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The ''Dissertatio de arte combinatoria'' ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
published in 1666 in
Leipzig Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as we ...
. It is an extended version of his first
doctoral dissertation A thesis ( : theses), or dissertation (abbreviated diss.), is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings.International Standard ISO 7144: ...
, written before the author had seriously undertaken the study of mathematics.
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
. ''Hauptschriften zur Grundlegung der Philosophie. Zur allgemeinen Charakteristik.'' ''Philosophische Werke'' 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.
The booklet was reissued without Leibniz' consent in 1690, which prompted him to publish a brief explanatory notice in the '' Acta Eruditorum''. During the following years he repeatedly expressed regrets about its being circulated as he considered it immature.Leibniz complained to various correspondents, e.g., to Morell (1 October 1697) or to Meier (23 January 1699); see ''Akademie'' I.14, p. 548 or I.16, p. 540. Nevertheless it was a very original work and it provided the author the first glimpse of fame among the scholars of his time.


Summary

The main idea behind the text is that of an alphabet of human thought, which is attributed to Descartes. 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 * Find all the predicates which are appropriate to a given subject, or * Find all the subjects which are convenient to a given predicate. For this, Leibniz was inspired in the '' Ars Magna'' of
Ramon Llull Ramon Llull (; c. 1232 – c. 1315/16) was a philosopher, theologian, poet, missionary, and Christian apologist from the Kingdom of Majorca. He invented a philosophical system known as the ''Art'', conceived as a type of universal logic to p ...
, although he criticized this author because of the arbitrariness of his categories and his indexing. Leibniz discusses in this work some combinatorial concepts. He had read Clavius' comments to
Sacrobosco Johannes de Sacrobosco, also written Ioannes de Sacro Bosco, later called John of Holywood or John of Holybush ( 1195 – 1256), was a scholar, monk, and astronomer who taught at the University of Paris. He wrote a short introduction to the Hi ...
's ''
De sphaera mundi ''De sphaera mundi'' ( Latin title meaning ''On the Sphere of the World'', sometimes rendered ''The Sphere of the Cosmos''; the Latin title is also given as ''Tractatus de sphaera'', ''Textus de sphaera'', or simply ''De sphaera'') is a medieva ...
'', and some other contemporary works. He introduced the term ''variationes ordinis'' for the permutations, ''combinationes'' for the combinations of two elements, ''con3nationes'' (shorthand for ''conternationes'') for those of three elements, etc. His general term for combinations was ''complexions''. He found the formula : = + which he thought was original. The first examples of use of his ''ars combinatoria'' are taken from law, the musical registry of an organ, and the Aristotelian theory of generation of elements from the four primary qualities. But philosophical applications are of greater importance. He cites the idea of
Thomas Hobbes Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book ''Leviathan'', in which he expounds an influent ...
that all reasoning is just a computation. The most careful example is taken from geometry, from where we shall give some definitions. He introduces the Class I concepts, which are primitive. ;Class I: 1 point, 2 space, 3 included, ..9 parts, 10 total, ..14 number, 15 various .. Class II contains simple combinations. ;Class II.1: Quantity is 14 των 9 Where των means "of the" (from grc, τῶν). Thus, "Quantity" is the number of the parts. Class III contains the ''con3nationes'': ;Class III.1: Interval is 2.3.10 Thus, "Interval" is the space included in total. Of course, concepts deriving from former classes may also be defined. ;Class IV.1: Line is 1/3 των 2 Where 1/3 means the first concept of class III. Thus, a "line" is the interval of (between) points. 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
Characteristica Universalis The Latin term ''characteristica universalis'', commonly interpreted as ''universal characteristic'', or ''universal character'' in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scienti ...
, the perfect language which would provide a direct representation of ideas along with a calculus for the philosophical reasoning. As a preface, the work begins with a proof of the existence of God, cast in geometrical form, and based on the argument from motion.


Notes


References

*E. J. Aiton, ''Leibniz: A Biography''. Hilger, Bristol, 1985. .


External links


''De Arte Combinatoria''
the original Latin-language text
Partial English translation


{{Gottfried Wilhelm Leibniz Combinatorics Philosophy of language literature 1666 books Works by Gottfried Wilhelm Leibniz 17th-century Latin books