De Arte Combinatoria

The Dissertatio de arte combinatoria ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended version of his first doctoral dissertation, written before the author had seriously undertaken the study of mathematics. 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. Nevertheless it was a very original work and it provided the author the first glimpse of fame among the scholars of his time.

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

For this, Leibniz was inspired in the Ars Magna of Ramon Llull, 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 the Tractatus de Sphaera of Sacrobosco, 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.

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