4. The distinction between Greek mathematics and modern mathematical analysis allegedly turns on certain discoveries of properties of infinite series. What this characterization obscures, however, is that we need not think of the problem of number as one of enumeration or, more generally, that the problem of multiplicity be confused with that of a series. The work from Bolzano to Cantor recognized the latter fact with the well-known consequence that there are perfectly good ways to speak of actual infinities. But the mortgage that set theory had to pay—and here the original problem returns—is, broadly speaking, an account of the structure of multiplicity, toward which we cannot remain indifferent and which has both logical and ontological consequences (the former, for example, being an effect of the reflexive problem exposed by Löwenheim-Skolem and the latter simply a consequence of the trivial fact that there is no reflection arrow for the empty set).
The turn toward intuitionism in mathematics, viewed in a certain light, is a return to the problem of Platonism not only in the ontological (Brouwer) but also the epistemological sense, which is the explicit difference in the treatment of number between, for example, Plotinus and Proclus (but which remains a difference in aspect only). The question of number takes place not at the level of unity and multiplicity (one and many) in the order of being(s) but, rather, in the passage from being to non-being where the latter is understood not as the negation of being already counted as one under the category of quantity but as a transcendence of being (i.e., the non-being of the One, for example, is already a double negation: a negation of the first negation of being as nothing). The typical theological mistake has been to conflate Platonic cosmogony/ology with ontology.*
*Here Heidegger’s account of onto-theology has severely limited our capacity to understand the terms of anti-Aristotelian metaphysics.
Proclus’ ideal (eidetikos) number or Plotinus’ substantial (ousiodes) number are principles of the intellect understood as the ontological expression of what is prior to being and nothing other than the activity (energeia) of being. Proclus in a sense ‘domesticates’ Plotinus’ account of substantial number in the intelligible by locating it as a sort of category in the soul; but this account nevertheless is supposed to explain how mathematical number is possible within the Platonic account of number as substance against the Aristotelians. The significance of the monad in Platonic metaphysics is that it is the principle not only of the unity but also the limit in being: the monad is not itself (counted-as-)one, which explains how the dyad participates in the monad in different ways (i.e., how the dyad is both clearly discrete and continuous). The persistent mistake of Aristotelianism has been to insist that the difference between number and monad be quantitative and to fails to understand that substantial number does not count substance.
5. The ambiguity of the substantial and the mathematical one is, however, necessary insofar as it expresses the duality of thought and being; or that thought and being are expressions of substance considered under different attributes à la Spinoza; or that thought is the reverse of being and vice versa. Modern mathematics has simply given rigorous formulation to the perennial Pythagorean proposition: not only that being is number but that being is number as structure. The absence of structure has been nominated variously as One (Plotinus), as zero (Peirce), or as void (Badiou). Everything turns, however, on how we interpret the nature of this absence and that we should not be misled neither by the nomination of the transcendental,** the confusion of number with enumeration, nor the conflation of the One with the “all” (as universe, the whole, the set of all sets, etc).
**Badiou is exemplary here: “I say ‘void’ rather than ‘nothing’, because the ‘nothing’ is the name of the void correlative to the global effect of structure. … The name I have chosen, the void, indicates precisely that nothing is presented [emphasis added], no term, and also that the designation of that nothing occurs ‘emptily’, it does not locate it structurally”.
6. The symbol of the monad is the circle since it “preserves the specific identity of any number with which it is conjoined” (Iamblichus), just as the void can be added to (and/or subtracted from) any set. For the Pythagoreans, the monad was also the intellect insofar as it was seminally (“potentially”) all beings; the circle has therefore long been the geometric expression or symbol of infinity.***
***See, for example, Augustine’s famous image of God or, more interestingly, Spinoza’s curious remark that “number is not applicable to the nature of the space between two non-concentric circles. Therefore if anyone sought to express all those inequalities by a definite number, he would also have to being it about that a circle should not be a circle”.
While the monad is often characterized as stability (monad is derived from “menein”, “to be stable”), stability is distinguished from nothingness as nascence (or, as before, harmony is only possible by forgetting a fundamental disharmony):
“[T]his incessant movement and progression which all things partake could never become sensible to us but by contrast to some principle of fixture or stability in the soul. Whilst the eternal generation of circles proceeds, the eternal generator abides. That central life is somewhat superior to creation … and contains all its circles. For ever it labors to create a life and thought as large and excellent as itself; but in vain; for that which is made instructs how to make a better.” (Emerson, emphasis added)
This is the real (ethical) meaning of transcendence or the “moral fact of the Universe”: that the given is never sufficient and that “every ultimate fact is only the first of a new series”. The very condition of possibility for thought is its inadequacy to being, which thus constitutes its fundamental imperative: to recognize this deficiency not in itself but in what is given to it. “Beware when the great God lets loose a thinker on this planet. Then all things are at risk.” The weakness of thought—its inadequacy—calls not for its mystical renunciation but a persistent refusal of that temptation toward cessation, whether in its annihilation or defeat by the overwhelming burden of totality or its pacification by the illusory satisfaction of identity—“I’m just me” or “I’m only human, after all”. The Pythagorean monad is the limit of being only as a self-limitation (which is the only way to account for the priority of the monad with respect to the dyad) and in a certain sense thought is nothing other than the (reflexive) expression of this “self”. This expression, however, betrays itself only through negation: just as the Pythagoreans called the One “Apollo” (from a-pollon, “not many”) and harmony requires the impossibility of complete unity, “the one thing which we seek with insatiable desire is to forget ourselves, to be surprised out of our propriety, to lose our sempiternal memory, and to do something without knowing how or why; in short, to draw a new circle” (emphasis added). Thought fulfills its destiny not only when it ventures into the unknown but takes the leap into what, in principle, it can never know.
[Cf. the previous post from March 2010 "Dialectics at a standstill".]