LEIBNIZ ON BODIES AND INFINITIES:RERUM NATURAAND MATHEMATICAL FICTIONS

Abstract

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

Three case studies in current Leibniz scholarship

“…(Leibniz [45] as translated by Ariew in [46], p. 230) However, Bassler's interpretation of the letter to Masson involves a conflation of mathematics and nature. A similar conflation occurs in the work of Rabouin and Arthur [51]; for details see [30,Section 2.2] as well as Section 4.2 below.…”

“…As in Mediaeval logic syncategoremata were the parts of speech lacking a definite reference of their own and depending instead on other parts (categoremata) in order to acquire a definite reference, syncategorematic infinitesimals depend on other mathematical entities, namely the ones involved in the method of exhaustion. See also [30,Section 3].…”

“…Esquisabel and Raffo Quintana, while acknowledging the fictional nature of Leibnizian infinitesimals, nonetheless accord to them the status of mathematical entities. Bascelli et al [13], Błaszczyk et al [17], and Katz et al [30] similarly accord to Leibniz's fictional infinitesimals the status of mathematical entities that do not idealize anything in nature. Blåsjö's approach in [15] is largely compatible with such a reading.…”

“…([14], p. 872; emphasis on 'wholes' added) The claim suffers from a conflation of quantity and multitude. 1 Leibniz held that infinite multitude taken as a whole (i.e., infinita interminata) contradicts the part-whole principle (see [30,Section 4.1]). However, Bassler provides no evidence that Leibniz viewed infinitesimals and infinite quantities (infinita terminata) as contradictory.…”

“…Knobloch's attribution of such a reading to Leibniz as early as 1673 clashes with Arthur's claim that the switch to syncategorematics occurred between february and april of 1676. 7 For additional details see [30,Section 3.1].…”