The set of the Gods is not formed by the class characteristic ‘God’, but by that of uniqueness, that is, by being units or henads, while the character of ‘godhood’ comes from the position of this set, the set of absolutely unique individuals, relative to all that is. The character of godhood in the henadic manifold thus expresses in the purest form Proclus’ maxim (discussed here) that ‘Gods’ are whatever things, in a given ontology, are “first according to nature”.1
In connection with the present discussion, I’d like to particularly note the deployment here of the notion of ‘firstness’, inasmuch as it invokes the distinction between ordinal and cardinal numbers. Ordinal numbers order things in a series—first, second, third—whereas cardinal numbers state how many of a thing there are—one, two, three. With respect to the term arithmos, ‘number’ in Greek, Jacob Klein has noted that in its basic sense arithmos “never means anything other than a definite number of definite objects”.2 The abstract or absolute sense of number seems to emerge from this concrete usage in a fashion not too far off from Gottlob Frege’s logical derivation of number from the relation of equinumerosity between sets. Hence, for example, there are three shotglasses on the table, and three tumblers of icewater, with the number three arising as something in itself from the capacity to say, e.g., “For every shotglass on the table there is a tumbler of icewater,” and the like, extending into the countability of sets themselves, so that there is one bottle of whiskey on the table corresponding to the one set of three shotglasses, and so forth.
Frege’s foundation of arithmetic on substantial set characters in this fashion led notoriously to a crisis in the very project of logicizing arithmetic due to the possibility, in dealing with sets and their memberships (or ‘extensions’), of generating sets based on paradoxical characters such as ‘The barber who shaves everyone in town who does not shave himself’. A member of such a set is a member precisely by not belonging to it. Paradoxes such as this were known to the ancients, the classical form arising from a remark attributed to Epimenides of Knossos (in Crete), to the effect that ‘All Cretans are liars’, which must be false in order to be true. And yet, as I have said, I see Frege’s procedure for deriving number from the power of forming sets of objects with some common character (being a shotglass on the table, being a tumbler of icewater on the table) as embodying a genuine insight.
The henadic manifold is the set of absolutely unique individuals, that is, the set of individuals who do not, in the ultimate sense, share any character in common. This arises from the integrity of henadic individuals, each of whom possess all her traits in an irreducibly peculiar fashion, so that while we may loosely speak of all sorts of common traits among Gods, in the strict sense no God has these traits in precisely the same manner as any other. More strongly, because of the ontological primacy of henadic unity over any other trait as such, each trait belonging to a God is, in the primary sense, not the same trait as the similar trait of any other God, not because the trait possesses some distinguishing qualitative character (though this will be the case as well), but simply insofar as it is Hers.
This property of the henadic manifold bears a clear formal resemblance to the paradoxical sets generated by the Barber or the Liar, but in the context of Platonic thought this is, as the saying goes, ‘not a bug, but a feature’. For the existence of such a set of ultimate units—whatever else we thought about the nature of those units—forms the precondition for reality as such, so that there is something, and hence something countable, rather than nothing at all. Furthermore, the unique individuals must be the first units counted, as it were, since any other set depends upon characteristics of greater semantic complexity.
In this fashion we see how cardinal number is grounded in the henadic manifold. What about ordinal number? (I am not concerned here with the question of whether cardinal or ordinal numbers are ontologically primary.) Here I believe we can usefully turn again to Damascius’ twin determinations of the henad as ‘One-all’ and ‘All-one’. We have seen in previous installments of this essay that these determinations encompass each henad’s potential to be either at the ‘center’ or ‘periphery’ of the henadic manifold; and from this I believe that we can establish ordinal number. Ordinal number is based upon the primary relationship of ‘succession’ or, so to speak, ‘nextness’. Now, it can be argued either that any unit counted as ‘next’ after another is at the periphery of the former, or vice versa, and hence that this relation, however construed, is at any rate one of the ways of a unit’s being peripheral to another.
A number of other relationships can presumably be derived from the ‘One-all’ and ‘All-one’ dimension of the henad, including, as suggested in a previous installment, the proto-temporal relationship of ‘now’ and ‘then’, as well as the proto-spatial relationship of ‘here’ and ‘there’; and obviously any hierarchical relationship can be reduced to these basic determinations of henadic existence as well. The categories of ‘firstness’, ‘secondness’ and ‘thirdness’ propounded by Charles Sanders Peirce as fundamental modes of being can also be derived from these henadic properties as (1) immediacy, (2) retention or primary association, and (3) mediation of experience respectively.
Traditionally, however, for Platonists the generation of figure is prior, in any event, to that of temporality, and perhaps to spatiality as constituted by ‘here’ and ‘there’.3 The primary henadic manifold cannot be determined as to number beyond the determination that its number is not infinite.4 Our remarks earlier about the nature of numerability show why: since there is no characteristic beyond ultimate uniqueness that defines membership in the henadic manifold, this set cannot be counted in advance, so to speak, of its constitution. This is the essential facticity of the henadic manifold: there are as many Gods as there turn out to be, though, in a transcendental move, Platonists assert that there cannot be fewer Gods than there are distinct processions of beings. This follows from the necessity of the existence of countable units prior to any counting of them, insofar as the latter involves forming sets based on characteristics of the units. (We may treat processions for our present purposes as categories in a kind of successor relation to one another.)
Figures may thus be understood as diagrams of the basic relations and as the primary ontic sets (i.e., the simplest sets after the henadic manifold itself). The line, hence, is the diagram of the relationship between two units in general and hence of the number two, the triangle the diagram of the relationship between three units in general, and so forth. In this sense, one may say that the ontologically primary figure of any type is the theophanic presentation of a relationship between the requisite number of henads; and I have argued elsewhere that Giordano Bruno, in his own speculative mathematics, presented just such a theory by means of complex diagrams each line of which embodied some mythic relationship between Gods.5 Bruno’s diagrams, insofar as they incorporate narrative, express the fullest procession of figure, which must includes projecting the figure in time, ‘drawing’ it, so to speak, in some given order so that one line succeeds another; and this is one way of stating the necessity of mythic narrative with respect to the henadic manifold.
- Platonic Theology I, 3, 12.12 Saffrey-Westerink. ↩
- Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (Cambridge, MA: M.I.T. Press, 1968), p. 7. ↩
- On divine locality, see “Universality and Locality in Platonic Polytheism,” Walking the Worlds: A Biannual Journal of Polytheism and Spiritwork Vol. 1, No. 2, Summer 2015, pp. 106-118. ↩
- Proclus, Elements of Theology prop. 149. ↩
- “Toward a Magical Enlightenment: Notes on Bruno’s Magic,” Walking the Worlds: A Biannual Journal of Polytheism and Spiritwork Vol. 2, No. 1, Winter 2015, pp. 96-109. ↩