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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg/300px-Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg" decoding="async" width="300" height="298" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg/450px-Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg/600px-Fotothek_df_tg_0005486_Mathematik_%5E_Kombinatorik.jpg 2x" data-file-width="800" data-file-height="795" /></a><figcaption>Frontispiece of the book printed in 1690</figcaption></figure> <p>The <i><b>Dissertatio de arte combinatoria</b></i> ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Leibniz</a> published in 1666 in <a href="/wiki/Leipzig" title="Leipzig">Leipzig</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is an extended version of his first <a href="/wiki/Doctoral_dissertation" class="mw-redirect" title="Doctoral dissertation">doctoral dissertation</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> written before the author had seriously undertaken the study of mathematics.<sup id="cite_ref-Philosophie_page_32_3-0" class="reference"><a href="#cite_note-Philosophie_page_32-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The booklet was reissued without Leibniz' consent in 1690, which prompted him to publish a brief explanatory notice in the <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i>.<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> During the following years he repeatedly expressed regrets about its being circulated as he considered it immature.<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> Nevertheless it was a very original work and it provided the author the first glimpse of fame among the scholars of his time. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Summary">Summary</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=De_Arte_Combinatoria&action=edit&section=1" title="Edit section: Summary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The main idea behind the text is that of an <a href="/wiki/Alphabet_of_human_thought" title="Alphabet of human thought">alphabet of human thought</a>, which is attributed to <a href="/wiki/Descartes" class="mw-redirect" title="Descartes">Descartes</a>. All concepts are nothing but combinations of a relatively small number of simple concepts, just as words are combinations of letters. All truths may be expressed as appropriate combinations of concepts, which can in turn be decomposed into simple ideas, rendering the analysis much easier. Therefore, this alphabet would provide a logic of invention, opposed to that of demonstration which was known so far. Since all sentences are composed of a subject and a predicate, one might </p> <ul><li>Find all the predicates which are appropriate to a given subject, or</li> <li>Find all the subjects which are convenient to a given predicate.</li></ul> <p>For this, Leibniz was inspired in the <i><a href="/wiki/Ars_Magna_(Ramon_Llull)" class="mw-redirect" title="Ars Magna (Ramon Llull)">Ars Magna</a></i> of <a href="/wiki/Ramon_Llull" title="Ramon Llull">Ramon Llull</a>, although he criticized this author because of the arbitrariness of his categories indexing. </p><p>Leibniz discusses in this work some combinatorial concepts. He had read <a href="/wiki/Clavius" class="mw-redirect" title="Clavius">Clavius'</a> comments to <a href="/wiki/Sacrobosco" class="mw-redirect" title="Sacrobosco">Sacrobosco</a>'s <i><a href="/wiki/De_sphaera_mundi" title="De sphaera mundi">De sphaera mundi</a></i>, and some other contemporary works. He introduced the term <i>variationes ordinis</i> for the permutations, <i>combinationes</i> for the combinations of two elements, <i>con3nationes</i> (shorthand for <i>conternationes</i>) for those of three elements, etc. His general term for combinations was <i>complexions</i>. He found the formula </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 {n \choose r}={n-1 \choose r}+{n-1 \choose r-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>r</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>r</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {n \choose r}={n-1 \choose r}+{n-1 \choose r-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5c4ee98e7acb26ab2bdb799e4afb47030546e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.392ex; height:6.176ex;" alt="{\displaystyle {n \choose r}={n-1 \choose r}+{n-1 \choose r-1}}"></span></dd></dl> <p>which he thought was original. </p><p>The first examples of use of his <i><b>ars combinatoria</b></i> are taken from law, the musical registry of an <a href="/wiki/Organ_(music)" title="Organ (music)">organ</a>, and the <a href="/wiki/Aristotle" title="Aristotle">Aristotelian</a> theory of generation of elements from the four primary qualities. But philosophical applications are of greater importance. He cites the idea of <a href="/wiki/Thomas_Hobbes" title="Thomas Hobbes">Thomas Hobbes</a> that all reasoning is just a computation. </p><p>The most careful example is taken from geometry, from where we shall give some definitions. He introduces the Class I concepts, which are primitive. </p> <dl><dt>Class I</dt> <dd>1 point, 2 space, 3 included, [...] 9 parts, 10 total, [...] 14 number, 15 various [...]</dd></dl> <p>Class II contains simple combinations. </p> <dl><dt>Class II.1</dt> <dd>Quantity is 14 των 9</dd></dl> <p>Where των means "of the" (from <a href="/wiki/Ancient_Greek_language" class="mw-redirect" title="Ancient Greek language">Ancient Greek</a>: <span lang="grc"><a href="https://en.wiktionary.org/wiki/%CF%84%E1%BF%B6%CE%BD" class="extiw" title="wikt:τῶν">τῶν</a></span>). Thus, "Quantity" is the number of the parts. Class III contains the <i>con3nationes</i>: </p> <dl><dt>Class III.1</dt> <dd>Interval is 2.3.10</dd></dl> <p>Thus, "Interval" is the space included in total. Of course, concepts deriving from former classes may also be defined. </p> <dl><dt>Class IV.1</dt> <dd>Line is 1/3 των 2</dd></dl> <p>Where 1/3 means the first concept of class III. Thus, a "line" is the interval of (between) points. </p><p>Leibniz compares his system to the Chinese and Egyptian languages, although he did not really understand them at this point. For him, this is a first step towards the <a href="/wiki/Characteristica_Universalis" class="mw-redirect" title="Characteristica Universalis">Characteristica Universalis</a>, the perfect language which would provide a direct representation of ideas along with a calculus for the philosophical reasoning. </p><p>As a preface, the work begins with a <a href="/wiki/Proof_of_the_existence_of_God" class="mw-redirect" title="Proof of the existence of God">proof of the existence of God</a>, cast in geometrical form, and based on the <a href="/wiki/Argument_from_motion" class="mw-redirect" title="Argument from motion">argument from motion</a>. </p> <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=De_Arte_Combinatoria&action=edit&section=2" title="Edit section: Notes"><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: 21em;"> <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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">G.W. Leibniz, <i>Dissertatio de arte combinatoria</i>, 1666, <i>Sämtliche Schriften und Briefe</i> (Berlin: <i>Akademie Verlag</i>, 1923), A VI 1, p. 163; <i>Philosophische Schriften</i> (edited by Carl Immanuel Gerhardt), Bd. IV, p. 30;</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">The first part of the book was his <a href="/wiki/Doctoral_thesis" class="mw-redirect" title="Doctoral thesis">doctoral thesis</a> in Philosophy at <a href="/wiki/Leipzig_University" title="Leipzig University">Leipzig University</a>. Leibniz defended his thesis in March 1666 (see Richard T. W. Arthur, <i>Leibniz</i>, John Wiley & Sons, 2014, p. x).</span> </li> <li id="cite_note-Philosophie_page_32-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Philosophie_page_32_3-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>. <i>Hauptschriften zur Grundlegung der Philosophie. Zur allgemeinen Charakteristik.</i> <i>Philosophische Werke</i> Band 1. p. 32. Translated in German by Artur Buchenau. Published, reviewed and added an introduction and notes by Ernst Cassirer. Hamburg: Felix Meiner, 1966, p. 32.</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">G.G.L. <i>Ars Combinatoria</i>, <i><a href="/wiki/Acta_Eruditorum" title="Acta Eruditorum">Acta Eruditorum</a></i>, Feb. 1691, pp. 63–64</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">Leibniz complained to various correspondents, e.g., to Morell (1 October 1697) or to Meier (23 January 1699); see <i>Akademie</i> I.14, p. 548 or I.16, p. 540.</span> </li> </ol></div></div> </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=De_Arte_Combinatoria&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>E. J. Aiton, <i>Leibniz: A Biography</i>. Hilger, Bristol, 1985. <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 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href="/wiki/Calculus_ratiocinator" title="Calculus ratiocinator">Calculus ratiocinator</a></i></li> <li><i><a href="/wiki/Characteristica_universalis" title="Characteristica universalis">Characteristica universalis</a></i></li> <li><a href="/wiki/Compossibility" title="Compossibility">Compossibility</a></li> <li><a href="/wiki/Difference_(philosophy)" title="Difference (philosophy)">Difference</a></li> <li><a href="/wiki/Dynamism_(metaphysics)" title="Dynamism (metaphysics)">Dynamism</a></li> <li><a href="/wiki/Identity_of_indiscernibles" title="Identity of indiscernibles">Identity of indiscernibles</a></li> <li><a href="/wiki/Individuation" title="Individuation">Individuation</a></li> <li><a href="/wiki/Law_of_continuity" title="Law of continuity">Law of continuity</a></li> <li><a href="/wiki/Leibniz_wheel" title="Leibniz wheel">Leibniz wheel</a></li> <li><a href="/wiki/Leibniz%27s_gap" title="Leibniz's gap">Leibniz's gap</a></li> <li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><i><a href="/wiki/Lingua_generalis" title="Lingua generalis">Lingua generalis</a></i></li> <li><i><a href="/wiki/Mathesis_universalis" title="Mathesis universalis">Mathesis universalis</a></i></li> <li><a href="/wiki/Pre-established_harmony" class="mw-redirect" title="Pre-established harmony">Pre-established harmony</a></li> <li><a href="/wiki/Principle_of_plenitude" title="Principle of plenitude">Plenitude</a></li> <li><a href="/wiki/Principle_of_sufficient_reason" title="Principle of sufficient reason">Sufficient reason</a></li> <li><i><a href="/wiki/Salva_veritate" title="Salva veritate">Salva veritate</a></i></li> <li><a href="/wiki/Theodicy" title="Theodicy">Theodicy</a></li> <li><a href="/wiki/Transcendental_law_of_homogeneity" title="Transcendental law of homogeneity">Transcendental law of homogeneity</a></li> <li><a href="/wiki/Rationalism" title="Rationalism">Rationalism</a></li> <li><a href="/wiki/Universal_science" title="Universal science">Universal science</a></li> <li><i><a href="/wiki/Vis_viva" title="Vis viva">Vis viva</a></i></li> <li><a href="/wiki/Well-founded_phenomenon" class="mw-redirect" title="Well-founded phenomenon">Well-founded phenomenon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a class="mw-selflink selflink">De Arte Combinatoria</a></i> <span style="font-size:85%;">(1666)</span></li> <li><i><a href="/wiki/Discourse_on_Metaphysics" title="Discourse on Metaphysics">Discourse on Metaphysics</a></i> <span style="font-size:85%;">(1686)</span></li> <li><i><a href="/wiki/New_Essays_on_Human_Understanding" title="New Essays on Human Understanding">New Essays on Human Understanding</a></i> <span style="font-size:85%;">(1704)</span></li> <li><i><a href="/wiki/Th%C3%A9odic%C3%A9e" title="Théodicée">Théodicée</a></i> <span style="font-size:85%;">(1710)</span></li> <li><i><a href="/wiki/Monadology" title="Monadology">Monadology</a></i> <span style="font-size:85%;">(1714)</span></li> <li><a href="/wiki/Leibniz%E2%80%93Clarke_correspondence" title="Leibniz–Clarke correspondence">Leibniz–Clarke correspondence</a> <span style="font-size:85%;">(1715–1716)</span></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" 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