Friday, December 23, 2005

Signing off

After a lot of long, hard thinking, I've decided that I just don't have enough left to keep up Internet theology. I think I've done some good (I hope that I have anyway), but with the upcoming holidays and a new little boy on the way soon after, I just don't have anything left to give. I hope I've left enough of an impression for people to carry on where I'm leaving off, and I appreciate all of the thoughtful comments and questions that I have received from various readers.

Grace and peace,

Thursday, December 15, 2005

Divine Infinity: St. Clement of Alexandria

Continuing with my theme of extracting mathematics from theological study, I now turn my attention to a subject of particularly mathematical interest: divine infinity. Happily, there have been a few intrepid souls who have examined the ideas of various Saints on the subject, and I will be reporting on some of their ideas. It will turn out, as I hope to show, that the concept of divine infinity is inextricably bound in the Western tradition with divine incomprehensibility and divine simplicity, so if God wills, this examination may actually prove helpful for putting into context just what the West means by these concepts.

To begin, we turn to Arkadi Choufrine's analysis of Clement in Gnosis, Theophany, Theosis, Chapter III. Choufrine notes that the attribution of infinity in the sense of apeiron (lit. without limit) to the highest metaphysical principle is often thought to begin with Plotinus, as Plato considered the term pejorative, reflecting the thought that things were good to the extent they were limited by form. Having no limit, no form, was therefore not a characteristic that would have been considered desirable in Greek metaphysics.

In endorsing a kind of infinity, later authors drew on pre-Socratic insight for their ideas, and in particular, on the problem of infinity as examined in Plato's Parminides. In distinction from Plotinus, however, Clement introduces an original idea by bringing in a separate concept from Aristotle, that of the unending end, in place of Plato's infinity of multiplicity ("And so the one, if it is, must be infinite in multiplicity? Clearly."). Plato's concept really captures the basic Greek prejudice that formless things are inferior. The basic line of the Clementine response is this. An infinity of the sort when steps are reached and surpassed is impossible for Aristotle (and rightly so, I should imagine), but an infinity in which the end is never achieved (the unending end, an asymptotic progression) is not. This corresponds to what Aristotle calls an infinite magnitude, and Aristotle uses this term interchangeably with Plato's term for infinity of parts. In this case, the infinite is still technically without parts, as the end is not included within the asymptotic series, but it is not without limit (apeiron) in the sense of having no limit whatsoever (as opposed to having no self-limiting principle). This allows Clement to conceive of the "magnitude of Christ" as a bridge to the infinite, that which is spanned in divinization.

With respect to the Father's own infinity, Clement adheres relatively closely also to Plato's presentation of apeiron, but with one major difference: he rejects the idea that a multiplicity of parts is necessary for wholeness (which is logical, since Aristotle has shown the ability to use the same term without requiring the parts/limits to actually be included in the thing itself, overriding the default Greek prejudice), and on this account, concludes that infinity can then be inferred directly from indivisibility into parts (adiastaton). This equation of infinity and simplicity at least foreshadows (if not anticipates) the usage of infinity in several later Christian authors, as I hope to show soon. With the addition of infinite magnitude of the monad allowing a bridge between the infinity of the One (the Father) and the monad (Christ), Clement has managed to circumvent the rather notorious problem of how the One can be truly infinite, in the sense of apeiron, without being entirely inaccessible to multiplicity or motion, in that Aristotle's concepts of the unending end and infinite magnitude show how both can be truly infinite (apeiron) while still being distinct and in intimate relation with one another. Plotinus, apparently lacking the insight from Aristotle, ends up having to resort to one or another subordinationist solution (as indeed, Middle Platonists were wont to do even among Christians, leading to struggles with subordination as evident in Justin Martyr, Origen, et al.).

I suspect that the Stoic influence on Clement had some bearing on his receptivity to this solution. Clement was a fan of the concept of logos spermatikos, particularly in the context of the wisdom of God delivered to man, and it may be that this concept of Christ as the end of a process implanted in creation helped his receptivity to the Aristotelian concept of infinity.

It is difficult to say how much direct influence Clement had on his successors in this regard. With Augustine, it was probably little, but the common threads of influence (particularly Aristotelian and Stoic) appear to emerge in Augustine's own original synthesis. Origen seems to be more rigorously Platonic, although there are certainly shades of these ideas in Origen's notion of apokatastasis, particularly its relationship with paidagagos (thinking of the divine process as received teaching, much like Clement's idea of wisdom as logos spermatikos dropped down onto humanity). Gregory of Nyssa is a more likely candidate for direct adoption of Clement's ideas, something a parallel discussion of his doctrine of infinity should illustrate. In any event, Clement's own views illustrate a parallel concept of divine infinity and simplicity distinct from that of Plotinus and Origen within the Christian tradition. Confusion of the two can lead to misguided charges of "Origenism," but I believe that a rigorous survey of historical development supports the thesis that the Clementine and Plotinian ideas are independent of one another. I will argue, therefore, that the essential concept of Clement (i.e., that of an unending progress toward an end of infinite magnitude) will appear over and over again in Christian history and, furthermore, that this represents an orthodox solution to the problem of the One and the Many, parallel to the Neoplatonic solution adopted by Pseudo-Dionysius and more thoroughly Christianized by St. Maximus Confessor.

Wednesday, December 14, 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.

Tuesday, December 06, 2005

Making myself sick again

Took the opportunity to reread David Bradshaw's Aristotle East and West, and in light of the reading I have done recently, some of the incredibly stupid things he says about Augustine, entirely bereft of footnotes, were almost enough to make me ill all over again. For starters, he might ought to think about the connection between infinite being (it's not just act; it's infinite act) and simplicity in the energeiai. Then, rather than trying to dig things that either aren't or are barely there out of Palamas, he'd give some attention to the fact that the ideas he advocates regarding the identification of Trinitarian love with the participation of the soul in God are developed quite explicitly in the Western theology of the beatific vision, and far more to the point than the fumblings of Patriarch Gregory at Blachernae. More pointedly, maybe he would discern that identifying infinity and simplicity as energies, so that God is "even beyond them," represents a fundamental mistake in Triadology.

More proof that the most triumphalist anti-Westerners are those who are least willing to attempt to understand the Western Fathers on their own terms. The sham of "fairness," as if Bradshaw actually showed any such thing, makes it even worse. If Bradshaw would even have answered his own (sarcastic) rhetorical questions, he might have shown some insight, but this is one of the most poorly-veiled attempts at polemics in the guise of scholarly objectivity that I have seen since Romanides. The sad thing was that I actually thought this book was reasonable until I started reading up on Augustine. For Bradshaw to calumniate Augustine as he does on such thin evidence from the scholarship is inexcusable. Had he stuck to defending the Eastern view, this might have been worthwhile, but the half-truths Bradshaw ends up peddling are worse than no truth at all.

Stupid bronchitis

I've been laid up all weekend with a fever that was bad enough to render me barely ambulatory and mostly incapable of focusing my eyes on black squiggles on a printed page without some pain. Antibiotics are slowly helping, but it will be a couple of days before I'm back to normal. Should be recovered in a couple of days; talk to you then.

Thursday, December 01, 2005

Yet another reading break

I have been reading tons lately, and I plan on reporting soon. The highlight has been Augustine: Presbyter Factus Sum, which actually had a math article on theology that I just loved (and hope to report on shortly, and I may just go article by article, because the collection has so many good ones), but the others have also been solid, including: The Trinity: An Interdisciplinary Symposium on the Trinity, Augustine and His Critics, Byzantine Christ, Origen of Alexandria and St. Maximus the Confessor: An Analysis and Critical Evaluation of Their Eschatological Doctrines, Clement of Alexandria: A Study in Christian Platonism and Gnosticism. Nice to have so many good options; sad to be limited in time!