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An exposition of Maximus' understanding of the Trinity. The Trinity is a Triad in Monad and a Monad in Triad. The Triad-in-Monad is at once both united and distinguished and there is no confusion or separation in it. A balance between essence and hypostasis is at the heart of this doctrine.

This chapter presents a third characterization of adjunctions. This one has the virtue of being entirely equational. Topics discussed include the triangle identities, monads and adjoints, algebras for a monad, comonads and coalgebras, and algebras for endofunctors. Exercises are given in the last part of the chapter.

While some 17th-century critics of Epicureanism, including Margaret Cavendish, focused on the impossibility of a structured and orderly world emerging from the purposeless collision of atoms, other critics challenged the coherence of the notion of a material particle as a fundamental building block, hoping to extirpate atheism and materialism at their source. Leibniz evolved an unusual scheme of immaterial atoms, which he termed ‘monads’. Monads were mind-like entities, dimensionless, devoid of physical properties such as shape, impenetrability, and location in absolute space, and differentiated by their experiences. Berkeley went further in proclaiming matter an incoherent idea and insisting that only ideas, God, and the human will really existed, and that the ‘external’ world was in fact in the mind.

This chapter analyzes Plotinus’ refutation of the Aristotle’s criticism of Plato’s view of number in the Parmenides. By rejecting any quantitative value of number in the intelligible realm, Plotinus specifically focuses on Aristotle’s inability to understand the Monad and the Indefinite Dyad as the principles of creation and order of the intelligible. The chapter shows that Plotinus not only follows the steps of his Platonic and Neopythagorean predecessors in defence of Plato’s position, but cleverly uses Aristotle’s own ideas in arguing that number in the intelligible is activity and a property of primary substance (ousia). The result is an original and ontologically elaborate theory of substantial number which offers a new and more successful defence of Plato’s “true numbers” against Aristotle’s criticism and explains the relationship between substantial non-quantitative number and monadic quantitative number as that between intelligible paradigm and its material copy.

This chapter examines the relationship of Substantial Number and all intelligible entities: Absolute Being as “unified number;” Intellect as “number moving in itself”; beings as “unfolded number;” and the Complete Living Being as “encompassing number.” A closer examination reveals that the four aspects of substantial number correspond to Plato’s primary kinds of rest, movement, otherness, and sameness respectively. The chapter concludes that the properties of substantial number enact the above four primary kinds, while the fifth primary kind, that of being, is the common denominator which represents them all. This conclusion also elucidates the relationship of substantial number with the One and Soul, as Plotinus does not identify them with a particular property of number. Since the properties of substantial number order Intellect on the inside and since both the One and Soul stand on the outside of Intellect, they do not inherit a particular property of number. Substantial number is an ontological expression of the One “beyond Being,” which does not possess any particular characteristic. Soul, however, as an image of Intellect, possesses all the properties of substantial number.

This chapter looks specifically at Leibniz's conception of mathematical structure—an ideal structure underlying the very possibility of the actual world. Both his early and late views are considered. The chapter shows how it is that Leibniz employed mathematics as a metaphor aimed at explicating his metaphysical views. The metaphysical concepts of unity and harmony, for example, are made clearer by looking at how Leibniz conceived them via the mathematical concepts of function and determinant.

This chapter begins with a discussion of real unities, entities without parts, and simple substances, the latter terms which seem to have entered Leibniz's vocabulary in 1695 or so. It then turns to Leibniz's discovery of monads. By 1700 or so, Leibniz had clearly committed himself to the new metaphysics of simple substances, substances without parts, what he had come to call monads. This chapter traces how the terms simple substance and monad are introduced, and how Leibniz moved from the metaphysics of corporeal substance in his middle years to the monadological metaphysics of his last years. Special attention is paid to the ‘Système nouveau’ of 1695 and the correspondence with Burcher de Volder.

This chapter examines Leibniz's treatment of body in the context of a world grounded in monads. It examines a variety of ways in which Leibniz tries to integrate bodies into the world of monads, and a variety of ways in which he treats the notion of corporeal substance in the context of the monadological metaphysics. These views range all the way from views that seem to eliminate bodies altogether to views in which monads seem to be eliminated in favour of bodies and corporeal substances. In the end, it is proposed that Leibniz left behind many loose ends in his thought, questions he considered but did not resolve. In his later years, he was struggling with the problems and struggling toward a considered view on these issues. But he died before he got there.

In the prologue of his commentary on Plato’s Timaeus Proclus defines Nature as the last of the productive causes of the corporeal, as an incorporeal essence inseparable from bodies. Nature is both an immanent (following Aristotle) ontological principle and a transcendent one (following Plato). It is an intermediate hypostasis between soul and the corporeal, possessing the reason-principles of sensible bodies—an hypostasis which Proclus identifies with the Divisible Essence that becomes in bodies, mentioned at Timaeus 35a2f. Immanence and transcendence can here be reconciled by the distinction between Nature as monad and the many natures dependent from this monad. Another important point is the distinction between nature qua nature and nature qua soul, intellect, One. Nature is so defined as ‘divine art’ and ‘instrument of the gods’—an instrument which is not deprived of self-motion and is integrated in the divine order.

On a number of occasions Leibniz claims that the physical world is the realm of efficient causation, and that monads are the realm of final causation. Recent scholars have asked what Leibniz has in mind when endorsing final causation, especially in light of the general rejection of final causation by other early moderns. This chapter illuminates his notion of final causation in the context of his use of the notion of substantial form, and what that notion means in his thought.

Xenocrates, who had accompanied Plato on one of his visits to Sicily, became head of the Academy in 339 B.C. Xenocrates stays close to what he takes to be the cosmological doctrine of Plato's Timaeus; indeed Xenocrates’ doctrine may be seen as something of a retreat from Speusippus’ radical position, perhaps in response to Aristotle's criticisms. Dillon reconstructs Xenocrates's cosmological or metaphysical scheme as comprising a pair of first principles, the Monad, or Nous, and the Dyad, or the ‘Everflowing’, to which the Pythagorean tetraktys corresponds as the active counterpart; and a World‐Soul, which receives the forms from the Supreme God's mind, and projects them upon the physical plane. In Logic, Xenocrates remained faithful to Platonic logic, rejecting the Aristotelian categories, although he did argue that the species was prior to the genus; in Ethics, while keen to formalize Plato's teachings, Xenocrates ends up with a position very similar to Aristotle's, in that he emphasizes the needs of the body as well as those of the soul. Xenocrates had a dominant effect on the development of Platonism, because he systematized what he took to be Plato's philosophical system, thus laying the foundation for the ‘Platonic’ system of philosophy; it is Xenocrates’ definition of Form, for instance, which became the standard definition of a Platonic Form.

This chapter addresses an aspect of Leibniz’s metaphysics that has not received much attention from scholars, namely, the doctrine of monadic domination. More specifically, it offers an account of what monadic domination, understood as a relation obtaining exclusively among monads, amounts to in the philosophy of Leibniz. It argues that Leibniz’s conception of monadic domination is usefully understood by appeal to the Aristotelian notion of a hierarchy of activities. It offers a fairly detailed account of what goes on in the monads that make up an animal when that animal performs some physical action. And it further argues that an understanding of Leibniz’s doctrine of monadic domination affords us some insight into his conception of how God ordains an end for his creation.

Leibniz often speaks of “corporeal substances.” According to many texts a monad and its organic body are both constituents of a single corporeal substance. This chapter explores the relations among them, and argues for an interpretation of Leibniz's corporeal substances as exhaustively constituted by relations of harmony among simple substances, and thus as consistent with his idealism. It argues also that after 1706, when it came to seem doubtful to both Leibniz and his contemporaries that such an interpretation could provide adequately for the strong unity that all agreed a corporeal substance must have, he tended to back away from the notion of corporeal substance rather than abandon his view of the world as composed ultimately of monads.