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<span>Paradoxes</span> </div> </a> <button aria-controls="toc-Paradoxes-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 Paradoxes subsection</span> </button> <ul id="toc-Paradoxes-sublist" class="vector-toc-list"> <li id="toc-Paradoxes_of_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Paradoxes_of_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Paradoxes of motion</span> </div> </a> <ul id="toc-Paradoxes_of_motion-sublist" class="vector-toc-list"> <li id="toc-Dichotomy_paradox" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dichotomy_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Dichotomy paradox</span> </div> </a> <ul id="toc-Dichotomy_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Achilles_and_the_tortoise" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Achilles_and_the_tortoise"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Achilles and the tortoise</span> </div> </a> <ul id="toc-Achilles_and_the_tortoise-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arrow_paradox" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Arrow_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Arrow paradox</span> </div> </a> <ul id="toc-Arrow_paradox-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_paradoxes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_paradoxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Other paradoxes</span> </div> </a> <ul id="toc-Other_paradoxes-sublist" class="vector-toc-list"> <li id="toc-Paradox_of_place" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Paradox_of_place"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Paradox of place</span> </div> </a> <ul id="toc-Paradox_of_place-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paradox_of_the_grain_of_millet" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Paradox_of_the_grain_of_millet"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Paradox of the grain of millet</span> </div> </a> <ul id="toc-Paradox_of_the_grain_of_millet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_moving_rows_(or_stadium)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_moving_rows_(or_stadium)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>The moving rows (or stadium)</span> </div> </a> <ul id="toc-The_moving_rows_(or_stadium)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Proposed_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proposed_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Proposed solutions</span> </div> </a> <button aria-controls="toc-Proposed_solutions-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 Proposed solutions subsection</span> </button> <ul id="toc-Proposed_solutions-sublist" class="vector-toc-list"> <li id="toc-In_classical_antiquity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_classical_antiquity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>In classical antiquity</span> </div> </a> <ul id="toc-In_classical_antiquity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_modern_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_modern_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>In modern mathematics</span> </div> </a> <ul id="toc-In_modern_mathematics-sublist" class="vector-toc-list"> <li id="toc-Henri_Bergson" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Henri_Bergson"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Henri Bergson</span> </div> </a> <ul id="toc-Henri_Bergson-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Peter_Lynds" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Peter_Lynds"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Peter Lynds</span> </div> </a> <ul id="toc-Peter_Lynds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bertrand_Russell" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bertrand_Russell"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Bertrand Russell</span> </div> </a> <ul id="toc-Bertrand_Russell-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermann_Weyl" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hermann_Weyl"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Hermann Weyl</span> </div> </a> <ul id="toc-Hermann_Weyl-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Quantum_Zeno_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_Zeno_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Quantum Zeno effect</span> </div> </a> <ul id="toc-Quantum_Zeno_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zeno_behaviour" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zeno_behaviour"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Zeno behaviour</span> </div> </a> <ul id="toc-Zeno_behaviour-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Similar_paradoxes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Similar_paradoxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Similar paradoxes</span> </div> </a> <button aria-controls="toc-Similar_paradoxes-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 Similar paradoxes subsection</span> </button> <ul id="toc-Similar_paradoxes-sublist" class="vector-toc-list"> <li id="toc-School_of_Names" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#School_of_Names"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>School of Names</span> </div> </a> <ul id="toc-School_of_Names-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lewis_Carroll's_"What_the_Tortoise_Said_to_Achilles"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lewis_Carroll's_"What_the_Tortoise_Said_to_Achilles""> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Lewis Carroll's "What the Tortoise Said to Achilles"</span> </div> </a> <ul id="toc-Lewis_Carroll's_"What_the_Tortoise_Said_to_Achilles"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Zeno's paradoxes</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 49 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-49" 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">49 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/Zenon_se_paradokse" title="Zenon se paradokse – Afrikaans" lang="af" hreflang="af" data-title="Zenon se paradokse" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%81%D8%A7%D8%B1%D9%82%D8%A7%D8%AA_%D8%B2%D9%8A%D9%86%D9%88%D9%86" title="مفارقات زينون – Arabic" lang="ar" hreflang="ar" data-title="مفارقات زينون" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Zenonun_aporialar%C4%B1" title="Zenonun aporiaları – Azerbaijani" lang="az" hreflang="az" data-title="Zenonun aporiaları" 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%9C%E0%A7%87%E0%A6%A8%E0%A7%8B%E0%A6%B0_%E0%A6%AA%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B0%E0%A6%BE%E0%A6%A1%E0%A6%95%E0%A7%8D%E0%A6%B8" title="জেনোর প্যারাডক্স – Bangla" lang="bn" hreflang="bn" data-title="জেনোর প্যারাডক্স" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81%D0%B8_%D0%BD%D0%B0_%D0%97%D0%B5%D0%BD%D0%BE%D0%BD" title="Парадокси на Зенон – Bulgarian" lang="bg" hreflang="bg" data-title="Парадокси на Зенон" 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/Zenonovi_paradoksi" title="Zenonovi paradoksi – Bosnian" lang="bs" hreflang="bs" data-title="Zenonovi paradoksi" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Paradokso%C3%B9_Zenon" title="Paradoksoù Zenon – Breton" lang="br" hreflang="br" data-title="Paradoksoù Zenon" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Paradoxes_de_Zen%C3%B3" title="Paradoxes de Zenó – Catalan" lang="ca" hreflang="ca" data-title="Paradoxes de Zenó" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Zen%C3%B3novy_paradoxy" title="Zenónovy paradoxy – Czech" lang="cs" hreflang="cs" data-title="Zenónovy paradoxy" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Zenons_paradokser" title="Zenons paradokser – Danish" lang="da" hreflang="da" data-title="Zenons paradokser" 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/Zenons_Paradoxien_der_Vielheit" title="Zenons Paradoxien der Vielheit – German" lang="de" hreflang="de" data-title="Zenons Paradoxien der Vielheit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%AC%CE%B4%CE%BF%CE%BE%CE%B1_%CF%84%CE%BF%CF%85_%CE%96%CE%AE%CE%BD%CF%89%CE%BD%CE%B1" title="Παράδοξα του Ζήνωνα – Greek" lang="el" hreflang="el" data-title="Παράδοξα του Ζήνωνα" 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/Paradojas_de_Zen%C3%B3n" title="Paradojas de Zenón – Spanish" lang="es" hreflang="es" data-title="Paradojas de Zenón" 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-Q70894304 mw-list-item" title=""><a href="https://eo.wikipedia.org/wiki/Paradokso_de_Zenono" title="Paradokso de Zenono – Esperanto" lang="eo" hreflang="eo" data-title="Paradokso de Zenono" 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/Zenonen_paradoxak" title="Zenonen paradoxak – Basque" lang="eu" hreflang="eu" data-title="Zenonen paradoxak" 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/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%D9%88%DA%A9%D8%B3%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%B2%D9%86%D9%88%D9%86" title="پارادوکس‌های زنون – Persian" lang="fa" hreflang="fa" data-title="پارادوکس‌های زنون" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Paradoxes_de_Z%C3%A9non" title="Paradoxes de Zénon – French" lang="fr" hreflang="fr" data-title="Paradoxes de Zénon" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Paradacsa_Zeno" title="Paradacsa Zeno – Irish" lang="ga" hreflang="ga" data-title="Paradacsa Zeno" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Paradoxos_de_Zen%C3%B3n" title="Paradoxos de Zenón – Galician" lang="gl" hreflang="gl" data-title="Paradoxos de Zenón" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EB%85%BC%EC%9D%98_%EC%97%AD%EC%84%A4" title="제논의 역설 – Korean" lang="ko" hreflang="ko" data-title="제논의 역설" 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/%D4%B6%D5%A5%D5%B6%D5%B8%D5%B6%D5%AB_%D5%A1%D5%BA%D5%B8%D6%80%D5%AB%D5%A1%D5%B6%D5%A5%D6%80" title="Զենոնի ապորիաներ – Armenian" lang="hy" hreflang="hy" data-title="Զենոնի ապորիաներ" 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%9C%E0%A4%BC%E0%A5%80%E0%A4%A8%E0%A5%8B_%E0%A4%AA%E0%A4%B0%E0%A5%8B%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%95" title="ज़ीनो परोक्षक – Hindi" lang="hi" hreflang="hi" data-title="ज़ीनो परोक्षक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Paradoks_Zeno" title="Paradoks Zeno – Indonesian" lang="id" hreflang="id" data-title="Paradoks Zeno" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Everst%C3%A6%C3%B0ur_Zenons" title="Þverstæður Zenons – Icelandic" lang="is" hreflang="is" data-title="Þverstæður Zenons" 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/Paradossi_di_Zenone" title="Paradossi di Zenone – Italian" lang="it" hreflang="it" data-title="Paradossi di Zenone" 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%A4%D7%A8%D7%93%D7%95%D7%A7%D7%A1%D7%99%D7%9D_%D7%A9%D7%9C_%D7%96%D7%A0%D7%95%D7%9F" title="הפרדוקסים של זנון – Hebrew" lang="he" hreflang="he" data-title="הפרדוקסים של זנון" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%97%D0%B5%D0%BD%D0%BE%D0%BD_%D0%BF%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81%D1%82%D0%B5%D1%80%D1%96" title="Зенон парадокстері – Kazakh" lang="kk" hreflang="kk" data-title="Зенон парадокстері" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Zenono_paradoksai" title="Zenono paradoksai – Lithuanian" lang="lt" hreflang="lt" data-title="Zenono paradoksai" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Z%C3%A9n%C3%B3n_paradoxonjai" title="Zénón paradoxonjai – Hungarian" lang="hu" hreflang="hu" data-title="Zénón paradoxonjai" 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%9F%D0%B0%D1%80%D0%B0%D0%B4%D0%BE%D0%BA%D1%81%D0%B8%D1%82%D0%B5_%D0%BD%D0%B0_%D0%97%D0%B5%D0%BD%D0%BE%D0%BD" title="Парадоксите на Зенон – Macedonian" lang="mk" hreflang="mk" data-title="Парадоксите на Зенон" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Zeno%27s_paradoxen" title="Zeno's paradoxen – Dutch" lang="nl" hreflang="nl" data-title="Zeno's paradoxen" 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/%E3%82%BC%E3%83%8E%E3%83%B3%E3%81%AE%E3%83%91%E3%83%A9%E3%83%89%E3%83%83%E3%82%AF%E3%82%B9" title="ゼノンのパラドックス – Japanese" lang="ja" hreflang="ja" data-title="ゼノンのパラドックス" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Zenon_aporiyalari" title="Zenon aporiyalari – Uzbek" lang="uz" hreflang="uz" data-title="Zenon aporiyalari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9C%E0%A8%BC%E0%A9%87%E0%A8%A8%E0%A9%8B%E0%A8%82_%E0%A8%A6%E0%A9%87_%E0%A8%B5%E0%A8%BF%E0%A8%B0%E0%A9%8B%E0%A8%A7%E0%A8%BE%E0%A8%AD%E0%A8%BE%E0%A8%B8" title="ਜ਼ੇਨੋਂ ਦੇ ਵਿਰੋਧਾਭਾਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਜ਼ੇਨੋਂ ਦੇ ਵਿਰੋਧਾਭਾਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Paradoksy_Zenona_z_Elei" title="Paradoksy Zenona z Elei – Polish" lang="pl" hreflang="pl" data-title="Paradoksy Zenona z Elei" 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/Paradoxos_de_Zen%C3%A3o" title="Paradoxos de Zenão – Portuguese" lang="pt" hreflang="pt" data-title="Paradoxos de Zenão" 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/Paradoxurile_lui_Zenon" title="Paradoxurile lui Zenon – Romanian" lang="ro" hreflang="ro" data-title="Paradoxurile lui Zenon" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BF%D0%BE%D1%80%D0%B8%D0%B8_%D0%97%D0%B5%D0%BD%D0%BE%D0%BD%D0%B0" title="Апории Зенона – Russian" lang="ru" hreflang="ru" data-title="Апории Зенона" 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/Paradokset_e_Zenonit" title="Paradokset e Zenonit – Albanian" lang="sq" hreflang="sq" data-title="Paradokset e Zenonit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes – Simple English" lang="en-simple" hreflang="en-simple" data-title="Zeno's paradoxes" 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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B1%D8%A7%D8%AF%DB%86%DA%A9%D8%B3%DB%95%DA%A9%D8%A7%D9%86%DB%8C_%D8%B2%DB%8C%D9%86%DB%86%D9%86" title="پارادۆکسەکانی زینۆن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پارادۆکسەکانی زینۆن" 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/Zenonovi_paradoksi" title="Zenonovi paradoksi – Serbian" lang="sr" hreflang="sr" data-title="Zenonovi paradoksi" 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/Zenonovi_paradoksi" title="Zenonovi paradoksi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Zenonovi paradoksi" 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/Zenonin_paradoksit" title="Zenonin paradoksit – Finnish" lang="fi" hreflang="fi" data-title="Zenonin paradoksit" 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/Zenons_paradoxer" title="Zenons paradoxer – Swedish" lang="sv" hreflang="sv" data-title="Zenons paradoxer" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Zenon_paradokslar%C4%B1" title="Zenon paradoksları – Turkish" lang="tr" hreflang="tr" data-title="Zenon paradoksları" 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%90%D0%BF%D0%BE%D1%80%D1%96%D1%97_%D0%97%D0%B5%D0%BD%D0%BE%D0%BD%D0%B0" title="Апорії Зенона – Ukrainian" lang="uk" hreflang="uk" data-title="Апорії Зенона" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ngh%E1%BB%8Bch_l%C3%BD_Zeno" title="Nghịch lý Zeno – Vietnamese" lang="vi" hreflang="vi" data-title="Nghịch lý Zeno" 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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%8A%9D%E8%AF%BA%E6%82%96%E8%AE%BA" title="芝诺悖论 – Chinese" lang="zh" hreflang="zh" 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Zeno%27s_Paradox&redirect=no" class="mw-redirect" title="Zeno's Paradox">Zeno's Paradox</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">"Arrow paradox" redirects here. For other uses, see <a href="/wiki/Arrow_paradox_(disambiguation)" class="mw-disambig" title="Arrow paradox (disambiguation)">Arrow paradox (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Set of philosophical problems</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Primary_sources plainlinks metadata ambox ambox-content ambox-Primary_sources" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies excessively on <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">references</a> to <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">primary sources</a></b>.<span class="hide-when-compact"> Please improve this article by adding <a href="/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources" title="Wikipedia:No original research">secondary or tertiary sources</a>. <br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Zeno%27s+paradoxes%22">"Zeno's paradoxes"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Zeno%27s+paradoxes%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Zeno%27s+paradoxes%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Zeno%27s+paradoxes%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Zeno%27s+paradoxes%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Zeno%27s+paradoxes%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Zeno's paradoxes</b> are a series of <a href="/wiki/Philosophy" title="Philosophy">philosophical</a> <a href="/wiki/Argument" title="Argument">arguments</a> presented by the <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greek</a> philosopher <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a> (c. 490–430 BC),<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> primarily known through the works of <a href="/wiki/Plato" title="Plato">Plato</a>, <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, and later commentators like <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius of Cilicia</a>.<sup id="cite_ref-:1_2-1" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Zeno devised these paradoxes to support his teacher <a href="/wiki/Parmenides" title="Parmenides">Parmenides</a>'s philosophy of <a href="/wiki/Monism" title="Monism">monism</a>, which posits that despite our sensory experiences, reality is singular and unchanging. The paradoxes famously challenge the notions of plurality (the <a href="/wiki/Existence" title="Existence">existence</a> of many things), motion, space, and time by suggesting they lead to <a href="/wiki/Contradiction" title="Contradiction">logical contradictions</a>. </p><p>Zeno's work, primarily known from <a href="/wiki/Secondary_source" title="Secondary source">second-hand accounts</a> since his <a href="/wiki/Primary_source" title="Primary source">original texts</a> are lost, comprises forty "paradoxes of plurality," which argue against the <a href="/wiki/Consistency" title="Consistency">coherence</a> of believing in multiple existences, and several arguments against motion and change.<sup id="cite_ref-:1_2-2" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility in <a href="/wiki/Space" title="Space">space</a> and <a href="/wiki/Time" title="Time">time</a>.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-3" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In this paradox, Zeno argues that a swift runner like <a href="/wiki/Achilles" title="Achilles">Achilles</a> cannot overtake a slower moving <a href="/wiki/Tortoise" title="Tortoise">tortoise</a> with a head start, because the <a href="/wiki/Distance" title="Distance">distance</a> between them can be infinitely subdivided, implying Achilles would require an <a href="/wiki/Infinity" title="Infinity">infinite</a> number of steps to catch the tortoise.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-4" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>These paradoxes have stirred extensive philosophical and mathematical discussion throughout <a href="/wiki/History" title="History">history</a>,<sup id="cite_ref-:0_1-3" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-5" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> particularly regarding the nature of infinity and the continuity of space and time. Initially, <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s interpretation, suggesting a potential rather than actual infinity, was widely accepted.<sup id="cite_ref-:0_1-4" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, modern solutions leveraging the mathematical framework of <a href="/wiki/Calculus" title="Calculus">calculus</a> have provided a different perspective, highlighting Zeno's significant early insight into the complexities of infinity and continuous motion.<sup id="cite_ref-:0_1-5" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Zeno's paradoxes remain a pivotal reference point in the philosophical and mathematical exploration of reality, motion, and the infinite, influencing both ancient thought and modern scientific understanding.<sup id="cite_ref-:0_1-6" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-6" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The origins of the paradoxes are somewhat unclear, but they are generally thought to have been developed to support <a href="/wiki/Parmenides" title="Parmenides">Parmenides</a>' doctrine of <a href="/wiki/Monism" title="Monism">monism</a>, that all of reality is one, and that <i>all change is impossible</i>, that is, that nothing ever <a href="/wiki/Motion" title="Motion">changes in location</a> or in any other respect.<sup id="cite_ref-:0_1-7" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-7" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Diogenes_La%C3%ABrtius" class="mw-redirect" title="Diogenes Laërtius">Diogenes Laërtius</a>, citing <a href="/wiki/Favorinus" title="Favorinus">Favorinus</a>, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/University" title="University">Modern academics</a> attribute the paradox to Zeno.<sup id="cite_ref-:0_1-8" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-8" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Many of these paradoxes argue that contrary to the evidence of one's senses, <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a> is nothing but an <a href="/wiki/Illusion" title="Illusion">illusion</a>.<sup id="cite_ref-:0_1-9" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-9" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Plato" title="Plato">Plato's</a> <a href="/wiki/Parmenides_(dialogue)" title="Parmenides (dialogue)"><i>Parmenides</i></a> (128a–d), Zeno is characterized as taking on the project of creating these <a href="/wiki/Paradoxes" class="mw-redirect" title="Paradoxes">paradoxes</a> because other philosophers claimed paradoxes arise when considering Parmenides' view. Zeno's arguments may then be early examples of a method of proof called <i><a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a></i>, also known as <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one."<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> Plato has <a href="/wiki/Socrates" title="Socrates">Socrates</a> claim that Zeno and Parmenides were essentially arguing exactly the same point.<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> They are also credited as a source of the <a href="/wiki/Dialectic" title="Dialectic">dialectic</a> method used by Socrates.<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> <div class="mw-heading mw-heading2"><h2 id="Paradoxes">Paradoxes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=2" title="Edit section: Paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of Zeno's nine surviving paradoxes (preserved in <a href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)">Aristotle's <i>Physics</i></a><sup id="cite_ref-aristotle_7-0" class="reference"><a href="#cite_note-aristotle-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><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> and <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius's</a> commentary thereon) are essentially equivalent to one another. Aristotle offered a response to some of them.<sup id="cite_ref-aristotle_7-1" class="reference"><a href="#cite_note-aristotle-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite.<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> However, none of the original ancient sources has Zeno discussing the sum of any infinite series. <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a> has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the <i>sum</i>, but rather with <i>finishing</i> a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?<sup id="cite_ref-KBrown_10-0" class="reference"><a href="#cite_note-KBrown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FMoorcroft_11-0" class="reference"><a href="#cite_note-FMoorcroft-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Papa-G_12-0" class="reference"><a href="#cite_note-Papa-G-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><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> <div class="mw-heading mw-heading3"><h3 id="Paradoxes_of_motion">Paradoxes of motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=3" title="Edit section: Paradoxes of motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three of the strongest and most famous—that of Achilles and the tortoise, the <a href="/wiki/Dichotomy" title="Dichotomy">Dichotomy</a> argument, and that of an arrow in flight—are presented in detail below. </p> <div class="mw-heading mw-heading4"><h4 id="Dichotomy_paradox">Dichotomy paradox</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=4" title="Edit section: Dichotomy paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zeno_Dichotomy_Paradox_alt.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Zeno_Dichotomy_Paradox_alt.png/220px-Zeno_Dichotomy_Paradox_alt.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Zeno_Dichotomy_Paradox_alt.png/330px-Zeno_Dichotomy_Paradox_alt.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Zeno_Dichotomy_Paradox_alt.png/440px-Zeno_Dichotomy_Paradox_alt.png 2x" data-file-width="1050" data-file-height="500" /></a><figcaption>The dichotomy</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p> That which is in locomotion must arrive at the half-way stage before it arrives at the goal.</p><div class="templatequotecite">— <cite>as recounted by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, <a href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)"><i>Physics</i></a> VI:9, 239b10</cite></div></blockquote> <p>Suppose <a href="/wiki/Atalanta" title="Atalanta">Atalanta</a> wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on. </p> <div class="timeline-wrapper"><map name="timeline_ssd4pqi61f7h6pkzopmhesc1ow8h6iq"></map><img usemap="#timeline_ssd4pqi61f7h6pkzopmhesc1ow8h6iq" src="//upload.wikimedia.org/wikipedia/en/timeline/ssd4pqi61f7h6pkzopmhesc1ow8h6iq.png" /></div> <p>The resulting sequence can be represented as: </p> <dl><dd><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\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo>⋯</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ea66ea6ee1580aeb96de307d55a7cb078b79f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.085ex; height:6.176ex;" alt="{\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}"></span></dd></dl> <p>This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.<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> </p><p>This sequence also presents a second problem in that it contains no first distance to run, for any possible (<a href="https://en.wiktionary.org/wiki/finite" class="extiw" title="wikt:finite">finite</a>) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an <a href="/wiki/Illusion" title="Illusion">illusion</a>.<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> </p><p>This argument is called the "<a href="/wiki/Dichotomy" title="Dichotomy">Dichotomy</a>" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in an <a href="/wiki/Asymptote" title="Asymptote">asymptote</a>. It is also known as the <b>Race Course</b> paradox. </p> <div class="mw-heading mw-heading4"><h4 id="Achilles_and_the_tortoise">Achilles and the tortoise</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=5" title="Edit section: Achilles and the tortoise"><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">"Achilles and the Tortoise" redirects here. For other uses, see <a href="/wiki/Achilles_and_the_Tortoise_(disambiguation)" class="mw-disambig" title="Achilles and the Tortoise (disambiguation)">Achilles and the Tortoise (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable selfref">See also: <a href="/wiki/Infinity#Zeno:_Achilles_and_the_tortoise" title="Infinity">Infinity § Zeno: Achilles and the tortoise</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zeno_Achilles_Paradox.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Zeno_Achilles_Paradox.png/220px-Zeno_Achilles_Paradox.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Zeno_Achilles_Paradox.png/330px-Zeno_Achilles_Paradox.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Zeno_Achilles_Paradox.png/440px-Zeno_Achilles_Paradox.png 2x" data-file-width="1050" data-file-height="1050" /></a><figcaption>Achilles and the tortoise</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p> In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.</p><div class="templatequotecite">— <cite>as recounted by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, <a href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)"><i>Physics</i></a> VI:9, 239b15</cite></div></blockquote> <p>In the paradox of <b>Achilles and the tortoise</b>, <a href="/wiki/Achilles" title="Achilles">Achilles</a> is in a footrace with a tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy.<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> It lacks, however, the apparent conclusion of motionlessness. </p> <div class="mw-heading mw-heading4"><h4 id="Arrow_paradox">Arrow paradox</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=6" title="Edit section: Arrow paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zeno_Arrow_Paradox.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Zeno_Arrow_Paradox.png/220px-Zeno_Arrow_Paradox.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Zeno_Arrow_Paradox.png/330px-Zeno_Arrow_Paradox.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Zeno_Arrow_Paradox.png/440px-Zeno_Arrow_Paradox.png 2x" data-file-width="1050" data-file-height="500" /></a><figcaption>The arrow</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Arrow_paradox_(disambiguation)" class="mw-disambig" title="Arrow paradox (disambiguation)"><i>other paradoxes of the same name</i></a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.<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="templatequotecite">— <cite>as recounted by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, <a href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)"><i>Physics</i></a> VI:9, 239b5</cite></div></blockquote> <p>In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that at any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.<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> It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible. </p><p>Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.<sup id="cite_ref-HuggettArrow_19-0" class="reference"><a href="#cite_note-HuggettArrow-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_paradoxes">Other paradoxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=7" title="Edit section: Other paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aristotle gives three other paradoxes. </p> <div class="mw-heading mw-heading4"><h4 id="Paradox_of_place">Paradox of place</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=8" title="Edit section: Paradox of place"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From Aristotle: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>If everything that exists has a place, place too will have a place, and so on <i><a href="/wiki/Ad_infinitum" title="Ad infinitum">ad infinitum</a></i>.<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></blockquote> <div class="mw-heading mw-heading4"><h4 id="Paradox_of_the_grain_of_millet">Paradox of the grain of millet</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=9" title="Edit section: Paradox of the grain of millet"><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">See also: <a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites paradox</a></div> <p>Description of the paradox from the <i>Routledge Dictionary of Philosophy</i>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The argument is that a single grain of <a href="/wiki/Millet" title="Millet">millet</a> makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion.<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></p></blockquote> <p>Aristotle's response: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.<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></blockquote> <p>Description from Nick Huggett: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>This is a <a href="/wiki/Parmenides" title="Parmenides">Parmenidean</a> argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.<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></p></blockquote> <div class="mw-heading mw-heading4"><h4 id="The_moving_rows_(or_stadium)"><span id="The_moving_rows_.28or_stadium.29"></span>The moving rows (or stadium)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=10" title="Edit section: The moving rows (or stadium)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zeno_Moving_Rows_Paradox.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Zeno_Moving_Rows_Paradox.png/220px-Zeno_Moving_Rows_Paradox.png" decoding="async" width="220" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Zeno_Moving_Rows_Paradox.png/330px-Zeno_Moving_Rows_Paradox.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Zeno_Moving_Rows_Paradox.png/440px-Zeno_Moving_Rows_Paradox.png 2x" data-file-width="1050" data-file-height="800" /></a><figcaption>The moving rows</figcaption></figure> <p>From Aristotle: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>... concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.<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></p></blockquote> <p>An expanded account of Zeno's arguments, as presented by Aristotle, is given in <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius's</a> commentary <i>On Aristotle's Physics</i>.<sup id="cite_ref-:2_25-0" class="reference"><a href="#cite_note-:2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-10" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_1-10" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>According to Angie Hobbs of Sheffield university, this paradox is intended to be considered together with the paradox of Achilles and the Tortoise, problematizing the concept of discrete space & time where the other problematizes the concept of infinitely divisible space & time.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Proposed_solutions">Proposed solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=11" title="Edit section: Proposed solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_classical_antiquity">In classical antiquity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=12" title="Edit section: In classical antiquity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Simplicius_of_Cilicia" title="Simplicius of Cilicia">Simplicius</a>, <a href="/wiki/Diogenes_the_Cynic" class="mw-redirect" title="Diogenes the Cynic">Diogenes the Cynic</a> said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions.<sup id="cite_ref-:2_25-1" class="reference"><a href="#cite_note-:2-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_2-11" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Throughout history several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes. </p><p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384 BC–322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.<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><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="In the section cited, Aristotle says nothing about the distance decreasing (October 2019)">failed verification</span></a></i>]</sup><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> Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").<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> Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."<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> <a href="/wiki/Thomas_Aquinas" title="Thomas Aquinas">Thomas Aquinas</a>, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_modern_mathematics">In modern mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=13" title="Edit section: In modern mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some mathematicians and historians, such as <a href="/wiki/Carl_Boyer" class="mw-redirect" title="Carl Boyer">Carl Boyer</a>, hold that Zeno's paradoxes are simply mathematical problems, for which modern <a href="/wiki/Calculus" title="Calculus">calculus</a> provides a mathematical solution.<sup id="cite_ref-boyer_34-0" class="reference"><a href="#cite_note-boyer-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the <a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">epsilon-delta</a> definition of <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a>, <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a> and <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Cauchy</a> developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.<sup id="cite_ref-Lee_35-0" class="reference"><a href="#cite_note-Lee-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-russell_36-0" class="reference"><a href="#cite_note-russell-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>Some <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosophers</a>, however, say that Zeno's paradoxes and their variations (see <a href="/wiki/Thomson%27s_lamp" title="Thomson's lamp">Thomson's lamp</a>) remain relevant <a href="/wiki/Metaphysics" title="Metaphysics">metaphysical</a> problems.<sup id="cite_ref-KBrown_10-1" class="reference"><a href="#cite_note-KBrown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FMoorcroft_11-1" class="reference"><a href="#cite_note-FMoorcroft-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Papa-G_12-1" class="reference"><a href="#cite_note-Papa-G-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown<sup id="cite_ref-KBrown_10-2" class="reference"><a href="#cite_note-KBrown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and Francis Moorcroft<sup id="cite_ref-FMoorcroft_11-2" class="reference"><a href="#cite_note-FMoorcroft-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> hold that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise. Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of '<a href="/wiki/Rorschach_test" title="Rorschach test">Rorschach image</a>' onto which people can project their most fundamental phenomenological concerns (if they have any)."<sup id="cite_ref-KBrown_10-3" class="reference"><a href="#cite_note-KBrown-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Henri_Bergson">Henri Bergson</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=14" title="Edit section: Henri Bergson"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An alternative conclusion, proposed by <a href="/wiki/Henri_Bergson" title="Henri Bergson">Henri Bergson</a> in his 1896 book <i><a href="/wiki/Matter_and_Memory" title="Matter and Memory">Matter and Memory</a></i>, is that, while the path is divisible, the motion is not.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Peter_Lynds">Peter Lynds</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=15" title="Edit section: Peter Lynds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2003, Peter Lynds argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Time’s_Up_Einstein_41-0" class="reference"><a href="#cite_note-Time’s_Up_Einstein-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. Nick Huggett argues that Zeno is <a href="/wiki/Begging_the_question" title="Begging the question">assuming the conclusion</a> when he says that objects that occupy the same space as they do at rest must be at rest.<sup id="cite_ref-HuggettArrow_19-1" class="reference"><a href="#cite_note-HuggettArrow-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Bertrand_Russell">Bertrand Russell</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=16" title="Edit section: Bertrand Russell"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Based on the work of <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>,<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> offered a solution to the paradoxes, what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is just change in position over time.<sup id="cite_ref-HuggettBook_43-0" class="reference"><a href="#cite_note-HuggettBook-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Hermann_Weyl">Hermann Weyl</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=17" title="Edit section: Hermann Weyl"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According to <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "<a href="/wiki/Weyl%27s_tile_argument" title="Weyl's tile argument">tile argument</a>" or "distance function problem".<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-atomism_uni_of_washington_46-0" class="reference"><a href="#cite_note-atomism_uni_of_washington-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. <a href="/wiki/Jean_Paul_Van_Bendegem" title="Jean Paul Van Bendegem">Jean Paul Van Bendegem</a> has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.<sup id="cite_ref-boyer_34-1" class="reference"><a href="#cite_note-boyer-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=18" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Quantum_Zeno_effect">Quantum Zeno effect</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=19" title="Edit section: Quantum Zeno effect"><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/Quantum_Zeno_effect" title="Quantum Zeno effect">Quantum Zeno effect</a></div> <p>In 1977,<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> physicists <a href="/wiki/E._C._George_Sudarshan" title="E. C. George Sudarshan">E. C. George Sudarshan</a> and B. Misra discovered that the dynamical evolution (<a href="/wiki/Motion" title="Motion">motion</a>) of a <a href="/wiki/Quantum_system" class="mw-redirect" title="Quantum system">quantum system</a> can be hindered (or even inhibited) through <a href="/wiki/Observation" title="Observation">observation</a> of the <a href="/wiki/System" title="System">system</a>.<sup id="cite_ref-u0_49-0" class="reference"><a href="#cite_note-u0-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> This effect is usually called the "<a href="/wiki/Quantum_Zeno_effect" title="Quantum Zeno effect">Quantum Zeno effect</a>" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Zeno_behaviour">Zeno behaviour</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=20" title="Edit section: Zeno behaviour"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the field of verification and design of <a href="/wiki/Timed_event_system" title="Timed event system">timed</a> and <a href="/wiki/Hybrid_system" title="Hybrid system">hybrid systems</a>, the system behaviour is called <i>Zeno</i> if it includes an infinite number of discrete steps in a finite amount of time.<sup id="cite_ref-Fishwick2007_51-0" class="reference"><a href="#cite_note-Fishwick2007-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> Some <a href="/wiki/Formal_verification" title="Formal verification">formal verification</a> techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Systems_design" title="Systems design">systems design</a> these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Similar_paradoxes">Similar paradoxes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=21" title="Edit section: Similar paradoxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="School_of_Names">School of Names</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=22" title="Edit section: School of Names"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Paradox_of_the_stick.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Paradox_of_the_stick.png/220px-Paradox_of_the_stick.png" decoding="async" width="220" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Paradox_of_the_stick.png/330px-Paradox_of_the_stick.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Paradox_of_the_stick.png/440px-Paradox_of_the_stick.png 2x" data-file-width="900" data-file-height="394" /></a><figcaption>Diagram of Hui Shi's stick paradox</figcaption></figure><p> Roughly contemporaneously during the <a href="/wiki/Warring_States_period" title="Warring States period">Warring States period</a> (475–221 BCE), <a href="/wiki/History_of_Science_and_Technology_in_China" class="mw-redirect" title="History of Science and Technology in China">ancient Chinese</a> philosophers from the <a href="/wiki/School_of_Names" title="School of Names">School of Names</a>, a school of thought similarly concerned with logic and dialectics, developed paradoxes similar to those of Zeno. The works of the School of Names have largely been lost, with the exception of portions of the <a href="/wiki/Gongsun_Long" title="Gongsun Long"><i>Gongsun Longzi</i></a>. The second of the Ten Theses of <a href="/wiki/Hui_Shi" title="Hui Shi">Hui Shi</a> suggests knowledge of infinitesimals:<i>That which has no thickness cannot be piled up; yet it is a thousand li in dimension.</i> Among the many puzzles of his recorded in the <a href="/wiki/Zhuangzi_(book)" title="Zhuangzi (book)"><i>Zhuangzi</i></a> is one very similar to Zeno's Dichotomy: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><div class="poem"> <p>"If from a stick a foot long you every day take the half of it, in a myriad ages it will not be exhausted." </p> </div><div class="templatequotecite">— <cite><i>Zhuangzi</i>, chapter 33 (Legge translation)<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></cite></div></blockquote> <p><a href="/wiki/Mozi_(book)" title="Mozi (book)">The Mohist canon</a> appears to propose a solution to this paradox by arguing that in moving across a measured length, the distance is not covered in successive fractions of the length, but in one stage. Due to the lack of surviving works from the School of Names, most of the other paradoxes listed are difficult to interpret.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading3"><h3 id="Lewis_Carroll's_"What_the_Tortoise_Said_to_Achilles""><span id="Lewis_Carroll.27s_.22What_the_Tortoise_Said_to_Achilles.22"></span>Lewis Carroll's "What the Tortoise Said to Achilles"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=23" title="Edit section: Lewis Carroll's "What the Tortoise Said to Achilles""><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/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a></div> <p>"What the Tortoise Said to Achilles",<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> written in 1895 by <a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a>, describes a paradoxical infinite regress argument in the realm of pure logic. It uses Achilles and the Tortoise as characters in a clear reference to Zeno's paradox of Achilles.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </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=Zeno%27s_paradoxes&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Incommensurable_magnitudes" class="mw-redirect" title="Incommensurable magnitudes">Incommensurable magnitudes</a></li> <li><a href="/wiki/Infinite_regress" title="Infinite regress">Infinite regress</a></li> <li><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of space and time</a></li> <li><a href="/wiki/Renormalization" title="Renormalization">Renormalization</a></li> <li><a href="/wiki/Ross%E2%80%93Littlewood_paradox" title="Ross–Littlewood paradox">Ross–Littlewood paradox</a></li> <li><a href="/wiki/Supertask" title="Supertask">Supertask</a></li> <li><a href="/wiki/Zeno_machine" title="Zeno machine">Zeno machine</a></li> <li><a href="/wiki/List_of_paradoxes" title="List of paradoxes">List of paradoxes</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Zeno%27s_paradoxes&action=edit&section=25" title="Edit section: Notes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:0_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:0_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:0_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:0_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-:0_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-:0_1-10"><sup><i><b>k</b></i></sup></a></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://iep.utm.edu/zenos-paradoxes/">"Zeno's Paradoxes | Internet Encyclopedia of Philosophy"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Zeno%27s+Paradoxes+%7C+Internet+Encyclopedia+of+Philosophy&rft_id=https%3A%2F%2Fiep.utm.edu%2Fzenos-paradoxes%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:1_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:1_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-:1_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-:1_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-:1_2-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-:1_2-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-:1_2-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-:1_2-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-:1_2-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2024" class="citation cs2">Huggett, Nick (2024), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno/">"Zeno's Paradoxes"</a>, in Zalta, Edward N.; Nodelman, Uri (eds.), <i>The Stanford Encyclopedia of Philosophy</i> (Spring 2024 ed.), Metaphysics Research Lab, Stanford University<span class="reference-accessdate">, retrieved <span class="nowrap">2024-03-25</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.edition=Spring+2024&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2024&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2024%2Fentries%2Fparadox-zeno%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Diogenes Laërtius, <i>Lives</i>, 9.23 and 9.29.</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"><i>Parmenides</i> 128d</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"><i>Parmenides</i> 128a–b</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">([fragment 65], Diogenes Laërtius. <a rel="nofollow" class="external text" href="http://classicpersuasion.org/pw/diogenes/dlzeno-eleatic.htm">IX</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101212095647/http://classicpersuasion.org/pw/diogenes/dlzeno-eleatic.htm">Archived</a> 2010-12-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> 25ff and VIII 57).</span> </li> <li id="cite_note-aristotle-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-aristotle_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-aristotle_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.html">Aristotle's <i>Physics</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110106095547/http://classics.mit.edu/Aristotle/physics.html">Archived</a> 2011-01-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080516213308/http://remacle.org/bloodwolf/philosophes/Aristote/physique6gr.htm#144">"Greek text of "Physics" by Aristotle (refer to §4 at the top of the visible screen area)"</a>. Archived from <a rel="nofollow" class="external text" href="http://remacle.org/bloodwolf/philosophes/Aristote/physique6gr.htm#144">the original</a> on 2008-05-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Greek+text+of+%22Physics%22+by+Aristotle+%28refer+to+%C2%A74+at+the+top+of+the+visible+screen+area%29&rft_id=http%3A%2F%2Fremacle.org%2Fbloodwolf%2Fphilosophes%2FAristote%2Fphysique6gr.htm%23144&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" 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="CITEREFBenson1999" class="citation book cs1">Benson, Donald C. (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/momentofproofmat00bens"><i>The Moment of Proof : Mathematical Epiphanies</i></a></span>. New York: Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/momentofproofmat00bens/page/14">14</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0195117219" title="Special:BookSources/978-0195117219"><bdi>978-0195117219</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Moment+of+Proof+%3A+Mathematical+Epiphanies&rft.place=New+York&rft.pages=14&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0195117219&rft.aulast=Benson&rft.aufirst=Donald+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmomentofproofmat00bens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-KBrown-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-KBrown_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-KBrown_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-KBrown_10-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-KBrown_10-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown" class="citation web cs1">Brown, Kevin. <a rel="nofollow" class="external text" href="https://archive.today/20121205030717/http://www.mathpages.com/rr/s3-07/3-07.htm">"Zeno and the Paradox of Motion"</a>. <i>Reflections on Relativity</i>. Archived from <a rel="nofollow" class="external text" href="http://www.mathpages.com/rr/s3-07/3-07.htm">the original</a> on 2012-12-05<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-06-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Reflections+on+Relativity&rft.atitle=Zeno+and+the+Paradox+of+Motion&rft.aulast=Brown&rft.aufirst=Kevin&rft_id=http%3A%2F%2Fwww.mathpages.com%2Frr%2Fs3-07%2F3-07.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-FMoorcroft-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-FMoorcroft_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FMoorcroft_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FMoorcroft_11-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoorcroft" class="citation web cs1">Moorcroft, Francis. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100418141459/http://www.philosophers.co.uk/cafe/paradox5.htm">"Zeno's Paradox"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.philosophers.co.uk/cafe/paradox5.htm">the original</a> on 2010-04-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Zeno%27s+Paradox&rft.aulast=Moorcroft&rft.aufirst=Francis&rft_id=http%3A%2F%2Fwww.philosophers.co.uk%2Fcafe%2Fparadox5.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-Papa-G-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Papa-G_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Papa-G_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapa-Grimaldi1996" class="citation journal cs1">Papa-Grimaldi, Alba (1996). <a rel="nofollow" class="external text" href="http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf">"Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition"</a> <span class="cs1-format">(PDF)</span>. <i>The Review of Metaphysics</i>. <b>50</b>: 299–314. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120609113959/http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2012-06-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-03-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Review+of+Metaphysics&rft.atitle=Why+Mathematical+Solutions+of+Zeno%27s+Paradoxes+Miss+the+Point%3A+Zeno%27s+One+and+Many+Relation+and+Parmenides%27+Prohibition&rft.volume=50&rft.pages=299-314&rft.date=1996&rft.aulast=Papa-Grimaldi&rft.aufirst=Alba&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2F2304%2F1%2Fzeno_maths_review_metaphysics_alba_papa_grimaldi.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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="CITEREFHuggett2010" class="citation encyclopaedia cs1">Huggett, Nick (2010). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/paradox-zeno/#ZenInf">"Zeno's Paradoxes: 5. Zeno's Influence on Philosophy"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#ZenInf">Archived</a> from the original on 2022-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes%3A+5.+Zeno%27s+Influence+on+Philosophy&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2010&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F%23ZenInf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLindberg2007" class="citation book cs1">Lindberg, David (2007). <i>The Beginnings of Western Science</i> (2nd ed.). University of Chicago Press. p. 33. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-48205-7" title="Special:BookSources/978-0-226-48205-7"><bdi>978-0-226-48205-7</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+Beginnings+of+Western+Science&rft.pages=33&rft.edition=2nd&rft.pub=University+of+Chicago+Press&rft.date=2007&rft.isbn=978-0-226-48205-7&rft.aulast=Lindberg&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2010" class="citation encyclopaedia cs1">Huggett, Nick (2010). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/paradox-zeno/#Dic">"Zeno's Paradoxes: 3.1 The Dichotomy"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#Dic">Archived</a> from the original on 2022-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes%3A+3.1+The+Dichotomy&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2010&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F%23Dic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2010" class="citation encyclopaedia cs1">Huggett, Nick (2010). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/paradox-zeno/#AchTor">"Zeno's Paradoxes: 3.2 Achilles and the Tortoise"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#AchTor">Archived</a> from the original on 2022-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes%3A+3.2+Achilles+and+the+Tortoise&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2010&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F%23AchTor&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAristotle" class="citation web cs1">Aristotle. <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.6.vi.html#752">"Physics"</a>. <i>The Internet Classics Archive</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html#752">Archived</a> from the original on 2008-05-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-08-21</span></span>. <q>Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Internet+Classics+Archive&rft.atitle=Physics&rft.au=Aristotle&rft_id=http%3A%2F%2Fclassics.mit.edu%2FAristotle%2Fphysics.6.vi.html%23752&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaërtiusc._230" class="citation book cs1"><a href="/wiki/Diogenes_La%C3%ABrtius" class="mw-redirect" title="Diogenes Laërtius">Laërtius, Diogenes</a> (c. 230). <a class="external text" href="https://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho">"Pyrrho"</a>. <a href="/wiki/Lives_and_Opinions_of_Eminent_Philosophers" class="mw-redirect" title="Lives and Opinions of Eminent Philosophers"><i>Lives and Opinions of Eminent Philosophers</i></a>. Vol. IX. passage 72. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-116-71900-2" title="Special:BookSources/1-116-71900-2"><bdi>1-116-71900-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110822084058/http://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho">Archived</a> from the original on 2011-08-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Pyrrho&rft.btitle=Lives+and+Opinions+of+Eminent+Philosophers&rft.pages=passage+72&rft.isbn=1-116-71900-2&rft.aulast=La%C3%ABrtius&rft.aufirst=Diogenes&rft_id=http%3A%2F%2Fen.wikisource.org%2Fwiki%2FLives_of_the_Eminent_Philosophers%2FBook_IX%23Pyrrho&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-HuggettArrow-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-HuggettArrow_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HuggettArrow_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2010" class="citation encyclopaedia cs1">Huggett, Nick (2010). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/paradox-zeno/#Arr">"Zeno's Paradoxes: 3.3 The Arrow"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#Arr">Archived</a> from the original on 2022-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes%3A+3.3+The+Arrow&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2010&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F%23Arr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" 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">Aristotle <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.4.iv.html"><i>Physics</i> IV:1, 209a25</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080509083946/http://classics.mit.edu//Aristotle/physics.4.iv.html">Archived</a> 2008-05-09 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></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">The Michael Proudfoot, A.R. Lace. Routledge Dictionary of Philosophy. Routledge 2009, p. 445</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">Aristotle <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.7.vii.html"><i>Physics</i> VII:5, 250a20</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080511153804/http://classics.mit.edu//Aristotle/physics.7.vii.html">Archived</a> 2008-05-11 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></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">Huggett, Nick, "Zeno's Paradoxes", The Stanford Encyclopedia of Philosophy (Winter 2010 Edition), Edward N. Zalta (ed.), <a rel="nofollow" class="external free" href="http://plato.stanford.edu/entries/paradox-zeno/#GraMil">http://plato.stanford.edu/entries/paradox-zeno/#GraMil</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#GraMil">Archived</a> 2022-03-01 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</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">Aristotle <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.6.vi.html"><i>Physics</i> VI:9, 239b33</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html">Archived</a> 2008-05-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-:2-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimplikiosKonstanSimplikios1989" class="citation book cs1">Simplikios; Konstan, David; Simplikios (1989). <i>Simplicius on Aristotle's Physics 6</i>. Ancient commentators on Aristotle. Ithaca N.Y: Cornell Univ. Pr. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8014-2238-6" title="Special:BookSources/978-0-8014-2238-6"><bdi>978-0-8014-2238-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Simplicius+on+Aristotle%27s+Physics+6&rft.place=Ithaca+N.Y&rft.series=Ancient+commentators+on+Aristotle&rft.pub=Cornell+Univ.+Pr&rft.date=1989&rft.isbn=978-0-8014-2238-6&rft.au=Simplikios&rft.au=Konstan%2C+David&rft.au=Simplikios&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://digitalmedia.sheffield.ac.uk/media/Zeno%27s+ParadoxesA+The+Moving+Rows/1_e2yi73na">"Zeno's Paradoxes: The Moving Rows"</a>. <i>The University of Sheffield Kaltura Digital Media Hub</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-28</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+University+of+Sheffield+Kaltura+Digital+Media+Hub&rft.atitle=Zeno%27s+Paradoxes%3A+The+Moving+Rows&rft_id=https%3A%2F%2Fdigitalmedia.sheffield.ac.uk%2Fmedia%2FZeno%2527s%2BParadoxesA%2BThe%2BMoving%2BRows%2F1_e2yi73na&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" 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">Aristotle. Physics 6.9</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"> Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>, while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a>, the sum of which has no limit. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research"><span title="The material near this tag possibly contains original research. (October 2020)">original research?</span></a></i>]</sup> Archimedes developed a more explicitly mathematical approach than Aristotle.</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">Aristotle. Physics 6.9; 6.2, 233a21-31</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="CITEREFAristotle" class="citation book cs1">Aristotle. <a rel="nofollow" class="external text" href="http://classics.mit.edu/Aristotle/physics.6.vi.html"><i>Physics</i></a>. Vol. VI. Part 9 verse: 239b5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-585-09205-2" title="Special:BookSources/0-585-09205-2"><bdi>0-585-09205-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html">Archived</a> from the original on 2008-05-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics&rft.pages=Part+9+verse%3A+239b5&rft.isbn=0-585-09205-2&rft.au=Aristotle&rft_id=http%3A%2F%2Fclassics.mit.edu%2FAristotle%2Fphysics.6.vi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Aquinas. Commentary on Aristotle's Physics, Book 6.861</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKiritsis2020" class="citation book cs1">Kiritsis, Paul (2020-04-01). <i>A Critical Investigation into Precognitive Dreams</i> (1 ed.). Cambridge Scholars Publishing. p. 19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1527546332" title="Special:BookSources/978-1527546332"><bdi>978-1527546332</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Critical+Investigation+into+Precognitive+Dreams&rft.pages=19&rft.edition=1&rft.pub=Cambridge+Scholars+Publishing&rft.date=2020-04-01&rft.isbn=978-1527546332&rft.aulast=Kiritsis&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: date and year (<a href="/wiki/Category:CS1_maint:_date_and_year" title="Category:CS1 maint: date and year">link</a>)</span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAquinas" class="citation web cs1"><a href="/wiki/Thomas_Aquinas" title="Thomas Aquinas">Aquinas, Thomas</a>. <a rel="nofollow" class="external text" href="https://aquinas.cc/la/en/~Phys.Bk6.L11">"Commentary on Aristotle's Physics"</a>. <i>aquinas.cc</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=aquinas.cc&rft.atitle=Commentary+on+Aristotle%27s+Physics&rft.aulast=Aquinas&rft.aufirst=Thomas&rft_id=https%3A%2F%2Faquinas.cc%2Fla%2Fen%2F~Phys.Bk6.L11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-boyer-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-boyer_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-boyer_34-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1959" class="citation book cs1">Boyer, Carl (1959). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofcalculu0000boye"><i>The History of the Calculus and Its Conceptual Development</i></a></span>. Dover Publications. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofcalculu0000boye/page/295">295</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60509-8" title="Special:BookSources/978-0-486-60509-8"><bdi>978-0-486-60509-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2010-02-26</span></span>. <q>If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+History+of+the+Calculus+and+Its+Conceptual+Development&rft.pages=295&rft.pub=Dover+Publications&rft.date=1959&rft.isbn=978-0-486-60509-8&rft.aulast=Boyer&rft.aufirst=Carl&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofcalculu0000boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-Lee-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lee_35-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee1965" class="citation journal cs1">Lee, Harold (1965). 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Oxford University Press: 563–570. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmind%2FLXXIV.296.563">10.1093/mind/LXXIV.296.563</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2251675">2251675</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mind&rft.atitle=Are+Zeno%27s+Paradoxes+Based+on+a+Mistake%3F&rft.volume=74&rft.issue=296&rft.pages=563-570&rft.date=1965&rft_id=info%3Adoi%2F10.1093%2Fmind%2FLXXIV.296.563&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2251675%23id-name%3DJSTOR&rft.aulast=Lee&rft.aufirst=Harold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-russell-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-russell_36-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">B Russell</a> (1956) <i>Mathematics and the metaphysicians</i> in "The World of Mathematics" (ed. <a href="/wiki/James_R._Newman" title="James R. Newman">J R Newman</a>), pp 1576-1590.</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBergson1896" class="citation book cs1"><a href="/wiki/Henri_Bergson" title="Henri Bergson">Bergson, Henri</a> (1896). <a rel="nofollow" class="external text" href="https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf"><i>Matière et Mémoire</i></a> [<i>Matter and Memory</i>] <span class="cs1-format">(PDF)</span>. Translation 1911 by Nancy Margaret Paul & W. Scott Palmer. George Allen and Unwin. pp. 77–78 of the PDF. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191015184719/https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2019-10-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-10-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mati%C3%A8re+et+M%C3%A9moire&rft.pages=77-78+of+the+PDF&rft.pub=Translation+1911+by+Nancy+Margaret+Paul+%26+W.+Scott+Palmer.+George+Allen+and+Unwin&rft.date=1896&rft.aulast=Bergson&rft.aufirst=Henri&rft_id=https%3A%2F%2Fantilogicalism.com%2Fwp-content%2Fuploads%2F2017%2F07%2Fmatter-and-memory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMassumi2002" class="citation book cs1">Massumi, Brian (2002). <i>Parables for the Virtual: Movement, Affect, Sensation</i> (1st ed.). Durham, NC: Duke University Press Books. pp. 5–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0822328971" title="Special:BookSources/978-0822328971"><bdi>978-0822328971</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Parables+for+the+Virtual%3A+Movement%2C+Affect%2C+Sensation&rft.place=Durham%2C+NC&rft.pages=5-6&rft.edition=1st&rft.pub=Duke+University+Press+Books&rft.date=2002&rft.isbn=978-0822328971&rft.aulast=Massumi&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</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="http://philsci-archive.pitt.edu/1197/">"Zeno's Paradoxes: A Timely Solution"</a>. January 2003. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120813040121/http://philsci-archive.pitt.edu/1197/">Archived</a> from the original on 2012-08-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-07-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Zeno%27s+Paradoxes%3A+A+Timely+Solution&rft.date=2003-01&rft_id=http%3A%2F%2Fphilsci-archive.pitt.edu%2F1197%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"> Lynds, Peter. Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. 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Retrieved <span class="nowrap">2020-01-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=plato.stanford.edu&rft.atitle=School+of+Names+%3E+Miscellaneous+Paradoxes+%28Stanford+Encyclopedia+of+Philosophy%29&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fschool-names%2Fparadoxes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll1895" class="citation journal cs1">Carroll, Lewis (1895-04-01). <a rel="nofollow" class="external text" href="https://academic.oup.com/mind/article/IV/14/278/1046872">"What the Tortoise Said to Achilles"</a>. <i>Mind</i>. <b>IV</b> (14): 278–280. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fmind%2FIV.14.278">10.1093/mind/IV.14.278</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-4423">0026-4423</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200720045852/https://academic.oup.com/mind/article/IV/14/278/1046872">Archived</a> from the original on 2020-07-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mind&rft.atitle=What+the+Tortoise+Said+to+Achilles&rft.volume=IV&rft.issue=14&rft.pages=278-280&rft.date=1895-04-01&rft_id=info%3Adoi%2F10.1093%2Fmind%2FIV.14.278&rft.issn=0026-4423&rft.aulast=Carroll&rft.aufirst=Lewis&rft_id=https%3A%2F%2Facademic.oup.com%2Fmind%2Farticle%2FIV%2F14%2F278%2F1046872&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTsilipakos2021" class="citation book cs1">Tsilipakos, Leonidas (2021). <i>Clarity and confusion in social theory: taking concepts seriously</i>. Philosophy and method in the social sciences. Abingdon New York (N.Y.): Routledge. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-032-09883-8" title="Special:BookSources/978-1-032-09883-8"><bdi>978-1-032-09883-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Clarity+and+confusion+in+social+theory%3A+taking+concepts+seriously&rft.place=Abingdon+New+York+%28N.Y.%29&rft.series=Philosophy+and+method+in+the+social+sciences&rft.pages=48&rft.pub=Routledge&rft.date=2021&rft.isbn=978-1-032-09883-8&rft.aulast=Tsilipakos&rft.aufirst=Leonidas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></span> </li> </ol></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=Zeno%27s_paradoxes&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Geoffrey_Kirk" title="Geoffrey Kirk">Kirk, G. S.</a>, <a href="/wiki/John_Raven" title="John Raven">J. E. Raven</a>, M. Schofield (1984) <i>The Presocratic Philosophers: A Critical History with a Selection of Texts, 2nd ed.</i> <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-27455-9" title="Special:BookSources/0-521-27455-9">0-521-27455-9</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuggett2010" class="citation encyclopaedia cs1">Huggett, Nick (2010). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/paradox-zeno/">"Zeno's Paradoxes"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/">Archived</a> from the original on 2022-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradoxes&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2010&rft.aulast=Huggett&rft.aufirst=Nick&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fparadox-zeno%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></li> <li><a href="/wiki/Plato" title="Plato">Plato</a> (1926) <i>Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias</i>, H. N. Fowler (Translator), <a href="/wiki/Loeb_Classical_Library" title="Loeb Classical Library">Loeb Classical Library</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-674-99185-0" title="Special:BookSources/0-674-99185-0">0-674-99185-0</a>.</li> <li>Sainsbury, R.M. (2003) <i>Paradoxes</i>, 2nd ed. Cambridge University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48347-6" title="Special:BookSources/0-521-48347-6">0-521-48347-6</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSkyrms1983" class="citation book cs1"><a href="/wiki/Brian_Skyrms" title="Brian Skyrms">Skyrms, Brian</a> (1983). "Zeno's Paradox of Measure". In Cohen, R. S.; <a href="/wiki/Larry_Laudan" title="Larry Laudan">Laudan, L.</a> (eds.). <i>Physics, Philosophy, and Psychoanalysis</i>. Dordrecht: Reidel. pp. 223–254. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-277-1533-5" title="Special:BookSources/90-277-1533-5"><bdi>90-277-1533-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno%27s+Paradox+of+Measure&rft.btitle=Physics%2C+Philosophy%2C+and+Psychoanalysis&rft.place=Dordrecht&rft.pages=223-254&rft.pub=Reidel&rft.date=1983&rft.isbn=90-277-1533-5&rft.aulast=Skyrms&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></li></ul> </div> <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=Zeno%27s_paradoxes&action=edit&section=27" 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 .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></span></span></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has original text related to this article: <div style="margin-left: 10px;"><b><a href="https://en.wikisource.org/wiki/Catholic_Encyclopedia_(1913)/Zeno_of_Elea" class="extiw" title="wikisource:Catholic Encyclopedia (1913)/Zeno of Elea">Zeno of Elea</a></b></div></div></div> </div> <ul><li>Dowden, Bradley. "<a rel="nofollow" class="external text" href="http://www.iep.utm.edu/zeno-par/">Zeno’s Paradoxes</a>." Entry in the <a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Antinomy">"Antinomy"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Antinomy&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAntinomy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.coursera.org/course/mathphil">Introduction to Mathematical Philosophy</a>, Ludwig-Maximilians-Universität München</li> <li>Silagadze, Z. K. "<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0505042">Zeno meets modern science</a>,"</li> <li><i><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ZenosParadoxAchillesAndTheTortoise/">Zeno's Paradox: Achilles and the Tortoise</a></i> by Jon McLoone, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li> <li><a rel="nofollow" class="external text" href="https://archive.today/20121205030717/http://www.mathpages.com/rr/s3-07/3-07.htm">Kevin Brown on Zeno and the Paradox of Motion</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPalmer2008" class="citation encyclopaedia cs1">Palmer, John (2008). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/zeno-elea/">"Zeno of Elea"</a>. <i>Stanford Encyclopedia of Philosophy</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Zeno+of+Elea&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2008&rft.aulast=Palmer&rft.aufirst=John&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fzeno-elea%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></li> <li><i>This article incorporates material from Zeno's paradox on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrime" class="citation web cs1">Grime, James. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181003050912/http://www.numberphile.com/videos/zeno_paradox.html">"Zeno's Paradox"</a>. <i>Numberphile</i>. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a>. Archived from <a rel="nofollow" class="external text" href="http://www.numberphile.com/videos/zeno_paradox.html">the original</a> on 2018-10-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2013-04-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=Zeno%27s+Paradox&rft.aulast=Grime&rft.aufirst=James&rft_id=http%3A%2F%2Fwww.numberphile.com%2Fvideos%2Fzeno_paradox.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AZeno%27s+paradoxes" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Notable_paradoxes" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Paradoxes" title="Template:Paradoxes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Paradoxes" title="Template talk:Paradoxes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Paradoxes" title="Special:EditPage/Template:Paradoxes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Notable_paradoxes" style="font-size:114%;margin:0 4em">Notable <a href="/wiki/Paradox" title="Paradox">paradoxes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophical</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/Paradox_of_analysis" title="Paradox of analysis">Analysis</a></li> <li><a href="/wiki/Buridan%27s_bridge" title="Buridan's bridge">Buridan's bridge</a></li> <li><a href="/wiki/Dream_argument" title="Dream argument">Dream argument</a></li> <li><a href="/wiki/Epicurean_paradox" title="Epicurean paradox">Epicurean</a></li> <li><a href="/wiki/Paradox_of_fiction" title="Paradox of fiction">Fiction</a></li> <li><a href="/wiki/Fitch%27s_paradox_of_knowability" title="Fitch's paradox of knowability">Fitch's knowability</a></li> <li><a href="/wiki/Argument_from_free_will" title="Argument from free will">Free will</a></li> <li><a href="/wiki/New_riddle_of_induction" title="New riddle of induction">Goodman's</a></li> <li><a href="/wiki/Paradox_of_hedonism" title="Paradox of hedonism">Hedonism</a></li> <li><a href="/wiki/Liberal_paradox" title="Liberal paradox">Liberal</a></li> <li><a href="/wiki/Meno" title="Meno">Meno's</a></li> <li><a href="/wiki/Mere_addition_paradox" title="Mere addition paradox">Mere addition</a></li> <li><a href="/wiki/Moore%27s_paradox" title="Moore's paradox">Moore's</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Paradox_of_nihilism" title="Paradox of nihilism">Nihilism</a></li> <li><a href="/wiki/Omnipotence_paradox" title="Omnipotence paradox">Omnipotence</a></li> <li><a href="/wiki/Preface_paradox" title="Preface paradox">Preface</a></li> <li><a href="/wiki/Wittgenstein_on_Rules_and_Private_Language" title="Wittgenstein on Rules and Private Language">Rule-following</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites</a></li> <li><a href="/wiki/Ship_of_Theseus" title="Ship of Theseus">Theseus' ship</a></li> <li><a href="/wiki/White_Horse_Dialogue" title="White Horse Dialogue">White horse</a></li> <li><a class="mw-selflink selflink">Zeno's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Logical</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/Barber_paradox" title="Barber paradox">Barber</a></li> <li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry</a></li> <li><a href="/wiki/Bhartrhari%27s_paradox" class="mw-redirect" title="Bhartrhari's paradox">Bhartrhari's</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti</a></li> <li><a href="/wiki/Paradox_of_the_Court" title="Paradox of the Court">Court</a></li> <li><a href="/wiki/Crocodile_dilemma" title="Crocodile dilemma">Crocodile</a></li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's</a></li> <li><a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice paradox</a></li> <li><a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson</a></li> <li><a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser</a></li> <li><a href="/wiki/Liar_paradox" title="Liar paradox">Liar</a> <ul><li><a href="/wiki/Card_paradox" title="Card paradox">Card</a></li> <li><a href="/wiki/No%E2%80%93no_paradox" title="No–no paradox">No-no</a></li> <li><a href="/wiki/Pinocchio_paradox" title="Pinocchio paradox">Pinocchio</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's</a></li> <li><a href="/wiki/Yablo%27s_paradox" class="mw-redirect" title="Yablo's paradox">Yablo's</a></li></ul></li> <li><a href="/wiki/Opposite_Day" title="Opposite Day">Opposite Day</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's</a></li> <li><a href="/wiki/I_know_that_I_know_nothing" title="I know that I know nothing">Socratic</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's Hotel</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Barbershop_paradox" title="Barbershop paradox">Barbershop</a></li> <li><a href="/wiki/Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a></li> <li><a href="/wiki/Chicken_or_the_egg" title="Chicken or the egg">Chicken or the egg</a></li> <li><a href="/wiki/Drinker_paradox" title="Drinker paradox">Drinker</a></li> <li><a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">Entailment</a></li> <li><a href="/wiki/Lottery_paradox" title="Lottery paradox">Lottery</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Raven_paradox" title="Raven paradox">Raven</a></li> <li><a href="/wiki/Imperative_logic#Ross's_paradox" title="Imperative logic">Ross's</a></li> <li><a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging</a></li> <li>"<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>"</li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Olbers%27s_paradox" title="Olbers's paradox">Olbers's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economic</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/Allais_paradox" title="Allais paradox">Allais</a></li> <li><a href="/wiki/The_Antitrust_Paradox" title="The Antitrust Paradox">Antitrust</a></li> <li><a href="/wiki/Arrow_information_paradox" title="Arrow information paradox">Arrow information</a></li> <li><a href="/wiki/Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand</a></li> <li><a href="/wiki/Braess%27s_paradox" title="Braess's paradox">Braess's</a></li> <li><a href="/wiki/Paradox_of_competition" title="Paradox of competition">Competition</a></li> <li><a href="/wiki/Income_and_fertility" title="Income and fertility">Income and fertility</a></li> <li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson</a></li> <li><a href="/wiki/Easterlin_paradox" title="Easterlin paradox">Easterlin</a></li> <li><a href="/wiki/Edgeworth_paradox" title="Edgeworth paradox">Edgeworth</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/European_paradox" title="European paradox">European</a></li> <li><a href="/wiki/Gibson%27s_paradox" title="Gibson's paradox">Gibson's</a></li> <li><a href="/wiki/Giffen_good" title="Giffen good">Giffen good</a></li> <li><a href="/wiki/Icarus_paradox" title="Icarus paradox">Icarus</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons</a></li> <li><a href="/wiki/Leontief_paradox" title="Leontief paradox">Leontief</a></li> <li><a href="/wiki/Lerner_paradox" title="Lerner paradox">Lerner</a></li> <li><a href="/wiki/Lucas_paradox" title="Lucas paradox">Lucas</a></li> <li><a href="/wiki/Mandeville%27s_paradox" title="Mandeville's paradox">Mandeville's</a></li> <li><a href="/wiki/Mayfield%27s_paradox" title="Mayfield's paradox">Mayfield's</a></li> <li><a href="/wiki/Metzler_paradox" title="Metzler paradox">Metzler</a></li> <li><a href="/wiki/Resource_curse" title="Resource curse">Plenty</a></li> <li><a href="/wiki/Productivity_paradox" title="Productivity paradox">Productivity</a></li> <li><a href="/wiki/Paradox_of_prosperity" title="Paradox of prosperity">Prosperity</a></li> <li><a href="/wiki/Scitovsky_paradox" title="Scitovsky paradox">Scitovsky</a></li> <li><a href="/wiki/Service_recovery_paradox" title="Service recovery paradox">Service recovery</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg</a></li> <li><a href="/wiki/Paradox_of_thrift" title="Paradox of thrift">Thrift</a></li> <li><a href="/wiki/Paradox_of_toil" title="Paradox of toil">Toil</a></li> <li><a href="/wiki/Tullock_paradox" class="mw-redirect" title="Tullock paradox">Tullock</a></li> <li><a href="/wiki/Paradox_of_value" title="Paradox of value">Value</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision theory</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/Abilene_paradox" title="Abilene paradox">Abilene</a></li> <li><a href="/wiki/Apportionment_paradox" title="Apportionment paradox">Apportionment</a> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">Alabama</a></li> <li><a href="/wiki/Coherence_(fairness)" title="Coherence (fairness)">New states</a></li> <li><a href="/wiki/State-population_monotonicity" class="mw-redirect" title="State-population monotonicity">Population</a></li></ul></li> <li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's</a></li> <li><a href="/wiki/Buridan%27s_ass" title="Buridan's ass">Buridan's ass</a></li> <li><a href="/wiki/Chainstore_paradox" title="Chainstore paradox">Chainstore</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Condorcet's</a></li> <li><a href="/wiki/Decision-making_paradox" title="Decision-making paradox">Decision-making</a></li> <li><a href="/wiki/Paradox_of_voting" title="Paradox of voting">Downs</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/Fenno%27s_paradox" title="Fenno's paradox">Fenno's</a></li> <li><a href="/wiki/Fredkin%27s_paradox" title="Fredkin's paradox">Fredkin's</a></li> <li><a href="/wiki/The_Green_Paradox" title="The Green Paradox">Green</a></li> <li><a href="/wiki/Hedgehog%27s_dilemma" title="Hedgehog's dilemma">Hedgehog's</a></li> <li><a href="/wiki/Inventor%27s_paradox" title="Inventor's paradox">Inventor's</a></li> <li><a href="/wiki/Kavka%27s_toxin_puzzle" title="Kavka's toxin puzzle">Kavka's toxin puzzle</a></li> <li><a href="/wiki/Morton%27s_fork" title="Morton's fork">Morton's fork</a></li> <li><a href="/wiki/Navigation_paradox" title="Navigation paradox">Navigation</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Parrondo%27s_paradox" title="Parrondo's paradox">Parrondo's</a></li> <li><a href="/wiki/Preparedness_paradox" title="Preparedness paradox">Preparedness</a></li> <li><a href="/wiki/Prevention_paradox" title="Prevention paradox">Prevention</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Paradox_of_tolerance" title="Paradox of tolerance">Tolerance</a></li> <li><a href="/wiki/Willpower_paradox" title="Willpower paradox">Willpower</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" 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