I want to focus now on a specific aspect of the tri-unity matrix: reflexive relations.
[Update: I've added an illustration of what reflexive relations look like below.]
Much as with the numerous relative identify formulations introduced in analytic philosophy, there may be a sense that this is too sophisticated a concept for the various historical figures to have held. This modern sophistication conveys the impression that these early forays into Triadology were a bit embarrassing and that contemporary Christian philosophers are covering their fathers' nakedness, at least in the sense of correcting some logical mistakes that they made. I myself accept that possibility on certain matters such as the creation of the universe, the evolution of man, and even theological matters such as the the spiritual status of Judaism after the first century. What is in common in these matters is that they are factual judgments based on principles in revelation. But when it comes to fundamental matters about theology and Christology, we are dealing with principles of revelation. For those matters, "the faith that was once for all delivered to the saints" (Jude 1:3) is all we have; this is the substance of the apostolic deposit. We can only learn it from the Fathers; we cannot correct them.
This might seem a little rich from someone offering group theory and matrix representation as an explanation for what the Fathers were saying. But what Thomas Ryba offered in his article "Trinitology and the Theory of Groups" and what I offered in the tri-unity matrix are merely logical representations of reflexive relations. Given that is the case, the question is whether reflexive relations themselves are the right description of the concept they had in mind. I believe that we have good reason to think so.
A reflexive relation is, as one might expect, it is a relation that looks back on itself (Lat. "re-flectere" meaning "to bend back"). Formally, it is a relation that maps a set back to itself. The most basic such relation is equality of a real number with itself, the reflexive property, i.e., for any real number a, a = a. In most cases, reflexive relations are trivial in their application simply because they are taken for granted as axioms of any system. But in the case of the Trinity, we are already dealing with relative identify, where Leibniz's Law cannot apply, so it is not surprising that reflexive relations would end up being more a useful tool.
The function that reflexive relations serve is to make copies of sets. The name is not coincidental; mirrors bend back (reflect) images of their prototypes. That is certainly not a concept that ancient people would have had difficulty understanding. Nor is this idea of copying foreign to Scripture. Hebrews 1:3a calls Jesus "the radiance of the glory of God and the exact imprint of his nature." Colossians 1:15 describes Jesus as "the image of the invisible God."
And did the Fathers use this concept for the Persons of the Trinity? Unquestionably!
St. Gregory Nazianzen does so explicitly in the Fourth Theological Oration:
And Truth, as being in nature One and not many (for truth is one and falsehood is manifold), and as the pure Seal of the Father and His most unerring Impress. And the Image as of one substance with Him, and because He is of the Father, and not the Father of Him. For this is of the Nature of an Image, to be the reproduction of its Archetype, and of that whose name it bears; only that there is more here. For in ordinary language an image is a motionless representation of that which has motion; but in this case it is the living reproduction of the Living One, and is more exactly like than was Seth to Adam, or any son to his father. For such is the nature of simple Existences, that it is not correct to say of them that they are Like in one particular and Unlike in another; but they are a complete resemblance, and should rather be called Identical than Like.
Regarding the Holy Spirit, there is a similar passage from St. John Damascene that I saw referenced in an article by Vladimir Lossky. This quotation is found in Part III of the APOLOGIA OF ST JOHN DAMASCENE AGAINST THOSE WHO DECRY HOLY IMAGES:
The Son is the first natural and unchangeable image of the invisible God, the Father, showing the Father in Himself. ‘For no man has seen God.’ Again, ‘Not that any one has seen the Father.’ The apostle says that the Son is the image of the Father, ‘Who is the image of the invisible God,’ and to the Hebrews, ‘Who being the brightness of His glory, and the figure of His substance.’ In the Gospel of St John we find that He does show the Father in Himself. When Philip said to Him, ‘Show us the Father and it is enough for us,’ our Lord replied, ‘Have I been so long with you and have you not known Me, Philip? He who sees Me, sees the Father.’ For the Son is the natural image of the Father, unchangeable, in everything like to the Father, except that He is begotten, and that He is not the Father. The Father begets, being unbegotten. The Son is begotten, and is not the Father, and the Holy Spirit is the image of the Son. For no one can say the Lord Jesus, except in the Holy Spirit. Through the Holy Spirit we know Christ, the Son of God and God, and in the Son we look upon the Father. For in things that are conceived by nature, language is the interpreter, and spirit is the interpreter of language. The Holy Spirit is the perfect and unchangeable image of the Son, differing only in His procession. The Son is begotten, but does not proceed. And the son of any father is his natural image. Thus, the natural is the first kind of image.
What is critical to perceive about these passages is that they refer to the image of the Person producing a copy according to nature, a natural image, as a child reproduces the nature of his father. The Theologian's reference here to "simple existences" can be nothing other than the hypostatic existence of the Persons with the divine essence. In the context of what Gregory says about the names of the Persons being relations, the imaging establishes a reflexive relation, in the way a son reflects a father or an impress reflects a seal. And because the divine essence is simple, the image is not merely like but absolutely identical.
This kind of process is already implicit in Latin theology, which is why it seemingly appears to "start with the essence." That is simply another way of saying that the Latin approach focuses on the divine essence being copied from the Father in the Persons. Rather than collecting examples, I will point to a characteristic Trinitarian picture of the West: Augustine's so-called "psychological analogy." In that model, the Father thinks upon Himself, which though is the Word (by mode of intellection) and the mutual love of the Father and that Word is the Holy Spirit (by mode of willing). The number of scholarly interpretations of this concept is so voluminous that the pages are uncountable. For our purposes, it doesn't matter. The point is that the self-reflective nature of the model definitely places it in the category of reflexive relations. It is, moreover, exactly the same set of reflexive relations that Ryba outlines for Augustine's Trinitarian formalism, even though he did not specifically mention the psychological analogy.
St. Thomas Aquinas made the same assessment in the Disputed Questions on the Power of God, Part 9:
On the other hand God does not acquire knowledge from things, but by his knowledge is the cause of things: nor by his will does he tend to anything external as his end, but he directs all external things to himself as their end. Accordingly both in us and in God there is a certain rotation in the acts of the intellect and will: for the will returns to that whence came the beginning of understanding: but whereas in us the circle ends in that which is external, the external good moving the intellect and the intellect moving the will, and the will by appetite and love tending to the external good; in God, on the other hand, the circle ends in him. For God, by understanding himself, conceives his word which is the type of all things understood by him, inasmuch as he understands all things by understanding himself, and from this word he proceeds to love of all things and of himself. Thus someone has said that a “monad engendered an atom and reflected its own beat upon itself.” And the circle being closed nothing more can be added, so that a third procession within the divine nature is impossible, although there follows a procession towards external nature. Hence in God there must be but one person that does not proceed, and only two persons that proceed, one of whom proceeds as love, the other as word: and thus the persons in God are three in number.
Aquinas neatly summarizes how the psychological analogy is reflexive. God knows Himself as Father, begetting the Son, then loves Himself as Father and Son, from which the Spirit proceeds.
It is important to distinguish reflexive relations from the Neoplatonic exitus-reditus, with which it could easily be confused. Neoplatonism starts from how the One can communicate itself, which has nothing to do with the One having any manner of relationship with itself. Rather, it describes how the One relates outside of itself through Nous and Soul. There is a certain harmony between this idea and the economy, the activity of the Trinity in the world, which is why there can be Christian Neoplatonism in St. Dionysius the Areopagite and similar influences in many others. But the relations defining the Trinity is always ad intra; the relations are reflexive within God and not in any way directed out of God.
To understand the difference, one can look at the difference between the Son as the true image of God when compared to man's creation in the image of God. The concept "image" appears in both places, but the true image of the divine nature itself is only within the Trinity. The "image" found in creation is different in kind, although it is still in some sense an image. While the Christian view is compatible with Neoplatonic emanations at the level of creation, it is thoroughly incompatible with the ad extra steps through Nous and Soul from the One. The Father, the Logos, and the Holy Spirit are instead the One God.
Much as there was a legitimate fear of the Neoplatonic Dyad reflected in Eastern resistance to the Latin formulation, there is a legitimate fear of the Neoplatonic Triad in the psychological analogy. Hearing this language of intellect and will immediately connotes the production of the Nous and the Soul to someone steeped in Neoplatonism, even though that concern simply wouldn't apply to reflexive relations at all for the reasons Aquinas gives above (the relations are directed solely ad intra).
It is entirely legitimate to view these reflexive relations as imaging the divine essence. At its root, the filioque says nothing that St. John Damascene has not already said: the Son is the Image of the Father, and the Holy Spirit is the Image of the Son, the Image of the Image of the Father. And based on what St. Gregory Nazianzen says, this true (which is to say, identical) imaging cannot be any kind of energetic procession, because the identity pertains to the "nature of simple Existences" rather than the plurality of the energies. As this pertains to the "invisible God," being the Image is likewise not a showing forth or manifestation of the Trinity in the economy, although like any aspect of the Trinity, it certainly has implications for economic activity. And while it is a natural imaging, it is both personal and relational; therefore, it pertains to the mode of existence of the Persons. In short, this double imaging is a matter of hypostatic origin, not eternal energetic procession.
The double imaging is also compatible with all of the typical Trinitarian formulae. Prototype, Image, and Image of the Image is no different than Father, Word, and Love/Bond; Uncaused, Immediately Caused, and Mediately Caused (the formula St. Gregory of Nyssa uses); or Principle Without Principle, Principle From Principle, and Proceeding. The copying aspect of being Image of the Image accounts for all of the references to what the Son has or receives and gives to the Spirit. As long as the Trinitarian relations are understood as reflexive, these are all the same.
More broadly, it accounts for a number of the "authorized" analogies for the Trinity that all have in common this feature of looking back to how we arrived at a certain point. St. Gregory of Nyssa provides a helpful analogy to a plant that could have grown as a seedling or could have been planted (from the Ad Ablabium, quoted in Giulio Maspero's Trinity and Man at p. 155-56). When we talk about how the plant got there, we aren't talking about the nature of the plant, which is common, but we can still talk meaningfully about its origin or cause. In the Trinity, these analogies involve the same process of looking back: pond to stream to source, third torch lit by second torch lit by first torch, warmth to ray to sun, leaf to branch to root ... all successful analogies that involve a reflexive view of something being passed in the manner of Image of the Image.
In short, by correctly conceiving the Trinitarian relations as reflexive, every objection to the filioque is disarmed. There is no eternal energetic procession needed to explain away relationships between the Persons, no fear of the Neoplatonic Dyad or Triad introducing itself into the idea of God, and no introduction of a tertium quid into between the Persons and the divine essence. That is not to say that an eternal energetic procession is wrong; on the contrary, it is hard to see how it could not be true. But it isn't needed in order to explain away a belief no one actually held. This seems to be a successful resolution of the discrepancy of the filioque with the virtue that it actually corresponds with the thought of the Fathers, as opposed to anachronistically transporting what seems to be a working solution today back into the past.
Now, as I mentioned previously, there are alternative formulations that could work. One is Basil Lourié's quasi-ordinal set that defines relations in a Triad. A similar one is Fr. Thomas Weinandy's spirituque proposal: "We believe in the Holy Spirit, the Lord the Giver of Life, who comes forth (ejkporeuvetai) from the Father as the one in whom the Son is begotten and who proceeds (proei'si) from and through the Son in communion with the Father, and together with the Father and the Son is worship and glorified." As with the eternal energetic processions, I do not believe that either is necessarily wrong. My friend Michael Liccione gives a good explanation as to why it is implicit in what we all believe and why there is even historical precedent for this give-and-take of the Son and Spirit (citing a passage from St. Symeon the New Theologian on the Spirit as Icon of the Icon, which fits in well here). Lourié has pointed to the same Ethiopian tradition that reversed the Son-Spirit causality.
My concern is that the resolution is anachronistic, and the historical precedent to which it cites is obscure and difficult to harmonize with the main communities in which this dialogue is taking place. Therefore, I do not believe that these approaches speak to the concerns and needs that each side is trying to express. They try to come up with a new language rather than allowing the traditions to speak. I would prefer to say "we have heard, and we proclaim the same." There may be skepticism about that claim, but it seems that reflexive relations have a long and successful pedigree on both sides, and that what I have proposed is only to make explicit what has been said implicitly for a very long time.