1. Along the way toward expressing the thoughts of God before the creation of the world, Hegel’s logic consumes the possibility of mathematics at the highest moment in the doctrine of being. Just before his explicit treatment of quantity, however, he includes a note on Leibniz’ monadology and observes that “plurality remains as a fixed fundamental determination, so that the connection between [monads] falls only in the monad of monads, or in the philosopher who contemplates them”. What Hegel has grasped only vaguely here is that for Leibniz mathematics and metaphysics express the same thought, i.e., that mathematics understands the world in the same way as the divine intellect (which is the real meaning of his remark at the determination of a maximum is the work of the divine mathematician who determines the greatest number of compossibles in a given world). Leibniz’ “new mathematics”, he says, “makes man commensurate with God”.
The problem of plurality to which Hegel refers is Leibniz’ notion that the infinite (number of) monads are representations of a single universe (Monadology §78) without thereby understanding this universe as substance.* Leibniz struggles to provide an adequate topological model of such a universe** and instead speaks of the “accommodation” or harmony of all things.
*One is tempted to say “Spinozist” substance were Spinoza’s definition of substance as “one” not problematic from a mathematical point of view and which would require extensive work in disambiguation. Rather, we might safely say here “Aristotelian” substance up to and including Heidegger’s interpretation of ousia.
**Elsewhere I have claimed that such a model would be something like a Klein bottle.
2. Yet we should remember that the essence of harmony is a fundamental gap or discontinuity in what the sensibility desires as unity. The law of the series that guarantees the immanence of the world in the monad (what Badiou calls the “absolute interiority” of the monad) allows us to speak of the monad as one in a strictly different sense than that of the universe.
Here we might benefit from recalling that this is the Platonic problem par excellence. Against the Aristotelian dictum that being is always a being (i.e., that unity follows immediately from being)—and Aristotle’s well-known confusion of the Indefinite Dyad as two “counted-as-one”—Plotinus’ account of substantial number accounts both for the ontogenetic differentiation of being (see, e.g., Enneads VI.6.15) and for the fact that the One is not enumerable. What is at stake, philosophically if not mathematically, in Platonist mathematics is precisely the capacity to distinguish the one in the order of intelligibility from the unity of any individual. Being, for Plotinus, exists only because it inherits unified number from the One and, conversely, multiplicity is not the division of the One but the intellect’s contemplation of the One. We might say that substantial number is the “form” of the monad—as the immediate image of the One—combined with the “matter” of the Indefinite Dyad or, in perhaps more precise language, the Indefinite Dyad is nothing other than the limitation of unity as apostasis (and reciprocally, according to the Neopythagorean conception of monadic number, the monad is the limit of quantity: the monadic number is a progression to and a regression from mulitiplicity), the intellect is nothing other than substantial number, which is why being is not itself number but number is the principle of being.
3. What does it mean, then, to be a thinker of the One? Or, perhaps more modestly, what is at stake is the character of our ethics. For a thinker of the One, ethics is beyond being, in a sort of pagan transcendence of that which cannot be counted-as-one, as opposed to an ethics of the void, which must resist, perhaps violently, the capacity for being named and that must tear itself away from the very conditions of its survival. Our choice, however, is not that between excess and subtraction since the Plotinian One is nothing other than a series of negations: not to move away and not to progress “even a little” to the two. If there is not a symmetry between these two orientations, our choice seems to be in what direction this negation operates: whether the difference that counts is a negation of the given (multiplicity) or in the (im)possibility of negating what does not exist (a double negation!).