Leibniz’s Rigorous Foundations of the Method of Indivisibles: Or How to Reason with Impossible Notions

International audienceIt has long been thought that Leibniz’s conceptions of infinitesimals were a little bit fuzzy—to say the least—and certainly not “rigorously” grounded. This could be seen as one of the reasons he was so cautious not to talk about infinitesimals when publicly introducing his differential algorithm in the 1680s. It is apparently only later, through the critiques put forward by authors such as Bernard Nieuwentijt and Michel Rolle in the debates of the 1690s–1700s, that he was pushed to justify the use of infinitesimals. These justifications were far from satisfactory (even according to Leibniz’s own supporters). Hence it is no surprise that he finally took refuge in a “fictionalist” point of view in which “infinitesimals” were just considered as “ways of speaking” (compendia loquendi). This well-known story has recently been challenged by the discovery of texts documenting the fact that Leibniz’s fictionalist conceptions were elaborated as early as 1676. The main discovery was the reconstitution of a complete treatise of infinitesimal techniques, which Leibniz wrote at the end of his stay in Paris: De Quadratura arithmetica circuli ellipseos et hyperbolae. In this treatise, Leibniz claimed to have given “the most solid foundations for the Method of indivisibles”. In this chapter, I would like to present this attempt to set the methodus indivisibilium on solid grounds and explain how it could change our views on Leibniz’s conceptions of infinitesimals