edward; <a href="/wiki/Klaus_Hulek" title="Klaus Hulek">Hulek, Klaus</a> (March 2020). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/notices/202003/rnoti-p410.pdf">"Mathematics Subject Classification 2020"</a> (PDF). Notices of the American Mathematical Society. 67 (3): 410–411. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti2052">10.1090/noti2052</a>. <a href="/wiki/EISSN_(identifier)" class="mw-redirect" title="EISSN (identifier)">eISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1088-9477">1088-9477</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9920">0002-9920</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/sf77000404">sf77000404</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1480366">1480366</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210803203928/https://www.ams.org/journals/notices/202003/rnoti-p410.pdf">Archived</a> (PDF) from the original on August 3, 2021. Retrieved February 3, 2024. <q>The new MSC contains 63 two-digit classifications, 529 three-digit classifications, and 6,006 five-digit classifications.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Mathematics+Subject+Classification+2020&rft.volume=67&rft.issue=3&rft.pages=410-411&rft.date=2020-03&rft.eissn=1088-9477&rft_id=info%3Adoi%2F10.1090%2Fnoti2052&rft_id=info%3Aoclcnum%2F1480366&rft_id=info%3Alccn%2Fsf77000404&rft.issn=0002-9920&rft.aulast=Dunne&rft.aufirst=Edward&rft.au=Hulek%2C+Klaus&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fnotices%2F202003%2Frnoti-p410.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-MSC-15">^ <a href="#cite_ref-MSC_15-0">a</a> <a href="#cite_ref-MSC_15-1">b</a> <a href="#cite_ref-MSC_15-2">c</a> <a href="#cite_ref-MSC_15-3">d</a> <a href="#cite_ref-MSC_15-4">e</a> <a href="#cite_ref-MSC_15-5">f</a> <a href="#cite_ref-MSC_15-6">g</a> <a href="#cite_ref-MSC_15-7">h</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://zbmath.org/static/msc2020.pdf">"MSC2020-Mathematics Subject Classification System"</a> (PDF). zbMath. Associate Editors of Mathematical Reviews and zbMATH. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240102023805/https://zbmath.org/static/msc2020.pdf">Archived</a> (PDF) from the original on January 2, 2024. Retrieved February 3, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=zbMath&rft.atitle=MSC2020-Mathematics+Subject+Classification+System&rft_id=https%3A%2F%2Fzbmath.org%2Fstatic%2Fmsc2020.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeVeque1977" class="citation book cs1"><a href="/wiki/William_J._LeVeque" title="William J. LeVeque">LeVeque, William J.</a> (1977). "Introduction". Fundamentals of Number Theory. <a href="/wiki/Addison-Wesley_Publishing_Company" class="mw-redirect" title="Addison-Wesley Publishing Company">Addison-Wesley Publishing Company</a>. pp. 1–30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-04287-8" title="Special:BookSources/0-201-04287-8"><bdi>0-201-04287-8</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/76055645">76055645</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/3519779">3519779</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118560854">118560854</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction&rft.btitle=Fundamentals+of+Number+Theory&rft.pages=1-30&rft.pub=Addison-Wesley+Publishing+Company&rft.date=1977&rft_id=info%3Aoclcnum%2F3519779&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118560854%23id-name%3DS2CID&rft_id=info%3Alccn%2F76055645&rft.isbn=0-201-04287-8&rft.aulast=LeVeque&rft.aufirst=William+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldman1998" class="citation book cs1">Goldman, Jay R. 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World Scientific. pp. 3–17, 63–70. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-02-3111-3" title="Special:BookSources/981-02-3111-3"><bdi>981-02-3111-3</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/97010349">97010349</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1474730">1474730</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36597731">36597731</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+and+Truth%3A+Putting+Chance+to+Work&rft.pages=3-17%2C+63-70&rft.edition=2nd&rft.pub=World+Scientific&rft.date=1997&rft_id=info%3Aoclcnum%2F36597731&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1474730%23id-name%3DMR&rft_id=info%3Alccn%2F97010349&rft.isbn=981-02-3111-3&rft.aulast=Rao&rft.aufirst=C.+Radhakrishna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-RaoOpt-64"><a href="#cite_ref-RaoOpt_64-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRao1981" class="citation book cs1"><a href="/wiki/C.R._Rao" class="mw-redirect" title="C.R. Rao">Rao, C. Radhakrishna</a> (1981). "Foreword". In Arthanari, T.S.; <a href="/wiki/Yadolah_Dodge" title="Yadolah Dodge">Dodge, Yadolah</a> (eds.). Mathematical programming in statistics. Wiley Series in Probability and Mathematical Statistics. New York: Wiley. pp. vii–viii. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-08073-2" title="Special:BookSources/978-0-471-08073-2"><bdi>978-0-471-08073-2</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/80021637">80021637</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0607328">0607328</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/6707805">6707805</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Foreword&rft.btitle=Mathematical+programming+in+statistics&rft.place=New+York&rft.series=Wiley+Series+in+Probability+and+Mathematical+Statistics&rft.pages=vii-viii&rft.pub=Wiley&rft.date=1981&rft_id=info%3Aoclcnum%2F6707805&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D607328%23id-name%3DMR&rft_id=info%3Alccn%2F80021637&rft.isbn=978-0-471-08073-2&rft.aulast=Rao&rft.aufirst=C.+Radhakrishna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-FOOTNOTEWhittle199410–11,_14–18-65"><a href="#cite_ref-FOOTNOTEWhittle199410–11,_14–18_65-0">^</a> <a href="#CITEREFWhittle1994">Whittle 1994</a>, pp. 10–11, 14–18. </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarchuk2020" class="citation web cs1">Marchuk, Gurii Ivanovich (April 2020). <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Extras/Computational_mathematics/">"G I Marchuk's plenary: ICM 1970"</a>. MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221113155409/https://mathshistory.st-andrews.ac.uk/Extras/Computational_mathematics/">Archived</a> from the original on November 13, 2022. Retrieved November 13, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MacTutor&rft.atitle=G+I+Marchuk%27s+plenary%3A+ICM+1970&rft.date=2020-04&rft.aulast=Marchuk&rft.aufirst=Gurii+Ivanovich&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FExtras%2FComputational_mathematics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnsonCavallini1991" class="citation conference cs1">Johnson, Gary M.; Cavallini, John S. (September 1991). Phua, Kang Hoh; Loe, Kia Fock (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jYNIDwAAQBAJ&pg=PA28">Grand Challenges, High Performance Computing, and Computational Science</a>. Singapore Supercomputing Conference'90: Supercomputing For Strategic Advantage. World Scientific. p. 28. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/91018998">91018998</a>. Retrieved November 13, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Grand+Challenges%2C+High+Performance+Computing%2C+and+Computational+Science&rft.pages=28&rft.pub=World+Scientific&rft.date=1991-09&rft_id=info%3Alccn%2F91018998&rft.aulast=Johnson&rft.aufirst=Gary+M.&rft.au=Cavallini%2C+John+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjYNIDwAAQBAJ%26pg%3DPA28&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethen2008" class="citation book cs1"><a href="/wiki/Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a> (2008). "Numerical Analysis". In <a href="/wiki/Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a>; <a href="/wiki/June_Barrow-Green" title="June Barrow-Green">Barrow-Green, June</a>; <a href="/wiki/Imre_Leader" title="Imre Leader">Leader, Imre</a> (eds.). <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/trefethen/NAessay.pdf">The Princeton Companion to Mathematics</a> (PDF). <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. 604–615. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-11880-2" title="Special:BookSources/978-0-691-11880-2"><bdi>978-0-691-11880-2</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2008020450">2008020450</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2467561">2467561</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/227205932">227205932</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230307054158/http://people.maths.ox.ac.uk/trefethen/NAessay.pdf">Archived</a> (PDF) from the original on March 7, 2023. Retrieved February 15, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Numerical+Analysis&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.pages=604-615&rft.pub=Princeton+University+Press&rft.date=2008&rft_id=info%3Aoclcnum%2F227205932&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2467561%23id-name%3DMR&rft_id=info%3Alccn%2F2008020450&rft.isbn=978-0-691-11880-2&rft.aulast=Trefethen&rft.aufirst=Lloyd+N.&rft_id=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Ftrefethen%2FNAessay.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li><a href="#CITEREFCresswell2021">Cresswell 2021</a>, § Mathematics</li><li><a href="#CITEREFPerisho1965">Perisho 1965</a>, p. 64</li></ul></div> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerisho1965" class="citation journal cs1">Perisho, Margaret W. (Spring 1965). "The Etymology of Mathematical Terms". <a href="/wiki/Pi_Mu_Epsilon_Journal" class="mw-redirect" title="Pi Mu Epsilon Journal">Pi Mu Epsilon Journal</a>. 4 (2): 62–66. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-952X">0031-952X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24338341">24338341</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/58015848">58015848</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1762376">1762376</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pi+Mu+Epsilon+Journal&rft.atitle=The+Etymology+of+Mathematical+Terms&rft.ssn=spring&rft.volume=4&rft.issue=2&rft.pages=62-66&rft.date=1965&rft_id=info%3Aoclcnum%2F1762376&rft.issn=0031-952X&rft_id=info%3Alccn%2F58015848&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24338341%23id-name%3DJSTOR&rft.aulast=Perisho&rft.aufirst=Margaret+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Boas-71"><a href="#cite_ref-Boas_71-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoas1995" class="citation book cs1"><a href="/wiki/Ralph_P._Boas_Jr." title="Ralph P. Boas Jr.">Boas, Ralph P.</a> (1995). "What Augustine Didn't Say About Mathematicians". In Alexanderson, Gerald L.; Mugler, Dale H. (eds.). Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. p. 257. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-323-8" title="Special:BookSources/978-0-88385-323-8"><bdi>978-0-88385-323-8</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/94078313">94078313</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/633018890">633018890</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+Augustine+Didn%27t+Say+About+Mathematicians&rft.btitle=Lion+Hunting+and+Other+Mathematical+Pursuits%3A+A+Collection+of+Mathematics%2C+Verse%2C+and+Stories&rft.pages=257&rft.pub=Mathematical+Association+of+America&rft.date=1995&rft_id=info%3Aoclcnum%2F633018890&rft_id=info%3Alccn%2F94078313&rft.isbn=978-0-88385-323-8&rft.aulast=Boas&rft.aufirst=Ralph+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <a href="/wiki/The_Oxford_Dictionary_of_English_Etymology" title="The Oxford Dictionary of English Etymology">The Oxford Dictionary of English Etymology</a>, <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, sub "mathematics", "mathematic", "mathematics". </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/dictionary/maths_n">"Maths (Noun)"</a>. <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. Retrieved January 25, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Oxford+English+Dictionary&rft.atitle=Maths+%28Noun%29&rft_id=https%3A%2F%2Fwww.oed.com%2Fdictionary%2Fmaths_n&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/dictionary/math_n3">"Math (Noun³)"</a>. <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200404201407/http://oed.com/view/Entry/114982">Archived</a> from the original on April 4, 2020. Retrieved January 25, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Oxford+English+Dictionary&rft.atitle=Math+%28Noun%C2%B3%29&rft_id=https%3A%2F%2Fwww.oed.com%2Fdictionary%2Fmath_n3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> See, for example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilder" class="citation book cs1"><a href="/wiki/Raymond_L._Wilder" class="mw-redirect" title="Raymond L. Wilder">Wilder, Raymond L.</a> Evolution of Mathematical Concepts; an Elementary Study. passim.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Evolution+of+Mathematical+Concepts%3B+an+Elementary+Study&rft.pages=passim&rft.aulast=Wilder&rft.aufirst=Raymond+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZaslavsky1999" class="citation book cs1"><a href="/wiki/Claudia_Zaslavsky" title="Claudia Zaslavsky">Zaslavsky, Claudia</a> (1999). Africa Counts: Number and Pattern in African Culture. Chicago Review Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61374-115-3" title="Special:BookSources/978-1-61374-115-3"><bdi>978-1-61374-115-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/843204342">843204342</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Africa+Counts%3A+Number+and+Pattern+in+African+Culture.&rft.pub=Chicago+Review+Press&rft.date=1999&rft_id=info%3Aoclcnum%2F843204342&rft.isbn=978-1-61374-115-3&rft.aulast=Zaslavsky&rft.aufirst=Claudia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKline1990Chapter_1-77"><a href="#cite_ref-FOOTNOTEKline1990Chapter_1_77-0">^</a> <a href="#CITEREFKline1990">Kline 1990</a>, Chapter 1. </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <a rel="nofollow" class="external text" href="https://www.ms.uky.edu/~dhje223/CrestOfThePeacockCh4-pages-2-21.pdf/">Mesopotamia</a> pg 10. Retrieved June 1, 2024 </li> <li id="cite_note-FOOTNOTEBoyer1991"Mesopotamia"_pp._24–27-79"><a href="#cite_ref-FOOTNOTEBoyer1991"Mesopotamia"_pp._24–27_79-0">^</a> <a href="#CITEREFBoyer1991">Boyer 1991</a>, "Mesopotamia" pp. 24–27. </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeath1981" class="citation book cs1"><a href="/wiki/Thomas_Heath_(classicist)" title="Thomas Heath (classicist)">Heath, Thomas Little</a> (1981) [1921]. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofgreekma0002heat/page/n14">A History of Greek Mathematics: From Thales to Euclid</a>. New York: Dover Publications. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-24073-2" title="Special:BookSources/978-0-486-24073-2"><bdi>978-0-486-24073-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Greek+Mathematics%3A+From+Thales+to+Euclid&rft.place=New+York&rft.pages=1&rft.pub=Dover+Publications&rft.date=1981&rft.isbn=978-0-486-24073-2&rft.aulast=Heath&rft.aufirst=Thomas+Little&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofgreekma0002heat%2Fpage%2Fn14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMueller1969" class="citation journal cs1">Mueller, I. (1969). "Euclid's Elements and the Axiomatic Method". 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Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98269-4" title="Special:BookSources/978-0-387-98269-4"><bdi>978-0-387-98269-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Out+of+Their+Minds%3A+The+Lives+and+Discoveries+of+15+Great+Computer+Scientists&rft.pages=228&rft.pub=Springer&rft.date=1998&rft.isbn=978-0-387-98269-4&rft.aulast=Shasha&rft.aufirst=Dennis+Elliot&rft.au=Lazere%2C+Cathy+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Nickles2013-112"><a href="#cite_ref-Nickles2013_112-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNickles2013" class="citation book cs1">Nickles, Thomas (2013). "The Problem of Demarcation". 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Springer Science & Business Media. pp. 235–268. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-34720-0" title="Special:BookSources/978-3-540-34720-0"><bdi>978-3-540-34720-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%CE%8C+%CE%98%CE%B5%E1%BD%B8%CF%82+%CE%86%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%AF%CE%B6%CE%B5%CE%B9%3A+The+Rise+of+Pure+Mathematics+as+Arithmetic+with+Gauss&rft.btitle=The+Shaping+of+Arithmetic+after+C.F.+Gauss%27s+Disquisitiones+Arithmeticae&rft.pages=235-268&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-3-540-34720-0&rft.aulast=Ferreir%C3%B3s&rft.aufirst=J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIUFTcOsMTysC%26pg%3DPA235&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuhn1976" class="citation journal cs1"><a href="/wiki/Thomas_Kuhn" title="Thomas Kuhn">Kuhn, Thomas S.</a> (1976). 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Forum of Mathematics, Pi. 5: e2. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Ffmp.2017.1">10.1017/fmp.2017.1</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2066%2F176365">2066/176365</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2050-5086">2050-5086</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216912822">216912822</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201204053232/https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/formal-proof-of-the-kepler-conjecture/78FBD5E1A3D1BCCB8E0D5B0C463C9FBC">Archived</a> from the original on December 4, 2020. Retrieved February 25, 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Forum+of+Mathematics%2C+Pi&rft.atitle=A+Formal+Proof+of+the+Kepler+Conjecture&rft.volume=5&rft.pages=e2&rft.date=2017&rft_id=info%3Ahdl%2F2066%2F176365&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216912822%23id-name%3DS2CID&rft.issn=2050-5086&rft_id=info%3Adoi%2F10.1017%2Ffmp.2017.1&rft.aulast=Hales&rft.aufirst=Thomas&rft.au=Adams%2C+Mark&rft.au=Bauer%2C+Gertrud&rft.au=Dang%2C+Tat+Dat&rft.au=Harrison%2C+John&rft.au=Hoang%2C+Le+Truong&rft.au=Kaliszyk%2C+Cezary&rft.au=Magron%2C+Victor&rft.au=Mclaughlin%2C+Sean&rft.au=Nguyen%2C+Tat+Thang&rft.au=Nguyen%2C+Quang+Truong&rft.au=Nipkow%2C+Tobias&rft.au=Obua%2C+Steven&rft.au=Pleso%2C+Joseph&rft.au=Rute%2C+Jason&rft.au=Solovyev%2C+Alexey&rft.au=Ta%2C+Thi+Hoai+An&rft.au=Tran%2C+Nam+Trung&rft.au=Trieu%2C+Thi+Diep&rft.au=Urban%2C+Josef&rft.au=Vu%2C+Ky&rft.au=Zumkeller%2C+Roland&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fjournals%2Fforum-of-mathematics-pi%2Farticle%2Fformal-proof-of-the-kepler-conjecture%2F78FBD5E1A3D1BCCB8E0D5B0C463C9FBC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-:2-143">^ <a href="#cite_ref-:2_143-0">a</a> <a href="#cite_ref-:2_143-1">b</a> <a href="#cite_ref-:2_143-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillstein2016" class="citation book cs1"><a href="/wiki/Roberta_Millstein" title="Roberta Millstein">Millstein, Roberta</a> (September 8, 2016). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf">"Probability in Biology: The Case of Fitness"</a> (PDF). In Hájek, Alan; Hitchcock, Christopher (eds.). The Oxford Handbook of Probability and Philosophy. pp. 601–622. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Foxfordhb%2F9780199607617.013.27">10.1093/oxfordhb/9780199607617.013.27</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230307054456/http://philsci-archive.pitt.edu/10901/1/Millstein-fitness-v2.pdf">Archived</a> (PDF) from the original on March 7, 2023. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Probability+in+Biology%3A+The+Case+of+Fitness&rft.btitle=The+Oxford+Handbook+of+Probability+and+Philosophy&rft.pages=601-622&rft.date=2016-09-08&rft_id=info%3Adoi%2F10.1093%2Foxfordhb%2F9780199607617.013.27&rft.aulast=Millstein&rft.aufirst=Roberta&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2F10901%2F1%2FMillstein-fitness-v2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> 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 </li> <li id="cite_note-FOOTNOTEBouleau1999282–283-145"><a href="#cite_ref-FOOTNOTEBouleau1999282–283_145-0">^</a> <a href="#CITEREFBouleau1999">Bouleau 1999</a>, pp. 282–283. </li> <li id="cite_note-FOOTNOTEBouleau1999285-146"><a href="#cite_ref-FOOTNOTEBouleau1999285_146-0">^</a> <a href="#CITEREFBouleau1999">Bouleau 1999</a>, p. 285. </li> <li id="cite_note-147"><a href="#cite_ref-147">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematical_Biology_(Chasnov)/01%3A_Population_Dynamics/1.04%3A_The_Lotka-Volterra_Predator-Prey_Model">"1.4: The Lotka-Volterra Predator-Prey Model"</a>. Mathematics LibreTexts. January 5, 2022. <a rel="nofollow" class="external text" href="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">Archived</a> from the original on December 29, 2022. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathematics+LibreTexts&rft.atitle=1.4%3A+The+Lotka-Volterra+Predator-Prey+Model&rft.date=2022-01-05&rft_id=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FApplied_Mathematics%2FMathematical_Biology_%28Chasnov%29%2F01%253A_Population_Dynamics%2F1.04%253A_The_Lotka-Volterra_Predator-Prey_Model&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-148"><a href="#cite_ref-148">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalsburg1992" class="citation journal cs1">Salsburg, David (August 17, 1992). <a rel="nofollow" class="external text" href="https://www.dfcm.utoronto.ca/sites/default/files/inline-files/salsburg_1.pdf">"Commentary"</a> (PDF). 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Retrieved May 19, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathnasium.com&rft.atitle=HOW+MATH+IS+USED+IN+WEATHER+FORECASTING&rft.date=2022-09-07&rft.aulast=Hill&rft.aufirst=Mullica&rft_id=https%3A%2F%2Fwww.mathnasium.com%2Fmath-centers%2Fmullicahill%2Fnews%2Fhow-math-used-weather-forecasting&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-153"><a href="#cite_ref-153">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://icp.giss.nasa.gov/education/modules/eccm/eccm_student_3.pdf">"Using Mathematical Models to Investigate Planetary Habitability"</a> (PDF). <a href="/wiki/NASA" title="NASA">NASA</a>. 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Annual Review of Sociology. 28 (1): 197–220. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.soc.28.110601.140942">10.1146/annurev.soc.28.110601.140942</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0360-0572">0360-0572</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Sociology&rft.atitle=Mathematics+in+Sociology&rft.volume=28&rft.issue=1&rft.pages=197-220&rft.date=2002&rft_id=info%3Adoi%2F10.1146%2Fannurev.soc.28.110601.140942&rft.issn=0360-0572&rft.aulast=Edling&rft.aufirst=Christofer+R.&rft_id=https%3A%2F%2Fwww.annualreviews.org%2Fdoi%2F10.1146%2Fannurev.soc.28.110601.140942&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-155"><a href="#cite_ref-155">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatchelder2015" class="citation cs1">Batchelder, William H. 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Retrieved January 3, 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Moral+Markets%3A+The+Critical+Role+of+Values+in+the+Economy&rft.pages=158&rft.pub=Princeton+University+Press&rft.date=2010&rft.isbn=978-1-4008-3736-6&rft.aulast=Zak&rft.aufirst=Paul+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6QrvmNo2qD4C%26pg%3DPA158&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-157"><a href="#cite_ref-157">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevinMilgrom2004" class="citation book cs1">Levin, Jonathan; Milgrom, Paul (September 2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~jdlevin/Econ%20202/Choice%20Theory.pdf">Introduction to Choice Theory</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Choice+Theory&rft.date=2004-09&rft.aulast=Levin&rft.aufirst=Jonathan&rft.au=Milgrom%2C+Paul&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~jdlevin%2FEcon%2520202%2FChoice%2520Theory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-158"><a href="#cite_ref-158">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKremer,_MichaelRao,_GautamSchilbach,_Frank2019" class="citation book cs1">Kremer, Michael; Rao, Gautam; Schilbach, Frank (2019). "Chapter 5 Behavioral development economics". <a rel="nofollow" class="external text" href="https://economics.mit.edu/sites/default/files/2022-09/behavioral-development-economics.pdf">Handbook of Behavioral Economics: Applications and Foundations</a> (PDF). Vol. 2.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+5+Behavioral+development+economics&rft.btitle=Handbook+of+Behavioral+Economics%3A+Applications+and+Foundations&rft.date=2019&rft.au=Kremer%2C+Michael&rft.au=Rao%2C+Gautam&rft.au=Schilbach%2C+Frank&rft_id=https%3A%2F%2Feconomics.mit.edu%2Fsites%2Fdefault%2Ffiles%2F2022-09%2Fbehavioral-development-economics.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-159"><a href="#cite_ref-159">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mdpi.com/journal/mathematics/special_issues/Mathematical_Modeling_Economics_Ecology_Environment">"Mathematics"</a>. mdpi.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mdpi.com&rft.atitle=Mathematics&rft_id=https%3A%2F%2Fwww.mdpi.com%2Fjournal%2Fmathematics%2Fspecial_issues%2FMathematical_Modeling_Economics_Ecology_Environment&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-160"><a href="#cite_ref-160">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/kondratiev-nikolai-dmitrievich">"Kondratiev, Nikolai Dmitrievich | Encyclopedia.com"</a>. www.encyclopedia.com. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160701224009/http://www.encyclopedia.com/doc/1G2-3404100667.html">Archived</a> from the original on July 1, 2016. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.encyclopedia.com&rft.atitle=Kondratiev%2C+Nikolai+Dmitrievich+%7C+Encyclopedia.com&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fhistory%2Fencyclopedias-almanacs-transcripts-and-maps%2Fkondratiev-nikolai-dmitrievich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-161"><a href="#cite_ref-161">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=ha010090244#:~:text=##+Math%C3%A9matique+de+l'histoire,org%E3%80%91">"Mathématique de l'histoire-géometrie et cinématique. Lois de Brück. Chronologie géodésique de la Bible., by Charles LAGRANGE et al. | The Online Books Page"</a>. onlinebooks.library.upenn.edu.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=onlinebooks.library.upenn.edu&rft.atitle=Math%C3%A9matique+de+l%27histoire-g%C3%A9ometrie+et+cin%C3%A9matique.+Lois+de+Br%C3%BCck.+Chronologie+g%C3%A9od%C3%A9sique+de+la+Bible.%2C+by+Charles+LAGRANGE+et+al.+%26%23124%3B+The+Online+Books+Page&rft_id=https%3A%2F%2Fonlinebooks.library.upenn.edu%2Fwebbin%2Fbook%2Flookupid%3Fkey%3Dha010090244%23%3A~%3Atext%3D%23%23%2BMath%25C3%25A9matique%2Bde%2Bl%27histoire%2Corg%25E3%2580%2591&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-162"><a href="#cite_ref-162">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/">"Cliodynamics: a science for predicting the future"</a>. ZDNet. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221229204104/https://www.zdnet.com/article/cliodynamics-a-science-for-predicting-the-future/">Archived</a> from the original on December 29, 2022. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Cliodynamics%3A+a+science+for+predicting+the+future&rft.pub=ZDNet&rft_id=https%3A%2F%2Fwww.zdnet.com%2Farticle%2Fcliodynamics-a-science-for-predicting-the-future%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-163"><a href="#cite_ref-163">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSokalJean_Bricmont1998" class="citation book cs1"><a href="/wiki/Alan_Sokal" title="Alan Sokal">Sokal, Alan</a>; <a href="/wiki/Jean_Bricmont" title="Jean Bricmont">Jean Bricmont</a> (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/fashionablenonse00soka">Fashionable Nonsense</a>. New York: Picador. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-312-19545-8" title="Special:BookSources/978-0-312-19545-8"><bdi>978-0-312-19545-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/39605994">39605994</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fashionable+Nonsense&rft.place=New+York&rft.pub=Picador&rft.date=1998&rft_id=info%3Aoclcnum%2F39605994&rft.isbn=978-0-312-19545-8&rft.aulast=Sokal&rft.aufirst=Alan&rft.au=Jean+Bricmont&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffashionablenonse00soka&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-164"><a href="#cite_ref-164">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.factcheck.org/2023/01/bidens-misleading-unemployment-statistic/">"Biden's Misleading Unemployment Statistic – FactCheck.org"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Biden%27s+Misleading+Unemployment+Statistic+%E2%80%93+FactCheck.org&rft_id=https%3A%2F%2Fwww.factcheck.org%2F2023%2F01%2Fbidens-misleading-unemployment-statistic%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-165"><a href="#cite_ref-165">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.minneapolisfed.org/article/2010/modern-macroeconomic-models-as-tools-for-economic-policy">"Modern Macroeconomic Models as Tools for Economic Policy | Federal Reserve Bank of Minneapolis"</a>. minneapolisfed.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=minneapolisfed.org&rft.atitle=Modern+Macroeconomic+Models+as+Tools+for+Economic+Policy+%26%23124%3B+Federal+Reserve+Bank+of+Minneapolis&rft_id=https%3A%2F%2Fwww.minneapolisfed.org%2Farticle%2F2010%2Fmodern-macroeconomic-models-as-tools-for-economic-policy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-SEP-Platonism-166"><a href="#cite_ref-SEP-Platonism_166-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBalaguer2016" class="citation encyclopaedia cs1">Balaguer, Mark (2016). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2016/entries/platonism">"Platonism in Metaphysics"</a>. In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Spring 2016 ed.). Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220130174043/https://plato.stanford.edu/archives/spr2016/entries/platonism/">Archived</a> from the original on January 30, 2022. Retrieved April 2, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Platonism+in+Metaphysics&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Spring+2016&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2016&rft.aulast=Balaguer&rft.aufirst=Mark&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2016%2Fentries%2Fplatonism&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-167"><a href="#cite_ref-167">^</a> See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhite1947" class="citation journal cs1">White, L. (1947). "The locus of mathematical reality: An anthropological footnote". <a href="/wiki/Philosophy_of_Science_(journal)" title="Philosophy of Science (journal)">Philosophy of Science</a>. 14 (4): 289–303. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F286957">10.1086/286957</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119887253">119887253</a>. 189303;</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophy+of+Science&rft.atitle=The+locus+of+mathematical+reality%3A+An+anthropological+footnote&rft.volume=14&rft.issue=4&rft.pages=289-303&rft.date=1947&rft_id=info%3Adoi%2F10.1086%2F286957&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119887253%23id-name%3DS2CID&rft.aulast=White&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> also in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewman1956" class="citation book cs1">Newman, J. R. (1956). The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+World+of+Mathematics&rft.place=New+York&rft.pages=2348-2364&rft.pub=Simon+and+Schuster&rft.date=1956&rft.aulast=Newman&rft.aufirst=J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-168"><a href="#cite_ref-168">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDorato2005" class="citation book cs1">Dorato, Mauro (2005). <a rel="nofollow" class="external text" href="https://www.academia.edu/download/52076815/2ch.pdf">"Why are laws mathematical?"</a> (PDF). The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature. Ashgate. pp. 31–66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7546-3994-7" title="Special:BookSources/978-0-7546-3994-7"><bdi>978-0-7546-3994-7</bdi></a>. <a rel="nofollow" class="external text" href="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">Archived</a> (PDF) from the original on August 17, 2023. Retrieved December 5, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Why+are+laws+mathematical%3F&rft.btitle=The+Software+of+the+Universe%2C+An+Introduction+to+the+History+and+Philosophy+of+Laws+of+Nature&rft.pages=31-66&rft.pub=Ashgate&rft.date=2005&rft.isbn=978-0-7546-3994-7&rft.aulast=Dorato&rft.aufirst=Mauro&rft_id=https%3A%2F%2Fwww.academia.edu%2Fdownload%2F52076815%2F2ch.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Mura-169"><a href="#cite_ref-Mura_169-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMura1993" class="citation journal cs1">Mura, Roberta (December 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics. 25 (4): 375–85. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01273907">10.1007/BF01273907</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3482762">3482762</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122351146">122351146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Educational+Studies+in+Mathematics&rft.atitle=Images+of+Mathematics+Held+by+University+Teachers+of+Mathematical+Sciences&rft.volume=25&rft.issue=4&rft.pages=375-85&rft.date=1993-12&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122351146%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3482762%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2FBF01273907&rft.aulast=Mura&rft.aufirst=Roberta&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Runge-170"><a href="#cite_ref-Runge_170-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTobiesNeunzert2012" class="citation book cs1"><a href="/wiki/Renate_Tobies" title="Renate Tobies">Tobies, Renate</a>; Neunzert, Helmut (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EDm0eQqFUQ4C&pg=PA9">Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry</a>. Springer. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-0229-1" title="Special:BookSources/978-3-0348-0229-1"><bdi>978-3-0348-0229-1</bdi></a>. Retrieved June 20, 2015. <q>[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.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Iris+Runge%3A+A+Life+at+the+Crossroads+of+Mathematics%2C+Science%2C+and+Industry&rft.pages=9&rft.pub=Springer&rft.date=2012&rft.isbn=978-3-0348-0229-1&rft.aulast=Tobies&rft.aufirst=Renate&rft.au=Neunzert%2C+Helmut&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEDm0eQqFUQ4C%26pg%3DPA9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-171"><a href="#cite_ref-171">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZieglerLoos2017" class="citation conference cs1"><a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a>; Loos, Andreas (November 2, 2017). Kaiser, G. (ed.). "What is Mathematics?" and why we should ask, where one should experience and learn that, and how to teach it. Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer. pp. 63–77. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-62597-3_5">10.1007/978-3-319-62597-3_5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-62596-6" title="Special:BookSources/978-3-319-62596-6"><bdi>978-3-319-62596-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=%22What+is+Mathematics%3F%22+and+why+we+should+ask%2C+where+one+should+experience+and+learn+that%2C+and+how+to+teach+it&rft.series=ICME-13+Monographs&rft.pages=63-77&rft.pub=Springer&rft.date=2017-11-02&rft_id=info%3Adoi%2F10.1007%2F978-3-319-62597-3_5&rft.isbn=978-3-319-62596-6&rft.aulast=Ziegler&rft.aufirst=G%C3%BCnter+M.&rft.au=Loos%2C+Andreas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> (Sections "What is Mathematics?" and "What is Mathematics, Really?") </li> <li id="cite_note-FOOTNOTEMura1993379,_381-172"><a href="#cite_ref-FOOTNOTEMura1993379,_381_172-0">^</a> <a href="#CITEREFMura1993">Mura 1993</a>, pp. 379, 381. </li> <li id="cite_note-FOOTNOTEBrownPorter1995326-173"><a href="#cite_ref-FOOTNOTEBrownPorter1995326_173-0">^</a> <a href="#CITEREFBrownPorter1995">Brown & Porter 1995</a>, p. 326. </li> <li id="cite_note-174"><a href="#cite_ref-174">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrauss2011" class="citation journal cs1">Strauss, Danie (2011). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/290955899">"Defining mathematics"</a>. Acta Academica. 43 (4): 1–28. Retrieved November 25, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Academica&rft.atitle=Defining+mathematics&rft.volume=43&rft.issue=4&rft.pages=1-28&rft.date=2011&rft.aulast=Strauss&rft.aufirst=Danie&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F290955899&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Franklin-175"><a href="#cite_ref-Franklin_175-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklin2009" class="citation book cs1"><a href="/wiki/James_Franklin_(philosopher)" title="James Franklin (philosopher)">Franklin, James</a> (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mbn35b2ghgkC&pg=PA104">Philosophy of Mathematics</a>. Elsevier. pp. 104–106. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-093058-9" title="Special:BookSources/978-0-08-093058-9"><bdi>978-0-08-093058-9</bdi></a>. Retrieved June 20, 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Mathematics&rft.pages=104-106&rft.pub=Elsevier&rft.date=2009&rft.isbn=978-0-08-093058-9&rft.aulast=Franklin&rft.aufirst=James&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dmbn35b2ghgkC%26pg%3DPA104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-Cajori-176"><a href="#cite_ref-Cajori_176-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1893" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1893). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mGJRjIC9fZgC&pg=PA285">A History of Mathematics</a>. American Mathematical Society (1991 reprint). pp. 285–286. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2102-2" title="Special:BookSources/978-0-8218-2102-2"><bdi>978-0-8218-2102-2</bdi></a>. Retrieved June 20, 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=285-286&rft.pub=American+Mathematical+Society+%281991+reprint%29&rft.date=1893&rft.isbn=978-0-8218-2102-2&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmGJRjIC9fZgC%26pg%3DPA285&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDevlin2018[httpsbooksgooglecombooksidgUb7CAAAQBAJpgPA3_3]-177"><a href="#cite_ref-FOOTNOTEDevlin2018[httpsbooksgooglecombooksidgUb7CAAAQBAJpgPA3_3]_177-0">^</a> <a href="#CITEREFDevlin2018">Devlin 2018</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gUb7CAAAQBAJ&pg=PA3">3</a>. </li> <li id="cite_note-178"><a href="#cite_ref-178">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSaunders_Maclane1986" class="citation book cs1">Saunders Maclane (1986). Mathematics, form and function. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%2C+form+and+function&rft.pub=Springer&rft.date=1986&rft.au=Saunders+Maclane&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988">, page 409 </li> <li id="cite_note-180"><a href="#cite_ref-180">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrownPorter1995" class="citation journal cs1"><a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Brown, Ronald</a>; Porter, Timothy (1995). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/280311">"The Methodology of Mathematics"</a>. The Mathematical Gazette. 79 (485): 321–334. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3618304">10.2307/3618304</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3618304">3618304</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:178923299">178923299</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230323164159/https://cds.cern.ch/record/280311">Archived</a> from the original on March 23, 2023. Retrieved November 25, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=The+Methodology+of+Mathematics&rft.volume=79&rft.issue=485&rft.pages=321-334&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A178923299%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3618304%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3618304&rft.aulast=Brown&rft.aufirst=Ronald&rft.au=Porter%2C+Timothy&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F280311&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-183"><a href="#cite_ref-183">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamami2022" class="citation journal cs1">Hamami, Yacin (June 2022). <a rel="nofollow" class="external text" href="https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf">"Mathematical Rigor and Proof"</a> (PDF). The Review of Symbolic Logic. 15 (2): 409–449. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1755020319000443">10.1017/S1755020319000443</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:209980693">209980693</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221205114343/https://www.yacinhamami.com/wp-content/uploads/2019/12/Hamami-2019-Mathematical-Rigor-and-Proof.pdf">Archived</a> (PDF) from the original on December 5, 2022. Retrieved November 21, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Review+of+Symbolic+Logic&rft.atitle=Mathematical+Rigor+and+Proof&rft.volume=15&rft.issue=2&rft.pages=409-449&rft.date=2022-06&rft_id=info%3Adoi%2F10.1017%2FS1755020319000443&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A209980693%23id-name%3DS2CID&rft.aulast=Hamami&rft.aufirst=Yacin&rft_id=https%3A%2F%2Fwww.yacinhamami.com%2Fwp-content%2Fuploads%2F2019%2F12%2FHamami-2019-Mathematical-Rigor-and-Proof.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-186"><a href="#cite_ref-186">^</a> <a href="#CITEREFPeterson1988">Peterson 1988</a>, p. 4: "A few complain that the computer program can't be verified properly." (in reference to the Haken–Apple proof of the <a href="/wiki/Four_Color_Theorem" class="mw-redirect" title="Four Color Theorem">Four Color Theorem</a>) </li> <li id="cite_note-187"><a href="#cite_ref-187">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerminov1988" class="citation journal cs1">Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs". Philosophy of Mathematics. 42 (167 (4)). 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International Journal of Educational Research. 93: 91–100. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijer.2018.10.002">10.1016/j.ijer.2018.10.002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:149753721">149753721</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Educational+Research&rft.atitle=Teachers%27+perceptions+of+the+official+curriculum%3A+Problem+solving+and+rigor&rft.volume=93&rft.pages=91-100&rft.date=2019&rft_id=info%3Adoi%2F10.1016%2Fj.ijer.2018.10.002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A149753721%23id-name%3DS2CID&rft.aulast=Davis&rft.aufirst=Jon+D.&rft.au=McDuffie%2C+Amy+Roth&rft.au=Drake%2C+Corey&rft.au=Seiwell%2C+Amanda+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-189"><a href="#cite_ref-189">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEndsley2021" class="citation book cs1">Endsley, Kezia (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1cEYEAAAQBAJ&pg=PA3">Mathematicians and Statisticians: A Practical Career Guide</a>. Practical Career Guides. Rowman & Littlefield. pp. 1–3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5381-4517-3" title="Special:BookSources/978-1-5381-4517-3"><bdi>978-1-5381-4517-3</bdi></a>. 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(1979). <a rel="nofollow" class="external text" href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf">"A proof that Euler missed... Apéry's Proof of the irrationality of ζ(3)"</a> (PDF). <a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a>. 1 (4): 195–203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03028234">10.1007/BF03028234</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121589323">121589323</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150906015716/http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf">Archived</a> (PDF) from the original on September 6, 2015. Retrieved November 22, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=A+proof+that+Euler+missed...+Ap%C3%A9ry%27s+Proof+of+the+irrationality+of+%CE%B6%283%29&rft.volume=1&rft.issue=4&rft.pages=195-203&rft.date=1979&rft_id=info%3Adoi%2F10.1007%2FBF03028234&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121589323%23id-name%3DS2CID&rft.aulast=van+der+Poorten&rft.aufirst=A.&rft_id=http%3A%2F%2Fpracownicy.uksw.edu.pl%2Fmwolf%2FPoorten_MI_195_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-203"><a href="#cite_ref-203">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetkovi2009" class="citation book cs1">Petkovi, Miodrag (September 2, 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AZlwAAAAQBAJ&pg=PR13">Famous Puzzles of Great Mathematicians</a>. American Mathematical Society. pp. xiii–xiv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4814-2" title="Special:BookSources/978-0-8218-4814-2"><bdi>978-0-8218-4814-2</bdi></a>. Retrieved November 25, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Famous+Puzzles+of+Great+Mathematicians&rft.pages=xiii-xiv&rft.pub=American+Mathematical+Society&rft.date=2009-09-02&rft.isbn=978-0-8218-4814-2&rft.aulast=Petkovi&rft.aufirst=Miodrag&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAZlwAAAAQBAJ%26pg%3DPR13&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-204"><a href="#cite_ref-204">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1940" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1940). <a rel="nofollow" class="external text" href="https://archive.org/details/hardy_annotated/">A Mathematician's Apology</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-42706-7" title="Special:BookSources/978-0-521-42706-7"><bdi>978-0-521-42706-7</bdi></a>. Retrieved November 22, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematician%27s+Apology&rft.pub=Cambridge+University+Press&rft.date=1940&rft.isbn=978-0-521-42706-7&rft.aulast=Hardy&rft.aufirst=G.+H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhardy_annotated%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> See also <a href="/wiki/A_Mathematician%27s_Apology" title="A Mathematician's Apology">A Mathematician's Apology</a>. </li> <li id="cite_note-205"><a href="#cite_ref-205">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlonGoldstonSárközySzabados2015" class="citation journal cs1">Alon, Noga; Goldston, Dan; Sárközy, András; Szabados, József; Tenenbaum, Gérald; Garcia, Stephan Ramon; Shoemaker, Amy L. (March 2015). Alladi, Krishnaswami; Krantz, Steven G. (eds.). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1223">"Reflections on Paul Erdős on His Birth Centenary, Part II"</a>. Notices of the American Mathematical Society. 62 (3): 226–247. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1223">10.1090/noti1223</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=Reflections+on+Paul+Erd%C5%91s+on+His+Birth+Centenary%2C+Part+II&rft.volume=62&rft.issue=3&rft.pages=226-247&rft.date=2015-03&rft_id=info%3Adoi%2F10.1090%2Fnoti1223&rft.aulast=Alon&rft.aufirst=Noga&rft.au=Goldston%2C+Dan&rft.au=S%C3%A1rk%C3%B6zy%2C+Andr%C3%A1s&rft.au=Szabados%2C+J%C3%B3zsef&rft.au=Tenenbaum%2C+G%C3%A9rald&rft.au=Garcia%2C+Stephan+Ramon&rft.au=Shoemaker%2C+Amy+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fnoti1223&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-206"><a href="#cite_ref-206">^</a> See, for example <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">History of Western Philosophy. 1919. p. 60.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Western+Philosophy&rft.pages=60&rft.date=1919&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-207"><a href="#cite_ref-207">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCazden1959" class="citation journal cs1">Cazden, Norman (October 1959). "Musical intervals and simple number ratios". Journal of Research in Music Education. 7 (2): 197–220. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F002242945900700205">10.1177/002242945900700205</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3344215">3344215</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220636812">220636812</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Research+in+Music+Education&rft.atitle=Musical+intervals+and+simple+number+ratios&rft.volume=7&rft.issue=2&rft.pages=197-220&rft.date=1959-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220636812%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3344215%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1177%2F002242945900700205&rft.aulast=Cazden&rft.aufirst=Norman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-208"><a href="#cite_ref-208">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBudden1967" class="citation journal cs1">Budden, F. J. (October 1967). "Modern mathematics and music". The Mathematical Gazette. 51 (377). Cambridge University Press ({CUP}): 204–215. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3613237">10.2307/3613237</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3613237">3613237</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126119711">126119711</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Gazette&rft.atitle=Modern+mathematics+and+music&rft.volume=51&rft.issue=377&rft.pages=204-215&rft.date=1967-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126119711%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3613237%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3613237&rft.aulast=Budden&rft.aufirst=F.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-209"><a href="#cite_ref-209">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnquistArak1994" class="citation journal cs1">Enquist, Magnus; Arak, Anthony (November 1994). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/372169a0">"Symmetry, beauty and evolution"</a>. Nature. 372 (6502): 169–172. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994Natur.372..169E">1994Natur.372..169E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F372169a0">10.1038/372169a0</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-4687">1476-4687</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/7969448">7969448</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4310147">4310147</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221228052049/https://www.nature.com/articles/372169a0">Archived</a> from the original on December 28, 2022. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Symmetry%2C+beauty+and+evolution&rft.volume=372&rft.issue=6502&rft.pages=169-172&rft.date=1994-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4310147%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1994Natur.372..169E&rft.issn=1476-4687&rft_id=info%3Adoi%2F10.1038%2F372169a0&rft_id=info%3Apmid%2F7969448&rft.aulast=Enquist&rft.aufirst=Magnus&rft.au=Arak%2C+Anthony&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2F372169a0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-210"><a href="#cite_ref-210">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHestenes1999" class="citation web cs1">Hestenes, David (1999). <a rel="nofollow" class="external text" href="https://davidhestenes.net/geocalc/pdf/SymmetryGroups.pdf">"Symmetry Groups"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Symmetry+Groups&rft.date=1999&rft.aulast=Hestenes&rft.aufirst=David&rft_id=https%3A%2F%2Fdavidhestenes.net%2Fgeocalc%2Fpdf%2FSymmetryGroups.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-211"><a href="#cite_ref-211">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBender2020" class="citation encyclopaedia cs1">Bender, Sara (September 2020). "The Rorschach Test". In Carducci, Bernardo J.; Nave, Christopher S.; Mio, Jeffrey S.; Riggio, Ronald E. (eds.). The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment. Wiley. pp. 367–376. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781119547167.ch131">10.1002/9781119547167.ch131</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-05751-2" title="Special:BookSources/978-1-119-05751-2"><bdi>978-1-119-05751-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Rorschach+Test&rft.btitle=The+Wiley+Encyclopedia+of+Personality+and+Individual+Differences%3A+Measurement+and+Assessment&rft.pages=367-376&rft.pub=Wiley&rft.date=2020-09&rft_id=info%3Adoi%2F10.1002%2F9781119547167.ch131&rft.isbn=978-1-119-05751-2&rft.aulast=Bender&rft.aufirst=Sara&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-212"><a href="#cite_ref-212">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl2015" class="citation book cs1"><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, Hermann</a> (2015). Symmetry. Princeton Science Library. Vol. 47. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?hl=en&lr=&id=GG1FCQAAQBAJ&pg=PA4">4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-7434-7" title="Special:BookSources/978-1-4008-7434-7"><bdi>978-1-4008-7434-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry&rft.series=Princeton+Science+Library&rft.pages=4&rft.pub=Princeton+University+Press&rft.date=2015&rft.isbn=978-1-4008-7434-7&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-213"><a href="#cite_ref-213">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/pages/part-i-mechanical-vibrations-and-waves/lecture-8/">"Lecture 8: Translation Symmetry | Physics III: Vibrations and Waves | Physics"</a>. MIT OpenCourseWare.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MIT+OpenCourseWare&rft.atitle=Lecture+8%3A+Translation+Symmetry+%26%23124%3B+Physics+III%3A+Vibrations+and+Waves+%26%23124%3B+Physics&rft_id=https%3A%2F%2Focw.mit.edu%2Fcourses%2F8-03sc-physics-iii-vibrations-and-waves-fall-2016%2Fpages%2Fpart-i-mechanical-vibrations-and-waves%2Flecture-8%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-214"><a href="#cite_ref-214">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradley2010" class="citation web cs1">Bradley, Larry (2010). <a rel="nofollow" class="external text" href="https://www.stsci.edu/~lbradley/seminar/fractals.html">"Fractals – Chaos & Fractals"</a>. stsci.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230307054609/https://www.stsci.edu/~lbradley/seminar/fractals.html">Archived</a> from the original on March 7, 2023. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=stsci.edu&rft.atitle=Fractals+%E2%80%93+Chaos+%26+Fractals&rft.date=2010&rft.aulast=Bradley&rft.aufirst=Larry&rft_id=https%3A%2F%2Fwww.stsci.edu%2F~lbradley%2Fseminar%2Ffractals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-215"><a href="#cite_ref-215">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.bu.edu/DYSYS/chaos-game/node5.html">"Self-similarity"</a>. math.bu.edu. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230302132911/http://math.bu.edu/DYSYS/chaos-game/node5.html">Archived</a> from the original on March 2, 2023. 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Fremantle, Western Australia: Australian Association of Mathematics Teachers. pp. 125–126. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230307054610/https://researchrepository.murdoch.edu.au/id/eprint/6242/">Archived</a> from the original on March 7, 2023. Retrieved December 29, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Popular+mathematics&rft.place=Fremantle%2C+Western+Australia&rft.pages=125-126&rft.pub=Australian+Association+of+Mathematics+Teachers&rft.date=2009-07&rft.aulast=Kissane&rft.aufirst=Barry&rft_id=https%3A%2F%2Fresearchrepository.murdoch.edu.au%2Fid%2Feprint%2F6242%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-217"><a href="#cite_ref-217">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteen2012" class="citation book cs1">Steen, L. A. 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Retrieved January 3, 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+Today+Twelve+Informal+Essays&rft.pages=2&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4613-9435-8&rft.aulast=Steen&rft.aufirst=L.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-d3TBwAAQBAJ%26pg%3DPA2%26dq%3D%2522%2522popular%2Bmathematics%2522%2Banalogies%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-218"><a href="#cite_ref-218">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPitici2017" class="citation book cs1">Pitici, Mircea (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9nGQDQAAQBAJ&pg=PA331&dq=%22%22popular+mathematics%22+analogies%22">The Best Writing on Mathematics 2016</a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-8560-2" title="Special:BookSources/978-1-4008-8560-2"><bdi>978-1-4008-8560-2</bdi></a>. 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(September 2000). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/4149">Wolf Prize in Mathematics</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F4149">10.1142/4149</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-3945-9" title="Special:BookSources/978-981-02-3945-9"><bdi>978-981-02-3945-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220221171351/https://www.worldscientific.com/worldscibooks/10.1142/4149">Archived</a> from the original on February 21, 2022. 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Logic and Computation in Philosophy series. Oxford University Press. pp. 3–27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-508030-8" title="Special:BookSources/978-0-19-508030-8"><bdi>978-0-19-508030-8</bdi></a>. 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Clay Mathematics Institute. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150703184941/http://www.claymath.org/millennium-problems/millennium-prize-problems">Archived</a> from the original on July 3, 2015. Retrieved January 23, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Millennium+Prize+Problems&rft.pub=Clay+Mathematics+Institute&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium-problems%2Fmillennium-prize-problems&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> <li id="cite_note-234"><a href="#cite_ref-234">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.claymath.org/millennium-problems">"Millennium Problems"</a>. 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Retrieved January 23, 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Millennium+Problems&rft.pub=Clay+Mathematics+Institute&rft_id=http%3A%2F%2Fwww.claymath.org%2Fmillennium-problems&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBouleau1999" class="citation book cs1"><a href="/wiki/Nicolas_Bouleau" title="Nicolas Bouleau">Bouleau, Nicolas</a> (1999). Philosophie des mathématiques et de la modélisation: Du chercheur à l'ingénieur. L'Harmattan. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-7384-8125-2" title="Special:BookSources/978-2-7384-8125-2"><bdi>978-2-7384-8125-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophie+des+math%C3%A9matiques+et+de+la+mod%C3%A9lisation%3A+Du+chercheur+%C3%A0+l%27ing%C3%A9nieur&rft.pub=L%27Harmattan&rft.date=1999&rft.isbn=978-2-7384-8125-2&rft.aulast=Bouleau&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1991" class="citation book cs1"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl Benjamin</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/n3/mode/2up">A History of Mathematics</a> (2nd ed.). New York: <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">Wiley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+Benjamin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye%2Fpage%2Fn3%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCresswell2021" class="citation book cs1">Cresswell, Julia (2021). Oxford Dictionary of Word Origins (3 ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-886875-0" title="Special:BookSources/978-0-19-886875-0"><bdi>978-0-19-886875-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+Dictionary+of+Word+Origins&rft.edition=3&rft.pub=Oxford+University+Press&rft.date=2021&rft.isbn=978-0-19-886875-0&rft.aulast=Cresswell&rft.aufirst=Julia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2018" class="citation book cs1">Devlin, Keith (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gUb7CAAAQBAJ&pg=PA3">Sets, Functions, and Logic: An Introduction to Abstract Mathematics</a> (3 ed.). CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-8602-1" title="Special:BookSources/978-1-4822-8602-1"><bdi>978-1-4822-8602-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+Functions%2C+and+Logic%3A+An+Introduction+to+Abstract+Mathematics&rft.edition=3&rft.pub=CRC+Press&rft.date=2018&rft.isbn=978-1-4822-8602-1&rft.aulast=Devlin&rft.aufirst=Keith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgUb7CAAAQBAJ%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard</a> (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-029558-4" title="Special:BookSources/978-0-03-029558-4"><bdi>978-0-03-029558-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+History+of+Mathematics&rft.edition=6th&rft.pub=Saunders&rft.date=1990&rft.isbn=978-0-03-029558-4&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner2007" class="citation book cs1"><a href="/wiki/Israel_Kleiner_(mathematician)" title="Israel Kleiner (mathematician)">Kleiner, Israel</a> (2007). Kleiner, Israel (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RTLRBK-wj6wC">A History of Abstract Algebra</a>. Springer Science & Business Media. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-4685-1">10.1007/978-0-8176-4685-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4684-4" title="Special:BookSources/978-0-8176-4684-4"><bdi>978-0-8176-4684-4</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2007932362">2007932362</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/76935733">76935733</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117392219">117392219</a>. Retrieved February 8, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Abstract+Algebra&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117392219%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-4685-1&rft_id=info%3Aoclcnum%2F76935733&rft_id=info%3Alccn%2F2007932362&rft.isbn=978-0-8176-4684-4&rft.aulast=Kleiner&rft.aufirst=Israel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRTLRBK-wj6wC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1990" class="citation book cs1"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1990). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalthou00klin">Mathematical Thought from Ancient to Modern Times</a>. New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-506135-2" title="Special:BookSources/978-0-19-506135-2"><bdi>978-0-19-506135-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+from+Ancient+to+Modern+Times&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1990&rft.isbn=978-0-19-506135-2&rft.aulast=Kline&rft.aufirst=Morris&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalthou00klin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonastyrsky2001" class="citation journal cs1">Monastyrsky, Michael (2001). <a rel="nofollow" class="external text" href="http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf">"Some Trends in Modern Mathematics and the Fields Medal"</a> (PDF). CMS – Notes – de la SMC. 33 (2–3). Canadian Mathematical Society. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060813224844/http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf">Archived</a> (PDF) from the original on August 13, 2006. 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New York: Penguin Random House. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-399-16524-5" title="Special:BookSources/978-0-399-16524-5"><bdi>978-0-399-16524-5</bdi></a>. <q>A Mind for Numbers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mind+For+Numbers%3A+How+to+Excel+at+Math+and+Science+%28Even+If+You+Flunked+Algebra%29&rft.place=New+York&rft.pub=Penguin+Random+House&rft.date=2014&rft.isbn=978-0-399-16524-5&rft.aulast=Oakley&rft.aufirst=Barbara&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780399165245&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a href="/wiki/Benjamin_Peirce" title="Benjamin Peirce">Peirce, Benjamin</a> (1881). <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles Sanders</a> (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=De0GAAAAYAAJ&pg=PA1&q=Peirce+Benjamin+Linear+Associative+Algebra">"Linear associative algebra"</a>. American Journal of Mathematics. 4 (1–4) (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.): 97–229. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2369153">10.2307/2369153</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fhvd.32044030622997">2027/hvd.32044030622997</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2369153">2369153</a>. 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 <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221">Eprint</a> and as an extract, D. Van Nostrand, 1882, Google <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_De0GAAAAYAAJ">Eprint</a>. 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Freeman and Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-1953-3" title="Special:BookSources/0-7167-1953-3"><bdi>0-7167-1953-3</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/87033078">87033078</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/17202382">17202382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Tourist%3A+Snapshots+of+Modern+Mathematics&rft.pub=W.+H.+Freeman+and+Company&rft.date=1988&rft_id=info%3Aoclcnum%2F17202382&rft_id=info%3Alccn%2F87033078&rft.isbn=0-7167-1953-3&rft.aulast=Peterson&rft.aufirst=Ivars&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Karl_Popper" title="Karl Popper">Popper, Karl R.</a> (1995). "On knowledge". <a rel="nofollow" class="external text" href="https://archive.org/details/insearchofbetter00karl">In Search of a Better World: Lectures and Essays from Thirty Years</a>. 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Notices of the AMS. 49 (7): 778–782. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061026000014/http://www.ams.org/notices/200207/comm-riehm.pdf">Archived</a> (PDF) from the original on October 26, 2006. Retrieved October 2, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+Early+History+of+the+Fields+Medal&rft.volume=49&rft.issue=7&rft.pages=778-782&rft.date=2002-08&rft.aulast=Riehm&rft.aufirst=Carl&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200207%2Fcomm-riehm.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSevryuk2006" class="citation journal cs1">Sevryuk, Mikhail B. (January 2006). <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf">"Book Reviews"</a> (PDF). <a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>. 43 (1): 101–109. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-05-01069-4">10.1090/S0273-0979-05-01069-4</a>. <a rel="nofollow" class="external text" href="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">Archived</a> (PDF) from the original on July 23, 2006. Retrieved June 24, 2006.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Book+Reviews&rft.volume=43&rft.issue=1&rft.pages=101-109&rft.date=2006-01&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-05-01069-4&rft.aulast=Sevryuk&rft.aufirst=Mikhail+B.&rft_id=https%3A%2F%2Fwww.ams.org%2Fbull%2F2006-43-01%2FS0273-0979-05-01069-4%2FS0273-0979-05-01069-4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittle1994" class="citation book cs1"><a href="/wiki/Peter_Whittle_(mathematician)" title="Peter Whittle (mathematician)">Whittle, Peter</a> (1994). <a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair">"Almost home"</a>. In <a href="/wiki/Frank_Kelly_(mathematician)" title="Frank Kelly (mathematician)">Kelly, F.P.</a> (ed.). Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory up to 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-94829-2" title="Special:BookSources/978-0-471-94829-2"><bdi>978-0-471-94829-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131219080017/http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair">Archived</a> from the original on December 19, 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Almost+home&rft.btitle=Probability%2C+statistics+and+optimisation%3A+A+Tribute+to+Peter+Whittle&rft.place=Chichester&rft.pages=1-28&rft.edition=previously+%22A+realised+path%3A+The+Cambridge+Statistical+Laboratory+up+to+1993+%28revised+2002%29%22&rft.pub=John+Wiley&rft.date=1994&rft.isbn=978-0-471-94829-2&rft.aulast=Whittle&rft.aufirst=Peter&rft_id=http%3A%2F%2Fwww.statslab.cam.ac.uk%2FHistory%2F2history.html%236._1966--72%3A_The_Churchill_Chair&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box metadata side-box-right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> <a href="/wiki/Wikipedia:The_Wikipedia_Library" title="Wikipedia:The Wikipedia Library">Library resources</a> about Mathematics <hr /></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Mathematics&library=OLBP">Online books</a></li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Mathematics">Resources in your library</a></li> <li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Mathematics&library=0CHOOSE0">Resources in other libraries</a></li> </ul></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Benson, Donald C. (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/momentofproofmat00bens/page/n5/mode/2up">The Moment of Proof: Mathematical Epiphanies</a>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-513919-8" title="Special:BookSources/978-0-19-513919-8"><bdi>978-0-19-513919-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Moment+of+Proof%3A+Mathematical+Epiphanies&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-513919-8&rft.aulast=Benson&rft.aufirst=Donald+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmomentofproofmat00bens%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Philip_J._Davis" title="Philip J. Davis">Davis, Philip J.</a>; <a href="/wiki/Reuben_Hersh" title="Reuben Hersh">Hersh, Reuben</a> (1999). <a href="/wiki/The_Mathematical_Experience" title="The Mathematical Experience">The Mathematical Experience</a> (Reprint ed.). Boston; New York: Mariner Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-395-92968-1" title="Special:BookSources/978-0-395-92968-1"><bdi>978-0-395-92968-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Experience&rft.place=Boston%3B+New+York&rft.edition=Reprint&rft.pub=Mariner+Books&rft.date=1999&rft.isbn=978-0-395-92968-1&rft.aulast=Davis&rft.aufirst=Philip+J.&rft.au=Hersh%2C+Reuben&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> Available <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalexpe0000davi/page/n5/mode/2up">online</a> (registration required).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Richard_Courant" title="Richard Courant">Courant, Richard</a>; <a href="/wiki/Herbert_Robbins" title="Herbert Robbins">Robbins, Herbert</a> (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/whatismathematic0000cour/page/n5/mode/2up">What Is Mathematics?: An Elementary Approach to Ideas and Methods</a> (2nd ed.). New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-510519-3" title="Special:BookSources/978-0-19-510519-3"><bdi>978-0-19-510519-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=What+Is+Mathematics%3F%3A+An+Elementary+Approach+to+Ideas+and+Methods&rft.place=New+York&rft.edition=2nd&rft.pub=Oxford+University+Press&rft.date=1996&rft.isbn=978-0-19-510519-3&rft.aulast=Courant&rft.aufirst=Richard&rft.au=Robbins%2C+Herbert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fwhatismathematic0000cour%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Jan_Gullberg" title="Jan Gullberg">Gullberg, Jan</a> (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull/page/n5/mode/2up">Mathematics: From the Birth of Numbers</a>. W.W. Norton & Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04002-9" title="Special:BookSources/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+From+the+Birth+of+Numbers&rft.pub=W.W.+Norton+%26+Company&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull%2Fpage%2Fn5%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, ed. (2000). <a href="/wiki/Encyclopaedia_of_Mathematics" class="mw-redirect" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a>. Kluwer Academic Publishers.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopaedia+of+Mathematics&rft.pub=Kluwer+Academic+Publishers&rft.date=2000&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"> – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and <a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Special:AllPages">online</a>. <a rel="nofollow" class="external text" href="https://archive.today/20121220135247/http://www.encyclopediaofmath.org/">Archived</a> December 20, 2012, at <a href="/wiki/Archive.today" title="Archive.today">archive.today</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgkin2005" class="citation book cs1">Hodgkin, Luke Howard (2005). A History of Mathematics: From Mesopotamia to Modernity. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-152383-0" title="Special:BookSources/978-0-19-152383-0"><bdi>978-0-19-152383-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics%3A+From+Mesopotamia+to+Modernity&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-152383-0&rft.aulast=Hodgkin&rft.aufirst=Luke+Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Philip_Jourdain" title="Philip Jourdain">Jourdain, Philip E. B.</a> (2003). "The Nature of Mathematics". In James R. Newman (ed.). The World of Mathematics. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43268-7" title="Special:BookSources/978-0-486-43268-7"><bdi>978-0-486-43268-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Nature+of+Mathematics&rft.btitle=The+World+of+Mathematics&rft.pub=Dover+Publications&rft.date=2003&rft.isbn=978-0-486-43268-7&rft.aulast=Jourdain&rft.aufirst=Philip+E.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Theoni_Pappas" title="Theoni Pappas">Pappas, Theoni</a> (1986). <a rel="nofollow" class="external text" href="https://archive.org/details/joyofmathematics0000papp_t0z1/page/n3/mode/2up">The Joy Of Mathematics</a>. San Carlos, California: Wide World Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-933174-65-8" title="Special:BookSources/978-0-933174-65-8"><bdi>978-0-933174-65-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Joy+Of+Mathematics&rft.place=San+Carlos%2C+California&rft.pub=Wide+World+Publishing&rft.date=1986&rft.isbn=978-0-933174-65-8&rft.aulast=Pappas&rft.aufirst=Theoni&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjoyofmathematics0000papp_t0z1%2Fpage%2Fn3%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="no" class="citation book cs1"><a href="/wiki/Wolfgang_Sartorius_von_Waltershausen" title="Wolfgang Sartorius von Waltershausen">Waltershausen, Wolfgang Sartorius von</a> (1965) [1856]. Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-253-01702-5" title="Special:BookSources/978-3-253-01702-5"><bdi>978-3-253-01702-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gauss+zum+Ged%C3%A4chtniss&rft.pub=S%C3%A4ndig+Reprint+Verlag+H.+R.+Wohlwend&rft.date=1965&rft.isbn=978-3-253-01702-5&rft.aulast=Waltershausen&rft.aufirst=Wolfgang+Sartorius+von&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematics" class="Z3988"></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a class="mw-selflink selflink">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a href="/wiki/Mathematical_optimization" title="Mathematical optimization">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /> <a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link 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//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Newton%27s_reflecting_telescope.jpg/34px-Newton%27s_reflecting_telescope.jpg 2x" data-file-width="1140" data-file-height="1276" /> <a href="/wiki/Portal:History_of_science" title="Portal:History of science">History of science</a></li><li class="portal-bar-item"><a href="/wiki/File:Nuvola_apps_kalzium.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/19px-Nuvola_apps_kalzium.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/29px-Nuvola_apps_kalzium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Nuvola_apps_kalzium.svg/38px-Nuvola_apps_kalzium.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Science" title="Portal:Science">Science</a></li></ul></div><style 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.sister-bar-content{flex:1;border-top:1px solid #a2a9b1;margin:0;list-style:none}.mw-parser-output .sister-bar-item{flex:0 0 20em;min-width:20em}}.mw-parser-output .navbox+link+.sister-bar,.mw-parser-output .navbox+style+.sister-bar,.mw-parser-output .portal-bar+link+.sister-bar,.mw-parser-output .portal-bar+style+.sister-bar,.mw-parser-output .sister-bar+.navbox-styles+.navbox,.mw-parser-output .sister-bar+.navbox-styles+.portal-bar{margin-top:-1px}@media print{body.ns-0 .mw-parser-output .sister-bar{display:none!important}}</style><div class="noprint metadata sister-bar" role="navigation" aria-label="sister-projects"><div class="sister-bar-header">Mathematics at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects">sister projects</a>:</div><ul class="sister-bar-content"><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/19px-Wiktionary-logo-v2.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/29px-Wiktionary-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/0/06/Wiktionary-logo-v2.svg/38px-Wiktionary-logo-v2.svg.png 2x" data-file-width="391" data-file-height="391" /><a href="https://en.wiktionary.org/wiki/Special:Search/Mathematics" class="extiw" title="wikt:Special:Search/Mathematics">Definitions</a> from Wiktionary</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/14px-Commons-logo.svg.png" decoding="async" width="14" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/21px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/28px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="c:Category:Mathematics">Media</a> from Commons</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/21px-Wikinews-logo.svg.png" decoding="async" width="21" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/32px-Wikinews-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wikinews-logo.svg/42px-Wikinews-logo.svg.png 2x" data-file-width="759" data-file-height="415" /><a href="https://en.wikinews.org/wiki/Category:Mathematics" class="extiw" title="n:Category:Mathematics">News</a> from Wikinews</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/16px-Wikiquote-logo.svg.png" decoding="async" width="16" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/24px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/32px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /><a href="https://en.wikiquote.org/wiki/Mathematics" class="extiw" title="q:Mathematics">Quotations</a> from Wikiquote</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/18px-Wikisource-logo.svg.png" decoding="async" width="18" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/28px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/36px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /><a href="https://en.wikisource.org/wiki/Special:Search/Mathematics" class="extiw" title="s:Special:Search/Mathematics">Texts</a> from Wikisource</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/19px-Wikibooks-logo.svg.png" decoding="async" width="19" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/29px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/38px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /><a href="https://en.wikibooks.org/wiki/Shelf:Mathematics" class="extiw" title="b:Shelf:Mathematics">Textbooks</a> from Wikibooks</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/21px-Wikiversity_logo_2017.svg.png" decoding="async" width="21" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/42px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /><a href="https://en.wikiversity.org/wiki/Special:Search/Mathematics" class="extiw" title="v:Special:Search/Mathematics">Resources</a> from Wikiversity</li><li class="sister-bar-item"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/21px-Wikidata-logo.svg.png" decoding="async" width="21" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/32px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/42px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /><a href="https://www.wikidata.org/wiki/Q395" class="extiw" title="d:Q395">Data</a> from Wikidata</li></ul></div>    </div><noscript><img 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4.48% 133.640 1 Template:Langx"," 3.92% 116.856 1 Template:Math_topics_TOC"," 3.91% 116.402 1 Template:Subject_bar"," 3.26% 97.045 1 Template:Sidebar"]},"scribunto":{"limitreport-timeusage":{"value":"1.992","limit":"10.000"},"limitreport-memusage":{"value":19036903,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"Branches_of_mathematics\"] = 1,\n [\"CITEREFAlexander2011\"] = 1,\n [\"CITEREFAlonGoldstonSárközySzabados2015\"] = 1,\n [\"CITEREFArchibald1949\"] = 1,\n [\"CITEREFAsper2009\"] = 1,\n [\"CITEREFAtiyah1990\"] = 1,\n [\"CITEREFAvigadReck2001\"] = 1,\n [\"CITEREFBalaguer2016\"] = 1,\n [\"CITEREFBatchelder2015\"] = 1,\n [\"CITEREFBell1945\"] = 1,\n [\"CITEREFBellomoPreziosi1994\"] = 1,\n [\"CITEREFBender2020\"] = 1,\n [\"CITEREFBernardProustRoss2014\"] = 1,\n [\"CITEREFBerntjes\"] = 1,\n [\"CITEREFBiggs1979\"] = 1,\n [\"CITEREFBishop1991\"] = 1,\n [\"CITEREFBoas1995\"] = 1,\n [\"CITEREFBorel1983\"] = 1,\n [\"CITEREFBouleau1999\"] = 1,\n [\"CITEREFBoyer1991\"] = 1,\n [\"CITEREFBoyer2004\"] = 1,\n [\"CITEREFBradley2010\"] = 1,\n [\"CITEREFBrownPorter1995\"] = 1,\n [\"CITEREFBudden1967\"] = 1,\n [\"CITEREFCajori1893\"] = 1,\n [\"CITEREFCazden1959\"] = 1,\n [\"CITEREFChenMaza2014\"] = 1,\n [\"CITEREFChernHirzebruch2000\"] = 1,\n [\"CITEREFChristianidisOaks2013\"] = 1,\n [\"CITEREFCorry2004\"] = 1,\n [\"CITEREFCresswell2021\"] = 1,\n [\"CITEREFDavisMcDuffieDrakeSeiwell2019\"] = 1,\n [\"CITEREFDevlin2018\"] = 1,\n [\"CITEREFDorato2005\"] = 1,\n [\"CITEREFDouglasHeadleyHaddenLeFevre2020\"] = 1,\n [\"CITEREFDudley2002\"] = 1,\n [\"CITEREFDunneHulek2020\"] = 1,\n [\"CITEREFEdling2002\"] = 1,\n [\"CITEREFEndsley2021\"] = 1,\n [\"CITEREFEnquistArak1994\"] = 1,\n [\"CITEREFEwald2018\"] = 1,\n [\"CITEREFFaruqi2006\"] = 1,\n [\"CITEREFFeferman1998\"] = 1,\n [\"CITEREFFerreirós2001\"] = 1,\n [\"CITEREFFerreirós2007\"] = 1,\n [\"CITEREFFerreirós2020\"] = 1,\n [\"CITEREFFranklin2009\"] = 1,\n [\"CITEREFFranklin2017\"] = 1,\n [\"CITEREFFriggHartmann2020\"] = 1,\n [\"CITEREFGinammi2016\"] = 1,\n [\"CITEREFGoldman1998\"] = 1,\n [\"CITEREFGozwamiSingh2019\"] = 1,\n [\"CITEREFGuicciardini2017\"] = 1,\n [\"CITEREFHales2014\"] = 1,\n [\"CITEREFHalesAdamsBauerDang2017\"] = 1,\n [\"CITEREFHalpernHarperImmermanKolaitis2001\"] = 1,\n [\"CITEREFHamami2022\"] = 1,\n [\"CITEREFHamilton1982\"] = 1,\n [\"CITEREFHanson1961\"] = 1,\n [\"CITEREFHardy1940\"] = 1,\n [\"CITEREFHartshorne2000\"] = 1,\n [\"CITEREFHeath1981\"] = 1,\n [\"CITEREFHennig2010\"] = 1,\n [\"CITEREFHestenes1999\"] = 1,\n [\"CITEREFHilbert1902\"] = 1,\n [\"CITEREFHill2022\"] = 1,\n [\"CITEREFHipólito2015\"] = 1,\n [\"CITEREFHodgkin2005\"] = 1,\n [\"CITEREFJansenMarriottYelland2000\"] = 1,\n [\"CITEREFJohnsonCavallini1991\"] = 1,\n [\"CITEREFJones1967\"] = 1,\n [\"CITEREFJoyner2008\"] = 1,\n [\"CITEREFKent2022\"] = 1,\n [\"CITEREFKissane2009\"] = 1,\n [\"CITEREFKleiner1991\"] = 1,\n [\"CITEREFKleiner2000\"] = 1,\n [\"CITEREFKleiner2007\"] = 1,\n [\"CITEREFKline1990\"] = 1,\n [\"CITEREFKolachanaMaheshRamasubramanian2019\"] = 1,\n [\"CITEREFKremer,_MichaelRao,_GautamSchilbach,_Frank2019\"] = 1,\n [\"CITEREFKrömer2007\"] = 1,\n [\"CITEREFKuhn1976\"] = 1,\n [\"CITEREFLeVeque1977\"] = 1,\n [\"CITEREFLetourneauWright_Sharp2017\"] = 1,\n [\"CITEREFLevinMilgrom2004\"] = 1,\n [\"CITEREFLim2018\"] = 1,\n [\"CITEREFLin1976\"] = 1,\n [\"CITEREFLorch2001\"] = 1,\n [\"CITEREFLuttenbergerWimmerPaechter2018\"] = 1,\n [\"CITEREFLützen2011\"] = 1,\n [\"CITEREFMackay1991\"] = 1,\n [\"CITEREFMaddy2008\"] = 1,\n [\"CITEREFMarchuk2020\"] = 1,\n [\"CITEREFMarker1996\"] = 1,\n [\"CITEREFMaurer1997\"] = 1,\n [\"CITEREFMcCarty2006\"] = 1,\n [\"CITEREFMillstein2016\"] = 1,\n [\"CITEREFMonastyrsky2001\"] = 1,\n [\"CITEREFMoschovakis2018\"] = 1,\n [\"CITEREFMueller1969\"] = 1,\n [\"CITEREFMukunth2015\"] = 1,\n [\"CITEREFMura1993\"] = 1,\n [\"CITEREFMusielak2022\"] = 1,\n [\"CITEREFNadjafikhahYaftian2013\"] = 1,\n [\"CITEREFNational_Research_Council2003\"] = 1,\n [\"CITEREFNewman1956\"] = 1,\n [\"CITEREFNickles2013\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson1996\"] = 1,\n [\"CITEREFO\u0026#039;ConnorRobertson1998\"] = 1,\n [\"CITEREFOaks2018\"] = 1,\n [\"CITEREFOre1988\"] = 1,\n [\"CITEREFOverbaySchorerConger\"] = 1,\n [\"CITEREFParshall2022\"] = 1,\n [\"CITEREFPeressini1999\"] = 1,\n [\"CITEREFPerisho1965\"] = 1,\n [\"CITEREFPerminov1988\"] = 1,\n [\"CITEREFPeterson1988\"] = 1,\n [\"CITEREFPetkovi2009\"] = 1,\n [\"CITEREFPigliucci2014\"] = 1,\n [\"CITEREFPitici2017\"] = 1,\n [\"CITEREFPérez-EscobarSarikaya2021\"] = 1,\n [\"CITEREFRaatikainen2005\"] = 1,\n [\"CITEREFRao1981\"] = 1,\n [\"CITEREFRao1997\"] = 1,\n [\"CITEREFRestivo1992\"] = 1,\n [\"CITEREFRiche2007\"] = 1,\n [\"CITEREFRiehm2002\"] = 1,\n [\"CITEREFRobson2009\"] = 1,\n [\"CITEREFRossi2006\"] = 1,\n [\"CITEREFRouaud2017\"] = 1,\n [\"CITEREFRowan-KenyonSwanCreager2012\"] = 1,\n [\"CITEREFSaliba1994\"] = 1,\n [\"CITEREFSalsburg1992\"] = 1,\n [\"CITEREFSarukkai2005\"] = 1,\n [\"CITEREFSaunders_Maclane1986\"] = 1,\n [\"CITEREFSchubringFuringhettiSiu2012\"] = 1,\n [\"CITEREFSevryuk2006\"] = 1,\n [\"CITEREFShane2022\"] = 1,\n [\"CITEREFShashaLazere1998\"] = 1,\n [\"CITEREFSilver2017\"] = 2,\n [\"CITEREFSingh1936\"] = 1,\n [\"CITEREFSipser1992\"] = 1,\n [\"CITEREFSiu2004\"] = 1,\n [\"CITEREFSnapper1979\"] = 1,\n [\"CITEREFSokalJean_Bricmont1998\"] = 1,\n [\"CITEREFSteen2012\"] = 1,\n [\"CITEREFStewart2018\"] = 1,\n [\"CITEREFStolz2002\"] = 1,\n [\"CITEREFStraume2014\"] = 1,\n [\"CITEREFStrauss2011\"] = 1,\n [\"CITEREFStump1997\"] = 1,\n [\"CITEREFSubramarian\"] = 1,\n [\"CITEREFTakase2014\"] = 1,\n [\"CITEREFTanswell2024\"] = 1,\n [\"CITEREFTiwari1992\"] = 1,\n [\"CITEREFTobiesNeunzert2012\"] = 1,\n [\"CITEREFTrefethen2008\"] = 1,\n [\"CITEREFWaghDeshpande2012\"] = 1,\n [\"CITEREFWagstaff2021\"] = 1,\n [\"CITEREFWang2002\"] = 1,\n [\"CITEREFWarner\"] = 1,\n [\"CITEREFWeil1983\"] = 1,\n [\"CITEREFWeyl2015\"] = 1,\n [\"CITEREFWhite1947\"] = 1,\n [\"CITEREFWhittle1994\"] = 1,\n [\"CITEREFWigner1960\"] = 1,\n [\"CITEREFWilder\"] = 1,\n [\"CITEREFWilsonLewis1912\"] = 1,\n [\"CITEREFWise\"] = 1,\n [\"CITEREFWolchover2013\"] = 1,\n [\"CITEREFWolfram2000\"] = 1,\n [\"CITEREFYaftian2015\"] = 1,\n [\"CITEREFZak2010\"] = 1,\n [\"CITEREFZaslavsky1999\"] = 1,\n [\"CITEREFZhuang\"] = 1,\n [\"CITEREFZieglerLoos2017\"] = 1,\n [\"CITEREFvan_der_Poorten1979\"] = 1,\n [\"CITEREFvon_DavierFoyMartinMullis2020\"] = 1,\n [\"no\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 3,\n [\"Abbr\"] = 1,\n [\"Anchor\"] = 1,\n [\"Areas of mathematics\"] = 1,\n [\"Authority control\"] = 1,\n [\"Blockquote\"] = 2,\n [\"C.\"] = 1,\n [\"CS1 config\"] = 1,\n [\"Circa\"] = 2,\n [\"Citation\"] = 1,\n [\"Cite arXiv\"] = 1,\n [\"Cite book\"] = 92,\n [\"Cite conference\"] = 10,\n [\"Cite encyclopedia\"] = 6,\n [\"Cite journal\"] = 63,\n [\"Cite magazine\"] = 1,\n [\"Cite web\"] = 52,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Efn\"] = 8,\n [\"Em\"] = 3,\n [\"Emdash\"] = 17,\n [\"Free access\"] = 1,\n [\"Further\"] = 2,\n [\"GBurl\"] = 32,\n [\"Gloss\"] = 2,\n [\"Harvnb\"] = 3,\n [\"Lang\"] = 3,\n [\"Langx\"] = 1,\n [\"Library resources box\"] = 1,\n [\"Literal translation\"] = 1,\n [\"Main\"] = 19,\n [\"Main category\"] = 1,\n [\"Math\"] = 4,\n [\"Math topics TOC\"] = 1,\n [\"Mdash\"] = 2,\n [\"Multiple image\"] = 1,\n [\"Multiref\"] = 1,\n [\"Notelist\"] = 1,\n [\"Open access\"] = 1,\n [\"Portal\"] = 1,\n [\"Pp\"] = 1,\n [\"Pp-move\"] = 1,\n [\"Redirect2\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 1,\n [\"See also\"] = 1,\n [\"Sfn\"] = 20,\n [\"Short description\"] = 1,\n [\"Subject bar\"] = 1,\n [\"TOC limit\"] = 1,\n [\"Use American English\"] = 1,\n [\"Use mdy dates\"] = 1,\n [\"Verification failed\"] = 1,\n [\"Webarchive\"] = 2,\n [\"Wikt-lang\"] = 1,\n}\narticle_whitelist = table#1 {\n}\ntable#1 {\n [\"size\"] = 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