Tuesday, December 13, 2005

Zummary: Augustine's Trinitology and the Theory of Groups by Thomas Ryba

At heart, I'm a math guy, and you don't get much of that in theology, so I've been focusing lately on the scraps of mathematics that I can get where I find them. One of the reasons I love Zubiri so much is that he was trained in physics, so what he says makes sense to a quantum mechanics junkie. As a result, I've decided to write my first Zummary (rather than Zoobie review, because I'm mostly reporting rather than reviewing) to this article in Augustine: Presbyter Factus Sum, pp. 151-168.

In this article, Thomas Ryba takes on A.C. Lloyd's classic critique, "On Augustine's Concept of Person," in which Lloyd argues that Augustine's account of persons as relations gores him on the horns of a dilemma between modalism and tritheism:

Not only can God be said of the Son, but the Son is identical with God; for according to Augustine himself, diversity of substance implies plurality of gods; from this it follows that singleness of gods is also, according to Augustine, a necessary fact, so that God is at least (i.e., if he is not a species) a member of a necessary one-member class; and whatever has the same substance as such an object must be identical with that object. Since therefore God is identical with the Son, a fortiori, Son can be said of God. The attempt to have three persons entails a non-symmetric identity [A is identical to X, but X is not identical to A].

Where Ryba takes issue with Lloyd is in his identification of relations with the substance, something that Augustine never did, at least according to Ryba. Ryba makes some textual arguments to the effect that Augustine used the non-accidental Aristotelian category of relations in a way that renders them neither accidental nor identical to the substance (a point which I happen to think is a valid one, and one that I wish more critics would note). But the real meat of Ryba's argument is his formal mathematical demonstration that it is possible to have reflexive relations with the divine substance that are nonetheless non-symmetrical and non-transitive. More or less, Lloyd argues that the denial of symmetry (if A is identical to X, then X is identical to A) and transitivity (if A is identical to X and B is identical to X and C is identical to X, then A is identical to B is identical to C) makes Augustine inconsistent, so Ryba's demonstration would render Lloyd's argument ineffective.

Ryba conceives Augustine's relations as mappings between a set x and itself, where x is taken as domain and range (origin and terminus) of the respective reflexive mappings (relations). The mapping G (for generation) maps the set F (the domain of G) to the set S (the range of G), and it is a bijective mapping (there exists an inverse mapping of S to F). However, spiration S, which maps both F and S to H, is not so invertible, although it may be represented as the product of two mappings of the same set [S(F, G(F)]. At this point, though, we are simply at the level of mathematical curiosity, having formalized the relations as such.

The real question is whether one can coherently define interesting identity relations, and here Ryba exploits the fact that only the generation relation is bijective to show that the identity mapping can reproduce any element or relation but cannot apply across relations (one of which is bijective and one of which is not). In other words, even though the mapping G between F and S is reflexive on x and the mapping S between (F,S) and H is also reflexive on x, there is no identity between the mappings or between the elements qua domains and ranges of the relations (viz., the identity relation between the mappings and the element x is neither transitive nor symmetric). Moreover, Ryba also demonstrates that one can have an additive associative relation that reproduces a unit element which is its own inverse, so that the reflexive mappings do not entail either an increase or a change in the substance.

Ryba summarizes the conclusion as follows:

Augustine is maintaining that the trinitarian relations themselves define the personal properties. These relations are reflexive with respect to the substance but directional. Put in modern logical terms this means that the substance standing as the one element of the domain of the relation of generation defines the property of Fatherhood but standing as it also does in the range of that same relation it defines the property of Sonship. But this identification of Fatherhood with the range [sic] of the relation of generation is absolute as is Sonship with the domain [sic] of the same relation. Sonship cannot be associated with the domain (or point of origin) any more than Fatherhood could be associated with the range of the same relation. The order which the direction of the relation of generation imposes on the substance defines a simple serial order. Because this order is formally absolute, the property of being in the domain of the relation of generation is not identical to the property of being in the range of the relation of generation -- in the language of the Trinity, the Father is not the Son....

Ryba closes with a physical analogy:

Imagine you have a bipolar bar magnet and you place it on a white flat surface. Then, you spill iron filings around it. The iron filings will line up in concentric arcs whose end points are the poles of the magnet. Now imagine you could pare down the magnet almost to a point source. The shapes of the arcs would change to form two circles with a commonly shared point (the fragment of the magnet). The area atop the point soutce would be one pole of the magnet and the area below the point source would be its opposite. If you let the magnet stand for the trinitarian substance, the arcs of magnetic force for the relations of generation and spiration and the poles for the trinitarian persons, then you have a rough idea of the structures Augustine was describing. Of course, in Augustine's Trinity there are three poles, two relations, and one substance, all of which are metaphysical and beyond physical categories of space, time, and force as we understand them. Neveretheless, they do have formal similarities to things we do understand, like sets, relations, and groups.

Ryba is not, of course, arguing that Augustine had anything like set theory in mind, but that isn't the point. The point is that so long as there is a formal mathematical structure that is isomorphic with the logic of someone's argument, it doesn't matter whether that person actually thought of it or not, so long as it exists. In many instances, the conclusions that someone drew can be supported even if the resources available were not adequate to justify those conclusions, and in those cases, we should not neglect them, particularly when there is good Scriptural or traditional reason to think that those conclusions are true. Or as Ryba put it, paraphrasing Maritain, we should not confuse what is unimaginable for us with what is logically impossible for everyone, particularly when dealing with divinity that defies every finite analogy our feeble minds can apply.