(Christian) theology as mathesis universalis

The Spinozist heresy is to have violated the hierarchy of the Aristotelian categories: God is not one being among many but Being itself. But there is more than one way to blur the ontological difference, i.e., as many ways as there are to count. There is, for example, the dialectic of the one and the nothing in Neoplatonic mathematics by which infinite progression telescopes to the one. It was the Christians, however, who taught us how to count directly from one to three: “we do not say that union is begotten from oneness or from equality of oneness, since union is not from oneness either through repetition or through multiplication. And although equality of oneness is begotten from oneness and although union proceeds from both [of these], nevertheless oneness, equality of oneness, and the union proceeding from both are one and the same thing …” (Cusanus).

The trinity is not only an ontological but a mathematical mystery: the simplicity and unicity of God is also the unicity of order. God is not only the infinite geometer, according to Plutarch, but infinitely arithmetizes; creation proceeds not from the word but from the number. “Number was the principal exemplar in the mind of the creator”, Boethius says (long before Leibniz’ “divine mathematician”), which is in itself a substance to which no other substance is joined (which is thus how number is then the measure of all things but not of itself). The echoes of Neoplatonic mathematics are clear: the unity of a being is at once its limit.

Cusanus gives us a clue to the passage from the ontological to the mathematical: “God is the being of things; for He is the Form of things and, hence, is also being”. For Plotinus, being consists of emanation from the one. Cusanus, however, following Thierry of Chartes (who was himself inspired by Boethius), introduces the concept of the fold into philosophy and mathematics:

a point is the enfolding of a line as oneness is the enfolding of a number. For anywhere in a line is found nothing but a point, even as in number there is nowhere found anything but oneness … Movement is the unfolding of rest, because in movement there is found nothing but rest. Similarly, the now is unfolded by way of time, because in time there is found nothing but the now.

All of these are images of the enfoldings of the Infinite Simplicity; in other words, Cusanus explains divine simplicity as nothing other than the enfolding of all things. Since, moreover, divine simplicity is the infinite mind, such that the thought of the divine mind is the creation of all things, our thought is an image of the eternal unfolding, hence guaranteeing the unity of thought and being.

The fold places multiplicity at the heart of being such that “God is so one that He is, actually, everything which is”. Cusanus is explicit in denying that oneness is number, “for number, which can be comparatively greater, cannot at all be either an unqualifiedly minimum or an unqualifiedly maximum. Rather, oneness is the beginning of all number, because it is the minimum; and it is the end of all number, because it is the maximum”. This proposition supports the paradoxes of De Docta Ignorantia: the coincidence of the absolute maximum and minimum and the assertion that “if there were an infinite line, it would be a straight line, a triangle, a circle, and a sphere” (so too Cusanus invokes an image of the divine trinity as a triangle whose angles are all right angles). More importantly, like Conway’s notion of the “intimate presence” of God to all creatures (“without any increase” in their being), the union of oneness and multiplicity folds all things in the divine without reducing being to the being of the divine (God is not-other). Against the Aristotelian convertibility of being and unity, then, Platonism in mathematics asserts not the being of number but the subordination of being to number. “The whole of nature is akin” (Meno 81d) only if the being of beings proceeds from the equality of one to one.

One and nothing: free variations (continued)

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”.]

One and nothing: free variations

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!).

A "fundamental" perplexity

A century before Hilbert, in his Beiträge, Bolzano proposed with astonishing prescience the autonomy of mathematics from transcendental philosophy. In a few brief, lucid paragraphs, Bolzano proposes a simple criticism of the Kantian project: not all objects that appear (to us) must have a form but only those that appear as external. Couple this observation with his definition of mathematics as the “science which deals with the general laws (forms) to which things must conform [sich richten nach] in their existence” and mathematics is effectively inoculated from the grounding mechanisms of transcendental philosophy from Kant to Heidegger.

In one sense, then, it should not be surprising that around 1961 the man who made Hilbert’s program so problematic should declare that there is something fundamentally correct about Kantian philosophy: i.e., that the construction of new mathematical theorems that cannot be derived from a finite number of axioms requires new intuitions. Yet Gödel avers here not to a Kantian notion of intuition—which he admits is unclear at best and, as Bolzano had already noted, simply false for a large part of mathematics outside geometry—but to Husserl and claims that in phenomenology philosophy for the first time meets the desiderata established by Kant. That is what should be surprising since the gulf between Kantian and Husserlian intuition seems too wide for the easy leap Gödel wishes to make.

Perhaps the missing link may in fact be Bolzano. Objects of perceptual experience, Bolzano claims, must have a form but also—unlike, for example, mathematical objects—sensible matter (as he says, something which “occupies [erfüllt] this form). Instead of the usual word “matter”, however, Bolzano asserts that these are also a priori forms (as space and time are for Kant), “except that the range to which the former relate is narrower than that of the latter, just as the form of space has a narrower range than that of time”. We are here well on the way to Husserlian hylomorphism; yet the later genetic phenomenology abandons the constitution of sense hylomorphically. As Henry has shown, for example, and as Husserl himself declares in the lectures on active and passive syntheses, hyle is ejected from its status as the blind content of the real into the life of the monad within which “a unitary nature and a world in general is constituted genetically … according to a constant process of attestation” (Husserl). Is this not the pathos of truth and the impossible ethical problem explored by Sartre insofar as, in his language, the “essence” of the for-itself is nothing other than relatedness (relation to itself, to being, and to others as three aspects of the same transcendental structure)?