Last time we finally reached Deleuze's articulation of the full fold. The soul is folded and matter is folded and the two sides are folded together into a single world which the soul in-folds and matter un-folds. The concept of "resemblance" served as a hinge that carried us from one side to the other because. The folds of matter are completely separate from the folds of the soul, and yet they resemble one another because of their pre-established harmony. Consider again the fractal curve that illustrates the fold -- each in-curve of one side is perfectly matched by an out-curve of the other in smaller and smaller details to infinity. While the distinction between the two sides is completely real, it is nevertheless a consequence of the fact that they are two sides of a single curve. Recall that resemblance is really nothing more than the differential relations of the folding itself. Thus the fold is non-dual -- distinction and inseparability go hand in hand. Matter and the soul are two sides of a single and same world that matter realizes and the soul actualizes. These operations are distinct, but the bear on the same world. This explains why Deleuze insists (F, 149) that the sides are two floors of a single house, and not two separate worlds. The screen or wall or membrane or curve which divides the two floors is contiguous with the rooftop and foundation which holds the whole structure together. This feature is part of the architecture of the Baroque Fold. In going to infinity (and in both directions) it brings the sides into such intimate contact that we can hardly tell where one ends and the other begins. At the limit, they converge to the same world, which can frame both of them only via a "torsion" or topological twist that fuses interior and exterior, and makes the curve cover everything.
I've stated this initial summary in architectural terms because Deleuze begins chapter 8 by returning to a reference he made to Bernard Cache back in chapter 2. Bear with me though, as this connection is rather winding.
The difference between the two floors is like the difference between a curve of infinite inflection defined immanently (what would be called a mani-fold in two dimensions), and the same curve embedded in an exterior coordinate system. As we recall from our discussion of Cache, points of inflection are primitive or intrinsic or immanent singularities of a curve. These do not depend on the orientation of the curve for their definition. By contrast, the singularities of minimum and maximum of a curve do depend on this embedding -- or framing-- of a curve within a two dimensional coordinate system. The same curve can be looked at from the inside or the outside.
And yet it is indeed a question of two halves, as we have seen in the case of the infinitesmial calculus. In effect, if we assimilate the object—that is to say, the world—to the primitive equation of a curve with an infinite inflection, we obtain the position or the respective point of view of monads as primitive forces, through a simple rule of tangents (vectors of concavity); from the equation, we can extract differential relations between the minute perceptions that are present in every monad, in such a manner that every monad expresses the entire curve from its point of view. This is thus a first part, a first moment, of the object—the object as perceived, or the world as the expressed. But there still remains the question of knowing what the other part is, which now corresponds to the initial equation: these are no longer pure relations, perception—that is, which concern matter and the bodies that perception resembles. This is the second moment of the object—no longer expression, but content. These are no longer decrees, but the maxims or empirical laws of second Nature. They are no longer singularities of inflection, but singularities of extremum, because the curve is now—and only now—related to coordinates that allow us to determine minima or maxima. (F, 128)
Here Deleuze delivers on the promise he made back on pg. 117 to articulate a fourth type of singularity. 1) Inflection 2) Center of curvature or point of view 3) The 'remarkable' differential relations between micro-perceptions that fill this point of view with various macro-perceptions and 4) Singularities of extrema.
It's interesting to compare these four singularities to the four classes of beings Deleuze arrived at Back in chapter 4. I don't think that these should map one-to-one, but there's still some clear parallels. Back then, it seemed obvious that the monad corresponded to the fourth class of beings -- individuals. These were closely related to the third class of beings -- things -- because both things and monads were defined by the convergence of an infinite series of differential relations. In the case of things, this convergence was analogous to the converge of dy/dx to the derivative at a point. We saw, in that case, that the limit was effectively 'folded into' the point; even though the terms of the series vanished, their relation at a particular point on the curve remained determinate (F, 21). Later though, we saw Deleuze use a very similar analogy to explain the relation of minute perceptions and conscious perceptions in the monad (F, 113). In that case, the conscious perceptions that constitute the clear zone inside the monad arise when a set of differentials of consciousness (micro-perceptions) enter into a relation that defines a "remarkable" point or singularity. These descriptions of the type of convergence of a thing and of a monad's perception are not exactly the same, but they seem to resemble one another. Especially in the context of the quote above, they're enough to make us wonder whether the third type of singularity -- the "thing" -- is meant to refer to both real objects and objects as actually perceived. These are the two "moments" of the object or thing. Are thoughts things? This in turn suggests that all of the first three singularities could be paired with the first three classes of being and thought of as belonging entirely on the side of the soul or monad, the inside of the curve. After all, these fall into what Deleuze called the "domain of essences" (F, 73) Inflections are the primitive or simple singularities which 'condense' into a monad (F, 87), providing it with a point of view that includes not only these particular singularities of inflection but the entire curve (I imagine a Taylor series approximation to a complete function that involves only higher order derivatives at a single point), and within this point of view arise perceptions that select and relate certain differentials that define a singular clear zone. These internal building blocks of the monad seem to correspond reasonably well to the first three singularities.
Whatever we think of this parallel between things and perceptions, what interests me most here is the the way they both contrast to the type of convergence necessary to define the full monad or fourth class of "unitarily simple" being. The difference is that the limit which defines the monad itself (as distinct from its perceptions) is exterior to the monad and not localizable within it. Essentially this limit is the world that all the monads converge to -- the best world. While we don't want to confuse the two, this is a moment when the monad effectively 'becomes' the whole world, its moment of closure. In this case, perhaps, the fourth class of beings is as much the world as it is the monad -- or rather it is the completion of the monad through the world, and of the world through the monad. In a way, these two both lie 'outside' of one another and serve as the convergent limit of one another. Does this suggest that the world itself is also an example of the fourth type of singularity -- a singularity of extremum?
It's only at this point that we can come back to our initial idea of framing. Cache pointed out that it is only the frame which establishes the coordinates for a curve and allows for minima and maxima to be calculated. The process is like coupling a weightless curve to the force of gravity that operates in a particular direction. In fact, this seems to be exactly how the monad and the world relate. The world is the closure of the monad, its exterior limit, and is precisely what allows it to coordinate or harmonize with the other monads. In short, the world is a frame that the monad passes through to become a complete individual. And this frame is an example of the fourth type of singularity -- the world is singularity of extremum because it is the best possible world.
It is as if the equation of the world had to be written twice in the form of the two calculi—a first time in the minds that conceive it more or less distinctly, and a second time in a Nature that effectuates it. And no doubt the two calculi implicate and continue each other; they are complementary and must be homogeneized. This is why Leibniz can present the choice between the world and the monads as already operating through a calculus of maximum and minimum. Yet the difference between the two halves subsists, since in one case it is differential relations that determine a maximum of the quantity of being, while in the other case it is the maximum (or minimum) that determines the relations in the equation. We have seen the diversity of singular points in Leibniz: the properties of the extremum indeed governs the constitution of the world chosen in Nature, but the choice itself refers first of all to other properties (of inflection) that bring into play the form of the whole [l'ensemble], at a higher level, as the property of being the limit of a convergent series [suite]. The world, as the great equation, thus has two levels, two moments, or two halves: in the first, it is enveloped or folded in the monads; in the second, it is inserted or refolded into matter. If we confuse the two, the whole system collapses, mathematically no less than metaphysically. On the upper floor, we have a line of variable curvature, without coordinates, a curve with an infinite inflection, whose internal vectors of concavity mark, for each branch, the position of individual monads without mass [en apesanteur]. But it is only in the lower level that we have coordinates that determine the extrema—extrema that define the stability of figures, figures that organize masses, and masses that follow an extrinsic vector of gravity [pesanteur] or the greatest incline. Such is the ogive, [the Gothic arch], which puts symmetry into the inflection, and realizes the figure capable of encountering the minimum of resistance in a fluid. Such is the organization of the Baroque house, with its division into two floors—one made up of weightless individuals, the other of heavy masses [l'un tout en apesanteur individuelle, l'autre en pesanteur de masse]—and the tension between the two, when the first rises or falls (spiritual elevation and physical gravity). (F, 129)
My reading of all these correspondences still feels somewhat speculative to me, but I'm pretty sure about equating Deleuze's World and Cache's Frame, since that is precisely what we need to make sense of this complicated passage. Because the interesting thing about Cache's frame is that it doesn't close off a space, without opening it up to new possibilities. It's a strange type of limit that doesn't provide a stopping point without relaunching us. Deleuze has already argued that this is how Leibniz's best possible world works as well. Fundamentally, it is not about calculating the maximum of a particular variable, but about maximizing the diversity and continuity of the world as a whole. Rather than the full curve being given in advance, the world is composed by joining up the singularities in the 'best' way. This is the odd type of extremum that Deleuze is trying to capture when he insists that there are two types of calculus. There is indeed a calculus of souls, a 'computation' of the best possible world composed by prolonging a singularity through the series of differential relations associated with it till it converges with the neighborhood of another singularity. But this is not the same calculus as a calculus of matter, which takes the curve of the world as given, as completely specified and provided with coordinates, and seeks to determine the extremes of certain variables by setting the differential relations equal to zero. The footnote to the passage above sheds more light on this question.
Insofar as the properties that make the selection possible are the properties of maximum or minimum, they confer on the obtained being an advantage of simplicity and, as it were, an appearance of finality, but this appearance disappears once we realize that what assures the passage to existence is not the fact that the properties in question are extremal properties, but that the selection they determine is implicated in the totality of the structure in question … The exceptional property that distinguishes it is no longer a property of extremum, but the property of being the limit of a convergent series. (F, 129)
So if the best world is a type of extreme, it is a moral extreme that opens the question of what should be optimized, instead of closing this question off by deciding in advance what the important variables are. The world and the frame can't close the monad, without opening its continuity with other monads. And, more importantly in this chapter, the world as a singularity of extremum doesn't close the monad without opening matter. The world is the infinitely nested curve that allows us to pass from the side of the soul to the side of matter. In fact, as we'll see, it's actually the matter of the world that allows us to pass from monad to monad in its composition. The real subject of this chapter is this passage from one side of the curve to the other, "a to-and-fro [un va-et-vient]" (F, 150) or what Cache called an oscillation (and related to Simondon's scheme of crystalline individuation).
Before discussing this oscillation though, Deleuze spends some more time sharpening the distinction between the two sides by invoking Ruyer's opposition between "true forms" and "mere aggregates". The soul knows itself directly and all-at-once, in a state of "self-survey" without distance, as if it were able to achieve a bird's eye view of itself as an absolute domain, one that does not require embedding in a higher dimensional space to be seen as a whole. It has a trans-spatial and trans-temporal unity, a co-presence to all the parts of itself that Deleuze calls the "distributive" unity of an "every". The monad is the world in a grain of sand, and every part or moment of the monad is a grain within it that includes the whole monad. By contrast, matter operates step-by-step, within space and time, and can only be grasped as a whole by some sort of integrating force that stands apart from it or hidden within it. Which is to say that matter is always an aggregation of separate units or "ones".
What must be radically distinguished are the folds [replis] of matter, which always consist in hiding something from the relative surface that they affect, and the folds of the form, which on the contrary reveal to itself the detail of an absolute surface, co-present to all its affections. (F, 132)
The ultimate question, however, is how these two radically distinct sides can be related. We saw part of the story last chapter. Since the monad contains both clear and obscure regions, it must have a spiritual body constituted by the condensation of obscure perceptions into clear perceptions. In effect, this amounts to a restatement of the fact that soul contains an infinity of folds within it that come together to form one overall fold. But then, because the structure of perception resembles something (= x), a fact guaranteed by Leibniz's pre-established harmony, we are surprised to discover that the structure internal to the monad must be matched by an external structure that fills it. The spiritual body of step one therefore has a real counterpart.
The world is a virtuality that is actualized in monads or souls, but it is also a possibility that must be realized in matter or bodies. It is curious, one might object, that the question of reality is posed with respect to bodies which, even if they are not appearances, are simple phenomena. Yet what is a phenomenon, properly speaking, is what is perceived in the monad. When we ask, by virtue of the resemblance of the perceived to something = x, if bodies do not act on each other in such a way that our inner perceptions correspond to them, we are thereby asking the question of a realization of the phenomenon or, better, of a "realizing" of the perceived, that is, of the transformation of the actually perceived world into an objectively real world, an objective Nature. It is not the body that realizes, but it is in the body that something is realized, through which the body itself becomes real or substantial. (F, 133)
So far, so good. The monad has a body. And we even know that this body must be organic, in the sense that it is "infinitely machined" (F, 6) with parts that are themselves organisms. This is the only way for a body to resemble the folds of a soul. The trouble begins with the fact that this resemblance between the folds of the soul and the folds of matter seems so perfect that it threatens to undermine the radical qualitative distinction between the two sides. The clear zone of the soul may be a condensation of certain folds, but the monad must nevertheless include the entire world. It's all at once unity is distributive and trans-spatio-temporal. So there's not really a sharp dividing line between the stuff inside the monad and the rest of the world; it's all down there somewhere. But this is not how matter works. It consists of limited and separated forms that are assembled piece by piece. There seems to be some tension here between resemblance and belonging.
But what allows us to speak of "the body of a monad" or "its body," since the monad is always an Each, an Every, while the body, always a body, is a One? What grounds the appurtenance of a body to each monad, despite the real distinction and the difference of floors or regimes? It is necessary that a One, without ceasing to be a One, must belong to each Every. (F, 134)
How can a spiritual whole resemble just a material part? Perhaps we can imagine a pre-established resemblance between the monad and all of matter taken as a whole (see my interpolation in the quote below). And this seems to be exactly what we need to fill the infinite obscure folds of the soul with "a" body = x. But this does not sound like a body that belongs to a monad in the sense we normally use this term. In fact, it would seem that every monad should have the same body in this case; the only body would be the world as a whole. What then would make this body mine? This is a question that goes beyond pre-established harmony and asks about the union of body and soul, or the incarnation of a particular soul.
In brief, the pre-established harmony is distinguished from Malebranche's occasionalism or Spinoza's parallelism not only in itself, but also by its consequences: far from replacing the problem of the union of soul and body, of the incarnation or the "immediate presence," it makes it all the more necessary, even if it is because it allows us to move from the first aspect to the second. In effect, harmony explains the correspondence between each [and every] soul and the [whole] material universe, but when it invokes the correspondence between the soul and its body, it cannot explain it through any relation in the body, because such a relation presupposes a prior belonging or "ownness" [appurtenance] (F, 134)
What follow from this question is a very complicated theory of belonging that centers on an interpretation of one of Leibniz's last and least clear concepts -- the vinculum or substantial link. I'm not sure how useful the details are here, so I will try to go just deep enough to appreciate the main point -- the curve of the world doesn't allow us to pass from the side of the soul to that of matter without carrying us back and forth across this boundary an infinite number of times. The more we zoom into the fractal curve, the more complexity we see, and the more rapidly we shuttle back and forth between increasingly tiny crenulations of body and soul. But something strange happens at this infinite limit, some sort of twist or torsion results in the smallest variations exploding to the size of the whole universe. We think we are plunging down, but somehow we end up rising all the way back to the surface of the world, just as if the center were nowhere and the periphery everywhere. Thus this question of belonging, of possessing, which at first seems rather arcane, turns out to be central to the deepest understanding of the fold. Belonging goes beyond the harmony of two separate floors and takes us to their inseparable union in a single house.
How can a body belong to me? How can I have a body? It becomes a strange question if you consider how it inverts the question "who am I?" For something to belong to me, it can't be me. Anything that I need to possess can't be essentially mine. So, while we know I must have a body, I am not identical to that body, and this is why I must posses it. In short, anything we can lose, we will. Deleuze includes a brilliant quote from Gabriel Tarde that illustrates this.
"The true opposite of the self is not the non-self, it is the mine; the true opposite of 'to be,' that is, having, is not non-being, but the had." Already in the monad, Leibniz had instituted "I am having diverse thoughts" as the correlate of "I am thinking": perceptions as included predicates, that is, as internal properties, came to replace the attributes. (F, 138)
Much of our discussion has covered things that are intrinsic to the monad -- there's no way it can lose its inflections or point of view or perceptions, ie. any of the first three singularities. We can say it "possesses" these, but only as internal predicates or definitions, not the way an inert substance possesses attributes. But the body is external to the monad, and only fills its folds by resemblance (F, 135). And yet, Deleuze says that the body, while distinct from the monad and outside it, cannot be separated from it. How do the two stay fixed together if they occupy qualitatively distinct sides that never touch?
We can gain an intuition for how this happens by once again considering the diagram of our curve. Because it contains the whole world yet is nevertheless finite, the monad must necessarily have both obscure and clear regions within it. The extent of these domains may grow or shrink, and in fact this is what defines the continuum of monads from rational to "naked", but every monad must contain some of both. In fact, while the perceptions of the clear region may be consciously denumerable, the obscure region, by its very nature contains a non-denumerable infinity of hidden microperceptions. No matter how much of the world is unfolded or exposed to our own internal point of view, there will always remain even more of it folded up invisibly inside of us. In a sense, this is just a restatement of the fact that the fold is an infinite fractal as seen from the concave, or soul, side of the curve. However, because of a pre-established harmony, the matter which fills this souls must then also be infinitely folded. In a sense, this is a restatement of the fact that the matter which fills the soul must be 'organic', it must have an infinite nested hierarchy of layers to match the folds of the soul. Each organism must be filled with and infinity of smaller organism, ad infinitum. The two sides fit perfectly on an infinity of levels. Even though they never quite touch, they share the common limit of the world as curve of variation.
Now, try to imagine pulling these nested fractals two apart. As soon as the curve of the world becomes a nested fractal, the two sides no longer simply match overall but become stuck to one another like velcro or puzzle pieces. You can no longer just slide them apart from one another and instead need an extra dimension to disentangle them. When the fold goes to infinity, we go from harmony, to inseparable union.
How can my monad have an extrinsic possession, outside of itself, on the lower floor? One of Leibniz's essential theses consists in positing both the real distinction and inseparability: it is not because two things are really distinct that they are separable. This is where Harmony and Union discover the principle of their distribution: the pre-established harmony of the soul and the body governs [régit] their real distinction, while union determines their inseparability. (F, 136)
When you fold a nested structure like our curve, you naturally produce a hidden substructure that tends to lock the fold in place, an interesting fact you can investigate with this origami pattern.
From the side of the soul, this nesting corresponds to my unconscious perceptions. From the side of the body, the nesting corresponds to all of the organic parts and sub-parts -- the organs -- that compose the whole body of my organism. That is, the details of my organs correspond to the unconscious microperceptions within my organism. To fulfill the requirements though, these organs must themselves be organic -- they must contain an infinity of sub-organs ad infinitum. Since we haven't reserved the term monad for human souls alone, each of these organic animal bodies has a corresponding monad of its own. So the obscure folds inside of me turn out to be the shadows of the other monads needed to compose the world I include and which belong to the hidden organs within my organic body. I contain multitudes, that is, crowds of other monads, not directly, but through the intermediary of my body.
Leibniz often insists on this point: God does not give the soul a body without furnishing this body with organs. Now what is it that makes a body organic, specific or generic? No doubt, it is made up of infinities of actual material parts, in conformity with infinite division, in conformity with the nature of masses or collections. But these infinities, in turn, would not compose organs if they were not inseparable from crowds of minute monads—monads of heart, liver, knee, eyes, hands (according to their privileged zone that corresponds to one infinity or another): animal monads that themselves belong to the material parts of "my" body, and which should not be confused with the monad to which my body belongs. These are only the requisites of my organic body, specific or generic; and there is no need to ask if matter thinks or perceives, but only whether or not it is separable from these minute souls capable of perception. (F, 136)
This is the process by which a very specific body gets 'stuck' to my monad. There's now a nesting defined from a particular point of view and not only as a whole, and it's this nesting that makes it impossible to pull what is now my body apart from my soul. The distributive unity of the soul requires parts that are hidden from it, and which operate like foreign agents embedded within it. The soul 'possess' matter the way a spirit possess a shaman, but this operation requires another soul to be effective.
Defining "belonging" in this way turns out to have some very weird effects. My body belongs to me, but its organs actually belong to other monads and not to my monad. There's an inversion or double directionality of belonging here which forces us to admit that a body belongs to a monad, but every sub-monad belongs to a body. The tight origami fit of the hierarchy of fractal folding makes us travel back and forth across the curve -- from my soul to its body, but then from the sub-bodies that compose it to their monads, and again from these sub-monads to their bodies, etc ... The direction of belonging flips with each iteration.
We can see, then, that Leibniz's theory of appurtenance brings about a fundamental inversion that will constantly be revived: monads that have a body, to which a body belongs, must be distinguishedfrom monads that are the specific requisites of this body, or which belong to parts of this body. And these second monads, these monads of bodies, themselves have a body that belongs to them, a body specifically other than the body whose requisites they are, and whose parts in turn possess crowds of tertiary monads. And these tertiary monads ...The soul and the body are always really distinct, but inseparability traces a coming-and-going between the two floors: my unique monad has a body; the parts of this body have crowds of monads; each of these monads has a body…. (F, 136)
Belonging takes us further than harmony. Its shuttling back and forth across the curve creates an inseparable union of the two sides, almost as if the curve of the world had gained a sort of "thickness" (F, 150) that led to their interpenetration and indiscernibility. While this resolves the question of how it can be my body, and how I can be immediately present within it without necessarily completely controlling it, it also seems to create major headaches for Leibniz's system. Because aren't the monads supposed to be completely separate from one another? Even though they all express the same world, they are not supposed to interact directly, since each already contains everything. And yet here, somehow, my body is filled with other monads, which are bound to me by what Leibniz calls a vinculum. Am I the only one around here who gives a shit about the rules?
But since no monad contains other monads, domination would remain a vague notion, having only a nominal definition, if Leibniz had not succeeded in defining it precisely by means of a "vinculum substantiale." It is a strange bond, a hook, a yoke, a knot, a complex relation that comprises variable terms and one constant term.
The constant term will be the dominant monad, since the vincular relation belongs to it or is "fixed" in it. Apparently, we can be all the more astonished that this relation, having as its variable terms other monads (henceforth dominated), cannot be a predicate contained in its subject. That is why the relation will be said to be "substantial," not being a predicate. Since every relation has a subject, the dominant monad is indeed the subject of the vinculum, but it is a subject of "adhesion," not of inherence or inhesion. This is an almost intolerable paradox in Leibnizianism ... (F, 139)
I don't want to delve into the details of the vinculum beyond what I've already said about the way it carries us back and forth across the curve. What I find more interesting here is the way this "intolerable paradox" relates to the concept of "torsion" we discussed earlier. Somehow the closure of the monad is the same as its opening to the continuity of the best world. What seals it off from all the other monads is paradoxically the same thing that connects it to them as well. Topologically speaking this paradox can only be resolved through a "twist" like that of a Möbius strip or a Klein bottle, which, unfortunately, makes it impossible to draw the resulting curve in two dimensions.
... the "unilaterality" of the monad implies, as its condition of closure, a torsion of the world, an infinite fold, which can be unfolded in conformity with the condition only by restoring the other side, not as exterior to the monad, but as the exterior or the outside of its own interiority: a wall [paroi], a supple and adherent membrane, coextensive with the whole inside. Such is the vinculum, the primary and unlocalizable link that lines [borde] the absolute interior. (F, 140)
This is the deepest aspect of the Fold -- it confuses inside and outside and makes it impossible to see exactly what two things are being folded together (F, 150). I think it also leads us back to the event of the world, "the expressible of all expressions, the realizible of all realizations, the Eventum tantum" (F, 133). This is the simultaneous closing of all the monads, the completion that requires they pass through the bodies that belong to them and the monads that belong to the bodies' bodies, and ... in an infinitely renewed cycle. We reach the world by plunging into the infinite series of folds between matter and soul, to the point where we reach a peculiar asymmetric indiscernibility where the curve fills all of space.
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