My recent “three principles” post arose from a lengthy discussion I had with reader StructureOfTruth in the comments of an earlier post on limitation. Thankfully, StructureOfTruth continues to keep me honest, and has raised the following concern in the comments on my “three principles” post:

The greatest difficulty I have with your line of thought in this post is (still) this idea of “infinite/unlimited/complete/perfect” vs. “finite/limited/incomplete/imperfect”. It seems to me that this is subtly wrong. Looking closely at your argument for the infinity principle, I believe the correct thing to say is not that unreceived act is infinite/unlimited, but rather that unreceived act is indefinite or indeterminate. This is what all of your examples actually support. If something is neither Y nor not-Y, it is not thereby less limited, but less definite.

As usual, I find his way of framing the concern to be very helpful, and it gives me an opportunity to explore more deeply a topic that I did not directly address in that post. In fact, in touching on this we are getting at an important feature of Thomistic metaphysics. As far as I know, if you asked Aristotle or Plato about what it means to be infinite, they would say that infinity involves radical incompleteness and indeterminacy. Aristotle’s matter is infinitely indeterminate between various material forms, the infinitude of numbers results from the fact that the collection of them is never completed, and the infinite of the continuum results from the indefinite divisions of the whole into (incomplete) parts. Infinity was not associated with notions like completeness, perfection, or determinacy, until the neo-Platonists. As W Norris Clarke explains:

The search for such a framework forces us now to leap five centuries down to Plotinus and Neoplatonism. It was in this profoundly different intellectual and spiritual atmosphere that there appeared for the first time in Western thought a doctrine of participation linked with a wholly new concept of infinite and finite, correlated now with the perfect and the imperfect respectively in a complete reversal of the age-old classical Greek tradition.^[1]

One of Aquinas’s important contributions to metaphysics is the unification of this Neoplatonic participation framework with the Aristotelian composition framework. In the former we study the limited “taking part” (participating) in the universal, while in the latter we study the composition of act and potency (as discussed in my “three principles” post). While Aristotle largely eschewed talk of participation, Aquinas showed that we can develop the Aristotelian theory of act and potency to the point where we can articulate the notion of participation within it, allowing us to reap the benefits and insights of both frameworks. Key to all of this was Aquinas’s own account of esse and its relationship to essence, which is no doubt why it seems to show up in so many Thomistic metaphysical discussions.

We will work our way towards the Thomistic account of participation, starting off with the StructureOfTruth’s concern. As a reminder, we have three things in view. There is (1) a chair in potential to being different colors, (2) an act itself, redness as such, which can be received into various things, and (3) the chair insofar as it has received this act, the red chair. Both redness as such and the chair in potential are in some sense open to various things: redness as such is open to being received into this or that object, shape, size, etc., while the chair is in potential to being this or that color. However, in my “three principles” post, the chair is described as indeterminate while the redness as such is described as (relatively) infinite. Why? The reason comes down to the fact that they are open in fundamentally different ways. Potentials make things indeterminate with respect to their being, so that when they receive act they thereby become something, as when the chair receives redness it becomes red. But when an act is received into this or that, the act itself does not thereby become anything new—redness as such does not become “chairful” when it is received into a chair, otherwise every red thing that would thereby become a chair. Rather, what happens is that a separate, more limited version of redness is realized, which is not redness as such but this or that redness.

Generalizing this, we can say that whenever X is open to Y and not-Y, then either (1) X itself is incomplete and completed by them, or (2) X itself is complete and not completed by them. In the former case X is in potential to Yness and X becomes Y when this potential is realized, as when a chair becomes red. In the latter case, X is in act apart from Y, and so does not become Y, as when redness as such does not become chairful. We may describe these cases respectively as (a) imperfect or incomplete infinity, because a thing in potential becomes more fully actual by receiving some act, and (b) perfect or complete infinity, because an act as such is already fully actual in the relevant sense. In general, infinity or unlimitedness is opposed to being bounded or constrained through the imposition of limits, whereby a thing becomes finite. Imperfect infinity was known to Plato and Aristotle, the latter of whom only recognized the limiting role that must therefore be played by act. The Neoplatonists discovered perfect infinity, which Aquinas was able to characterize in terms of act. This is what enables us to state the infinity principle (the third of the “three principles”), for otherwise it would make no sense to speak about unlimited (or infinite) act at all.

It seems to me that indeterminacy, unlike infinity, can only be used in connection with imperfection. After all, indeterminate things become determinate, and are therefore incomplete insofar as this is not the case. The chair is indeterminate between various colors, but redness as such is determinate, and that which determines the chair. Nevertheless, there are ways of conceptualizing redness in ways that are indeterminate. This happens, for instance, when we abstract away the details of a particular reception of red into this or that thing. There is this red chair, there are red chairs in general (which includes this and other chairs), there are red objects in general (which includes chairs and humans and paintings and more), and then finally there are red things general (which also includes waves of light, and other non-object forms of redness). What we are doing here is conceptualizing red Xs, where X becomes increasingly general as we conceptualize the possible subjects of redness in more indeterminate ways. And like all cases of indeterminacy, our conception becomes more determinate as we fill in the details in particular cases. This generalization through abstraction results in a less and less complete conception of redness applied to a thing through a less and less detailed conception of a thing. But we must not confuse redness as applied to a thing, at whatever level of generality we like, with redness as such. The former is a conception in potential to further specification while the latter is an act in itself. The former is a more abstract conception of something insofar as it has received this act, the red thing; the latter is an act itself, redness as such, which can be received into various things. The former, we said, is a separate and more limited version of redness as such, which we just conceptualize at different levels of abstraction.

Now, if redness as received is a separate and more limited version of redness as such (that which is received), then what is the relationship that obtains between them? Instead of composition—which obtains between redness as received and the chair which receives it—we now have participation: this or that redness “takes part in” the fullness of redness as such. In his exposition on Boethius’s “On the Hebdomads”, Aquinas articulates three ways in which one thing (the participant) is said to participate in another (the participated). While he does seem to consider this division as mutually exclusive and jointly exhaustive, there is unfortunately much disagreement over how to understand it, since he only gestures at the underlying principles which generate it. Here I will offer what I take to be the most plausible account, and perhaps in a future post I will lay out the details of the exegesis which leads me to this.

To participate, says Aquinas, is “to grasp a part”. Going into a bit more detail, Koterski says that it is “to receive only part of what belongs to another fully, and so merely to share in it without exhausting it.”^[2] In his commentary on Colossians, Aquinas goes further:

A thing belongs by participation to another if it surpasses the nature of the thing which has it, although the thing participates to a certain extent in it, although imperfectly; thus man is intellectual by participation, while to be intellectual, which is superior to being rational, is in the angels essentially. (Commentary on Colossians, §41)

It is because some X of the participated surpasses the nature of the participant in some way that the latter cannot attain it fully, completely, or perfectly. Neither, however, is it the case that the participant can in no way attain it. Rather, we have a middle ground, where the participant can attain it partially, incompletely, or imperfectly. What we’re talking about here is not exactly mereology, although there is some overlap. Whether or not a specific participation amounts to parthood will depend on the X in which it shares, and the manner in which the participant has X in relation to the X of the participated. Something which is highlighted in Koterski’s definition, and which becomes clear in Aquinas’s discussion, is that the participant receives the X of the participated, and is therefore in potential to it (the reception principle). This gives us some insight into what it means for the participant to attain the X imperfectly, since whatever is received is received according to the mode of the receiver (the modality principle). The mode of the receiver is what imposes a  qualification, or limitation, on that which is received, so that the result is partial, incomplete, imperfect.

The first mode of participation is abstract-abstract, by which I mean that both the participant and the participated are abstract conceptions. With Aristotle, Aquinas agrees that abstract objects as such do not exist outside of our minds, but rather result when the intellect attends to some features of a thing to the exclusion of (or, in abstraction from) others. In short, abstract objects are abstracted objects. As we abstract more away, the resulting intelligible content (ie. the content of the concept) becomes more universal, and conversely the more we add to it the more particular it becomes. Thus, abstract-abstract participation has to do with the degree to which the intelligible content of an abstract concept is universal, or common to many things. Aquinas gives two examples: “human being is said to participate in animal because it does not possess the intelligible structure of animal according to its total commonality; and in the same way, Socrates participates in human.”^[3] Two things are worth noting here.

First, we shouldn’t be misled by the example of Socrates participating in human, as if Aquinas has something other than abstract-abstract participation in mind. Even though individuals like Socrates are concrete things, they may nevertheless be considered in the abstract, as we do when we consider the character of Sherlock Holmes or Batman in abstraction from incarnations in this or that story. In the present case, the abstraction occurs by virtue of the fact that we are focusing specifically on the scope of the individual’s intelligible content—insofar as Socrates’s humanity is this or that humanity rather than humanity as such. As human is to animal, the intelligible content of Socrates is less than the full scope of human, the latter of which covers all humans whatsoever.

The second thing to note is that this mode of participation is the inverse to the way we would usually think about abstract concepts. It is natural to think that the more abstract a concept is, the more potential it is. A genus, for instance, is in potential to various species, each of which is actualized when the genus receives its specific difference. However, in abstract-abstract participation, the species is what receives intelligible content from the genus, and therefore is in potential to it. Since we’re operating within the freer domain of our conceptions, there’s no problem with this sort of symmetry—it’s a straightforward consequence of the freedom that we have when abstracting different aspects of a concept. I raise it simply to highlight what might otherwise be a source of confusion when thinking about this mode of participation.

The second mode of participation is concrete-abstract participation. Aquinas: “a subject participates in accident, and matter in form, because a substantial form, or an accidental one, which is common by virtue of its own intelligible structure, is determined to this or that subject.” As a thing open to receiving the form, the concrete participant is not intelligible in itself, but merely serves to concretize and individuate the intelligibility inherent in that which it receives. Of course, both subject and (secondary) matter will have some intelligible content, insofar as they are in act. But in this second mode we are considering them specifically insofar as they receive something from the participated, and therefore insofar as they are in potential rather than act.

Now, it might seem a bit strange to speak of concrete things (which are real) participating in abstract things (which are not), but this really comes down to the two ways in which an accident or form can be understood. This wooden square and that metal square are really two distinct things, each with its own squareness. But there is also something they have in common, which we grasp by abstracting away the individual details in each square, and seeing the same intelligible content arising in both cases. Thus, there is square in concrete, which is received by the wood or metal, and the square in abstract, which is common to both. There is the concrete actuality which is the ontological ground for the intelligible content we grasp abstractly. It is this actuality which serves as the “concrete intelligibility” that the otherwise unintelligible matter receives, and thereby participates in the more common abstract intelligibility. This is essentially the Aristotelian account of immanent forms, now put to work in the Thomistic account of participation.

The third mode of participation is concrete-concrete participation. Up until this point the participated has always been something abstract, so we cannot rely on the previous modes to tell us what to expect with this one. Instead we must rely on the account of participation in terms of reception: the participant receives act from the participated, and since both act and potential are concrete, this would amount to the actualization of the participant by the participated, which is just another way of describing  efficient causation. Thus, we might wish to describe this mode as effect-cause participation. Thus, says Aquinas, “an effect is said ‘to participate’ in its own cause.” Of course, participation also requires that the participated have the X which is received more fully than the participant which receives it. Thus, Aquinas adds that this mode of participation occurs, “especially when [the effect] is not equal to the power of its cause, as for example, if we should say that ‘air participates in the light of the sun’ because it does not receive that light with the brilliance it has in the sun.” It is with regard to the power of the cause, then, that we judge an effect to be impartial, incomplete, and imperfect. That is, the cause is capable of bringing about more than what is in the effect, but the modality of the effect as receiver only grasps some part of this. So, a painting participates in the skill of the painter, and an instrument participates in the activity of its agent.

In both of these cases we also see how the cause surpasses the nature of the effect, for the painting cannot paint itself and the instrument cannot move itself. But what about when the effect is equal to the power of the cause, as when one fire ignites another? It seems to me that this is still a case of effect-cause participation, albeit a lesser version of it. Once the second fire is lit it has the same power as the first, but while it is being lit it is only passive with respect to this power, and to this extent participates in it. That is, it participates inasmuch as it is an effect of the first fire. In fact, looking back at the painting and instrument, we see that they also differ from one another in this respect. The instrument as such is an effect of the agent throughout its use, and therefore participates in its agent as long as it is an instrument. The painting most truly participates in the painter’s skill only while it is coming into existence, for that is when it is receiving this skill into itself. Once the painting is complete, we continue to count it as a participant in the painter’s skill because of how it came into existence and because it does not have the same power in itself. The second fire is one more step in this direction, where it does have the same power in itself, and so the only sense in which it participates in the first fire is insofar as it was brought into existence by it.

These three modes of participation are certainly mutually exclusive, since they each involve different types of things participating in different types of things. They are also jointly exhaustive. The only option left would be abstract-concrete participation, but the abstract will always cover more than the concrete, since we arrive at it by abstracting away details which otherwise restrict it. Perhaps we could consider the collection of features belonging to a concrete thing, in which case excluding some of these features by abstraction would result in a smaller collection, which we could then say participates in the bigger collection. But by considering these collections we have moved away from abstract-concrete participation toward abstract-abstract participation, the first mode. As we noted when discussing this, it is not enough that we are considering some concrete individual, because we can do this in the abstract. So, these three modes are mutually exclusive and jointly exhaustive. This does not mean, however, that this is the only way of dividing modes of participation. There might be an equally valid way of dividing them up which is orthogonal to the abstract/concrete rubric used by Aquinas here, but having one is valuable enough for our purposes.

Earlier, I distinguished redness as such from redness as received. Let me close by articulating the relationship between these in more detail, now that we understand a bit more about participation. We arrive at redness as such by abstracting away the subject as such, which serves to concretize the accident. We therefore have the second mode of participation in view. The chair in potential participates in redness as such, by concretizing the intelligible content of the latter, grounded in the redness as received into the chair. The intelligible content is not made more specific or determinate, since the subject as such does not contribute any intelligibility of its own but is made intelligible by the act received. Thus, in terms of abstract-concrete participation, the redness as received is what grounds the subject’s grasping in part of the intelligibility of the redness as such.

We also mentioned that we can consider the redness as received at different levels of abstraction. In this case, we must have the first mode of participation in view, because with redness as received we include the subject in our conception, since this is what makes the redness received in the first place. Another consideration in favor of thinking this is the first mode of participation is the graded nature of the participation: as with Aquinas’s example of the species human, we have things which are both participant and participated, something not possible with the second mode. As we move further up in the levels of generality, viewed through the lens of participation we are isolating the contributions to intelligible content made by each concept: an individual red chair receives the intelligible content of chairness, chairness receives the content of objectness, and so on. As mentioned in the discussion on the first mode, there is a flexibility in abstract concepts about what we consider in potential to what. Earlier we spoke of the more general being in potential to specification, but in participation we speak of the more specific being in potential to the intelligible content it receives from the more general.

Hopefully all of this clarifies some of the things left unsaid in my “three principles” post about, especially how the reasoning for the infinity principle works. I realize that I have not discussed unreceived esse in this post, but much of the reasoning I’ve covered applies there too. I plan on discussing esse in relation to participation in a future post.

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[1] W. Norris Clarke, “The Limitation of Act by Potency: Aristotelianism or Neoplatonism,” The New Scholasticism 26 (1952): 167–94.

[2] Joseph W. Koterski, S.J., “The Doctrine of Participation in Thomistic Metaphysics.”

[3] Thomas Aquinas, An Exposition of the “On the Hebdomads” of Boethius, trans. Janice L. Schultz and Edward A. Synan (Washington, DC: The Catholic University of America Press, 2001), 19–21.