Monads at the Bottom, Monads at the Top, Monads All Over

This chapter contests a widely accepted reading of the role monads play as the most fundamental elements of reality. Garber (2009) argues that simple monads—seen as mindlike atoms without parts and extension—replace the corporeal substance of Leibniz’s middle period. The author argues that, for Leibniz, monads function not only as building blocks at the bottom level of composition (for aggregates) but also at the top as grounding the unity, and hence the being of complete substances and organic unities. Since Leibniz sees organic unities as natural machines with a nested structure that develops ad infinitum, and since he likens monads to living beings, this would imply that the use of the concept “monad” holds not only at the bottom and not only at the top but also in the entire range in between.

Created Things as Infinite and Limited

The chapter starts with Leibniz’s characterization of God, the most perfect Being, as infinite in a hypercategorematic sense—i.e. a being beyond any determination. In contrast to this, creatures are determinate beings; they are determinate and thus limited and particular expressions of the divine essence. However, for Leibniz, creatures are also infinite; thus, creatures are seen as infinite and limited. This leads to taking creatures to be infinite in kind, in distinction from the absolute and hypercategorematic infinity of God. The author presents three lines of argument to substantiate this point: (1) understanding creatures as entailing a particular sequence of perfections and imperfections; (2) understanding creatures under the rubric of an intermediate degree of infinity and perfection that, in 1676, Leibniz calls maximum or infinite in kind; and (3) observing that primitive force, a defining feature of created substance, may be seen as infinite in a metaphysical sense.

Infinity and Life

This chapter aims to trace some of the major steps Leibniz takes before drawing the line between living and nonliving things. The author first presents a brief sketch of the way in which Leibniz uses infinity initially to describe, and ultimately to define, living beings. In so doing, the author traces the development of the way Leibniz uses two concepts—infinity and life—that initially seem disparate, until they come together in his distinction between natural and artificial machines. As will become clear in this brief survey, according to Leibniz, to be living and active turns out to be a prerequisite for being a real entity. In other words, Leibniz comes to associate being with being animate, or being activated by some soul-like thing—anima, entelechy, or substantial form, as he variously terms the source of activity and life in living beings.

Introducing the Main Characters

This first chapter introduces the central concepts and distinctions that Leibniz uses in articulating his view of infinity. In other words, the author introduces the main players in this book. These include: Leibniz’s rejection of infinite number; his distinction between infinite being and infinite number; degrees of infinity; the distinction between actual and potential infinity; indivisibility; his syncategorematic approach to infinite terms; his distinction between infinite number and infinite series; the law of the series; and the distinction between primitive force and derivative force. The chapter’s aim is to present at the outset some of the terminology and concepts used in the book in order to present Leibniz’s approach to infinity—that is, to clarify the major resources needed in order to present his complex views. At the same time, this serves as a sketch of (what the author takes to be) Leibniz’s approach to infinity.

Leibniz in Paris

The first section of this chapter presents Leibniz’s rejection of infinite number in response to Galileo’s paradox. The next section presents a problem that arises from his resolution of the paradox. The problem is this: if Leibniz regards the notion of infinite number as inconsistent, how is it that he regards the notion of infinite being as consistent? In the third section, the author considers a semantic solution to this problem and concludes that it is appealing but ultimately inadequate. In the fourth section, the author considers a more promising solution—namely that Leibniz distinguishes between different senses of infinity. The chapter concludes with a discussion of Leibniz’s attitude toward infinity vis-à-vis his critique of Descartes’s distinction between the infinite and the indefinite.

Conclusion

In conclusion, let us attend to Leibniz’s broad aims and motivation. I think that the best way to see this is to recall once more the historical context. What I have in mind is Leibniz’s subtle response to the Cartesian attempt to mechanize both nature in general and the domain of living things in particular. Descartes attempted to mechanize virtually all the functions that had traditionally been assigned to the vegetative and sensitive souls. His vision of nature, which made it particularly amenable to mechanization, consists of bits of matter in motion: matter is devoid of any powers, activity, or life, and thus involves mere extension. In this sense, Descartes attacked the ancient view of nature and replaced it with a thoroughly disenchanted view of nature....

Infinity and Unity

This chapter explores the connection between infinity and unity. According to Leibniz, any living being admits of both infinite complexity and strict unity. The author develops an analogy between numerical and metaphysical unity: while substantial unities are presupposed by aggregates, a substantial unity is also presupposed by a substance’s infinite qualities, or by its sequence of states and perceptions. This point is exemplified and developed through Leibniz’s use of a law of a series to define an individual substance. The author seeks to show that Leibniz’s qualification of a substance as “one being” is primarily intended to emphasize the essential unity and indivisibility of a substance. This claim can also be expressed by noting that unity per se (or an indivisible unity) implies numerical oneness but not vice versa.

Leibniz Reads Spinoza

Having argued in chapter 2 that Leibniz was preoccupied with the difference between the notion of infinite number and that of the infinite being, in this chapter the author examines Spinoza’s solution to a similar problem. The gist of “Spinoza’s solution” is to distinguish between various kinds of infinity and, in particular, between one that applies to substance and one that applies to numbers, seen as auxiliaries of the imagination. Leibniz, the author argues, accepts this kind of approach and adapts it to his own purposes. Leibniz recasts Spinoza’s distinctions between different types of infinity (A 6.3:282; LLC 114–15) in terms of degrees of infinity. These degrees are (1) Omnia (absolute infinity), which applies to God alone; (2) Omnia sui generis, or maximum in its own kind; and (3) Infinitum tantum, or mere infinity, which applies to numbers and other entia rationis (in a syncategorematic sense).

Life and Force

This chapter highlights an important analogy between life and force. Just as Leibniz invokes force to account for the phenomena of motion in physics, in his life sciences he invokes the soul (or anima or entelechy) that must be presupposed as the ground for the phenomena of life. Indeed, Leibniz’s motivation in invoking the notion of a natural machine is precisely to limit the extension of mechanical philosophy and draw a line between living and nonliving things. This distinction does not turn primarily on physiological grounds but, rather, involves metaphysical considerations. Likewise, both the notion of force and the source of life, according to the author, must be understood metaphysically. In both cases, these metaphysical principles are supposed to be compatible with a mechanical explanation of the phenomena of life. Nevertheless, the principle of life is a metaphysical principle that cannot be observed; only its consequences, the phenomena of life, are observable.

Living Mirrors and Mites

This chapter examines Leibniz’s comment on fragment 22 of Pascal’s Pensées in the Port Royal Edition (currently Lafuma §199). Leibniz responds to Pascal’s employment of the infinitely large and infinitely small, and to the way he uses infinity to describe living beings, through the example of a mite (ciron). In contrast, Leibniz invokes the image of a living mirror (miroir vivant). The author argues that, in spite of superficial similarities, Leibniz’s use of infinity to define living beings stands in stark contrast to Pascal’s use of infinity, in that it stresses unity and harmony rather than divisibility and disparity. Leibniz’s use of infinity through the notion of a living mirror suggests that each individual forms an integral part of a well-connected and harmonious system. While Pascal uses infinity to highlight our alienation and incomprehension of the world, for Leibniz, infinity serves instead as a mark of unity, connectedness, and belonging.