By delaying this issue until Question 11 in the First Part of Summa Theologiae (and chapter 42 in the SCG, I), Thomas is indicating that it is easier to prove that there is at least one God than it is to prove that there is no more than one. It is also important to realize that the ‘one’ that we appeal to in proving God’s unity is not the number 1, but the concept of oneness that is convertible with being. God is supremely one precisely because He is supremely a being. We don’t count gods in the same way that we can count apples or doors. As we shall see, the nature of God excludes the very possibility of there being two or more gods. At the same time, since the oneness involved is not that of the numerical one, Thomas is leaving open the door for God’s being, in some sense, a plurality or multitude. This will help in working through both the problem of God’s ideas  and the Trinity.
 
In Summa Theologiae I, Q11, article 1, Thomas explains that the focal meaning of ‘one’ in ‘there is one God’ is that of being undivided. God is supremely indivisible, and this follows from His simplicity. Thomas argues that if oneness were not convertible with being, an infinite regress would result. If oneness is not equivalent to being, then it would have to be something that is added to being. But then we can ask what makes this addition one addition, and an infinite regress follows.
 
But what about multitudes, like crowds of people? If they exist, they must also be one. And, indeed, a crowd is one crowd. But how can something be both one and many? Isn’t that inconsistent? Thomas anticipates here an answer given much later by the German logician Gottlob Frege. A multitude is one in one way, and many in another. It is, for example, one crowd but many people. Thomas elaborates this point in article 2, again distinguishing between one as the principle of number and one as convertible with being. We can ask for the number of a crowd, and this question must appeal to some way of dividing the crowd—into families, or individuals, or human cells.  At the same time, a crowd must be, like anything that exists, in some more basic sense one thing. Some more recent metaphysicians, following the work of American logician George Boolos, have suggested that multitudes can exist without being one thing at all. Very large proper classes, like the class of all sets, for example, seems to be a real multitude that is not in any sense one thing. If this is right, it could create some difficulty for some of Thomas’s proofs (in article 3) for God’s oneness. Or, it might simply point to the fact that the oneness of God is consistent with His comprising a kind of multiplicity, so long as these things are not parts or attributes of God.
 
In objection 4, Thomas addresses the problem of the definition of one as undivided. Being undivided is a negative notion, signifying the absence of division. But being is perfectly positive. So, how can being be convertible with oneness? Thomas replies that division is prior to one only in the order of our understanding. Ontologically, being undivided is prior to being divided. It’s just that we are first aware of composite things before we are aware of their simple parts.
 
In article 3, Thomas offers three arguments for the oneness of God. The first argument appeals to God’s simplicity. God is made to be God by His divine nature, and that divine nature also makes Him exist as a particular being. For there to be two gods, there would have to be two divine natures, each of the same species. But for two natures to exist with the same species, there would have to be something responsible for making each distinct from the other. So, for example, two men can be two by virtue of being combined with two packets of prime matter. Two packets of prime matter have no actual nature of their own, and so they can be fundamentally or primitively distinct. The divine nature is an actual nature (it is maximally actual), and so two divine natures cannot be fundamentally distinct. Since God is identical to His own nature, there cannot be two instances of the divine nature, just as there cannot be two instances of a single angelic species.
 
In the second argument, Thomas appeals to God’s infinity. (This is a new argument, not present in the SCG.) In fact, he appeals to the infinity of God’s perfection, by which he means that nothing can be superior to God in perfection. Suppose that there were two such maximally perfect beings. In this argument, Thomas concedes (for the sake of argument) that there could be two distinct species of god. If there were two such species, something would have to differentiate them. One would have to have something that the other did not have. But this means that one would have to have some form of perfection that was lacking in the other. But God has all perfections. So, in fact, this argument actually appeals to God’s perfection, rather than His infinity.
 
Third, Thomas appeals to the apparent unity of the world. This is one of the relatively few cases in which Thomas appeals to some form of the Fifth Way—pointing to God as the cause of the world’s systematic harmony, the fact that the active and passive powers of the world’s created substances fit together in order to make a stable, scientifically intelligible universe. Thomas gives a more detailed version of this argument in Summa Contra Gentiles 1.42, paragraph 7.
 
In Summa Contra Gentiles 1.42, paragraph 5, Thomas also appeals to some details of Aristotle’s natural philosophy, especially the assumption that the movement of the heavenly spheres are regular and continuous. It’s not clear to me whether any of this can be salvaged, given the falsity of Aristotle’s astronomy.
 
In paragraph 8, Thomas argues that if there were two gods, at least one would have to be composite. But no composite being could be necessary through itself (as established in the Third Way). It’s not obvious here why one of the gods would have to be composite, but the two arguments that Thomas gives in the Summa Theologiae support this premise: either the two gods would belong to the same species, in which case each would have to have a part that individuates it from the other, or they would belong to two species, in which case each would have to add some differentia to their common genus.
 
Paragraphs 9-11 contain another interesting argument that appeals to God’s necessity of being. This argument involves a complex dilemma: if there were two necessary beings (each necessary per se), then either (1) the two differ by something required for the completion of the necessity of being, or (2) not.
 

• Assuming (2). (Paragraph 9) If not, then they are different by something accidental. This accident must have a cause (since all accidental additions must be caused), which is either (a) the essence of the necessary thing, or (b) some external thing. We can rule out (a), since both necessary beings would have the same accident, if it is a proper accident determined by their common essence. If (b), then the existence of two necessary beings depends on the action of this third thing. But then the existence of each thing would also depend on the third thing, and so neither could be necessary in itself.
•  
• Assuming (1). (Paragraph 10) That which differentiates the two necessary beings would be either (a) something included in the common nature of necessary being, or (b) something that differentiates the two natures into two species. Case (a) is impossible because in that case, the factor will be present in both necessary beings. In case (b), the common nature must be made actual by each of the two differentiae, in which case, each necessary being would derive its existence from something else. Hence, the two things cannot be necessary in themselves, nor could each be simple and identical to its own act of existence. This complex argument seems to be mirrored in the first two arguments in the Summa Theologiae. The SCG helps us to see that these are really two parts of a single argument, based on a dilemma (either one species of god or two).

 
Paragraph 12 involves a very simple argument, based on the thesis that God is identical to His nature and to His act of existence. If there were two gods, then each god would have the same divine nature. But this divine nature would then have to be identical to two distinct acts of existence. But nothing can be identical to two distinct things.
 
Paragraphs 13 and 14 contain another complex argument.
 

1. If x is a per se necessary being, then x’s necessary being belongs to x insofar as it is x. That is, the necessity of x is grounded by x’s being x.
2. Suppose there were two per se necessary beings, x and y. Then x’s being x would also consist in x’s not being y, since if x and y are distinct, x’s being x depends on its not being y.
3. Consequently, x’s necessity would belong to it insofar as x is distinct from y.
4. This means that x’s necessity depends on the existence of y.
5. Consequently, x’s necessity would not be per se. Contradiction
6. So, there cannot be two per se necessary beings.

 
Thomas defends premise 1 in paragraph 14. If x’s necessary being depends only on x, then x’s necessary being must belong to x insofar as it is x.
 
In paragraph 18, Thomas appeals to the fact that God is supreme being. Since being is convertible with oneness, God must be supremely one, and so undivided. This applies equally well to the divine nature.
 
Thomas also appeals in paragraph 20 to the superiority of monarchy as the form of government. Since God is the perfect governor of the universe, God must be one.

As I explained in an earlier post, the use of prime matter as a way of grounding numerical distinctions between substances in the same infima species results in an attractive form of "moderate realism," with a powerful explanation both for specific sameness and for intra-specific plurality. There is, however, an obvious alternative: the use of Duns Scotus's haecceities, which are properties of primitive thisness. On Scotus's picture, what individuates Socrates and Plato is not their respective prime matters but two special properties, one of Socrateity and another of Platonity, properties that, as a matter of metaphysical necessity, can only be instantiated by Socrates and by Plato (respectively), and which Socrates and Plato instantiate whenever they exist.
 
Is there any way to evaluate the two proposals? I think there is: we have to look at the problem, not only of individuating whole substances, but also of individuating all of the quantitative or material parts (both actual and potential) of substances. Consider, for example, a perfectly symmetrical starfish. Each sector (one-fifth) of the starfish is (let's suppose) qualitatively and functionally interchangeable with the other four sectors. The sectors are conspecific as parts of the whole. Thus, we must individuate them --find something that can ground their mutual distinction. For Thomists, this is no problem. The prime matter that individuates the whole starfish is infinitely divisible: it contains five non-overlapping parts that can also individuate the five sectors from one another. There is no redundancy involved: the same thing that (as a whole) individuates the whole starfish from other starfish in the same species also individuates (as the sum of five parts) the five sectors from each other.
 
Scotists will have to posit haecceities, not only for the whole starfish, but also for each of the five sectors (with six haecceities in total). Consider a perfectly homogeneous bronze sphere. We will need haecceities for every part of the sphere--for every hemisphere that it contains, for example. Thus, we will need an infinite number of different haecceities for each such sphere (assuming the continuity of matter).
 
So far, this seems like a tie. Both Thomists and Scotists require an infinite number of individuators. In fact both require the same infinite cardinality, probably aleph-one, the number of connected, smoothly-bounded spatial regions. There is, however, a crucial difference. The Scotist account involves an infinite number of redundancies that are avoided entirely by the Thomist account. Consider again our starfish. Once we have the five haecceities for the five sectors, the sixth haecceity (for the whole) is entirely superfluous. The five sector-haecceities suffice to individuate the whole starfish from all other starfish. In contrast, the Thomists have just one thing (the whole mass of prime matter) that simultaneously individuates the five sectors from each other and the whole starfish from other starfish.
 
To match the elegance of the Thomist approach, the Scotist would have to posit that each haecceity is composed of an infinite number of "smaller" haecceities, each corresponding to a proper part of the whole. However, it is far from clear that properties can be composed in this way. And, if we suppose that they can, then the whole difference between haecceities and prime matter seems to disappear. The difference threatens to be merely verbal.

 As I mentioned in my last post, I do not think that parcels of prime matter persist for longer than an instant. How then can prime matter fulfill its traditional role (from Physics, Book II) of being the enduring substrate for substantial change: that is, being the persisting subject that is successively informed by one substantial form and then by another? In my view, it is not prime matter as such that persists, but there is a derived entity, which we can call "persisting matter" that does do so. Persisting matter is an example of what Roderick Chisholm called an 'ens successivum'. That is, a gob of persisting matter is fundamentally a series of packets of instantaneous prime matter that are tied together by a series of causal connections. These causal connections constitute what Hans Reichenbach termed a 'genidentity' relation between pairs of packets of prime matter. Two packets of prime matter count as successive versions or temporal parts of persisting matter when they are linked together by an appropriate causal relation.
 
When one substance is corrupted and another generated in its place, there is always a third entity involved: the agent of the corruption/generation. This third entity must be disjoint from the other two--neither generated nor corrupted at the moment of substantial change, but persisting through it. The agent acts upon the prime-matter packet that constitutes the corrupting substance, replacing it with a new prime-matter packet that constitutes the new, generated substance. This causal relation ties the two packets together into a single stream of "persisting matter". In the natural order, corruption and generation cannot occur unless both the old and new substance contain prime matter, and some external agent causes the replacement of the old matter by the new.
 
Let's consider a less drastic case of substantial change: substance A grows by incorporating part of substance B (which shrinks as a result). In this case, there are just two substances involved, since A can be the agent of its own growth, acting on part of the prime matter of B, causing it to be replaced by new prime matter that constitutes the new part of A. Again, this causal connection will tie together the old packet of prime matter and the new. We don't need a fundamentally persisting entity at the level of prime matter.

One of the more controversial and difficult topics in Aristotelian/Thomist philosophy of nature is that of prime matter. According to A&T, material substances are in some sense "composed" of substantial form and prime matter. Prime matter is somewhat mysterious stuff. Thomas tells us (in De Ente et Essentia) that it cannot be defined or understood in itself but only in relation to form and the composite substance. Thomists often refer to prime matter as "pure potentiality", as though it had no actuality at all. Moreover, prime matter is supposed to play some central role in individuating material substances (that is, in grounding the numerical distinctness of distinct members of the same infima species), even though prime matter is not itself an individual, and even though all the world's prime matter is in some sense numerically one. How to make sense of all this?
 
In my own view, the primary role of prime matter is that of individuation Two "packets" of prime matter are primitively, fundamentally numerically distinct. Their distinctness is not grounded in anything else. Thus, prime matter has something actual (even necessary) about it--namely, its numerical identity to itself and its numerical distinctness from all other packets of prime matter (whether actual or merely potential, past, present, or future). In addition, the mereological facts about packets of prime matter (i.e., which packets are proper parts of which, which pairs of packets overlap) are themselves necessary truths (and hence, actual truths). Prime matter is a mass of gunky bare particularity. There are no atomic bits of prime matter--each packet of prime matter is divisible into smaller bits.
 
Thomas explains, in De Ente, that it is signate matter that individuates--prime matter considered under "determinate dimensions". What can this mean? Some commentators suppose that packets of signate matter are individuated by their accidents of spatial extension (shape, size, location). However, this leads to a vicious circularity: signate matter individuated by quantitative accidents of space, accidents of space individuated by composite substances, and substances individuated by signate matter. I would propose that a packet of signate matter is simply a packet of prime matter, and that packets of signate/prime matter are primitively individual. Remember that prime matter can only be understood relationally. So, a packet of signate matter is simply some prime matter considered as the individuator of some quantitative accident of space. It is the packet of matter that individuates the accident, not vice versa. But it is in relation to that individuated accident that we are able to define and understand a particular packet of prime matter. Individuation is a metaphysical relation, while understanding is a rational or epistemic relation. The packet of prime matter is metaphysically prior to the spatial accident, but the spatial accident is prior in the order of understanding.
 
St. Thomas does not say this, but I think that the "determinate dimensions" of a packet of signate matter must include a temporal dimension as well as a spatial one. That is, I propose that all packets of prime matter exist only for a single instant. This is because substantial form is the principle of rest/motion, and so it is also the principle of persistence. Prime matter as such cannot persist.