Book Lambda begins with an overview of philosophy that stands somewhat independently of the rest of the Metaphysics. Aristotle re-emphasizes the primacy of substance and explains that there are three kinds of substance: two kinds of perceptible substances, perishable or imperishable, which are the subject of natural science, and substance that is immune to change, which is the subject of logic and mathematics.
Theology investigates the question of whether there is some common source to all substance, and Aristotle identifies this common source as a divine “prime mover.” There must be some kind of eternal, unchanging substance because the earth and time are not perishable, so there must be some substance within them that is also imperishable. This eternal substance has no potentiality, but only actuality, and its perpetual actuality makes the world eternal as well. This eternal substance must also be the prime mover, the source of all movement and change in the cosmos. To be the primer mover, this substance must itself be unmoving. The prime mover is an object of desire for the heavenly bodies, causing them to move. The prime mover is an object of desire only because it is supremely desirable, so it must enjoy the best possible life. Aristotle hence identifies this prime mover with a benign God, who spends his time in contemplation of contemplation itself. Aristotle wavers between saying there is a single prime mover or multiple prime movers. If there are many, their number, based on astronomical calculations, is either 47 or 55. The prime mover contemplates contemplation because anything lower would be unworthy of it and anything higher would imply that there is something more desirable than the existence of the prime mover himself. Because the prime mover is good, this means the universe as a whole is good.
Books Mu and Nu consider the metaphysical status of mathematics, and Aristotle concludes that mathematical entities are not substances. Aristotle attacks in particular Plato’s view that each number corresponds to a Form, primarily because this view obscures the relationships between numbers and fails to explain the relationship between numbers and sensible particulars. Aristotle suggests instead that numbers are physical objects considered in abstraction from their physical and accidental properties. For example, the number five is the same thing as five cats once we factor out everything that makes the cats cats instead of something else. Aristotle concludes by rejecting the idea that numbers can play a causal role in nature, reaffirming his view that substance is at the foundation of nature.
In arguing that actuality is more fundamental than potentiality, Aristotle effectively argues that the chicken comes before the egg, as one commentator puts it. He is telling us that an object can only be a potential something if there is already an actual something for that object to become. This claim has the paradoxical result that, for instance, the chicken must already exist for the egg to be a potential chicken. Of course, it is obviously false that individual chickens precede individual eggs: every chicken that now exists must have been an egg at some point. However, according to book Zeta, individual chickens are not substances. The species of chicken is a substance, and there can be no chicken eggs until there is a species called “chicken” for those eggs to become. After all, we cannot point to an object and say, “that is a chicken egg” if there is no such thing as a chicken. Substance is the most fundamental thing there is, so substance must be an actuality. Since, as Aristotle has argued earlier, nothing can exist unless substance exists, that means that potentialities cannot exist unless their actualities as substances already exist.
Aristotle’s discussion of actuality and potentiality in book Theta provides an important link between the discussion of substance in books Zeta and Eta and the discussion of theology in book Lambda. In book Zeta, Aristotle tells us that species are substances, so the universe is fundamentally made up of the sorts of things we find in the world around us. In book Theta, he explains that substances are fundamental because they have actuality: they are what other things are trying to become. With actuality added as a new and important criterion for substancehood, we can infer that the most fundamental substances are completely actual, with no potentiality. Substances such as humans and chickens have potentiality in the form of fetuses and eggs, so they are not completely actual. In book Lambda, Aristotle suggests that there are also eternal substances and that these are more important and more fundamental than the species of the world around us because they are only actuality, with no potentiality. The concept of actuality, then, points to a way in which the material substances discussed in book Zeta fall a bit short of the fundamental role Aristotle wants them to play, and the theological discussion of book Lambda must make up for this shortfall.
When Aristotle talks about a prime mover or a first cause, he means that this mover or cause comes first conceptually rather than chronologically. That is, we should not imagine a universe at rest however many billions of years ago that is then set in motion by the prime movers: they do not come first in the sense that they are what first set everything in motion. On the contrary, Aristotle argues at a number of points in the Physics and Metaphysics that time is eternal, so that there is no beginning to time. Rather, we should think of the prime movers as first conceptually. For example, we could ask why the soccer ball is rolling and say that Ronaldo kicked it. We could then ask why Ronaldo moved his leg and say that he felt a certain desire. We could then explain the desire by appealing to certain causes in Ronaldo’s life, and so on. So the movement of the soccer ball could be explained by the movement of Ronaldo’s leg, by Ronaldo’s desire, by Ronaldo’s life story, and so on. The deepest explanation of any movement, says Aristotle, are these prime movers. They are not first in time so much as they are the ultimate explanation one can appeal to in explaining any movement.