If the form of multiplicity exhibited by the henads, namely, a multiplicity all of the members of which are in each one, is the primary and ultimate kind, then whatever other kinds of multiple there are must be derivable from it in their form. Two elementary kinds of multiplicity are known as homoiomerous (or ‘homoeomerous’) and anhomoiomerous (or ‘anomoeomerous’). (It is a problem to ascertain whether this is the only exhaustive division.)
Homoiomerous multiplicity is made up of parts that are alike (homoios), at least for structural purposes. An example of such a multiplicity is a body of water, insofar as we take it just as a collection of water molecules, or a flock of geese, insofar as we take it purely as a collection of individual geese. Anhomoiomerous multiplicity is made up of parts that are structurally unlike (anhomoios), and by virtue of this may be regarded as a structured multiplicity. An example of anhomoiomerous multiplicity is a face, insofar as we take it as made up of two eyes, two ears, a nose and a mouth, or a flock of geese, insofar as we take it as having the sort of structure we see when geese fly in formation.
The fact that the same multiplicity can be treated, now as homoiomerous, now as anhomoiomerous, indicates that these forms of multiplicity have a common ground, and I have argued that this common ground is the ultimate, all-in-each or polycentric form of multiplicity exhibited by henads. The emergence of homoiomerous and anhomoiomerous forms of multiplicity must therefore be traceable to differences internal to henadic individuals.
Stepping back from this technical analysis, we do see that divine organizations exhibit both of these structural potentials, but always interwoven with one another. Thus, a pantheon is a group of Gods, and in this respect homoiomerous, but Gods within a pantheon also perform differentiated functions, rendering the group anhomoiomerous. We cannot eliminate conceptually either of these aspects of divine organization, any more than we can eliminate one of these modes of multiplicity generally.
What is it in the nature of the ultimate units that makes it an ineliminable potential for such units to form both kinds of group, and furthermore in such fashion that any group, generally speaking, can always be viewed in either of these respects? That is, we can expect that a sufficiently close investigation will find structures discernible in the body of water such that it is not characterizable merely as molecules of water, but as molecules of water in this or that state, e.g., of motion, and differentiable on account of this; and we can expect that sufficiently close examination of a hierarchically structured social organization will also find moments expressing equality among citizens.
We can see this potential for these two different kinds of multiplicity in those two fundamental characteristics of the ultimate units that were discussed in part II of the present essay, and which were discussed by Platonists under various terms: Limit and Unlimited, Monad and Dyad, Existence and Power(s), among others, but which Damascius most helpfully treated as grounded in the basic character of the ultimate, polycentric manifold. Since all of the units of this manifold are in each one, each one has the character of being One-all and of being All-one. That is, each one is all the others, and all the others are it. Insofar as these conditions are not simply equivalent, but also distinct, depending upon whether we take a given unit now as the center, and hence as one-all, now on the periphery, and so all-one, they form the potential for the two kinds of groups discussed in the present essay, and with just the complex reciprocity we have already observed.
Insofar as each is ‘one-all’, henads form a homoiomerous multiplicity of units each of whom is the center. But since each can only be the center one at a time, each is also peripheral, or ‘all-one’. (We see at this point the potential for grounding temporality in the basic conditions of henadic existence, but will not pursue it at present.) But there is no single way of being peripheral, because the peripheries of different centers are different, even if it is a question of the same peripheral point, insofar as it is taken as the periphery now of this, now of that center. Each henad is also the center in a peculiar fashion, but this is given by the positivity of henadic individuality, not by the nature of centrality, because a center is not simultaneously many centers in the way that a periphery is simultaneously many peripheries. (Think of the difference between the many circles with a common point on their circumferences, and the many concentric circles around a single center.) We may say, in fact, that the all-one is the principle of differentiating centers according to their peripheries, and that through this principle, the henads form anhomoiomerous multiplicities of diverse kinds.
Centrality and periphery thus interpenetrate one another in a complex fashion. The nature of being-a-center is differentiated by the peripheries involved, and the nature of being-peripheral by the peculiar centers implied. It may be that units can be exhaustively determined by their relations, that is, from their periphery, but can they be given by their relations, however many or however much of their properties are explicable through these relations? For even a center, as an ideal, is posited through the circumference. In this fashion, we can imagine a circle whose circumference is everywhere, and its center nowhere. This is, indeed, what the intellect demands. But if every point is merely peripheral, then it is also fixed and arbitrarily determined to its position (or trajectory, velocity, et al.). The possibility of centering, of selfhood, can neither be eliminated from the system, nor restricted to a single one, because that single possible center could only be that point vanishing for itself, determined from and wholly dependent upon the points actually existing.
I think this is brilliant. I wonder how henadic multiplicity could be characterized in set-theory. Might have to hit the books.
This is a subject that interests me a great deal. The most sophisticated philosophical appropriation of set theory so far, in my opinion, is Badiou’s; but while Badiou’s synthesis is not without a refreshing clarity on certain aspects of Platonic doctrine—e.g., his recognition that there is no “One Itself”—Badiou displays no knowledge of the doctrine of henads. I would argue that the opposition of “Being” (as the closed system of ontology) and “Event” (as the open economy of agency and choice) in Badiou’s system parallels that between ontology and henology in a Proclus or Damascius. Through henadology, however, the Platonists bridge this opposition quite differently than Badiou, who in my opinion remains trapped within the confines of his set-theoretic ontology insofar as he conceives its exterior in a purely evental fashion. The question that arises, however, is whether the most direct path to formalizing henadology lies through set theory, or rather through a mathematics not itself reducible to set theory. The answer, in turn, will have a great deal of importance for the question of whether there is a limit to the project of “logicizing” mathematics, as has been argued particularly by Zalamea.