TheDissertatioDe Arte Combinatoria ('Dissertation on the Art of Combinations' or'Dissertation on the Combinatorial Art') is an early work byGottfried Leibniz published in 1666 inLeipzig.[1][2] It is an extended version of his firstdoctoral dissertation,[a] written before the author had seriously undertaken the study of mathematics.[4] The booklet was reissued without Leibniz' consent in 1690, which prompted him to publish a brief explanatory notice in theActa Eruditorum.[5] During the following years he repeatedly expressed regrets about its being circulated as he considered it immature.[b] Nevertheless it was a very original work and it provided the author the first glimpse of fame among the scholars of his time.
The book was created as part of Leibniz's efforts to create theCharacteristica Universalis, theperfect language which would provide a direct representation of ideas along with acalculus for the philosophical reasoning.
The main idea behind the text is that of analphabet of human thought, which is attributed toDescartes. All concepts are nothing butcombinations of a relatively small number of simple concepts, just as words are combinations of letters. Alltruths 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 asubject and apredicate, one might
For this, Leibniz was inspired in theArs Magna ofRamon Llull, although he criticized this author because of the arbitrariness of his categories indexing.
Leibniz discusses in this work somecombinatorial concepts. He had readClavius' comments toSacrobosco'sDe sphaera mundi, and some other contemporary works. He introduced the termvariationes ordinis for thepermutations,combinationes for the combinations of two elements,con3nationes (shorthand forconternationes) for those of three elements, etc. His general term for combinations wascomplexions. He found the formula
which he thought was original.
The first examples of use of hisars combinatoria[further explanation needed] are taken from law, the musical registry of anorgan, and theAristotelian theory of generation of elements from the four primary qualities. But philosophical applications are of greater importance. He cites the idea ofThomas Hobbes 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 II contains simple combinations.
Where των means "of the" (fromAncient Greek:τῶν). Thus, "Quantity" is the number of the parts. Class III contains thecon3nationes:
Thus, "Interval" is the space included in total. Of course, concepts deriving from former classes may also be defined.
Where 1/3 means the first concept of class III. Thus, a "line" is the interval of (between) points.
Leibniz compares his system to theChinese andEgyptian languages, although he did not really understand them at this point. For him, this is a first step towards theCharacteristica Universalis, theperfect language which would provide a direct representation of ideas along with a calculus for the philosophical reasoning.
As a preface, the work begins with aproof of the existence of God, cast in geometrical form, and based on theargument from motion.
