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<span>Nature and purpose</span> </div> </a> <ul id="toc-Nature_and_purpose-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods_of_proof" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Methods_of_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Methods of proof</span> </div> </a> <button aria-controls="toc-Methods_of_proof-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Methods of proof subsection</span> </button> <ul id="toc-Methods_of_proof-sublist" class="vector-toc-list"> <li id="toc-Direct_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Direct proof</span> </div> </a> <ul id="toc-Direct_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_by_mathematical_induction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_by_mathematical_induction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Proof by mathematical induction</span> </div> </a> <ul id="toc-Proof_by_mathematical_induction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_by_contraposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_by_contraposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Proof by contraposition</span> </div> </a> <ul id="toc-Proof_by_contraposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_by_contradiction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_by_contradiction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Proof by contradiction</span> </div> </a> <ul id="toc-Proof_by_contradiction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_by_construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_by_construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Proof by construction</span> </div> </a> <ul id="toc-Proof_by_construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_by_exhaustion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_by_exhaustion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Proof by exhaustion</span> </div> </a> <ul id="toc-Proof_by_exhaustion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed_chain_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed_chain_inference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Closed chain inference</span> </div> </a> <ul id="toc-Closed_chain_inference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probabilistic_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probabilistic_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Probabilistic proof</span> </div> </a> <ul id="toc-Probabilistic_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Combinatorial proof</span> </div> </a> <ul id="toc-Combinatorial_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonconstructive_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonconstructive_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Nonconstructive proof</span> </div> </a> <ul id="toc-Nonconstructive_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_proofs_in_pure_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_proofs_in_pure_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.11</span> <span>Statistical proofs in pure mathematics</span> </div> </a> <ul id="toc-Statistical_proofs_in_pure_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer-assisted_proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer-assisted_proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.12</span> <span>Computer-assisted proofs</span> </div> </a> <ul id="toc-Computer-assisted_proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Undecidable_statements" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Undecidable_statements"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Undecidable statements</span> </div> </a> <ul id="toc-Undecidable_statements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heuristic_mathematics_and_experimental_mathematics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Heuristic_mathematics_and_experimental_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Heuristic mathematics and experimental mathematics</span> </div> </a> <ul id="toc-Heuristic_mathematics_and_experimental_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Related concepts</span> </div> </a> <button aria-controls="toc-Related_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related concepts subsection</span> </button> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> <li id="toc-Visual_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visual_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Visual proof</span> </div> </a> <ul id="toc-Visual_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementary_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Elementary proof</span> </div> </a> <ul id="toc-Elementary_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two-column_proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-column_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Two-column proof</span> </div> </a> <ul id="toc-Two-column_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Colloquial_use_of_"mathematical_proof"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Colloquial_use_of_"mathematical_proof""> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Colloquial use of "mathematical proof"</span> </div> </a> <ul id="toc-Colloquial_use_of_"mathematical_proof"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_proof_using_data" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_proof_using_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Statistical proof using data</span> </div> </a> <ul id="toc-Statistical_proof_using_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inductive_logic_proofs_and_Bayesian_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductive_logic_proofs_and_Bayesian_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Inductive logic proofs and Bayesian analysis</span> </div> </a> <ul id="toc-Inductive_logic_proofs_and_Bayesian_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proofs_as_mental_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proofs_as_mental_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>Proofs as mental objects</span> </div> </a> <ul id="toc-Proofs_as_mental_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence_of_mathematical_proof_methods_outside_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Influence_of_mathematical_proof_methods_outside_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8</span> <span>Influence of mathematical proof methods outside mathematics</span> </div> </a> <ul id="toc-Influence_of_mathematical_proof_methods_outside_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ending_a_proof" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ending_a_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Ending a proof</span> </div> </a> <ul id="toc-Ending_a_proof-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">8</span> <span>See also</span> </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"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mathematical proof</span></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 95 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-95" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">95 languages</span> </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/Wiskundige_bewys" title="Wiskundige bewys – Afrikaans" lang="af" hreflang="af" data-title="Wiskundige bewys" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Beweis_(Mathematik)" title="Beweis (Mathematik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Beweis (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A8%D8%B1%D9%87%D8%A7%D9%86_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="برهان رياضي – Arabic" lang="ar" hreflang="ar" data-title="برهان رياضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Demostraci%C3%B3n_matematica" title="Demostración matematica – Aragonese" lang="an" hreflang="an" data-title="Demostración matematica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%A9%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A1%D5%BA%D5%A1%D6%81%D5%B8%D5%B5%D6%81" title="Մաթեմաթիկական ապացոյց – Western Armenian" lang="hyw" hreflang="hyw" data-title="Մաթեմաթիկական ապացոյց" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A7%B0%E0%A6%AE%E0%A6%BE%E0%A6%A3" title="গাণিতিক প্ৰমাণ – Assamese" lang="as" hreflang="as" data-title="গাণিতিক প্ৰমাণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Demostraci%C3%B3n_matem%C3%A1tica" title="Demostración matemática – Asturian" lang="ast" hreflang="ast" data-title="Demostración matemática" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_isbat" title="Riyazi isbat – Azerbaijani" lang="az" hreflang="az" data-title="Riyazi isbat" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%AE%E0%A6%BE%E0%A6%A3" title="গাণিতিক প্রমাণ – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক প্রমাণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%A8ng-b%C3%AAng" title="Chèng-bêng – Minnan" lang="nan" hreflang="nan" data-title="Chèng-bêng" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></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%B8%D2%AB%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Математик иҫбатлау – Bashkir" lang="ba" hreflang="ba" data-title="Математик иҫбатлау" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></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%D1%87%D0%BD%D1%8B_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Матэматычны доказ – Belarusian" lang="be" hreflang="be" data-title="Матэматычны доказ" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></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%D1%87%D0%BD%D1%8B_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Матэматычны доказ – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычны доказ" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></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%D1%87%D0%B5%D1%81%D0%BA%D0%BE_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%81%D1%82%D0%B2%D0%BE" title="Математическо доказателство – Bulgarian" lang="bg" hreflang="bg" data-title="Математическо доказателство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matemati%C4%8Dki_dokaz" title="Matematički dokaz – Bosnian" lang="bs" hreflang="bs" data-title="Matematički dokaz" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Demostraci%C3%B3_(matem%C3%A0tiques)" title="Demostració (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Demostració (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C4%94%D0%BD%D0%B5%D0%BD%D1%82%D0%B5%D1%80%D3%B3_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ĕнентерӳ (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Ĕнентерӳ (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%BD_d%C5%AFkaz" title="Matematický důkaz – Czech" lang="cs" hreflang="cs" data-title="Matematický důkaz" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Prawf_mathemategol" title="Prawf mathemategol – Welsh" lang="cy" hreflang="cy" data-title="Prawf mathemategol" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bevis_(matematik)" title="Bevis (matematik) – Danish" lang="da" hreflang="da" data-title="Bevis (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Beweis_(Mathematik)" title="Beweis (Mathematik) – German" lang="de" hreflang="de" data-title="Beweis (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/T%C3%B5estus" title="Tõestus – Estonian" lang="et" hreflang="et" data-title="Tõestus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></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%AE_%CE%B1%CF%80%CF%8C%CE%B4%CE%B5%CE%B9%CE%BE%CE%B7" title="Μαθηματική απόδειξη – Greek" lang="el" hreflang="el" data-title="Μαθηματική απόδειξη" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Demostraci%C3%B3n_matem%C3%A1tica" title="Demostración matemática – Spanish" lang="es" hreflang="es" data-title="Demostración matemática" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://eo.wikipedia.org/wiki/Matematika_pruvo" title="Matematika pruvo – Esperanto" lang="eo" hreflang="eo" data-title="Matematika pruvo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Froga_matematiko" title="Froga matematiko – Basque" lang="eu" hreflang="eu" data-title="Froga matematiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="اثبات ریاضی – Persian" lang="fa" hreflang="fa" data-title="اثبات ریاضی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof – Fiji Hindi" lang="hif" hreflang="hif" data-title="Mathematical proof" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9monstration_(logique_et_math%C3%A9matiques)" title="Démonstration (logique et mathématiques) – French" lang="fr" hreflang="fr" data-title="Démonstration (logique et mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Proba_matem%C3%A1tica" title="Proba matemática – Galician" lang="gl" hreflang="gl" data-title="Proba matemática" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明 – Gan" lang="gan" hreflang="gan" data-title="數學證明" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A6%9D%EB%AA%85_(%EC%88%98%ED%95%99)" title="증명 (수학) – Korean" lang="ko" hreflang="ko" data-title="증명 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></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%D5%AF%D5%A1%D5%B6_%D5%A1%D5%BA%D5%A1%D6%81%D5%B8%D6%82%D5%B5%D6%81" title="Մաթեմատիկական ապացույց – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական ապացույց" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></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%E0%A5%80%E0%A4%AF_%E0%A4%89%E0%A4%AA%E0%A4%AA%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A4%BF" title="गणितीय उपपत्ति – Hindi" lang="hi" hreflang="hi" data-title="गणितीय उपपत्ति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dki_dokaz" title="Matematički dokaz – Croatian" lang="hr" hreflang="hr" data-title="Matematički dokaz" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pembuktian_matematika" title="Pembuktian matematika – Indonesian" lang="id" hreflang="id" data-title="Pembuktian matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Prova_mathematic" title="Prova mathematic – Interlingua" lang="ia" hreflang="ia" data-title="Prova mathematic" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></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%B0ileg_s%C3%B6nnun" title="Stærðfræðileg sönnun – Icelandic" lang="is" hreflang="is" data-title="Stærðfræðileg sönnun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dimostrazione_matematica" title="Dimostrazione matematica – Italian" lang="it" hreflang="it" data-title="Dimostrazione matematica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%95%D7%9B%D7%97%D7%94" title="הוכחה – Hebrew" lang="he" hreflang="he" data-title="הוכחה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><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%A3%E1%83%A0%E1%83%98_%E1%83%93%E1%83%90%E1%83%9B%E1%83%A2%E1%83%99%E1%83%98%E1%83%AA%E1%83%94%E1%83%91%E1%83%90" title="მათემატიკური დამტკიცება – Georgian" lang="ka" hreflang="ka" data-title="მათემატიკური დამტკიცება" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D3%99%D0%BB%D0%B5%D0%BB%D0%B4%D0%B5%D1%83" title="Дәлелдеу – Kazakh" lang="kk" hreflang="kk" data-title="Дәлелдеу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Thibitisho_la_kihisabati" title="Thibitisho la kihisabati – Swahili" lang="sw" hreflang="sw" data-title="Thibitisho la kihisabati" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/D%C3%A9monstrasyon" title="Démonstrasyon – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Démonstrasyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Demonstratio_mathematica" title="Demonstratio mathematica – Latin" lang="la" hreflang="la" data-title="Demonstratio mathematica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tisks_pier%C4%81d%C4%ABjums" title="Matemātisks pierādījums – Latvian" lang="lv" hreflang="lv" data-title="Matemātisks pierādījums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matematinis_%C4%AFrodymas" title="Matematinis įrodymas – Lithuanian" lang="lt" hreflang="lt" data-title="Matematinis įrodymas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/cipra" title="cipra – Lojban" lang="jbo" hreflang="jbo" data-title="cipra" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target"><span>La .lojban.</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_bizony%C3%ADt%C3%A1s" title="Matematikai bizonyítás – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai bizonyítás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></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%D1%87%D0%BA%D0%B8_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Математички доказ – Macedonian" lang="mk" hreflang="mk" data-title="Математички доказ" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%86%E0%B4%B3%E0%B4%BF%E0%B4%B5%E0%B5%8D_%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B5%BD" title="തെളിവ് ഗണിതശാസ്ത്രത്തിൽ – Malayalam" lang="ml" hreflang="ml" data-title="തെളിവ് ഗണിതശാസ്ത്രത്തിൽ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></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%E0%A5%80%E0%A4%AF_%E0%A4%AA%E0%A5%81%E0%A4%B0%E0%A4%BE%E0%A4%B5%E0%A4%BE" title="गणितीय पुरावा – Marathi" lang="mr" hreflang="mr" data-title="गणितीय पुरावा" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bukti_matematik" title="Bukti matematik – Malay" lang="ms" hreflang="ms" data-title="Bukti matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundig_bewijs" title="Wiskundig bewijs – Dutch" lang="nl" hreflang="nl" data-title="Wiskundig bewijs" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A8%BC%E6%98%8E_(%E6%95%B0%E5%AD%A6)" title="証明 (数学) – Japanese" lang="ja" hreflang="ja" data-title="証明 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Bewis" title="Bewis – Northern Frisian" lang="frr" hreflang="frr" data-title="Bewis" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematisk_bevis" title="Matematisk bevis – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematisk bevis" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_bevis" title="Matematisk bevis – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk bevis" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Demostracion_matematica" title="Demostracion matematica – Occitan" lang="oc" hreflang="oc" data-title="Demostracion matematica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></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%E0%A8%95_%E0%A8%B8%E0%A8%AC%E0%A9%82%E0%A8%A4" title="ਗਣਿਤਕ ਸਬੂਤ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਣਿਤਕ ਸਬੂਤ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%DB%8C%D8%AA%DA%BE%D9%85%DB%8C%D9%B9%DB%8C%DA%A9%D9%84_%D8%AB%D8%A8%D9%88%D8%AA" title="میتھمیٹیکل ثبوت – Western Punjabi" lang="pnb" hreflang="pnb" data-title="میتھمیٹیکل ثبوت" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%DA%A9%D9%8A_%D8%AB%D8%A8%D9%88%D8%AA" title="ریاضیکي ثبوت – Pashto" lang="ps" hreflang="ps" data-title="ریاضیکي ثبوت" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Matimatikal_pruuf" title="Matimatikal pruuf – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Matimatikal pruuf" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Bewies_(Mathematik)" title="Bewies (Mathematik) – Low German" lang="nds" hreflang="nds" data-title="Bewies (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dow%C3%B3d_(matematyka)" title="Dowód (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Dowód (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Prova_matem%C3%A1tica" title="Prova matemática – Portuguese" lang="pt" hreflang="pt" data-title="Prova matemática" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Demonstra%C8%9Bie_matematic%C4%83" title="Demonstrație matematică – Romanian" lang="ro" hreflang="ro" data-title="Demonstrație matematică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></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%D1%87%D0%BD%D0%B5_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE" title="Математічне доказательство – Rusyn" lang="rue" hreflang="rue" data-title="Математічне доказательство" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></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%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D1%81%D1%82%D0%B2%D0%BE" title="Математическое доказательство – Russian" lang="ru" hreflang="ru" data-title="Математическое доказательство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Provat_matematikore" title="Provat matematikore – Albanian" lang="sq" hreflang="sq" data-title="Provat matematikore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Dimustrazzioni_matim%C3%A0tica" title="Dimustrazzioni matimàtica – Sicilian" lang="scn" hreflang="scn" data-title="Dimustrazzioni matimàtica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></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%B8%E0%B6%BA_%E0%B6%94%E0%B6%B4%E0%B7%8A%E0%B6%B4%E0%B7%94_%E0%B6%9A%E0%B7%92%E0%B6%BB%E0%B7%93%E0%B6%B8%E0%B7%8A" title="ගණිතමය ඔප්පු කිරීම් – Sinhala" lang="si" hreflang="si" data-title="ගණිතමය ඔප්පු කිරීම්" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_proof" title="Mathematical proof – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical proof" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://sk.wikipedia.org/wiki/D%C3%B4kaz_(matematika)" title="Dôkaz (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Dôkaz (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matemati%C4%8Dni_dokaz" title="Matematični dokaz – Slovenian" lang="sl" hreflang="sl" data-title="Matematični dokaz" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%8C%D8%B3%D9%BE%D8%A7%D8%AA_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ئیسپات (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئیسپات (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></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%D1%87%D0%BA%D0%B8_%D0%B4%D0%BE%D0%BA%D0%B0%D0%B7" title="Математички доказ – Serbian" lang="sr" hreflang="sr" data-title="Математички доказ" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dokaz_(matematika)" title="Dokaz (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Dokaz (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_todistus" title="Matemaattinen todistus – Finnish" lang="fi" hreflang="fi" data-title="Matemaattinen todistus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematiskt_bevis" title="Matematiskt bevis – Swedish" lang="sv" hreflang="sv" data-title="Matematiskt bevis" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Patibay_pangmatematika" title="Patibay pangmatematika – Tagalog" lang="tl" hreflang="tl" data-title="Patibay pangmatematika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></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%A8%E0%AE%BF%E0%AE%B1%E0%AF%81%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="கணித நிறுவல் – Tamil" lang="ta" hreflang="ta" data-title="கணித நிறுவல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></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%B8%D1%81%D0%B1%D0%B0%D1%82%D0%BB%D0%B0%D1%83" title="Математик исбатлау – Tatar" lang="tt" hreflang="tt" data-title="Математик исбатлау" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9E%E0%B8%B4%E0%B8%AA%E0%B8%B9%E0%B8%88%E0%B8%99%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%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="การพิสูจน์เชิงคณิตศาสตร์ – Thai" lang="th" hreflang="th" data-title="การพิสูจน์เชิงคณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_ispat" title="Matematiksel ispat – Turkish" lang="tr" hreflang="tr" data-title="Matematiksel ispat" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%BD%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Доведення (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Доведення (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></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%D8%A7%D8%AA%DB%8C_%D8%AB%D8%A8%D9%88%D8%AA" title="ریاضیاتی ثبوت – Urdu" lang="ur" hreflang="ur" data-title="ریاضیاتی ثبوت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ch%E1%BB%A9ng_minh_to%C3%A1n_h%E1%BB%8Dc" title="Chứng minh toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Chứng minh toán học" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%AD%89%E6%98%8E" title="證明 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="證明" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pruweba_panmatematika" title="Pruweba panmatematika – Waray" lang="war" hreflang="war" data-title="Pruweba panmatematika" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%AF%81%E6%98%8E" title="证明 – Wu" lang="wuu" hreflang="wuu" data-title="证明" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></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%A9%D7%A2%D7%A8_%D7%93%D7%A2%D7%A8%D7%95%D7%95%D7%99%D7%99%D7%96" title="מאטעמאטישער דערווייז – Yiddish" lang="yi" hreflang="yi" data-title="מאטעמאטישער דערווייז" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明 – Cantonese" lang="yue" hreflang="yue" data-title="數學證明" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Matemat%C4%97nis_iruod%C4%97ms" title="Matematėnis iruodėms – Samogitian" lang="sgs" hreflang="sgs" data-title="Matematėnis iruodėms" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明 – Chinese" lang="zh" hreflang="zh" data-title="數學證明" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/435px-P._Oxy._I_29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/P._Oxy._I_29.jpg/580px-P._Oxy._I_29.jpg 2x" data-file-width="1694" data-file-height="1032" /></a><figcaption><a href="/wiki/Papyrus_Oxyrhynchus_29" title="Papyrus Oxyrhynchus 29">P. Oxy. 29</a>, one of the oldest surviving fragments of <a href="/wiki/Euclid" title="Euclid">Euclid</a>'s <i><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</a></i>, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>A <b>mathematical proof</b> is a <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive</a> <a href="/wiki/Argument-deduction-proof_distinctions" class="mw-redirect" title="Argument-deduction-proof distinctions">argument</a> for a <a href="/wiki/Proposition" title="Proposition">mathematical statement</a>, showing that the stated assumptions <a href="/wiki/Logic" title="Logic">logically</a> guarantee the conclusion. The argument may use other previously established statements, such as <a href="/wiki/Theorem" title="Theorem">theorems</a>; but every proof can, in principle, be constructed using only certain basic or original assumptions known as <a href="/wiki/Axiom" title="Axiom">axioms</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nutsandbolts_3-0" class="reference"><a href="#cite_note-nutsandbolts-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> along with the accepted rules of <a href="/wiki/Inference" title="Inference">inference</a>. Proofs are examples of exhaustive <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a> which establish logical certainty, to be distinguished from <a href="/wiki/Empirical_evidence" title="Empirical evidence">empirical</a> arguments or non-exhaustive <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a> which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in <i>all</i> possible cases. A proposition that has not been proved but is believed to be true is known as a <a href="/wiki/Conjecture" title="Conjecture">conjecture</a>, or a hypothesis if frequently used as an assumption for further mathematical work. </p><p>Proofs employ <a href="/wiki/Logic" title="Logic">logic</a> expressed in mathematical symbols, along with <a href="/wiki/Natural_language" title="Natural language">natural language</a> which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of <a href="/wiki/Rigour#Mathematics" title="Rigour">rigorous</a> <a href="/wiki/Informal_logic" title="Informal logic">informal logic</a>. Purely <a href="/wiki/Formal_proof" title="Formal proof">formal proofs</a>, written fully in <a href="/wiki/Symbolic_language_(mathematics)" title="Symbolic language (mathematics)">symbolic language</a> without the involvement of natural language, are considered in <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>. The distinction between <a href="/wiki/Proof_theory#Formal_and_informal_proof" title="Proof theory">formal and informal proofs</a> has led to much examination of current and historical <a href="/wiki/Mathematical_practice" title="Mathematical practice">mathematical practice</a>, <a href="/wiki/Quasi-empiricism_in_mathematics" title="Quasi-empiricism in mathematics">quasi-empiricism in mathematics</a>, and so-called <a href="/wiki/Mathematical_folklore" title="Mathematical folklore">folk mathematics</a>, oral traditions in the mainstream mathematical community or in other cultures. The <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">philosophy of mathematics</a> is concerned with the role of language and logic in proofs, and <a href="/wiki/Mathematics_as_a_language" class="mw-redirect" title="Mathematics as a language">mathematics as a language</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_and_etymology">History and etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=1" title="Edit section: History and etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_logic" title="History of logic">History of logic</a></div> <p>The word "proof" comes from the Latin <i>probare</i> (to test). Related modern words are English "probe", "probation", and "probability", Spanish <i>probar</i> (to smell or taste, or sometimes touch or test),<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Italian <i>provare</i> (to try), and German <i>probieren</i> (to try). The legal term "probity" means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.<sup id="cite_ref-Krantz_7-0" class="reference"><a href="#cite_note-Krantz-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> It is likely that the idea of demonstrating a conclusion first arose in connection with <a href="/wiki/Geometry" title="Geometry">geometry</a>, which originated in practical problems of land measurement.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The development of mathematical proof is primarily the product of <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematics</a>, and one of its greatest achievements.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Thales" class="mw-redirect" title="Thales">Thales</a> (624–546 BCE) and <a href="/wiki/Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a> (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus</a> (408–355 BCE) and <a href="/wiki/Theaetetus_(mathematician)" title="Theaetetus (mathematician)">Theaetetus</a> (417–369 BCE) formulated theorems but did not prove them. <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. </p><p>Mathematical proof was revolutionized by <a href="/wiki/Euclid" title="Euclid">Euclid</a> (300 BCE), who introduced the <a href="/wiki/Axiomatic_method" class="mw-redirect" title="Axiomatic method">axiomatic method</a> still in use today. It starts with <a href="/wiki/Undefined_term" class="mw-redirect" title="Undefined term">undefined terms</a> and <a href="/wiki/Axiom" title="Axiom">axioms</a>, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). From this basis, the method proves theorems using <a href="/wiki/Deductive_logic" class="mw-redirect" title="Deductive logic">deductive logic</a>. Euclid's book, the <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a>, was read by anyone who was considered educated in the West until the middle of the 20th century.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In addition to theorems of geometry, such as the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>, the <i>Elements</i> also covers <a href="/wiki/Number_theory" title="Number theory">number theory</a>, including a proof that the <a href="/wiki/Square_root_of_two" class="mw-redirect" title="Square root of two">square root of two</a> is <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> and a proof that there are infinitely many <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>. </p><p>Further advances also took place in <a href="/wiki/Mathematics_in_medieval_Islam" class="mw-redirect" title="Mathematics in medieval Islam">medieval Islamic mathematics</a>. In the 10th century CE, the <a href="/wiki/Iraqi_people" class="mw-redirect" title="Iraqi people">Iraqi</a> mathematician <a href="/wiki/Al-Hashimi" class="mw-redirect" title="Al-Hashimi">Al-Hashimi</a> worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> An <a href="/wiki/Mathematical_induction" title="Mathematical induction">inductive proof</a> for <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic sequences</a> was introduced in the <i>Al-Fakhri</i> (1000) by <a href="/wiki/Al-Karaji" title="Al-Karaji">Al-Karaji</a>, who used it to prove the <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> and properties of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>. </p><p>Modern <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a> treats proofs as inductively defined <a href="/wiki/Data_structure" title="Data structure">data structures</a>, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">Axiomatic set theory</a> and <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Nature_and_purpose">Nature and purpose</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=2" title="Edit section: Nature and purpose"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As practiced, a proof is expressed in natural language and is a rigorous <a href="/wiki/Argument" title="Argument">argument</a> intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected. </p><p>The concept of proof is formalized in the field of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A <a href="/wiki/Formal_proof" title="Formal proof">formal proof</a> is written in a <a href="/wiki/Formal_language" title="Formal language">formal language</a> instead of natural language. A formal proof is a sequence of <a href="/wiki/Well-formed_formula" title="Well-formed formula">formulas</a> in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a> studies formal proofs and their properties, the most famous and surprising being that almost all axiomatic systems can generate certain <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">undecidable statements</a> not provable within the system. </p><p>The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated <a href="/wiki/Proof_assistant" title="Proof assistant">proof assistants</a>, this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are <a href="/wiki/Analytic_proposition" class="mw-redirect" title="Analytic proposition">analytic</a> or <a href="/wiki/Synthetic_proposition" class="mw-redirect" title="Synthetic proposition">synthetic</a>. <a href="/wiki/Immanuel_Kant" title="Immanuel Kant">Kant</a>, who introduced the <a href="/wiki/Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">analytic–synthetic distinction</a>, believed mathematical proofs are synthetic, whereas <a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine</a> argued in his 1951 "<a href="/wiki/Two_Dogmas_of_Empiricism" title="Two Dogmas of Empiricism">Two Dogmas of Empiricism</a>" that such a distinction is untenable.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Proofs may be admired for their <a href="/wiki/Mathematical_beauty" title="Mathematical beauty">mathematical beauty</a>. The mathematician <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book <i><a href="/wiki/Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a></i>, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing. </p> <div class="mw-heading mw-heading2"><h2 id="Methods_of_proof"><span class="anchor" id="Types_of_proof"></span>Methods of proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=3" title="Edit section: Methods of proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Direct_proof">Direct proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=4" title="Edit section: Direct proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Direct_proof" title="Direct proof">Direct proof</a></div> <p>In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> For example, direct proof can be used to prove that the sum of two <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> <a href="/wiki/Integer" title="Integer">integers</a> is always even: </p> <dl><dd>Consider two even integers <i>x</i> and <i>y</i>. Since they are even, they can be written as <i>x</i> = 2<i>a</i> and <i>y</i> = 2<i>b</i>, respectively, for some integers <i>a</i> and <i>b</i>. Then the sum is <i>x</i> + <i>y</i> = 2<i>a</i> + 2<i>b</i> = 2(<i>a</i>+<i>b</i>). Therefore <i>x</i>+<i>y</i> has 2 as a <a href="/wiki/Divisor" title="Divisor">factor</a> and, by definition, is even. Hence, the sum of any two even integers is even.</dd></dl> <p>This proof uses the definition of even integers, the integer properties of <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> under addition and multiplication, and the <a href="/wiki/Distributive_property" title="Distributive property">distributive property</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_by_mathematical_induction">Proof by mathematical induction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=5" title="Edit section: Proof by mathematical induction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Mathematical_induction" title="Mathematical induction">Mathematical induction</a></div> <p>Despite its name, mathematical induction is a method of <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deduction</a>, not a form of <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case <a href="/wiki/Material_conditional" title="Material conditional">implies</a> the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually <a href="/wiki/Infinite_set" title="Infinite set">infinitely</a> many) cases are provable.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This avoids having to prove each case individually. A variant of mathematical induction is <a href="/wiki/Proof_by_infinite_descent" title="Proof by infinite descent">proof by infinite descent</a>, which can be used, for example, to prove the <a href="/wiki/Square_root_of_2#Proofs_of_irrationality" title="Square root of 2">irrationality of the square root of two</a>. </p><p>A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>:<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Let <span class="texhtml"><b>N</b> = {1, 2, 3, 4, ...</span>} be the set of natural numbers, and let <span class="texhtml"><i>P</i>(<i>n</i>)</span> be a mathematical statement involving the natural number <span class="texhtml"><i>n</i></span> belonging to <span class="texhtml"><b>N</b></span> such that </p> <ul><li><b>(i)</b> <span class="texhtml"><i>P</i>(1)</span> is true, i.e., <span class="texhtml"><i>P</i>(<i>n</i>)</span> is true for <span class="texhtml"><i>n</i> = 1</span>.</li> <li><b>(ii)</b> <span class="texhtml"><i>P</i>(<i>n</i>+1)</span> is true whenever <span class="texhtml"><i>P</i>(<i>n</i>)</span> is true, i.e., <span class="texhtml"><i>P</i>(<i>n</i>)</span> is true implies that <span class="texhtml"><i>P</i>(<i>n</i>+1)</span> is true.</li> <li><b>Then <span class="texhtml"><i>P</i>(<i>n</i>)</span> is true for all natural numbers <span class="texhtml"><i>n</i></span>.</b></li></ul> <p>For example, we can prove by induction that all positive integers of the form <span class="texhtml">2<i>n</i> − 1</span> are <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a>. Let <span class="texhtml"><i>P</i>(<i>n</i>)</span> represent "<span class="texhtml">2<i>n</i> − 1</span> is odd": </p> <dl><dd><b>(i)</b> For <span class="texhtml"><i>n</i> = 1</span>, <span class="texhtml">2<i>n</i> − 1 = 2(1) − 1 = 1</span>, and <span class="texhtml">1</span> is odd, since it leaves a remainder of <span class="texhtml">1</span> when divided by <span class="texhtml">2</span>. Thus <span class="texhtml"><i>P</i>(1)</span> is true.</dd> <dd><b>(ii)</b> For any <span class="texhtml"><i>n</i></span>, if <span class="texhtml">2<i>n</i> − 1</span> is odd (<span class="texhtml"><i>P</i>(<i>n</i>)</span>), then <span class="texhtml">(2<i>n</i> − 1) + 2</span> must also be odd, because adding <span class="texhtml">2</span> to an odd number results in an odd number. But <span class="texhtml">(2<i>n</i> − 1) + 2 = 2<i>n</i> + 1 = 2(<i>n</i>+1) − 1</span>, so <span class="texhtml">2(<i>n</i>+1) − 1</span> is odd (<span class="texhtml"><i>P</i>(<i>n</i>+1)</span>). So <span class="texhtml"><i>P</i>(<i>n</i>)</span> implies <span class="texhtml"><i>P</i>(<i>n</i>+1)</span>.</dd> <dd><b>Thus</b> <span class="texhtml">2<i>n</i> − 1</span> is odd, for all positive integers <span class="texhtml"><i>n</i></span>.</dd></dl> <p>The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction".<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Proof_by_contraposition">Proof by contraposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=6" title="Edit section: Proof by contraposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Contraposition" title="Contraposition">Contraposition</a></div> <p><a href="/wiki/Proof_by_contrapositive" class="mw-redirect" title="Proof by contrapositive">Proof by contraposition</a> <a href="/wiki/Rule_of_inference" title="Rule of inference">infers</a> the statement "if <i>p</i> then <i>q</i>" by establishing the <a href="/wiki/Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a> <a href="/wiki/Contrapositive" class="mw-redirect" title="Contrapositive">contrapositive statement</a>: "if <i>not q</i> then <i>not p</i>". </p><p>For example, contraposition can be used to establish that, given an integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> is even, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is even: </p> <dl><dd>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is not even. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is odd. The product of two odd numbers is odd, hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=x\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=x\cdot x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a96338692144806b04481dce1d910a10b1b002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.821ex; height:2.676ex;" alt="{\displaystyle x^{2}=x\cdot x}"></span> is odd. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> is not even. Thus, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> <i>is</i> even, the supposition must be false, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> has to be even.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Proof_by_contradiction">Proof by contradiction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=7" title="Edit section: Proof by contradiction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Proof_by_contradiction" title="Proof by contradiction">Proof by contradiction</a></div> <p>In proof by contradiction, also known by the Latin phrase <i><a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a></i> (by reduction to the absurd), it is shown that if some statement is assumed true, a <a href="/wiki/Contradiction" title="Contradiction">logical contradiction</a> occurs, hence the statement must be false. A famous example involves the proof that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> is an <a href="/wiki/Irrational_number" title="Irrational number">irrational number</a>: </p> <dl><dd>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> were a rational number. Then it could be written in lowest terms as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}={a \over b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}={a \over b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e658177b307c22b9ce1ea75982b1810350527d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.263ex; height:4.843ex;" alt="{\displaystyle {\sqrt {2}}={a \over b}}"></span> where <i>a</i> and <i>b</i> are non-zero integers with <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">no common factor</a>. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b{\sqrt {2}}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b{\sqrt {2}}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de8ba3839b5eae053b764b1242ac9c2d25f6a38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.424ex; height:3.009ex;" alt="{\displaystyle b{\sqrt {2}}=a}"></span>. Squaring both sides yields 2<i>b</i><sup>2</sup> = <i>a</i><sup>2</sup>. Since the expression on the left is an integer multiple of 2, the right expression is by definition divisible by 2. That is, <i>a</i><sup>2</sup> is even, which implies that <i>a</i> must also be even, as seen in the proposition above (in <a href="#Proof_by_contraposition">#Proof by contraposition</a>). So we can write <i>a</i> = 2<i>c</i>, where <i>c</i> is also an integer. Substitution into the original equation yields 2<i>b</i><sup>2</sup> = (2<i>c</i>)<sup>2</sup> = 4<i>c</i><sup>2</sup>. Dividing both sides by 2 yields <i>b</i><sup>2</sup> = 2<i>c</i><sup>2</sup>. But then, by the same argument as before, 2 divides <i>b</i><sup>2</sup>, so <i>b</i> must be even. However, if <i>a</i> and <i>b</i> are both even, they have 2 as a common factor. This contradicts our previous statement that <i>a</i> and <i>b</i> have no common factor, so we must conclude that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> is an irrational number.</dd></dl> <p>To paraphrase: if one could write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> as a <a href="/wiki/Fraction" title="Fraction">fraction</a>, this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_by_construction">Proof by construction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=8" title="Edit section: Proof by construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Proof_by_construction" class="mw-redirect" title="Proof by construction">Proof by construction</a></div> <p>Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a>, for instance, proved the existence of <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a> by constructing an <a href="/wiki/Liouville_number" title="Liouville number">explicit example</a>. It can also be used to construct a <a href="/wiki/Counterexample" title="Counterexample">counterexample</a> to disprove a proposition that all elements have a certain property. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_by_exhaustion">Proof by exhaustion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=9" title="Edit section: Proof by exhaustion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Proof_by_exhaustion" title="Proof by exhaustion">Proof by exhaustion</a></div> <p>In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Closed_chain_inference">Closed chain inference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=10" title="Edit section: Closed chain inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Closed_chain_inference" class="mw-redirect" title="Closed chain inference">Closed chain inference</a></div> <p>A closed chain inference shows that a collection of statements are pairwise equivalent. </p><p>In order to prove that the statements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f074740e6950ab5d73f5208aba316943f92599e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.491ex; height:2.176ex;" alt="{\displaystyle \varphi _{1},\ldots ,\varphi _{n}}"></span> are each pairwise equivalent, proofs are given for the implications <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e80740451c00982f5c01b0701561c4fa15cb7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.763ex; height:2.343ex;" alt="{\displaystyle \varphi _{1}\Rightarrow \varphi _{2}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f80f7fb4889b96b445e7839821ce96ad59a87c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.763ex; height:2.343ex;" alt="{\displaystyle \varphi _{2}\Rightarrow \varphi _{3}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5411a9d9722322917df8faecb6e01b72e3ecede4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.723ex; height:0.843ex;" alt="{\displaystyle \dots }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5339e2fb29bcd3e16e6c069522490277abb03716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.192ex; height:2.343ex;" alt="{\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8a020f92fb9a247848db4386f6cc41cfee11da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.927ex; height:2.343ex;" alt="{\displaystyle \varphi _{n}\Rightarrow \varphi _{1}}"></span>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The pairwise equivalence of the statements then results from the <a href="/wiki/Transitive_relation" title="Transitive relation">transitivity</a> of the <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Probabilistic_proof">Probabilistic proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=11" title="Edit section: Probabilistic proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Probabilistic_method" title="Probabilistic method">Probabilistic method</a></div> <p>A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>. Probabilistic proof, like proof by construction, is one of many ways to prove <a href="/wiki/Existence_theorem" title="Existence theorem">existence theorems</a>. </p><p>In the probabilistic method, one seeks an object having a given property, starting with a large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one. </p><p>A probabilistic proof is not to be confused with an argument that a theorem is 'probably' true, a 'plausibility argument'. The work toward the <a href="/wiki/Collatz_conjecture" title="Collatz conjecture">Collatz conjecture</a> shows how far plausibility is from genuine proof, as does the disproof of the <a href="/wiki/Mertens_conjecture" title="Mertens conjecture">Mertens conjecture</a>. While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's <a href="/wiki/Probabilistic_algorithm" class="mw-redirect" title="Probabilistic algorithm">probabilistic algorithm</a> for <a href="/wiki/Primality_test" title="Primality test">testing primality</a>) are as good as genuine mathematical proofs.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_proof">Combinatorial proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=12" title="Edit section: Combinatorial proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Combinatorial_proof" title="Combinatorial proof">Combinatorial proof</a></div> <p>A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Often a <a href="/wiki/Bijection" title="Bijection">bijection</a> between two <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> is used to show that the expressions for their two sizes are equal. Alternatively, a <a href="/wiki/Double_counting_(proof_technique)" title="Double counting (proof technique)">double counting argument</a> provides two different expressions for the size of a single set, again showing that the two expressions are equal. </p> <div class="mw-heading mw-heading3"><h3 id="Nonconstructive_proof">Nonconstructive proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=13" title="Edit section: Nonconstructive proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Nonconstructive_proof" class="mw-redirect" title="Nonconstructive proof">Nonconstructive proof</a></div> <p>A nonconstructive proof establishes that a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> <i>a</i> and <i>b</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921151d29231ebd65eea7632a88215273a32234c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.167ex; height:2.676ex;" alt="{\displaystyle a^{b}}"></span> is a <a href="/wiki/Rational_number" title="Rational number">rational number</a>. This proof uses that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> is irrational (an easy proof is known since <a href="/wiki/Euclid" title="Euclid">Euclid</a>), but not that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span> is irrational (this is true, but the proof is not elementary). </p> <dl><dd>Either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span> is a rational number and we are done (take <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfa96b1d5656c4b847bb84ab943a71ab20bcd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.523ex; height:3.009ex;" alt="{\displaystyle a=b={\sqrt {2}}}"></span>), or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50fb2611d11dbc1741aefae7aea2bd2b317a768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.521ex; height:3.676ex;" alt="{\displaystyle {\sqrt {2}}^{\sqrt {2}}}"></span> is irrational so we can write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f6488178e1b6646218dd6a631bfd4f668f087e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.85ex; height:3.676ex;" alt="{\displaystyle a={\sqrt {2}}^{\sqrt {2}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ec1aee3feb4d5e7c32b13d22bbf9bb52d16c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.194ex; height:3.009ex;" alt="{\displaystyle b={\sqrt {2}}}"></span>. This then gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8344902e7dce191ed617d159374bc0ef205c7c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.232ex; height:5.676ex;" alt="{\displaystyle \left({\sqrt {2}}^{\sqrt {2}}\right)^{\sqrt {2}}={\sqrt {2}}^{2}=2}"></span>, which is thus a rational number of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{b}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{b}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/696aa08b8418270d3f204dc5f8101ff488e18974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.814ex; height:2.676ex;" alt="{\displaystyle a^{b}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Statistical_proofs_in_pure_mathematics">Statistical proofs in pure mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=14" title="Edit section: Statistical proofs in pure mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Statistical_proof" title="Statistical proof">Statistical proof</a></div> <p>The expression "statistical proof" may be used technically or colloquially in areas of <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematics</a>, such as involving <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>, <a href="/w/index.php?title=Chaotic_series&action=edit&redlink=1" class="new" title="Chaotic series (page does not exist)">chaotic series</a>, and <a href="/wiki/Probabilistic" class="mw-redirect" title="Probabilistic">probabilistic number theory</a> or <a href="/wiki/Analytic_number_theory" title="Analytic number theory">analytic number theory</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> It is less commonly used to refer to a mathematical proof in the branch of mathematics known as <a href="/wiki/Mathematical_statistics" title="Mathematical statistics">mathematical statistics</a>. See also the "<a href="#Colloquial_use,_Statistical_proof_using_data">Statistical proof using data</a>" section below. </p> <div class="mw-heading mw-heading3"><h3 id="Computer-assisted_proofs">Computer-assisted proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=15" title="Edit section: Computer-assisted proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></div> <p>Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity.<sup id="cite_ref-Krantz_7-1" class="reference"><a href="#cite_note-Krantz-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the <a href="/wiki/Four_color_theorem" title="Four color theorem">four color theorem</a> is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of a proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved. </p> <div class="mw-heading mw-heading2"><h2 id="Undecidable_statements">Undecidable statements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=16" title="Edit section: Undecidable statements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A statement that is neither provable nor disprovable from a set of <a href="/wiki/Axiom" title="Axiom">axioms</a> is called undecidable (from those axioms). One example is the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>, which is neither provable nor refutable from the remaining axioms of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. </p><p>Mathematicians have shown there are many statements that are neither provable nor disprovable in <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory_with_the_axiom_of_choice" class="mw-redirect" title="Zermelo–Fraenkel set theory with the axiom of choice">Zermelo–Fraenkel set theory with the axiom of choice</a> (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see <a href="/wiki/List_of_statements_undecidable_in_ZFC" class="mw-redirect" title="List of statements undecidable in ZFC">List of statements undecidable in ZFC</a>. </p><p><a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's (first) incompleteness theorem</a> shows that many axiom systems of mathematical interest will have undecidable statements. </p> <div class="mw-heading mw-heading2"><h2 id="Heuristic_mathematics_and_experimental_mathematics">Heuristic mathematics and experimental mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=17" title="Edit section: Heuristic mathematics and experimental mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Experimental_mathematics" title="Experimental mathematics">Experimental mathematics</a></div> <p>While early mathematicians such as <a href="/wiki/Eudoxus_of_Cnidus" title="Eudoxus of Cnidus">Eudoxus of Cnidus</a> did not use proofs, from <a href="/wiki/Euclid" title="Euclid">Euclid</a> to the <a href="/wiki/Foundational_mathematics" class="mw-redirect" title="Foundational mathematics">foundational mathematics</a> developments of the late 19th and 20th centuries, proofs were an essential part of mathematics.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> With the increase in computing power in the 1960s, significant work began to be done investigating <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> beyond the proof-theorem framework,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> in <a href="/wiki/Experimental_mathematics" title="Experimental mathematics">experimental mathematics</a>. Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of <a href="/wiki/Fractal_geometry" class="mw-redirect" title="Fractal geometry">fractal geometry</a>,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> which was ultimately so resolved. </p> <div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=18" title="Edit section: Related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Visual_proof">Visual proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=19" title="Edit section: Visual proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "<a href="/wiki/Proof_without_words" title="Proof without words">proof without words</a>". The left-hand picture below is an example of a historic visual proof of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> in the case of the (3,4,5) <a href="/wiki/Triangle" title="Triangle">triangle</a>. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Chinese_pythagoras.jpg" class="mw-file-description" title="Visual proof for the (3,4,5) triangle as in the Zhoubi Suanjing 500–200 BCE."><img alt="Visual proof for the (3,4,5) triangle as in the Zhoubi Suanjing 500–200 BCE." src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/120px-Chinese_pythagoras.jpg" decoding="async" width="120" height="65" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/180px-Chinese_pythagoras.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Chinese_pythagoras.jpg/240px-Chinese_pythagoras.jpg 2x" data-file-width="871" data-file-height="475" /></a></span></div> <div class="gallerytext">Visual proof for the (3,4,5) triangle as in the <a href="/wiki/Zhoubi_Suanjing" title="Zhoubi Suanjing">Zhoubi Suanjing</a> 500–200 BCE.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pythagoras-2a.gif" class="mw-file-description" title="Animated visual proof for the Pythagorean theorem by rearrangement."><img alt="Animated visual proof for the Pythagorean theorem by rearrangement." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/120px-Pythagoras-2a.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/180px-Pythagoras-2a.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Pythagoras-2a.gif/240px-Pythagoras-2a.gif 2x" data-file-width="590" data-file-height="590" /></a></span></div> <div class="gallerytext">Animated visual proof for the Pythagorean theorem by rearrangement.</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Pythag_anim.gif" class="mw-file-description" title="A second animated proof of the Pythagorean theorem."><img alt="A second animated proof of the Pythagorean theorem." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Pythag_anim.gif/120px-Pythag_anim.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Pythag_anim.gif/180px-Pythag_anim.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/65/Pythag_anim.gif 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext">A second animated proof of the Pythagorean theorem.</div> </li> </ul> <p>Some illusory visual proofs, such as the <a href="/wiki/Missing_square_puzzle" title="Missing square puzzle">missing square puzzle</a>, can be constructed in a way which appear to prove a supposed mathematical fact but only do so by neglecting tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated. </p> <div class="mw-heading mw-heading3"><h3 id="Elementary_proof">Elementary proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=20" title="Edit section: Elementary proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Elementary_proof" title="Elementary proof">Elementary proof</a></div> <p>An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in <a href="/wiki/Number_theory" title="Number theory">number theory</a> to refer to proofs that make no use of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>. For some time it was thought that certain theorems, like the <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a>, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques. </p> <div class="mw-heading mw-heading3"><h3 id="Two-column_proof">Two-column proof</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=21" title="Edit section: Two-column proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Twocolumnproof.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Twocolumnproof.png/220px-Twocolumnproof.png" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Twocolumnproof.png/330px-Twocolumnproof.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/13/Twocolumnproof.png 2x" data-file-width="437" data-file-height="404" /></a><figcaption>A two-column proof published in 1913</figcaption></figure> <p>A particular way of organising a proof using two parallel columns is often used as a <a href="/wiki/Mathematical_exercise" class="mw-redirect" title="Mathematical exercise">mathematical exercise</a> in elementary geometry classes in the United States.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Colloquial_use_of_"mathematical_proof""><span id="Colloquial_use_of_.22mathematical_proof.22"></span>Colloquial use of "mathematical proof"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=22" title="Edit section: Colloquial use of "mathematical proof""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data. </p> <div class="mw-heading mw-heading3"><h3 id="Statistical_proof_using_data">Statistical proof using data</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=23" title="Edit section: Statistical proof using data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Statistical_proof" title="Statistical proof">Statistical proof</a></div> <p>"Statistical proof" from data refers to the application of statistics, <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>, or <a href="/wiki/Bayesian_analysis" class="mw-redirect" title="Bayesian analysis">Bayesian analysis</a> to infer propositions regarding the <a href="/wiki/Probability" title="Probability">probability</a> of data. While <i>using</i> mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the <i>assumptions</i> from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized <i><a href="/wiki/Mathematical_methods_of_physics" class="mw-redirect" title="Mathematical methods of physics">mathematical methods of physics</a></i> applied to analyze data in a <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a> experiment or <a href="/wiki/Observational_study" title="Observational study">observational study</a> in <a href="/wiki/Physical_cosmology" title="Physical cosmology">physical cosmology</a>. "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as <a href="/wiki/Scatter_plot" title="Scatter plot">scatter plots</a>, when the data or diagram is adequately convincing without further analysis. </p> <div class="mw-heading mw-heading3"><h3 id="Inductive_logic_proofs_and_Bayesian_analysis">Inductive logic proofs and Bayesian analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=24" title="Edit section: Inductive logic proofs and Bayesian analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Inductive_logic" class="mw-redirect" title="Inductive logic">Inductive logic</a> and <a href="/wiki/Bayesian_analysis" class="mw-redirect" title="Bayesian analysis">Bayesian analysis</a></div> <p>Proofs using <a href="/wiki/Inductive_logic" class="mw-redirect" title="Inductive logic">inductive logic</a>, while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to <a href="/wiki/Probability" title="Probability">probability</a>, and may be less than full <a href="/wiki/Certainty" title="Certainty">certainty</a>. Inductive logic should not be confused with <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>. </p><p>Bayesian analysis uses <a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a> to update a person's <a href="/wiki/Bayesian_probability" title="Bayesian probability">assessment of likelihoods</a> of hypotheses when new <a href="/wiki/Evidence" title="Evidence">evidence</a> or information is acquired. </p> <div class="mw-heading mw-heading3"><h3 id="Proofs_as_mental_objects">Proofs as mental objects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=25" title="Edit section: Proofs as mental objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Psychologism" title="Psychologism">Psychologism</a> and <a href="/wiki/Language_of_thought" class="mw-redirect" title="Language of thought">Language of thought</a></div> <p>Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, and <a href="/wiki/Carnap" class="mw-redirect" title="Carnap">Carnap</a> have variously criticized this view and attempted to develop a semantics for what they considered to be the <a href="/wiki/Language_of_thought" class="mw-redirect" title="Language of thought">language of thought</a>, whereby standards of mathematical proof might be applied to <a href="/wiki/Empirical_science" class="mw-redirect" title="Empirical science">empirical science</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Influence_of_mathematical_proof_methods_outside_mathematics">Influence of mathematical proof methods outside mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=26" title="Edit section: Influence of mathematical proof methods outside mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Philosopher-mathematicians such as <a href="/wiki/Spinoza" class="mw-redirect" title="Spinoza">Spinoza</a> have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the <a href="/wiki/Certainty" title="Certainty">certainty</a> of propositions deduced in a mathematical proof, such as <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a>' <a href="/wiki/Cogito_ergo_sum" class="mw-redirect" title="Cogito ergo sum"><i>cogito</i></a> argument. </p> <div class="mw-heading mw-heading2"><h2 id="Ending_a_proof">Ending a proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=27" title="Edit section: Ending a proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Q.E.D." title="Q.E.D.">Q.E.D.</a></div> <p>Sometimes, the abbreviation <i>"Q.E.D."</i> is written to indicate the end of a proof. This abbreviation stands for <i>"quod erat demonstrandum"</i>, which is <a href="/wiki/Latin" title="Latin">Latin</a> for <i>"that which was to be demonstrated"</i>. A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a "<a href="/wiki/Tombstone_(typography)" title="Tombstone (typography)">tombstone</a>" or "halmos" after its <a href="/wiki/Eponym" title="Eponym">eponym</a> <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a>. Often, "which was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎) <small>(220E(hex) = 8718(dec))</small>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.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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/></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></span></li><li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><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/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-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/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li> <li><a href="/wiki/Invalid_proof" class="mw-redirect" title="Invalid proof">Invalid proof</a></li> <li><a href="/wiki/List_of_incomplete_proofs" title="List of incomplete proofs">List of incomplete proofs</a></li> <li><a href="/wiki/List_of_long_proofs" class="mw-redirect" title="List of long proofs">List of long proofs</a></li> <li><a href="/wiki/List_of_mathematical_proofs" title="List of mathematical proofs">List of mathematical proofs</a></li> <li><a href="/wiki/Nonconstructive_proof" class="mw-redirect" title="Nonconstructive proof">Nonconstructive proof</a></li> <li><a href="/wiki/Proof_by_intimidation" title="Proof by intimidation">Proof by intimidation</a></li> <li><a href="/wiki/Termination_analysis" title="Termination analysis">Termination analysis</a></li> <li><a href="/wiki/Thought_experiment" title="Thought experiment">Thought experiment</a></li> <li><i><a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a></i></li> <li><a href="/wiki/Zero-knowledge_proof" title="Zero-knowledge proof">Zero-knowledge proof</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=29" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFBill_Casselman" class="citation web cs1"><a href="/wiki/Bill_Casselman_(mathematician)" class="mw-redirect" title="Bill Casselman (mathematician)">Bill Casselman</a>. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html">"One of the Oldest Extant Diagrams from Euclid"</a>. University of British Columbia<span class="reference-accessdate">. Retrieved <span class="nowrap">September 26,</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=One+of+the+Oldest+Extant+Diagrams+from+Euclid&rft.pub=University+of+British+Columbia&rft.au=Bill+Casselman&rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cass%2FEuclid%2Fpapyrus%2Fpapyrus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClapham,_C.Nicholson,_J.N." class="citation book cs1">Clapham, C. & Nicholson, J.N. <i>The Concise Oxford Dictionary of Mathematics, Fourth edition</i>. <q>A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Concise+Oxford+Dictionary+of+Mathematics%2C+Fourth+edition&rft.au=Clapham%2C+C.&rft.au=Nicholson%2C+J.N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-nutsandbolts-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-nutsandbolts_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCupillari2005" class="citation book cs1"><a href="/wiki/Antonella_Cupillari" title="Antonella Cupillari">Cupillari, Antonella</a> (2005) [2001]. <i>The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs</i> (Third ed.). <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-088509-1" title="Special:BookSources/978-0-12-088509-1"><bdi>978-0-12-088509-1</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+Nuts+and+Bolts+of+Proofs%3A+An+Introduction+to+Mathematical+Proofs&rft.pages=3&rft.edition=Third&rft.pub=Academic+Press&rft.date=2005&rft.isbn=978-0-12-088509-1&rft.aulast=Cupillari&rft.aufirst=Antonella&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGossett2009" class="citation book cs1">Gossett, Eric (July 2009). <i>Discrete Mathematics with Proof</i>. <a href="/wiki/Wiley_(publisher)" title="Wiley (publisher)">John Wiley & Sons</a>. p. 86. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0470457931" title="Special:BookSources/978-0470457931"><bdi>978-0470457931</bdi></a>. <q>Definition 3.1. Proof: An Informal Definition</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+with+Proof&rft.pages=86&rft.pub=John+Wiley+%26+Sons&rft.date=2009-07&rft.isbn=978-0470457931&rft.aulast=Gossett&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">"proof" New Shorter Oxford English Dictionary, 1993, OUP, Oxford.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHacking1984" class="citation book cs1"><a href="/wiki/Ian_Hacking" title="Ian Hacking">Hacking, Ian</a> (1984) [1975]. <i>The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-31803-7" title="Special:BookSources/978-0-521-31803-7"><bdi>978-0-521-31803-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Emergence+of+Probability%3A+A+Philosophical+Study+of+Early+Ideas+about+Probability%2C+Induction+and+Statistical+Inference&rft.pub=Cambridge+University+Press&rft.date=1984&rft.isbn=978-0-521-31803-7&rft.aulast=Hacking&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-Krantz-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Krantz_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Krantz_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.wustl.edu/~sk/eolss.pdf">The History and Concept of Mathematical Proof</a>, Steven G. Krantz. 1. February 5, 2007</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnealeKneale1985" class="citation book cs1"><a href="/wiki/William_Kneale_(logician)" class="mw-redirect" title="William Kneale (logician)">Kneale, William</a>; Kneale, Martha (May 1985) [1962]. <i>The development of logic</i> (New ed.). <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-824773-9" title="Special:BookSources/978-0-19-824773-9"><bdi>978-0-19-824773-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+development+of+logic&rft.pages=3&rft.edition=New&rft.pub=Oxford+University+Press&rft.date=1985-05&rft.isbn=978-0-19-824773-9&rft.aulast=Kneale&rft.aufirst=William&rft.au=Kneale%2C+Martha&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoutsios-RentzosSpyrou2015" class="citation web cs1">Moutsios-Rentzos, Andreas; Spyrou, Panagiotis (February 2015). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-01281050/document">"The genesis of proof in ancient Greece The pedagogical implications of a Husserlian reading"</a>. <i>Archive ouverte HAL</i><span class="reference-accessdate">. Retrieved <span class="nowrap">October 20,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Archive+ouverte+HAL&rft.atitle=The+genesis+of+proof+in+ancient+Greece+The+pedagogical+implications+of+a+Husserlian+reading&rft.date=2015-02&rft.aulast=Moutsios-Rentzos&rft.aufirst=Andreas&rft.au=Spyrou%2C+Panagiotis&rft_id=https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-01281050%2Fdocument&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1990" class="citation book cs1"><a href="/wiki/Howard_Eves" title="Howard Eves">Eves, Howard W.</a> (January 1990) [1962]. <i>An Introduction to the History of Mathematics (Saunders Series)</i> (6th ed.). <a href="/wiki/Cengage" class="mw-redirect" title="Cengage">Brooks/Cole</a>. p. 141. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0030295584" title="Special:BookSources/978-0030295584"><bdi>978-0030295584</bdi></a>. <q>No work, except The Bible, has been more widely used...</q></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+%28Saunders+Series%29&rft.pages=141&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=1990-01&rft.isbn=978-0030295584&rft.aulast=Eves&rft.aufirst=Howard+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatvievskaya1987" class="citation cs2">Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", <i><a href="/wiki/New_York_Academy_of_Sciences" title="New York Academy of Sciences">Annals of the New York Academy of Sciences</a></i>, <b>500</b> (1): 253–77 [260], <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/1987NYASA.500..253M">1987NYASA.500..253M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1749-6632.1987.tb37206.x">10.1111/j.1749-6632.1987.tb37206.x</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:121416910">121416910</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+New+York+Academy+of+Sciences&rft.atitle=The+Theory+of+Quadratic+Irrationals+in+Medieval+Oriental+Mathematics&rft.volume=500&rft.issue=1&rft.pages=253-77+260&rft.date=1987&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121416910%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fj.1749-6632.1987.tb37206.x&rft_id=info%3Abibcode%2F1987NYASA.500..253M&rft.aulast=Matvievskaya&rft.aufirst=Galina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuss1998" class="citation cs2"><a href="/wiki/Samuel_Buss" title="Samuel Buss">Buss, Samuel R.</a> (1998), "An introduction to proof theory", in <a href="/wiki/Samuel_Buss" title="Samuel Buss">Buss, Samuel R.</a> (ed.), <i>Handbook of Proof Theory</i>, Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, pp. 1–78, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-053318-6" title="Special:BookSources/978-0-08-053318-6"><bdi>978-0-08-053318-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+introduction+to+proof+theory&rft.btitle=Handbook+of+Proof+Theory&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.pages=1-78&rft.pub=Elsevier&rft.date=1998&rft.isbn=978-0-08-053318-6&rft.aulast=Buss&rft.aufirst=Samuel+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA3">p. 3</a>: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine1961" class="citation web cs1">Quine, Willard Van Orman (1961). <a rel="nofollow" class="external text" href="https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf">"Two Dogmas of Empiricism"</a> <span class="cs1-format">(PDF)</span>. <i>Universität Zürich – Theologische Fakultät</i>. p. 12<span class="reference-accessdate">. Retrieved <span class="nowrap">October 20,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Universit%C3%A4t+Z%C3%BCrich+%E2%80%93+Theologische+Fakult%C3%A4t&rft.atitle=Two+Dogmas+of+Empiricism&rft.pages=12&rft.date=1961&rft.aulast=Quine&rft.aufirst=Willard+Van+Orman&rft_id=https%3A%2F%2Fwww.theologie.uzh.ch%2Fdam%2Fjcr%3Affffffff-fbd6-1538-0000-000070cf64bc%2FQuine51.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Cupillari, p. 20.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Cupillari, p. 46.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://zimmer.csufresno.edu/~larryc/proofs/proofs.mathinduction.html">Examples of simple proofs by mathematical induction for all natural numbers</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/AEAhelp/glossary/glossaryParser.php?glossaryFile=Proof%20by%20induction.htm">Proof by induction</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120218033011/http://www.warwick.ac.uk/AEAhelp/glossary/glossaryParser.php?glossaryFile=Proof%20by%20induction.htm">Archived</a> February 18, 2012, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, University of Warwick Glossary of Mathematical Terminology</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">See <a href="/wiki/Four_color_theorem#Simplification_and_verification" title="Four color theorem">Four color theorem#Simplification and verification</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlaueScherfner2019" class="citation book cs1 cs1-prop-foreign-lang-source">Plaue, Matthias; Scherfner, Mike (February 11, 2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-WCHDwAAQBAJ"><i>Mathematik für das Bachelorstudium I: Grundlagen und Grundzüge der linearen Algebra und Analysis</i></a> [<i>Mathematics for the Bachelor's degree I: Fundamentals and basics of linear algebra and analysis</i>] (in German). Springer-Verlag. p. 26. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-58352-4" title="Special:BookSources/978-3-662-58352-4"><bdi>978-3-662-58352-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematik+f%C3%BCr+das+Bachelorstudium+I%3A+Grundlagen+und+Grundz%C3%BCge+der+linearen+Algebra+und+Analysis&rft.pages=26&rft.pub=Springer-Verlag&rft.date=2019-02-11&rft.isbn=978-3-662-58352-4&rft.aulast=Plaue&rft.aufirst=Matthias&rft.au=Scherfner%2C+Mike&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-WCHDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStruckmannWätjen2016" class="citation book cs1 cs1-prop-foreign-lang-source">Struckmann, Werner; Wätjen, Dietmar (October 20, 2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1epNDQAAQBAJ"><i>Mathematik für Informatiker: Grundlagen und Anwendungen</i></a> [<i>Mathematics for Computer Scientists: Fundamentals and Applications</i>] (in German). Springer-Verlag. p. 28. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-49870-5" title="Special:BookSources/978-3-662-49870-5"><bdi>978-3-662-49870-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematik+f%C3%BCr+Informatiker%3A+Grundlagen+und+Anwendungen&rft.pages=28&rft.pub=Springer-Verlag&rft.date=2016-10-20&rft.isbn=978-3-662-49870-5&rft.aulast=Struckmann&rft.aufirst=Werner&rft.au=W%C3%A4tjen%2C+Dietmar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1epNDQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Davis, Philip J. (1972), "Fidelity in Mathematical Discourse: Is One and One Really Two?" <i>American Mathematical Monthly</i> 79:252–63.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Fallis, Don (1997), "The Epistemic Status of Probabilistic Proof." <i>Journal of Philosophy</i> 94:165–86.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">"in number theory and commutative algebra... in particular the statistical proof of the lemma." <a rel="nofollow" class="external autonumber" href="https://www.jstor.org/pss/2686395">[1]</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">"Whether constant π (i.e., pi) is normal is a confusing problem without any strict theoretical demonstration except for some <i>statistical</i> proof"" (Derogatory use.)<a rel="nofollow" class="external autonumber" href="https://doi.org/10.1007%2F978-3-540-74282-1_78">[2]</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">"these observations suggest a statistical proof of Goldbach's conjecture with very quickly vanishing probability of failure for large E" <a rel="nofollow" class="external autonumber" href="http://people.web.psi.ch/gassmann/eneseminare/abstracts/Goldbach1.pdf">[3]</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMumfordSeriesWright2002" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David B.</a>; <a href="/wiki/Caroline_Series" title="Caroline Series">Series, Caroline</a>; <a href="/wiki/David_Wright_(arranger)" title="David Wright (arranger)">Wright, David</a> (2002). <i><a href="/wiki/Indra%27s_Pearls_(book)" title="Indra's Pearls (book)">Indra's Pearls: The Vision of Felix Klein</a></i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-35253-6" title="Special:BookSources/978-0-521-35253-6"><bdi>978-0-521-35253-6</bdi></a>. <q>What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm and the conventions of that day dictated that journals only published theorems.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Indra%27s+Pearls%3A+The+Vision+of+Felix+Klein&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-35253-6&rft.aulast=Mumford&rft.aufirst=David+B.&rft.au=Series%2C+Caroline&rft.au=Wright%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090215114618/https://home.att.net/~fractalia/history.htm">"A Note on the History of Fractals"</a>. Archived from <a rel="nofollow" class="external text" href="https://home.att.net/~fractalia/history.htm">the original</a> on February 15, 2009. <q>Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Note+on+the+History+of+Fractals&rft_id=https%3A%2F%2Fhome.att.net%2F~fractalia%2Fhistory.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLesmoir-Gordon2000" class="citation book cs1">Lesmoir-Gordon, Nigel (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introducingfract0000lesm"><i>Introducing Fractal Geometry</i></a></span>. <a href="/wiki/Introducing..._(book_series)" title="Introducing... (book series)">Icon Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84046-123-7" title="Special:BookSources/978-1-84046-123-7"><bdi>978-1-84046-123-7</bdi></a>. <q>...brought home again to Benoit [Mandelbrot] that there was a 'mathematics of the eye', that visualization of a problem was as valid a method as any for finding a solution. Amazingly, he found himself alone with this conjecture. The teaching of mathematics in France was dominated by a handful of dogmatic mathematicians hiding behind the pseudonym 'Bourbaki'...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introducing+Fractal+Geometry&rft.pub=Icon+Books&rft.date=2000&rft.isbn=978-1-84046-123-7&rft.aulast=Lesmoir-Gordon&rft.aufirst=Nigel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroducingfract0000lesm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerbst2002" class="citation journal cs1">Herbst, Patricio G. (2002). <a rel="nofollow" class="external text" href="https://deepblue.lib.umich.edu/bitstream/2027.42/42653/1/10649_2004_Article_5096042.pdf">"Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Educational_Studies_in_Mathematics" title="Educational Studies in Mathematics">Educational Studies in Mathematics</a></i>. <b>49</b> (3): 283–312. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1020264906740">10.1023/A:1020264906740</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F42653">2027.42/42653</a></span>. <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:23084607">23084607</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=Establishing+a+Custom+of+Proving+in+American+School+Geometry%3A+Evolution+of+the+Two-Column+Proof+in+the+Early+Twentieth+Century&rft.volume=49&rft.issue=3&rft.pages=283-312&rft.date=2002&rft_id=info%3Ahdl%2F2027.42%2F42653&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A23084607%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1020264906740&rft.aulast=Herbst&rft.aufirst=Patricio+G.&rft_id=https%3A%2F%2Fdeepblue.lib.umich.edu%2Fbitstream%2F2027.42%2F42653%2F1%2F10649_2004_Article_5096042.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDr._Fisher_Burns" class="citation web cs1">Dr. Fisher Burns. <a rel="nofollow" class="external text" href="https://www.onemathematicalcat.org/Math/Geometry_obj/two_column_proof.htm">"Introduction to the Two-Column Proof"</a>. <i>onemathematicalcat.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">October 15,</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=onemathematicalcat.org&rft.atitle=Introduction+to+the+Two-Column+Proof&rft.au=Dr.+Fisher+Burns&rft_id=https%3A%2F%2Fwww.onemathematicalcat.org%2FMath%2FGeometry_obj%2Ftwo_column_proof.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=30" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPólya1954" class="citation cs2"><a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya, G.</a> (1954), <a href="/wiki/Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning"><i>Mathematics and Plausible Reasoning</i></a>, Princeton University Press, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.39015008206248">2027/mdp.39015008206248</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691080055" title="Special:BookSources/9780691080055"><bdi>9780691080055</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+and+Plausible+Reasoning&rft.pub=Princeton+University+Press&rft.date=1954&rft_id=info%3Ahdl%2F2027%2Fmdp.39015008206248&rft.isbn=9780691080055&rft.aulast=P%C3%B3lya&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFallis2002" class="citation cs2">Fallis, Don (2002), <a rel="nofollow" class="external text" href="http://dlist.sir.arizona.edu/1581/">"What Do Mathematicians Want? Probabilistic Proofs and the Epistemic Goals of Mathematicians"</a>, <i>Logique et Analyse</i>, <b>45</b>: 373–88</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Logique+et+Analyse&rft.atitle=What+Do+Mathematicians+Want%3F+Probabilistic+Proofs+and+the+Epistemic+Goals+of+Mathematicians&rft.volume=45&rft.pages=373-88&rft.date=2002&rft.aulast=Fallis&rft.aufirst=Don&rft_id=http%3A%2F%2Fdlist.sir.arizona.edu%2F1581%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklinDaoud2011" class="citation cs2"><a href="/wiki/James_Franklin_(philosopher)" title="James Franklin (philosopher)">Franklin, J.</a>; Daoud, A. (2011), <a rel="nofollow" class="external text" href="http://www.maths.unsw.edu.au/~jim/proofs.html"><i>Proof in Mathematics: An Introduction</i></a>, Kew Books, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-646-54509-7" title="Special:BookSources/978-0-646-54509-7"><bdi>978-0-646-54509-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+in+Mathematics%3A+An+Introduction&rft.pub=Kew+Books&rft.date=2011&rft.isbn=978-0-646-54509-7&rft.aulast=Franklin&rft.aufirst=J.&rft.au=Daoud%2C+A.&rft_id=http%3A%2F%2Fwww.maths.unsw.edu.au%2F~jim%2Fproofs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGold,_BonnieSimons,_Rogers_A.2008" class="citation book cs1"><a href="/wiki/Bonnie_Gold" title="Bonnie Gold">Gold, Bonnie</a>; Simons, Rogers A. (2008). <i>Proof and Other Dilemmas: Mathematics and Philosophy</i>. MAA.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proof+and+Other+Dilemmas%3A+Mathematics+and+Philosophy&rft.pub=MAA&rft.date=2008&rft.au=Gold%2C+Bonnie&rft.au=Simons%2C+Rogers+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSolow2004" class="citation cs2">Solow, D. (2004), <i>How to Read and Do Proofs: An Introduction to Mathematical Thought Processes</i>, <a href="/wiki/Wiley_Publishing" class="mw-redirect" title="Wiley Publishing">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-68058-1" title="Special:BookSources/978-0-471-68058-1"><bdi>978-0-471-68058-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+to+Read+and+Do+Proofs%3A+An+Introduction+to+Mathematical+Thought+Processes&rft.pub=Wiley&rft.date=2004&rft.isbn=978-0-471-68058-1&rft.aulast=Solow&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVelleman2006" class="citation cs2">Velleman, D. (2006), <i>How to Prove It: A Structured Approach</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-67599-4" title="Special:BookSources/978-0-521-67599-4"><bdi>978-0-521-67599-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+to+Prove+It%3A+A+Structured+Approach&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-67599-4&rft.aulast=Velleman&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHammack2018" class="citation cs2">Hammack, Richard (2018), <a rel="nofollow" class="external text" href="https://richardhammack.github.io/BookOfProof/"><i>Book of Proof</i></a>, Richard Hammack, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9894721-3-5" title="Special:BookSources/978-0-9894721-3-5"><bdi>978-0-9894721-3-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Book+of+Proof&rft.pub=Richard+Hammack&rft.date=2018&rft.isbn=978-0-9894721-3-5&rft.aulast=Hammack&rft.aufirst=Richard&rft_id=https%3A%2F%2Frichardhammack.github.io%2FBookOfProof%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+proof" class="Z3988"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_proof&action=edit&section=31" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output 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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:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">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%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Term_logic" title="Term logic">Traditional</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/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</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/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</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/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</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/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/Cardinality" title="Cardinality">cardinality</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/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/Binary_operation" title="Binary operation">binary</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</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/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li> <li><a href="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>of <a href="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> 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<li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</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/Classical_logic" title="Classical logic">Classical</a></li> <li><a href="/wiki/Informal_logic" title="Informal logic">Informal</a> <ul><li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li> <li><a href="/wiki/Reason" title="Reason">Reason</a></li></ul></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical</a></li> <li><a href="/wiki/Non-classical_logic" title="Non-classical 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