Thursday, September 16, 2021

The Tri-Unity Matrix?

Copyright 2021 Jonathan E. Prejean; all rights reserved.

A bit over five years ago, I had the idea of writing an alternative account of the disconnect on the filioque between East and West. There were a couple of significant hurdles to overcome at the time that made the idea of that work a daunting prospect. 

First, the state of historical research on the subject was quite contentious. Although it remains so to some extent, it does seem that moderate positions are becoming more accepted. I should say that nothing that I will present actually depends on the historical conflict over the filioque. My thesis is that there is an analytical framework that is consistent with both that obviates the basis of the dispute. But in the general realm of the discussion, a raging historical debate is a distraction, so that quieting the polemics is more conducive to dialogue.

Second, at the time, there were a number of analytical proposals on how to handle the paraconsistent logic of the Trinity, and I wanted to give some interaction to these. One article that was particularly interesting to me was Basil Lourié's "Nicephorus Blemmydes on the Holy Trinity and the Paraconsistent Notion of Numbers: A Logical Analysis of a Byzantine Approach to the Filioque," a draft at the time that was later published in Studia Humana 5:1 (2016), pp. 40—54.  In addition, there were some historical views of the Trinity that I wanted to compare and contrast with the formalism. For the most part, I've accepted that this is too much to take on in any comprehensive way, and I hope that presenting the proposal will trigger a fruitful dialogue that will provide such engagement on its own. Happily, I did come across another of Lourié's articles that is directly on point for this discussion, which I will cover later.

But mostly, I feel like we need something now. Where Christians, East and West, could be coming together, we are becoming as atomized as the society. In reflecting on how bad it was right now on social media, I thought that it wasn't so long ago when I participated in reasonable discussions about the filioque on the Internet. I went back and looked at one of those blog discussions initiated by my friend Scott Carson; it was nearly fifteen years ago. One good thing is that I am still friends with people in those discussions to this day, but I wish that the same feeling were spreading everywhere. So I write this in the hope that it might bring people together.

I've put a copyright notice at the top for that purpose. I would like to kick off a wider discussion, and I hope to be part of it. I don't even know how to count links or hits or whatever, but I figured it might help to have the discussions point back to the same place.

With that explanation setting the stage, I start with the fundamental problem of Leibnizian identity in the Trinity. The logical problem of any Triadology is first and foremost why, if the Father is identical to the One God, the Son is identical to the One God, and the Holy Spirit is identical to the One God, yet the Father, the Son, and the Holy Spirit are all not identical to one another. Of course, the Trinity is a mystery beyond comprehension; it is entirely inscrutable how the three divine Persons can be the One God. But it is worth asking if the statement that the three divine Persons can be the One God is nothing more than a contradiction in terms. In both East and West, this was the challenge presented by Arianism, and the two regions of the Christian world answered this challenge in different ways. 

St. Augustine's response is paradigmatic for the Western approach. In Augustine's formulation, each divine Person possesses and is identical with the divine essence as a result of divine simplicity. In order to explain this, St. Augustine posits that the Persons are distinguished (and named) by relations, a metaphysical concept which he has taken from Aristotle's Categories and modified for his own use. Augustine was criticized by some later commentators for failing at his task of defending the Trinity, since the category of relation must either be not real, in which case Augustine is a modalist, or real. If the relations defining the Persons are real, then Leibniz's Law means that either the Persons must be identical to one another or that they are not, which would entail tritheism.  

Thomas Ryba responds to that argument (as made by A.C. Lloyd) in his article "Augustine's Trinitology and the Theory of Groups"(from J. Lienhard, S.J., et al., eds., Augustine: Presbyter Factus Sum, Collectanea Augustiniana, Peter Lang (NY 1993), pp. 151-168), which is the foundation of my own thesis. In the article, Ryba uses group theory to show that reflexive relations on the same set need not be identical with one another, even while the set itself remains identical. This is not to suggest that Augustine had this argument in mind; as Ryba explains, "Augustine does not make use of a Latin synonym of the English word 'reflexive,' but the notion of reflexivity of the trinitarian relations is implicit in his notion that all of the persons are of the substance and yet related to one another." Although Augustine himself could never have had such a proof in mind, the formal structure he employs matches the group theory proof. In Ryba's formulation, the Father is the set defined as the divine essence. The Son is a set defined by a first reflexive relation (generation) that maps the Father set back onto itself and therefore is also the divine essence. The Holy Spirit is set defined by a second reflexive relation (spiration) that maps both the Father and Son sets, which are each the set of the divine essence, back onto itself and is therefore also the divine essence.

This is all well and good, in that the argument shows that Augustine's notion of relations (explicitly including the filioque) does not run into either tritheism or modalism. One could extend this further to Boethius's concept of the Persons as "subsistent relations" and even further to St. Thomas Aquinas's "relations of opposition," and the explanation works equally well. We might even go so far as to say that it vindicates the view of St. Gregory Nazianzus in several of his Orations, such as when he says the following in his Third Theological Oration: "Father is not a name either of an essence or of an action, most clever sirs. But it is the name of the Relation in which the Father stands to the Son, and the Son to the Father." Likewise, he says in his Fifth Theological Oration: "But the difference of manifestation, if I may so express myself, or rather of their mutual relations one to another, has caused the difference of their Names. For indeed it is not some deficiency in the Son which prevents His being Father (for Sonship is not a deficiency), and yet He is not Father." (More on Gregory the Theologian later.)

I believe that Ryba's concept can be taken a step further by transforming it into a matrix. The matrix formalism has been enormously helpful in the counterintuitive discipline of quantum mechanics by providing a framework for mathematical operations, even when the real-world analogue is difficult to perceive. For example, spinors and Pauli matrices are used to describe the properties of particle spin for fermions, which are subject to exclusion rules that are in some ways like the exclusive distinctions between the Persons of the Trinity. The example of particle spin also illustrates that even simple two-by-one and two-by-two matrices can describe phenomena that in some ways defy comprehension.

For this application, I would propose the following tri-unity matrix:

The tri-unity matrix T illustrates the formal structure of relations in the Trinity, the union of the three Persons. The sets (Persons) of the divine essence (Godhead G) are denoted by the indices alpha (A) and omega (Ω). Respectively, these indices indicate whether the Person is in the domain (0) or range (1) of the reflexive relation and whether it is the first reflexive relation (0) or the second reflexive relation (1).  The matrix is therefore the smallest possible matrix including the relevant degrees of freedom that uniquely specify each Person in Ryba's formalism. The Father is uniquely in the domain of the first reflexive relation (00), the Son is uniquely in the range of the first reflexive relation (10), and the Holy Spirit is uniquely in the range of the second reflexive relation (11).

As is often the case in a matrix formalism, there is a built-in excess of information, which we call degeneracy. In this case, given the information required to specify each of the other three Persons in the formalism, there must necessarily be a fourth element (01) that does not uniquely specify any Person. In fact, this position would correspond to each of the Father and the Son in the domain of the second reflexive relation. It is impossible to uniquely specify each of the three Persons without implicitly including this additional information; the fourth element provides redundant information about other Persons without actually defining them in any way. The tri-unity matrix is therefore the minimal formal structure needed to count the Persons of the Trinity based on reflexive relations, and this irreducible excess of information is needed to do so. That excess element in the two-by-two matrix is the filioque.

That last point about the filioque may seem like an abrupt conclusion, but let's retrace the path that took us here. All we have done is to specify the minimum formal content for a way to think coherently about the Trinity without modalism or tritheism. The only assumptions we made are that the Persons are defined by reflexive relations, that these relations map a domain to a range, and that the relations are discernible from one another. We don't even have to take a position on the metaphysics of relations or of the divine essence. As with spinors, we can meaningfully talk about their formal properties without necessarily having any clear idea about what the underlying reality is.

That this approach is purely logical and formal is why I have taken it. Augustine, Boethius, Aquinas, and Scotus all had different metaphysical principles in play. That's to say nothing of many Eastern saints: the Cappadocian Fathers, Dionysius the Areopagite, Maximus the Confessor, John Damascene, Photius, Gregory Palamas, Mark of Ephesus, Gregory of Cyprus ... one could go on for days just explaining how different they are. But none of those differences matter. We do not need to know what a relation is, what counts as a cause, or what the exact formulation of divine simplicity we should adopt; we need only know what logical content is required to coherently deny modalism and tritheism.

The formalism clearly fits with the Triadology of the West. To take the most developed example, which is Aquinas's account, there are four relations of opposition: generation, filiation, active spiration, and passive spiration, but active spiration does not in any sense define the Father and the Spirit. This simply restates the (01) element of T. In other words, it is a necessary logical consequence of distinguishing the relation of passive spiration (procession) from generation-filiation without saying anything more about the properties of the Father or the Son. Far from adding something, the filioque does nothing but to reproduce Ryba's relational structure in its simplest form. Aquinas recognizes both the formal content and the fact that it does not define a person, exactly as the (01) element in T shows. To put it another way, reciting the filioque is an affirmation that the relations of all three persons are inextricably linked in this four-element structure, not an attempt to somehow distinguish the Holy Spirit from the shared properties of the Father and the Son.

Looking East

It is no surprise that what Ryba says for Augustine goes for the West generally. What of the East? Here we turn to a perspicacious account by Basil Lourié titled "What Means ‘Tri-’ in ‘Trinity’? An Eastern Patristic Approach to the ‘Quasi-Ordinals’" in Journal of Applied Logics — IfCoLog Journal of Logics and their Applications 6:6 (2019), pp. 1093-1107. This follows the same approach to paraconsistent logic found in the article on Nicephorus Blemmydes. But while Blemmydes's formulation was not widely or explicitly adopted by the Eastern commentators on the filioque, the approach described in this later paper likely was. 

Specifically, Joseph Bryennios, whose approach Lourié analyzes, was a direct influence of Mark of Ephesus, the most prominent filioque opponent at the Council of Florence. In another article ("A Logical Scheme and Paraconsistent Topological Separation in Byzantium: Inter-Trinitarian Relations according to Hieromonk Hierotheos and Joseph Bryennios"), Lourié notes that "Mark of Ephesus in his epitaphion to Bryennios called him Ὁ τοῦ Λόγου πρόμαχος ἐνθέοις λόγοις ‘The front-fighter of the Logos with divine logoi.’" I say this to indicate that we are not talking about some fringe theologian, obscure scribe, or minor player in the filioque debate. Bryennios was directly motivating the theological doctrines that were being debated. Lourié further argues that Bryennios was an authentic inheritor of a much longer tradition through Hierotheos all the way back to Gregory of Nazianzus and Evagrius Ponticus, making Bryennios's arguments not only relevant for the filioque debate but also representative of a much broader Eastern perspective.

As Lourié outlines the situation, the concern in Eastern Christian thought was to avoid the idea that the Trinity was, in Evagrius's terms, "a Trinity of numbers." By this Evagrius means that we do not count the Persons of the Trinity as a first Person, a second Person, and a third Person but only as a mutually interrelated Trinity of Persons. Lourié also cites a passage from Gregory Nazianzen's Oration 23 about how the Dyad is "overstepped" in moving from the Monad to the Triad. Lourié convincingly argues that the Dyad is not referring here to the Son (which some Byzantine writers later misunderstood) but rather to the Dyad of the dialectic of opposition, which produces composition and division. In other words, if there were an ordinal counting for the Trinity, in the form of first Person, second Person, and third Person, it would necessarily involve some form of Arianism, subordinationism, or tritheism. Lourié notes that "[t]his idea of skipping the Dyad is an innovation peculiar to the Greek patristics," no doubt because Neoplatonism and the One-Nous-Soul triad was the philosophical opponent of Christianity. I'll mention here a very similar quote from Gregory of Nyssa in Ad Graecos: "No other Person is ever generated or proceeds from the Father or from one of the Persons, in such a way that the Trinity could ever be a group of four; nor does one of these Persons ever finish, as if in the blink of an eye, so that the Trinity would ever become a dyad, not even in thought" (reproduced in Giulio Maspero, Trinity and Man: Gregory of Nyssa's Ad Ablabium, p. 161).

Much like Augustine, Bryennios establishes a formalism to account for the distinction of the Persons based on relations, which can likewise be understood in terms of group theory. In this case, his method for avoiding counting in the manner of ordinal numbers, characterized by binary adding operations (1+1=2, 2+1=3, etc.), is to define relations that are only and always distinguished from one another in relation to two other second elements. This is the ternary exclusive OR operator, which defines each element relative to the other two second elements. The operation defines a sextet, so it includes symmetric degeneracy as well, such that each Person is defined in two different relation sets. For example, the Holy Spirit appears as the Spirit of the Son and the Person proceeding from the Father.

Most remarkably, the article includes a graphical illustration of the concept, based on a version first drawn by Hierotheos in in the 1270s and upon which Bryennios had further elaborated in the 1420s. To see the very logical structure that Lourié described being illustrated is nearly as surprising as it would have been to see a drawing of Ryba's reflexive relations in one of Augustine's letters. I am hard pressed to imagine what could be a more convincing piece of evidence that Lourié's formalism is what Bryennios, and likely Mark of Ephesus, had in mind. (The illustrator also used A and Ω to index two of the sets, which gave me the feeling that all readers of history will recognize: any idea you might consider clever has very likely already been taken.)

Reconciling the Two Views

I hope that it won't be be difficult to accept at this point that Ryba has the Western view described correctly and that Lourié has the Eastern view described correctly. Both are purely logical formalisms that defy modalism and tritheism while also avoiding the objection from Leibnizian identity. Using "reflexive" to refer to this type of identity, Lourié maintains that Bryennios has presented a "super-reflexive" logic that overcomes the limitations of reflexive identity, as contrasted with a merely "non-reflexive" logic that might bypass the limitations of reflexive identity. Ryba's logic of reflexive relations without reflexive identity is likewise "super-reflexive" in this sense. And in like manner to my observation about the tri-unity matrix earlier, Lourié found similar logical structures put to use in the counter-intuitive realm of quantum mechanics.

We have now reached the point where I hope my small expansion of Ryba's idea will come into service. Because Ryba's relations define two degrees of freedom, they can suitably be represented by a two-by-two matrix. The critically important observation is this: a two-by-two matrix is not a dyad! One does not count from a monad (one-by-one matrix) to a two-by-two matrix in the manner of ordered pairs; the relations in T are all simultaneous and inseparable. So if Ryba's relations accurately capture the Western view of the Trinity, which I have every reason to believe they do, then they do not run afoul of the legitimate Eastern concern about attempting to reach the Trinity through a dyad. They are not counted in sequence; rather, they all appear at once by mutual definition.

In case Ryba's account is unconvincing on its own, I can avail myself of Chekov's gun placed in the first act, to wit, Scott Carson's article on St. Anselm. In that article, Scott quotes from Anselm in Section 15 of On the Procession of the Holy Spirit, a treatise written on this very topic. Anselm says the following:

If we should consider the Father and the Son and the Holy Spirit in pairs, it is clear from what I have said that it is necessary either that one of the pair is from the other, since the other is not from that one, or that that one is not from the other, since the other is from that one. For example, if we should pair the Father and the Son, we perceive that the Son is from the Father, since the Father is not from him, and that the Father is not from the Son, since the Son is from the Father. And similarly, if we should consider the Father and the Holy Spirit, we find that the Holy Spirit is from the Father, since the Father is not from him, and that the Father is not from the Holy Spirit, since the Holy Spirit is from him. So also, if we should explore how the Son and the Holy Spirit are related to one another, we shall understand that the Holy Spirit is from the Son, since the Son is not from him, and that the Son is not from the holy Spirit, since the Holy Spirit is from the Son. Therefore, what I said before [in section 1] is evident, that the aforementioned relations, although they are in one thing, cannot let their plurality be absorbed in the unity nor can the unity let its uniqueness be absorbed in the relations.

I'll commend the article to you for the excellent explanation and borrow the conclusion: this is part of a purely logical structure to accomplish exactly what Ryba's and Lourié's "super-reflexive" logical structures accomplish. The only reason that Anselm's language seems suspicious to Eastern readers is that this idea of pairing sounds like a dyad. That is not the case; the pairings Anselm is describing here are the same relations that result from comparing elements in the matrix T. It is not an attempt to use pairings to produce the Trinity; rather, it is taking the structure of two relations and two positions in each relation (the two-by-two matrix) and looking at the elements in pairs.  
There is much more that could be written about the metaphysical and theological implications, but again, this explanation is purely logical. Based on the logic of the two formalisms, both consistently avoid modalism and tritheism, and neither reaches the Trinity by way of counting through a dyad, which is clearly proscribed among the Eastern Fathers. We have two valid "super-reflexive" formalisms that explain the Trinity as a set of relations inseparable and inextricable from one another. Strip away all of the confusion of language, all of the various metaphysical systems, and all of the cultural baggage, and you find two different approaches to reach the same end.


We have left only to apply the reality check; is it really possible that this whole thing came down to the inability of the two sides to express their implicit logical assumptions? I would say almost certainly. Matrices used to describe spatial degrees of freedom only date to the 19th century; how would we expect the Western theologians to have articulated the difference between "by twos" and "two-by-two" explicitly? Likewise, how could the East explain their objection to the dyad to people who had been speaking in this way without incident for centuries? Ryba didn't write his own paper until 1990, and he didn't explicitly use a matrix based on the degrees of freedom so that it could be contrasted with a dyad. Lourié's paper providing the background for Mark of Ephesus is only two years old, so much of the logical underpinning of the Eastern view remained unstated and potentially obscured by philosophical concepts. But there is very little excuse to remain at the impasse if we can abstract these explanations from the writings of these various authors. I hope that we will not.