Despite his commitment to freedom, Leibniz’ philosophy is also founded on pre-established harmony. Understanding the life of the individual as a spiritual automaton led Leibniz to refer to the puzzle of the way out of determinism as the Labyrinth of Freedom. Leibniz claimed that infinite complexity is the reason why it is impossible to prove a contingent truth. But by means of Leibniz’ calculus, it actually can be shown in a finite number of steps how to calculate a summation of (...) infinite parts. It appears that the analogy Leibniz drew between the mathematics of infinite series and the logic of contingent truths did more harm than good. A solution consistent with Leibniz’ perception of infinity is proposed. Alongside the existence of the aforementioned analogy, it is based on a disanalogy between the mathematics of infinite series and the logic of infinitely complex truths. This is a disanalogy that Leibniz had already used to solve the Labyrinth of Continuum which he declared more than once could, like the Labyrinth of Freedom, be solved by means of the nature of infinity. The solution of both labyrinths is based on the fictitiousness of the infinitesimal. (shrink)

Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz’s argument against the world soul relies on his rejection of infinite number, and, as such, (...) Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz’s rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’– collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts. (shrink)

This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to (...) three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status. (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)

In this paper we study the impact of Leibniz’s critique of infinite numbers in his metaphysics of bodies. After presenting the relation that the German philosopher establishes in his youth between the notions of body, extension and infinite quantities, we analyze his thoughts on the paradoxes of the infinite numbers and we claim that his defense of the inconsistency of such numbers is an inflexion point in his conception of body and mark the beginning of his offensive against the res (...) extensa. (shrink)