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class="vector-toc-numb">3.2</span> <span>Stimulus detection and discrimination</span> </div> </a> <ul id="toc-Stimulus_detection_and_discrimination-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stimulus_identification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stimulus_identification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Stimulus identification</span> </div> </a> <ul id="toc-Stimulus_identification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_decision" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_decision"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Simple decision</span> </div> </a> <ul id="toc-Simple_decision-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Memory_scanning,_visual_search" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Memory_scanning,_visual_search"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Memory scanning, visual search</span> </div> </a> <ul id="toc-Memory_scanning,_visual_search-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Error_response_times" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Error_response_times"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Error response times</span> </div> </a> <ul id="toc-Error_response_times-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sequential_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sequential_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Sequential effects</span> </div> </a> <ul id="toc-Sequential_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Learning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Learning"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Learning</span> </div> </a> <ul id="toc-Learning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Measurement theory</span> </div> </a> <ul id="toc-Measurement_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Developmental_psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Developmental_psychology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Developmental psychology</span> </div> </a> <ul id="toc-Developmental_psychology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Journals_and_organizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Journals_and_organizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Journals and organizations</span> </div> </a> <ul id="toc-Journals_and_organizations-sublist" class="vector-toc-list"> </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-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">6</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">7</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" title="Table of Contents" > <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">Mathematical psychology</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 17 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-17" 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">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%84%D9%85_%D8%A7%D9%84%D9%86%D9%81%D8%B3_%D8%A7%D9%84%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A" title="علم النفس الحسابي – Arabic" lang="ar" hreflang="ar" data-title="علم النفس الحسابي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mathematische_Psychologie" title="Mathematische Psychologie – German" lang="de" hreflang="de" data-title="Mathematische Psychologie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatiline_ps%C3%BChholoogia" title="Matemaatiline psühholoogia – Estonian" lang="et" hreflang="et" data-title="Matemaatiline psühholoogia" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CF%88%CF%85%CF%87%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Μαθηματική ψυχολογία – Greek" lang="el" hreflang="el" data-title="Μαθηματική ψυχολογία" 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/Psicolog%C3%ADa_matem%C3%A1tica" title="Psicología matemática – Spanish" lang="es" hreflang="es" data-title="Psicología matemática" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D9%88%D8%A7%D9%86%E2%80%8C%D8%B4%D9%86%D8%A7%D8%B3%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="روان‌شناسی ریاضی – Persian" lang="fa" hreflang="fa" data-title="روان‌شناسی ریاضی" 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/Psychologie_math%C3%A9matique" title="Psychologie mathématique – French" lang="fr" hreflang="fr" data-title="Psychologie mathématique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EB%A6%AC%EC%8B%AC%EB%A6%AC%ED%95%99" title="수리심리학 – Korean" lang="ko" hreflang="ko" data-title="수리심리학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Psikologi_matematika" title="Psikologi matematika – Indonesian" lang="id" hreflang="id" data-title="Psikologi matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E5%BF%83%E7%90%86%E5%AD%A6" title="数理心理学 – Japanese" lang="ja" hreflang="ja" data-title="数理心理学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematik_psixologiya" title="Matematik psixologiya – Uzbek" lang="uz" hreflang="uz" data-title="Matematik psixologiya" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BF%D1%81%D0%B8%D1%85%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Математическая психология – Russian" lang="ru" hreflang="ru" data-title="Математическая психология" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%89%E0%AE%B3%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="கணித உளவியல் – Tamil" lang="ta" hreflang="ta" data-title="கணித உளவியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_psikoloji" title="Matematiksel psikoloji – Turkish" lang="tr" hreflang="tr" data-title="Matematiksel psikoloji" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%BF%D1%81%D0%B8%D1%85%D0%BE%D0%BB%D0%BE%D0%B3%D1%96%D1%8F" title="Математична психологія – Ukrainian" lang="uk" hreflang="uk" data-title="Математична психологія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E7%90%86%E5%BF%83%E7%90%86%E5%AD%B8" title="數理心理學 – Cantonese" lang="yue" hreflang="yue" data-title="數理心理學" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E5%BF%83%E7%90%86%E5%AD%A6" title="数学心理学 – Chinese" lang="zh" hreflang="zh" data-title="数学心理学" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical modeling of psychological theories and phenomena</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the mathematical modeling of psychological theories and phenomena. For research design and methodology in psychology, see <a href="/wiki/Psychological_statistics" title="Psychological statistics">Psychological statistics</a>. For the theory and technique of measurement of psychological attributes, see <a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a>.</div> <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 ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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var(--color-base)"><a href="/wiki/Basic_science_(psychology)" title="Basic science (psychology)">Basic psychology</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="excerpt-block"><div class="excerpt"> <ul><li><a href="/wiki/Abnormal_psychology" title="Abnormal psychology">Abnormal</a></li> <li><a href="/wiki/Affective_neuroscience" title="Affective neuroscience">Affective neuroscience</a></li> <li><a href="/wiki/Affective_science" title="Affective science">Affective science</a></li> <li><a href="/wiki/Behavioural_genetics" title="Behavioural genetics">Behavioral genetics</a></li> <li><a href="/wiki/Behavioral_neuroscience" title="Behavioral neuroscience">Behavioral neuroscience</a></li> <li><a href="/wiki/Behaviorism" title="Behaviorism">Behaviorism</a></li> <li><a href="/wiki/Cognitive_psychology" title="Cognitive psychology">Cognitive</a>/<a href="/wiki/Cognitivism_(psychology)" title="Cognitivism (psychology)">Cognitivism</a></li> <li><a href="/wiki/Cognitive_neuroscience" title="Cognitive neuroscience">Cognitive neuroscience</a> <ul><li><a href="/wiki/Social_cognitive_neuroscience" title="Social cognitive neuroscience">Social</a></li></ul></li> <li><a href="/wiki/Comparative_psychology" title="Comparative psychology">Comparative</a></li> <li><a href="/wiki/Cross-cultural_psychology" title="Cross-cultural psychology">Cross-cultural</a></li> <li><a href="/wiki/Cultural_psychology" title="Cultural psychology">Cultural</a></li> <li><a href="/wiki/Developmental_psychology" title="Developmental psychology">Developmental</a></li> <li><a href="/wiki/Differential_psychology" title="Differential psychology">Differential</a></li> <li><a href="/wiki/Ecological_psychology" title="Ecological psychology">Ecological</a></li> <li><a href="/wiki/Evolutionary_psychology" title="Evolutionary psychology">Evolutionary</a></li> <li><a href="/wiki/Experimental_psychology" title="Experimental psychology">Experimental</a></li> <li><a href="/wiki/Gestalt_psychology" title="Gestalt psychology">Gestalt</a></li> <li><a href="/wiki/Intelligence" title="Intelligence">Intelligence</a></li> <li>Mathematical</li> <li><a href="/wiki/Moral_psychology" title="Moral psychology">Moral</a></li> <li><a href="/wiki/Neuropsychology" title="Neuropsychology">Neuropsychology</a></li> <li><a href="/wiki/Perception" title="Perception">Perception</a></li> <li><a href="/wiki/Personality_psychology" title="Personality psychology">Personality</a></li> <li><a href="/wiki/Psycholinguistics" title="Psycholinguistics">Psycholinguistics</a></li> <li><a href="/wiki/Psychophysiology" title="Psychophysiology">Psychophysiology</a></li> <li><a href="/wiki/Quantitative_psychology" title="Quantitative psychology">Quantitative</a></li> <li><a href="/wiki/Social_psychology" title="Social psychology">Social</a></li> <li><a href="/wiki/Theoretical_psychology" title="Theoretical psychology">Theoretical</a></li></ul></div></div></div></div></td> </tr><tr><td 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psychology">Media</a></li> <li><a href="/wiki/Medical_psychology" title="Medical psychology">Medical</a></li> <li><a href="/wiki/Military_psychology" title="Military psychology">Military</a></li> <li><a href="/wiki/Music_psychology" class="mw-redirect" title="Music psychology">Music</a></li> <li><a href="/wiki/Occupational_health_psychology" title="Occupational health psychology">Occupational health</a></li> <li><a href="/wiki/Pastoral_psychology" class="mw-redirect" title="Pastoral psychology">Pastoral</a></li> <li><a href="/wiki/Political_psychology" title="Political psychology">Political</a></li> <li><a href="/wiki/Positive_psychology" title="Positive psychology">Positive</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li> <li><a href="/wiki/Psychotherapy" title="Psychotherapy">Psychotherapy</a></li> <li><a href="/wiki/Psychology_of_religion" title="Psychology of religion">Religion</a></li> <li><a href="/wiki/School_psychology" title="School 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href="/wiki/Special:EditPage/Template:Psychology_sidebar" title="Special:EditPage/Template:Psychology sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Mathematical psychology</b> is an approach to <a href="/wiki/Psychology" title="Psychology">psychological</a> research that is based on <a href="/wiki/Mathematical_modeling" class="mw-redirect" title="Mathematical modeling">mathematical modeling</a> of perceptual, <a href="/wiki/Thought" title="Thought">thought</a>, <a href="/wiki/Cognition" title="Cognition">cognitive</a> and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior (in practice often constituted by task performance). The mathematical approach is used with the goal of deriving <a href="/wiki/Hypothesis" title="Hypothesis">hypotheses</a> that are more exact and thus yield stricter empirical validations. There are five major research areas in mathematical psychology: <a href="/wiki/Learning" title="Learning">learning</a> and <a href="/wiki/Memory" title="Memory">memory</a>, <a href="/wiki/Perception" title="Perception">perception</a> and <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a>, <a href="/wiki/Choice" title="Choice">choice</a> and <a href="/wiki/Decision-making" title="Decision-making">decision-making</a>, <a href="/wiki/Language" title="Language">language</a> and <a href="/wiki/Thought" title="Thought">thinking</a>, and <a href="/wiki/Measurement" title="Measurement">measurement</a> and <a href="/wiki/Feature_scaling" title="Feature scaling">scaling</a>.<sup id="cite_ref-Batchelder_808–815_1-0" class="reference"><a href="#cite_note-Batchelder_808–815-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Although psychology, as an independent subject of science, is a more recent discipline than <a href="/wiki/Physics" title="Physics">physics</a>,<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> the application of mathematics to psychology has been done in the hope of emulating the success of this approach in the <a href="/wiki/Physical_Sciences" class="mw-redirect" title="Physical Sciences">physical sciences</a>, which dates back to at least the <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">seventeenth century</a>.<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> Mathematics in psychology is used extensively roughly in two areas: one is the mathematical modeling of psychological theories and experimental phenomena, which leads to mathematical psychology; the other is the statistical approach of quantitative measurement practices in psychology, which leads to <a href="/wiki/Psychometrics" title="Psychometrics">psychometrics</a>.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>As quantification of behavior is fundamental in this endeavor, the <a href="/wiki/Theory_of_conjoint_measurement" title="Theory of conjoint measurement">theory of measurement</a> is a central topic in mathematical psychology. Mathematical psychology is therefore closely related to psychometrics. However, where psychometrics is concerned with individual differences (or population structure) in mostly static variables, mathematical psychology focuses on process models of perceptual, cognitive and motor processes as inferred from the 'average individual'. Furthermore, where psychometrics investigates the stochastic dependence structure between variables as observed in the population, mathematical psychology almost exclusively focuses on the modeling of data obtained from experimental paradigms and is therefore even more closely related to <a href="/wiki/Experimental_psychology" title="Experimental psychology">experimental psychology</a>, <a href="/wiki/Cognitive_psychology" title="Cognitive psychology">cognitive psychology</a>, and <a href="/wiki/Psychonomics" title="Psychonomics">psychonomics</a>. Like <a href="/wiki/Computational_neuroscience" title="Computational neuroscience">computational neuroscience</a> and <a href="/wiki/Econometrics" title="Econometrics">econometrics</a>, mathematical psychology theory often uses <a href="/wiki/Stochastic_programming" title="Stochastic programming">statistical optimality</a> as a guiding principle, assuming that the human brain has evolved to solve problems in an optimized way. Central themes from cognitive psychology (e.g., limited vs. unlimited processing capacity, serial vs. parallel processing) and their implications are central in rigorous analysis in mathematical psychology. </p><p>Mathematical psychologists are active in many fields of psychology, especially in psychophysics, <a href="/wiki/Sensation_and_perception_psychology" class="mw-redirect" title="Sensation and perception psychology">sensation</a> and <a href="/wiki/Perception" title="Perception">perception</a>, <a href="/wiki/Problem_solving" title="Problem solving">problem solving</a>, <a href="/wiki/Decision_theory" title="Decision theory">decision-making</a>, <a href="/wiki/Learning" title="Learning">learning</a>, <a href="/wiki/Memory" title="Memory">memory</a>, <a href="/wiki/Psycholinguistics" title="Psycholinguistics">language</a>, and the quantitative <a href="/wiki/Behavioural_sciences" title="Behavioural sciences">analysis of behavior</a>, and contribute to the work of other subareas of psychology such as <a href="/wiki/Clinical_psychology" title="Clinical psychology">clinical psychology</a>, <a href="/wiki/Social_psychology" title="Social psychology">social psychology</a>, <a href="/wiki/Educational_psychology" title="Educational psychology">educational psychology</a>, and <a href="/wiki/Music_psychology" class="mw-redirect" title="Music psychology">psychology of music</a>. </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=Mathematical_psychology&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Ernst_Heinrich_Weber.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Ernst_Heinrich_Weber.jpg/150px-Ernst_Heinrich_Weber.jpg" decoding="async" width="150" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Ernst_Heinrich_Weber.jpg/225px-Ernst_Heinrich_Weber.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Ernst_Heinrich_Weber.jpg/300px-Ernst_Heinrich_Weber.jpg 2x" data-file-width="515" data-file-height="674" /></a><figcaption><a href="/wiki/Ernst_Heinrich_Weber" title="Ernst Heinrich Weber">Ernst Heinrich Weber</a></figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Gustav_Fechner.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Gustav_Fechner.jpg/150px-Gustav_Fechner.jpg" decoding="async" width="150" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Gustav_Fechner.jpg/225px-Gustav_Fechner.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Gustav_Fechner.jpg/300px-Gustav_Fechner.jpg 2x" data-file-width="1389" data-file-height="1975" /></a><figcaption><a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Fechner</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Mathematics_and_psychology_before_the_19th_century">Mathematics and psychology before the 19th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=2" title="Edit section: Mathematics and psychology before the 19th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Choice and decision making theory are rooted in the development of statistics theory. In the mid 1600s, Blaise Pascal considered situations in gambling and further extended to Pascal's wager.<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> In the 18th century, Nicolas Bernoulli proposed the <a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg Paradox</a> in decision making, Daniel Bernoulli gave a solution and <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> proposed a modification to the solution later on. In 1763, <a href="/wiki/Thomas_Bayes" title="Thomas Bayes">Bayes</a> published the paper "<a href="/wiki/An_Essay_Towards_Solving_a_Problem_in_the_Doctrine_of_Chances" title="An Essay Towards Solving a Problem in the Doctrine of Chances">An Essay Towards Solving a Problem in the Doctrine of Chances</a>", which is the milestone of Bayesian statistics. </p><p><a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a> worked on modeling human memory, which is a precursor of the study of memory. </p> <div class="mw-heading mw-heading3"><h3 id="Mathematics_and_psychology_in_the_19th_century">Mathematics and psychology in the 19th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=3" title="Edit section: Mathematics and psychology in the 19th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The research developments in Germany and England in the 19th century made psychology a new academic subject. Since the German approach emphasized experiments in the investigation of the psychological processes that all humans share and the English approach was in the measurement of individual differences, the applications of mathematics were also different. </p><p>In Germany, <a href="/wiki/Wilhelm_Wundt" title="Wilhelm Wundt">Wilhelm Wundt</a> established the first experimental psychology laboratory. The math in German psychology is mainly applied in sensory and psychophysics. <a href="/wiki/Ernst_Heinrich_Weber" title="Ernst Heinrich Weber">Ernst Weber</a> (1795–1878) created the first mathematical law of the mind, <a href="/wiki/Weber%E2%80%93Fechner_law" title="Weber–Fechner law">Weber's law</a>, based on a variety of experiments. <a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Fechner</a> (1801–1887) contributed theories in sensations and perceptions and one of them is the <a href="/wiki/Weber%E2%80%93Fechner_law" title="Weber–Fechner law">Fechner's law</a>, which modifies Weber's law. </p><p>Mathematical modeling has a long history in psychology starting in the 19th century with <a href="/wiki/Ernst_Heinrich_Weber" title="Ernst Heinrich Weber">Ernst Weber</a> (1795–1878) and <a href="/wiki/Gustav_Fechner" title="Gustav Fechner">Gustav Fechner</a> (1801–1887) being among the first to apply <a href="/wiki/Functional_equations" class="mw-redirect" title="Functional equations">functional equations</a> to psychological processes. They thereby established the fields of <a href="/wiki/Experimental_psychology" title="Experimental psychology">experimental psychology</a> in general, and that of <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a> in particular. </p><p>Researchers in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a> in the 19th century were mapping distances between stars by denoting the exact time of a star's passing of a cross-hair on a telescope. For lack of the automatic registration instruments of the modern era, these time measurements relied entirely on human response speed. It had been noted that there were small systematic differences in the times measured by different astronomers, and these were first systematically studied by German astronomer <a href="/wiki/Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Friedrich Bessel</a> (1782–1846). Bessel constructed <i>personal equations</i> from measurements of basic response speed that would cancel out individual differences from the astronomical calculations. Independently, physicist <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> measured reaction times to determine nerve conduction speed, developed resonance theory of hearing and the <a href="/wiki/Young%E2%80%93Helmholtz_theory" title="Young–Helmholtz theory">Young-Helmholtz theory</a> of color vision. </p><p>These two lines of work came together in the research of Dutch physiologist <a href="/wiki/Franciscus_Donders" title="Franciscus Donders">F. C. Donders</a> and his student <a href="/w/index.php?title=Johan_Jacob_de_Jaager&action=edit&redlink=1" class="new" title="Johan Jacob de Jaager (page does not exist)">J. J. de Jaager</a>, who recognized the potential of reaction times for more or less objectively quantifying the amount of time elementary mental operations required. Donders envisioned the employment of his <i><a href="/wiki/Mental_chronometry" title="Mental chronometry">mental chronometry</a></i> to scientifically infer the elements of complex cognitive activity by measurement of <i>simple reaction time</i><sup id="cite_ref-Leahey1987_5-0" class="reference"><a href="#cite_note-Leahey1987-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Although there are developments in sensation and perception, <a href="/wiki/Johann_Friedrich_Herbart" title="Johann Friedrich Herbart">Johann Herbart</a> developed a system of mathematical theories in cognitive area to understand the mental process of consciousness. </p><p>The origin of English psychology can be traced to the theory of evolution by Darwin. But the emergence of English psychology is because of <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>, who interested in individual differences between humans on psychological variables. The math in English psychology is mainly statistics and the work and methods of Galton is the foundation of <a href="/wiki/Psychometrics" title="Psychometrics">psychometrics</a>. </p><p>Galton introduced bivariate normal distribution in modeling the traits of the same individual, he also investigated measurement error and built his own model, and he also developed a stochastic branching process to examine the extinction of family names. There is also a tradition of the interest in studying intelligence in English psychology started from Galton. <a href="/wiki/James_McKeen_Cattell" title="James McKeen Cattell">James McKeen Cattell</a> and <a href="/wiki/Alfred_Binet" title="Alfred Binet">Alfred Binet</a> developed tests of intelligence. </p><p>The first psychological laboratory was established in Germany by <a href="/wiki/Wilhelm_Wundt" title="Wilhelm Wundt">Wilhelm Wundt</a>, who amply used Donders' ideas. However, findings that came from the laboratory were hard to replicate and this was soon attributed to the method of <a href="/wiki/Introspection" title="Introspection">introspection</a> that Wundt introduced. Some of the problems resulted from individual differences in response speed found by astronomers. Although Wundt did not seem to take interest in these individual variations and kept his focus on the study of the <i>general human mind</i>, Wundt's U.S. student <a href="/wiki/James_McKeen_Cattell" title="James McKeen Cattell">James McKeen Cattell</a> was fascinated by these differences and started to work on them during his stay in England. </p><p>The failure of Wundt's method of introspection led to the rise of different schools of thought. Wundt's laboratory was directed towards conscious human experience, in line with the work of Fechner and Weber on the intensity of stimuli. In the United Kingdom, under the influence of the anthropometric developments led by <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>, interest focussed on individual differences between humans on psychological variables, in line with the work of Bessel. Cattell soon adopted the methods of Galton and helped laying the foundation of psychometrics. </p> <div class="mw-heading mw-heading3"><h3 id="20th_century">20th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=4" title="Edit section: 20th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many statistical methods were developed even before the 20th century: <a href="/wiki/Charles_Spearman" title="Charles Spearman">Charles Spearman</a> invented <a href="/wiki/Factor_analysis" title="Factor analysis">factor analysis</a> which studies individual differences by the variance and covariance. German psychology and English psychology have been combined and taken over by the United States. The statistical methods dominated the field during the beginning of the century. There are two important statistical developments: <a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural Equation Modeling</a> (SEM) and <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> (ANOVA). Since factor analysis unable to make causal inferences, the method of structural equation modeling was developed by <a href="/wiki/Sewall_Wright" title="Sewall Wright">Sewall Wright</a> to correlational data to infer causality, which is still a major research area today. Those statistical methods formed psychometrics. The Psychometric Society was established in 1935 and the journal <i>Psychometrika</i> was published since 1936. </p><p>In the United States, <a href="/wiki/Behaviorism" title="Behaviorism">behaviorism</a> arose in opposition to <a href="/wiki/Introspectionism" class="mw-redirect" title="Introspectionism">introspectionism</a> and associated reaction-time research, and turned the focus of psychological research entirely to learning theory.<sup id="cite_ref-Leahey1987_5-1" class="reference"><a href="#cite_note-Leahey1987-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In Europe introspection survived in <a href="/wiki/Gestalt_psychology" title="Gestalt psychology">Gestalt psychology</a>. Behaviorism dominated American psychology until the end of the <a href="/wiki/Second_World_War" class="mw-redirect" title="Second World War">Second World War</a>, and largely refrained from inference on mental processes. Formal theories were mostly absent (except for <a href="/wiki/Visual_perception" title="Visual perception">vision</a> and <a href="/wiki/Hearing_(sense)" class="mw-redirect" title="Hearing (sense)">hearing</a>). </p><p>During the war, developments in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> and <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, and the military need to understand <a href="/wiki/Human_factors" class="mw-redirect" title="Human factors">human performance and limitations</a>, brought together experimental psychologists, mathematicians, engineers, physicists, and economists. Out of this mix of different disciplines mathematical psychology arose. Especially the developments in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, <a href="/wiki/Linear_systems" class="mw-redirect" title="Linear systems">linear systems</a> and <a href="/wiki/Filter_theory" class="mw-redirect" title="Filter theory">filter theory</a>, <a href="/wiki/Game_theory" title="Game theory">game theory</a>, <a href="/wiki/Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic processes</a> and <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> gained a large influence on psychological thinking.<sup id="cite_ref-Leahey1987_5-2" class="reference"><a href="#cite_note-Leahey1987-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Batchelder2002_6-0" class="reference"><a href="#cite_note-Batchelder2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Two seminal papers on learning theory in <i><a href="/wiki/Psychological_Review" title="Psychological Review">Psychological Review</a></i> helped to establish the field in a world that was still dominated by behaviorists: A paper by Bush and Mosteller instigated the linear operator approach to learning,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and a paper by Estes that started the stimulus sampling tradition in psychological theorizing.<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> These two papers presented the first detailed formal accounts of data from learning experiments. </p><p>Mathematical modeling of learning process were greatly developed in the 1950s as the behavioral learning theory was flourishing. One development is the stimulus sampling theory by <a href="/wiki/William_Kaye_Estes" title="William Kaye Estes">Williams K. Estes</a>, the other is linear operator models by Robert R. Bush, and <a href="/wiki/Frederick_Mosteller" title="Frederick Mosteller">Frederick Mosteller</a>. </p><p>Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition. <a href="/wiki/John_von_Neumann" title="John von Neumann">Von Neumann</a>'s book <i><a href="/w/index.php?title=The_Theory_of_Games_and_Economic_Behavior&action=edit&redlink=1" class="new" title="The Theory of Games and Economic Behavior (page does not exist)">The Theory of Games and Economic Behavior</a></i> establish the importance of game theory and decision making. <a href="/wiki/R._Duncan_Luce" title="R. Duncan Luce">R. Duncan Luce</a> and <a href="/wiki/Howard_Raiffa" title="Howard Raiffa">Howard Raiffa</a> contributed to the choice and decision making area. </p><p>The area of language and thinking comes into the spotlight with the development of computer science and linguistics, especially information theory and computation theory. <a href="/wiki/Noam_Chomsky" title="Noam Chomsky">Chomsky</a> proposed the model of linguistics and computational hierarchy theory. <a href="/wiki/Allen_Newell" title="Allen Newell">Allen Newell</a> and <a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a> proposed the model of human solving problems. The development in artificial intelligence and human computer interface are active areas in both computer science and psychology. </p><p>Before the 1950s, psychometricians emphasized the structure of measurement error and the development of high-power statistical methods to the measurement of psychological quantities but little of the psychometric work concerned the structure of the psychological quantities being measured or the cognitive factors behind the response data. Scott and Suppes studied relationship between the structure of data and the structure of numerical systems that represent the data.<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> Coombs constructed formal cognitive models of the respondent in a measurement situation rather than statistical data processing algorithms, for example the unfolding model.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Another breakthrough is the development of a new form of the psychophysical scaling function along with new methods of collecting psychophysical data, like <a href="/wiki/Stevens%27s_power_law" title="Stevens's power law">Stevens' power law.</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The 1950s saw a surge in mathematical theories of psychological processes, including <a href="/wiki/Luce%27s_choice_axiom" title="Luce's choice axiom">Luce's theory of choice</a>, Tanner and Swets' introduction of <a href="/wiki/Signal_detection_theory" class="mw-redirect" title="Signal detection theory">signal detection theory</a> for human stimulus detection, and Miller's approach to information processing.<sup id="cite_ref-Batchelder2002_6-1" class="reference"><a href="#cite_note-Batchelder2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> By the end of the 1950s, the number of mathematical psychologists had increased from a handful by more than a tenfold, not counting psychometricians. Most of these were concentrated at the Indiana University, Michigan, Pennsylvania, and Stanford.<sup id="cite_ref-Batchelder2002_6-2" class="reference"><a href="#cite_note-Batchelder2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Estes2002_13-0" class="reference"><a href="#cite_note-Estes2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Some of these were regularly invited by the U.S. Social Science Research Counsel to teach in summer workshops in mathematics for social scientists at Stanford University, promoting collaboration. </p><p>To better define the field of mathematical psychology, the mathematical models of the 1950s were brought together in sequence of volumes edited by Luce, Bush, and Galanter: Two readings<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> and three handbooks.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This series of volumes turned out to be helpful in the development of the field. In the summer of 1963 the need was felt for a journal for theoretical and mathematical studies in all areas in psychology, excluding work that was mainly factor analytical. An initiative led by <a href="/wiki/Richard_C._Atkinson" title="Richard C. Atkinson">R. C. Atkinson</a>, <a href="/w/index.php?title=R._R._Bush&action=edit&redlink=1" class="new" title="R. R. Bush (page does not exist)">R. R. Bush</a>, <a href="/wiki/W._K._Estes" class="mw-redirect" title="W. K. Estes">W. K. Estes</a>, <a href="/wiki/R._Duncan_Luce" title="R. Duncan Luce">R. D. Luce</a>, and <a href="/wiki/P._Suppes" class="mw-redirect" title="P. Suppes">P. Suppes</a> resulted in the appearance of the first issue of the <i><a href="/wiki/Journal_of_Mathematical_Psychology" title="Journal of Mathematical Psychology">Journal of Mathematical Psychology</a></i> in January 1964.<sup id="cite_ref-Estes2002_13-1" class="reference"><a href="#cite_note-Estes2002-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling gravitated towards computational mechanisms and devices. Examples of the latter constitute so called <a href="/wiki/Cognitive_architectures" class="mw-redirect" title="Cognitive architectures">cognitive architectures</a> (e.g., <a href="/wiki/Production_system_(computer_science)" title="Production system (computer science)">production rule systems</a>, <a href="/wiki/ACT-R" title="ACT-R">ACT-R</a>) as well as <a href="/wiki/Connectionist" class="mw-redirect" title="Connectionist">connectionist</a> systems or <a href="/wiki/Neural_networks" class="mw-redirect" title="Neural networks">neural networks</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2022)">citation needed</span></a></i>]</sup> </p><p>Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are <a href="/wiki/Weber%E2%80%93Fechner_law" title="Weber–Fechner law">Weber–Fechner law</a>, <a href="/w/index.php?title=Ekman%27s_law&action=edit&redlink=1" class="new" title="Ekman's law (page does not exist)">Ekman's law</a>, <a href="/wiki/Stevens%27s_power_law" title="Stevens's power law">Stevens's power law</a>, Thurstone's <a href="/wiki/Law_of_comparative_judgment" title="Law of comparative judgment">law of comparative judgment</a>, the <a href="/wiki/Detection_theory#Psychology" title="Detection theory">theory of signal detection</a> (borrowed from radar engineering), the <a href="/wiki/Matching_law" title="Matching law">matching law</a>, and <a href="/wiki/Rescorla%E2%80%93Wagner_model" title="Rescorla–Wagner model">Rescorla–Wagner rule</a> for classical conditioning. While the first three laws are all <a href="/wiki/Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> in nature, later established relations are more fundamentally <a href="/wiki/Stochastic" title="Stochastic">stochastic</a>. This has been a general theme in the evolution in mathematical modeling of psychological processes: from deterministic relations as found in classical physics to inherently stochastic models.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Influential_mathematical_psychologists">Influential mathematical psychologists</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=5" title="Edit section: Influential mathematical psychologists"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 10em;"> <ul><li><a href="/wiki/John_Robert_Anderson_(psychologist)" title="John Robert Anderson (psychologist)">John Anderson</a></li> <li><a href="/wiki/Richard_C._Atkinson" title="Richard C. Atkinson">Richard C. Atkinson</a></li> <li><a href="/w/index.php?title=William_H._Batchelder&action=edit&redlink=1" class="new" title="William H. Batchelder (page does not exist)">William H. Batchelder</a></li> <li><a href="/wiki/Michael_H._Birnbaum" title="Michael H. Birnbaum">Michael H. Birnbaum</a></li> <li><a href="/wiki/Jerome_Busemeyer" title="Jerome Busemeyer">Jerome Busemeyer</a></li> <li><a href="/w/index.php?title=Hans_Colonius&action=edit&redlink=1" class="new" title="Hans Colonius (page does not exist)">Hans Colonius</a></li> <li><a href="/wiki/Clyde_Coombs" title="Clyde Coombs">C. H. Coombs</a></li> <li><a href="/wiki/Robyn_Dawes" title="Robyn Dawes">Robyn Dawes</a></li> <li>Adele Diederich</li> <li><a href="/w/index.php?title=Ehtibar_Dzhafarov&action=edit&redlink=1" class="new" title="Ehtibar Dzhafarov (page does not exist)">Ehtibar Dzhafarov</a></li> <li><a href="/wiki/William_Kaye_Estes" title="William Kaye Estes">William Kaye Estes</a></li> <li><a href="/wiki/Jean-Claude_Falmagne" title="Jean-Claude Falmagne">Jean-Claude Falmagne</a></li> <li><a href="/w/index.php?title=B._F._Green&action=edit&redlink=1" class="new" title="B. F. Green (page does not exist)">B. F. Green</a></li> <li><a href="/wiki/Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li> <li><a href="/w/index.php?title=Eric_Maris&action=edit&redlink=1" class="new" title="Eric Maris (page does not exist)">Eric Maris</a></li> <li><a href="/wiki/Roger_E._Kirk" title="Roger E. Kirk">Roger E. Kirk</a></li> <li><a href="/w/index.php?title=D._H._Krantz&action=edit&redlink=1" class="new" title="D. H. Krantz (page does not exist)">D. H. Krantz</a></li> <li><a href="/w/index.php?title=D._R._J._Laming&action=edit&redlink=1" class="new" title="D. R. J. Laming (page does not exist)">D. R. J. Laming</a></li> <li>Michael D. Lee</li> <li><a href="/wiki/Philip_Marcus_Levy" title="Philip Marcus Levy">Philip Marcus Levy</a></li> <li><a href="/wiki/R._Duncan_Luce" title="R. Duncan Luce">R. Duncan Luce</a></li> <li><a href="/wiki/David_Marr_(neuroscientist)" title="David Marr (neuroscientist)">David Marr</a></li> <li><a href="/wiki/James_McClelland_(psychologist)" title="James McClelland (psychologist)">James L. McClelland</a></li> <li><a href="/w/index.php?title=Jeff_Miller_(psychologist)&action=edit&redlink=1" class="new" title="Jeff Miller (psychologist) (page does not exist)">Jeff Miller</a></li> <li><a href="/w/index.php?title=Jay_Myung&action=edit&redlink=1" class="new" title="Jay Myung (page does not exist)">Jay Myung</a></li> <li><a href="/wiki/Louis_Narens" title="Louis Narens">Louis Narens</a></li> <li><a href="/wiki/Allen_Newell" title="Allen Newell">Allen Newell</a></li> <li><a href="/wiki/Robert_M._Nosofsky" class="mw-redirect" title="Robert M. Nosofsky">Robert M. Nosofsky</a></li> <li><a href="/w/index.php?title=Roger_Ratcliff&action=edit&redlink=1" class="new" title="Roger Ratcliff (page does not exist)">Roger Ratcliff</a></li> <li><a href="/wiki/David_E._Rumelhart" class="mw-redirect" title="David E. Rumelhart">David E. Rumelhart</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a></li> <li><a href="/wiki/Roger_Shepard" title="Roger Shepard">Roger Shepard</a></li> <li><a href="/wiki/Richard_Shiffrin" title="Richard Shiffrin">Richard Shiffrin</a></li> <li>Philip L. Smith</li> <li><a href="/wiki/Stanley_Smith_Stevens" title="Stanley Smith Stevens">Stanley S. Stevens</a></li> <li><a href="/wiki/George_Sperling" title="George Sperling">George Sperling</a></li> <li><a href="/wiki/Saul_Sternberg" title="Saul Sternberg">Saul Sternberg</a></li> <li><a href="/wiki/P._Suppes" class="mw-redirect" title="P. Suppes">Patrick Suppes</a></li> <li><a href="/wiki/John_A._Swets" title="John A. Swets">John A. Swets</a></li> <li><a href="/wiki/Joshua_Tenenbaum" title="Joshua Tenenbaum">Joshua Tenenbaum</a></li> <li><a href="/wiki/James_Townsend_(psychologist)" title="James Townsend (psychologist)">James T. Townsend</a></li> <li><a href="/wiki/Louis_L._Thurstone" class="mw-redirect" title="Louis L. Thurstone">Louis L. Thurstone</a></li> <li><a href="/wiki/Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li> <li><a href="/w/index.php?title=Rolf_Ulrich&action=edit&redlink=1" class="new" title="Rolf Ulrich (page does not exist)">Rolf Ulrich</a></li> <li><a href="/w/index.php?title=Dirk_Vorberg&action=edit&redlink=1" class="new" title="Dirk Vorberg (page does not exist)">Dirk Vorberg</a></li> <li><a href="/wiki/Eric-Jan_Wagenmakers" title="Eric-Jan Wagenmakers">Eric-Jan Wagenmakers</a></li> <li><a href="/wiki/Elke_U._Weber" title="Elke U. Weber">Elke U. Weber</a></li> <li><a href="/w/index.php?title=Thomas_D._Wickens&action=edit&redlink=1" class="new" title="Thomas D. Wickens (page does not exist)">Thomas D. Wickens</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Important_theories_and_models">Important theories and models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=6" title="Edit section: Important theories and models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Source:<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> </p> <div class="mw-heading mw-heading3"><h3 id="Sensation,_perception,_and_psychophysics"><span id="Sensation.2C_perception.2C_and_psychophysics"></span>Sensation, perception, and psychophysics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=7" title="Edit section: Sensation, perception, and psychophysics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Stevens%27_power_law" class="mw-redirect" title="Stevens' power law">Stevens' power law</a></li> <li><a href="/wiki/Weber%E2%80%93Fechner_law" title="Weber–Fechner law">Weber–Fechner law</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Stimulus_detection_and_discrimination">Stimulus detection and discrimination</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=8" title="Edit section: Stimulus detection and discrimination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Signal_detection_theory" class="mw-redirect" title="Signal detection theory">Signal detection theory</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Stimulus_identification">Stimulus identification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=9" title="Edit section: Stimulus identification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Accumulator models</li> <li>Diffusion models</li> <li><a href="/wiki/Neural_network" title="Neural network">Neural network</a>/connectionist models</li> <li>Race models</li> <li>Random walk models</li> <li>Renewal models</li></ul> <div class="mw-heading mw-heading3"><h3 id="Simple_decision">Simple decision</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=10" title="Edit section: Simple decision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Cascade model</li> <li>Level and change race model</li> <li>Recruitment model</li> <li><a href="/wiki/SPRT" class="mw-redirect" title="SPRT">SPRT</a></li> <li><a href="/wiki/Decision_field_theory" title="Decision field theory">Decision field theory</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Memory_scanning,_visual_search"><span id="Memory_scanning.2C_visual_search"></span>Memory scanning, visual search</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=11" title="Edit section: Memory scanning, visual search"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Push-down stack</li> <li>Serial exhaustive search (SES) model</li></ul> <div class="mw-heading mw-heading3"><h3 id="Error_response_times">Error response times</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=12" title="Edit section: Error response times"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Fast guess model</li></ul> <div class="mw-heading mw-heading3"><h3 id="Sequential_effects">Sequential effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=13" title="Edit section: Sequential effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Linear operator model</li></ul> <div class="mw-heading mw-heading3"><h3 id="Learning">Learning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=14" title="Edit section: Learning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Linear operator model</li> <li>Stochastic learning theory</li></ul> <div class="mw-heading mw-heading3"><h3 id="Measurement_theory">Measurement theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=15" title="Edit section: Measurement theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Theory_of_conjoint_measurement" title="Theory of conjoint measurement">Theory of conjoint measurement</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Developmental_psychology">Developmental psychology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=16" title="Edit section: Developmental psychology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Developmental_psychology#Mathematical_models" title="Developmental psychology">Developmental psychology § Mathematical models</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Developmental_psychology&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>Developmental psychology is concerned not only with describing the characteristics of psychological change over time but also seeks to explain the principles and internal workings underlying these changes. Psychologists have attempted to better understand these factors by using <a href="/wiki/Model_(abstract)" class="mw-redirect" title="Model (abstract)">models</a>. A model must simply account for the means by which a process takes place. This is sometimes done in reference to changes in the <a href="/wiki/Human_brain" title="Human brain">brain</a> that may correspond to changes in behavior over the course of the development. </p><p>Mathematical modeling is useful in developmental psychology for implementing theory in a precise and easy-to-study manner, allowing generation, explanation, integration, and prediction of diverse phenomena. Several modeling techniques are applied to development: <a href="/wiki/Symbol" title="Symbol">symbolic</a>, <a href="/wiki/Connectionist" class="mw-redirect" title="Connectionist">connectionist</a> (<a href="/wiki/Neural_net" class="mw-redirect" title="Neural net">neural network</a>), or <a href="/wiki/Cognitive_model#Dynamical_systems" title="Cognitive model">dynamical systems</a> models. </p> Dynamic systems models illustrate how many different features of a complex system may interact to yield emergent behaviors and abilities. Nonlinear dynamics has been applied to human systems specifically to address issues that require attention to temporality such as life transitions, human development, and behavioral or emotional change over time. Nonlinear dynamic systems is currently being explored as a way to explain discrete phenomena of human development such as affect,<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> second language acquisition,<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> and locomotion.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading2"><h2 id="Journals_and_organizations">Journals and organizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=17" title="Edit section: Journals and organizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Central journals are the <i><a href="/wiki/Journal_of_Mathematical_Psychology" title="Journal of Mathematical Psychology">Journal of Mathematical Psychology</a></i> and the <i><a href="/wiki/British_Journal_of_Mathematical_and_Statistical_Psychology" title="British Journal of Mathematical and Statistical Psychology">British Journal of Mathematical and Statistical Psychology</a></i>. There are three annual conferences in the field, the annual meeting of the <i>Society for Mathematical Psychology</i> in the U.S, the annual <a href="/wiki/European_Mathematical_Psychology_Group" title="European Mathematical Psychology Group">European Mathematical Psychology Group</a> meeting in Europe, and the <i>Australasian Mathematical Psychology</i> conference. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Computational_cognition" title="Computational cognition">Computational cognition</a></li> <li><a href="/wiki/Mathematical_models_of_social_learning" title="Mathematical models of social learning">Mathematical models of social learning</a></li> <li><a href="/wiki/Outline_of_psychology" title="Outline of psychology">Outline of psychology</a></li> <li><a href="/wiki/Psychological_statistics" title="Psychological statistics">Psychological statistics</a></li> <li><a href="/wiki/Quantitative_psychology" title="Quantitative psychology">Quantitative psychology</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_psychology&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Batchelder_808–815-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Batchelder_808–815_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and 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New York: Wiley. <a rel="nofollow" class="external text" href="https://www.archive.org/download/handbookofmathem017893mbp/handbookofmathem017893mbp.pdf">Volume II from Internet Archive</a></i></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="CITEREFLuce1986" class="citation book cs1">Luce, R. Duncan (1986). <i>Response Times: Their Role in Inferring Elementary Mental Organization</i>. Oxford Psychology Series. Vol. 8. 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Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1138951655" title="Special:BookSources/978-1138951655"><bdi>978-1138951655</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+social+psychology&rft.pub=Routledge&rft.date=2017&rft.isbn=978-1138951655&rft.aulast=Vallacher&rft.aufirst=RR&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+psychology" 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="CITEREFde_Bot2007" class="citation journal cs1">de Bot K (2007). "A dynamic systems theory approach to second language acquisition". <i>Bilingualism: Language and Cognition</i>. <b>10</b>: <span class="nowrap">7–</span>21. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1366728906002732">10.1017/S1366728906002732</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bilingualism%3A+Language+and+Cognition&rft.atitle=A+dynamic+systems+theory+approach+to+second+language+acquisition&rft.volume=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E7-%3C%2Fspan%3E21&rft.date=2007&rft_id=info%3Adoi%2F10.1017%2FS1366728906002732&rft.aulast=de+Bot&rft.aufirst=K&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+psychology" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRheaKieferD'AndreaWarren2014" class="citation journal cs1">Rhea CK, Kiefer AW, D'Andrea SE, Warren WH, Aaron RK (August 2014). <a rel="nofollow" class="external text" href="http://libres.uncg.edu/ir/uncg/f/C_Rhea_Entrainment_2014.pdf">"Entrainment to a real time fractal visual stimulus modulates fractal gait dynamics"</a> <span class="cs1-format">(PDF)</span>. <i>Human Movement Science</i>. <b>36</b>: <span class="nowrap">20–</span>34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.humov.2014.04.006">10.1016/j.humov.2014.04.006</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24911782">24911782</a>.</cite><span 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<ul><li><a href="/wiki/Anomalistic_psychology" title="Anomalistic psychology">Anomalistic</a></li> <li><a href="/wiki/Applied_behavior_analysis" title="Applied behavior analysis">Applied behavior analysis</a></li> <li><a href="/wiki/Psychological_testing" title="Psychological testing">Assessment</a></li> <li><a href="/wiki/Clinical_psychology" title="Clinical psychology">Clinical</a></li> <li><a href="/wiki/Coaching_psychology" title="Coaching psychology">Coaching</a></li> <li><a href="/wiki/Community_psychology" title="Community psychology">Community</a></li> <li><a href="/wiki/Consumer_behaviour" title="Consumer behaviour">Consumer</a></li> <li><a href="/wiki/Counseling_psychology" title="Counseling psychology">Counseling</a></li> <li><a href="/wiki/Critical_psychology" title="Critical psychology">Critical</a></li> <li><a href="/wiki/Educational_psychology" title="Educational psychology">Educational</a></li> <li><a href="/wiki/Ergonomics" title="Ergonomics">Ergonomics</a></li> <li><a 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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#CFE1FF;"><a href="/wiki/List_of_psychological_research_methods" title="List of psychological research methods">Methodologies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Animal_testing" title="Animal testing">Animal testing</a></li> <li><a href="/wiki/Archival_research" title="Archival research">Archival research</a></li> <li><a href="/wiki/Behavioral_epigenetics" title="Behavioral epigenetics">Behavior epigenetics</a></li> <li><a href="/wiki/Case_study" title="Case study">Case study</a></li> <li><a href="/wiki/Content_analysis" title="Content analysis">Content analysis</a></li> <li><a href="/wiki/Experimental_psychology" title="Experimental psychology">Experiments</a></li> <li><a href="/wiki/Human_subject_research" title="Human subject research">Human subject research</a></li> <li><a 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0.25em"><div class="excerpt-block"><div class="excerpt"> <ul><li><a href="/wiki/Behavior" title="Behavior">Behavior</a></li> <li><a href="/wiki/Applied_behavior_analysis" title="Applied behavior analysis">Behavioral engineering</a></li> <li><a href="/wiki/Behavioural_genetics" title="Behavioural genetics">Behavioral genetics</a></li> <li><a href="/wiki/Behavioral_neuroscience" title="Behavioral neuroscience">Behavioral neuroscience</a></li> <li><a href="/wiki/Cognition" title="Cognition">Cognition</a></li> <li><a href="/wiki/Competence_(polyseme)" title="Competence (polyseme)">Competence</a></li> <li><a href="/wiki/Consciousness" title="Consciousness">Consciousness</a></li> <li><a href="/wiki/Consumer_behaviour" title="Consumer behaviour">Consumer behavior</a></li> <li><a href="/wiki/Emotion" title="Emotion">Emotions</a></li> <li><a href="/wiki/Feeling" title="Feeling">Feelings</a></li> <li><a href="/wiki/Ergonomics" title="Ergonomics">Human factors and ergonomics</a></li> <li><a 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Watson">John B. Watson</a></li> <li><a href="/wiki/Clark_L._Hull" title="Clark L. Hull">Clark L. Hull</a></li> <li><a href="/wiki/Kurt_Lewin" title="Kurt Lewin">Kurt Lewin</a></li> <li><a href="/wiki/Jean_Piaget" title="Jean Piaget">Jean Piaget</a></li> <li><a href="/wiki/Gordon_Allport" title="Gordon Allport">Gordon Allport</a></li> <li><a href="/wiki/J._P._Guilford" title="J. P. Guilford">J. P. Guilford</a></li> <li><a href="/wiki/Carl_Rogers" title="Carl Rogers">Carl Rogers</a></li> <li><a href="/wiki/Erik_Erikson" title="Erik Erikson">Erik Erikson</a></li> <li><a href="/wiki/B._F._Skinner" title="B. F. Skinner">B. F. Skinner</a></li> <li><a href="/wiki/Donald_O._Hebb" title="Donald O. Hebb">Donald O. Hebb</a></li> <li><a href="/wiki/Ernest_Hilgard" title="Ernest Hilgard">Ernest Hilgard</a></li> <li><a href="/wiki/Harry_Harlow" title="Harry Harlow">Harry Harlow</a></li> <li><a href="/wiki/Raymond_Cattell" title="Raymond Cattell">Raymond Cattell</a></li> <li><a href="/wiki/Abraham_Maslow" title="Abraham Maslow">Abraham Maslow</a></li> <li><a href="/wiki/Neal_E._Miller" title="Neal E. Miller">Neal E. Miller</a></li> <li><a href="/wiki/Jerome_Bruner" title="Jerome Bruner">Jerome Bruner</a></li> <li><a href="/wiki/Donald_T._Campbell" title="Donald T. Campbell">Donald T. Campbell</a></li> <li><a href="/wiki/Hans_Eysenck" title="Hans Eysenck">Hans Eysenck</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a></li> <li><a href="/wiki/David_McClelland" title="David McClelland">David McClelland</a></li> <li><a href="/wiki/Leon_Festinger" title="Leon Festinger">Leon Festinger</a></li> <li><a href="/wiki/George_Armitage_Miller" title="George Armitage Miller">George A. Miller</a></li> <li><a href="/wiki/Richard_Lazarus" title="Richard Lazarus">Richard Lazarus</a></li> <li><a href="/wiki/Stanley_Schachter" title="Stanley Schachter">Stanley Schachter</a></li> <li><a href="/wiki/Robert_Zajonc" title="Robert Zajonc">Robert Zajonc</a></li> <li><a href="/wiki/Albert_Bandura" title="Albert Bandura">Albert Bandura</a></li> <li><a href="/wiki/Roger_Brown_(psychologist)" title="Roger Brown (psychologist)">Roger Brown</a></li> <li><a href="/wiki/Endel_Tulving" title="Endel Tulving">Endel Tulving</a></li> <li><a href="/wiki/Lawrence_Kohlberg" title="Lawrence Kohlberg">Lawrence Kohlberg</a></li> <li><a href="/wiki/Noam_Chomsky" title="Noam Chomsky">Noam Chomsky</a></li> <li><a href="/wiki/Ulric_Neisser" title="Ulric Neisser">Ulric Neisser</a></li> <li><a href="/wiki/Jerome_Kagan" title="Jerome Kagan">Jerome Kagan</a></li> <li><a href="/wiki/Walter_Mischel" title="Walter Mischel">Walter Mischel</a></li> <li><a href="/wiki/Elliot_Aronson" title="Elliot Aronson">Elliot Aronson</a></li> <li><a href="/wiki/Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li> <li><a href="/wiki/Paul_Ekman" title="Paul Ekman">Paul Ekman</a></li> <li><a href="/wiki/Michael_Posner_(psychologist)" title="Michael Posner (psychologist)">Michael Posner</a></li> <li><a href="/wiki/Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li> <li><a href="/wiki/Bruce_McEwen" title="Bruce McEwen">Bruce McEwen</a></li> <li><a href="/wiki/Larry_Squire" title="Larry Squire">Larry Squire</a></li> <li><a href="/wiki/Richard_E._Nisbett" title="Richard E. 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