In this paper, I address a topic that has been mostly neglected in Leibniz scholarship: Leibniz’s conception of number. I argue that Leibniz thinks of numbers as a certain kind of relation, and that as such, numbers have a privileged place in his metaphysical system as entities that express a certain kind of possibility. Establishing the relational view requires reconciling two seemingly inconsistent definitions of number in Leibniz’s corpus; establishing where numbers fit in Leibniz’s ontology requires confronting a challenge from the well-known nominalist reading of Leibniz most forcefully articulated in Mates (1986). While my main focus is limited to the positive integers, I also argue that Leibniz intends to subsume them under a more general conception of number.

The question of Leibniz’s relationship to mysticism has been a topic of some debate since the early part of the 20th Century. An initial wave of scholarship led by Jean Baruzi pre­sented Leibniz as a mystic. However, later in the 20th Century the mood turned against this view and the negative appraisal holds sway today. In this paper I do two things: First I provide a detailed account of the ways in which Leibniz is critical of mysticism; second, I argue that there is, nonetheless, an important sense in which Leibniz should be regarded as an advocate of mysticism. However, the approach that I take does not focus on an effort to overturn the kinds of considerations that led people to reject the views of Baruzi. Instead, I try to reframe the discussion and explore more complex and interesting relationships that exist between mysticism and Leibniz’s philosophical theology than have been articulated previously. Here I draw on some recent discussions of mysticism in the philosophical literature to illuminate Leibniz’s own distinction between “false mysticism” and “true mystical theology” and his assessment of the views of a number of other people who might plausibly be identified as mystics.

It is well-known that Leibniz ends and crowns the 1714 “Monadologie” with a version of his notion of jurisprudence universelle or “justice as the charity [love] of the wise:” for sections 83-90 of the Vienna manuscript claim that “the totality of all spirits must compose the City of God . . . this perfect government . . . the most perfect state that is possible . . . this truly universal monarchy [which is] a moral world in the natural world”—a moral world of iustitia in which “no good action would be unrewarded” for those “citizens” who “find pleasure . . . in the contemplation of [God’s] perfections, as is the way of genuine ‘pure love.’” But the opening four-fifths of the work offer Leibniz’ theory of “substance” (or monad) viewed as the necessary pre-condition of justice: for “on the knowledge of substance, and in the consequence of the soul, depends the knowledge of virtue and of justice” (to Pierre Coste, 1712). Thus without a complete and correct notion of substance/monad, no complete and correct notion commune de la justice would be conceivable. Hence the entire “Monadologie” can be understood as a theory of justice underpinned by a Grundlegung of moral “monads” or justice-loving rational “substances.” In this connection it is revelatory that Leibniz cites the relevant sections of the 1710 Théodicée in most of the 90 articles of the “Monadologie” (beginning indeed with article #1): for Théodicée (theos-dike) is (Leibniz says) “the justice of God,” and Leibniz makes that justice “appear” in the opening lines of the “Monadologie” (in effect) by referring the reader immediately to Théodicée #10 (“Preliminary Dissertation”) —which relates “im­mortal spirits” to a just God who is cherished through “genuine pure love.” This means that “the justice of God” as “higher love” colors the “Monadologie” instantly. Thus one need not “wait” for sections 83-90 to arrive in order for the “Monadologie” to be(come) a “theory of justice:” it is such ab initio.

Leibniz’s infinite-analysis theory of contingency says a truth is contingent if and only if it cannot be proved via analysis in finitely many steps. Some have argued that this theory faces the Problem of Lucky Proof—we might, by luck, complete our proof early in the analysis, and thus have a finite proof of a contingent truth—and the related Problem of Guaranteed Proof—even if we do not complete our proof early in the analysis, we are guaranteed to complete it in finitely many steps. I aim to solve both problems. For Leibniz, analysis is constrained by three rules: an analysis begins with the conclusion; subsequent steps replace a term by (part of) its real definition; and the analysis is finished only when an identity is reached. Furthermore, real definitions of complete concepts are infinitely complex, and Leibniz thinks infinities lack parts. From these observations, a solution to our problems follows: an analysis of a truth containing a complete concept cannot be completed in a finite number of steps—indeed, the first step of the analysis cannot be completed. I conclude by defusing some alleged counterexamples to my account.