Analysis Situs, the Foundations of Mathematics and a Geometry of Space
Résumé
Analysis situs is the name given by Leibniz to a cluster of mathematical and philosophical investigations into the foundations, development, and formalization of geometry. Leibniz devoted himself to this multifaceted project throughout his life: his earliest studies date back to the Parisian period (1672-1676), and his last drafts were composed at the end of his life (1714-1716). These forty years' worth of efforts yielded some hundreds of essays that range from short collections of notes to complex, long and highly developed manuscripts. None of these works, however, was published during Leibniz's lifetime. His contemporaries' dismissive attitudes towards the project, as well as Leibniz's own conviction that his new discipline had yielded nothing of great consequence for the field of geometry, 1 prevented Leibniz from seeking publication. 2 Some of his writings on analysis situs were printed in 19 th-century Leibniz editions, while others have only been published in recent years. 3 The great bulk of Leibniz's studies on the topic, however, remains unpublished in the Hannover library. Nevertheless, we have enough material to attempt to reconstruct-albeit partially and tentatively-the main aims, developments and outcomes of Leibniz's endeavor in a new geometry. Here some complaints from the well-known letter to L'Hospital from 27 December 1694 (in GM 2, 255-62): "My metaphysics is all mathematics, so to speak, or it can become so. At the present, I dare not publish my projects on the characteristica situs because, unless made believable through examples of some importance, it would be regarded as just a vision. Nonetheless I see in advance that it will not fail." 2 Leibniz was particularly influenced by Huygens's poor reception of the nascent characteristica geometrica in his letter of 22 November 1679 (A III, 2, 888-89). 3 Leibniz's papers on analysis situs are mostly to be found in volumes 5 (from 1858) and 7 (from 1863) of Gerhardt's edition of the Mathematische Schriften, but some of them are also printed in Couturat's Opuscules et fragments inédits. Further publications include an important text in the appendix of M. Mugnai, "Leibniz's Theory of Relations,"
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