What makes something Baroque?

It could be that when we call art, architecture, or music "baroque" we simply mean that the work was created during the baroque period.  Obviously though, this is just a nominal definition; we'd like to understand what holds together the various works of that period and what distinguishes them from the periods before and after.  In other words, we would like to understand the Baroque as a style or a concept that might apply to any artwork in any area at any time.  In fact, we already use the term colloquially in exactly this way -- something is baroque if it has a lot of curlicues and whirligigs, lotta ins lotta outs lotta whathaveyous.  When it is "bizarre, uselessly complicated ... associated with magic, complexity, confusion, and excess". It's intuitively clear that repeating the simple act of folding can produce this sort of baroque complexity.  In this third chapter, Deleuze tries to expand this intuition into a more precise definition by specifying exactly what type of fold is properly baroque.  The simple answer is that the baroque fold goes to infinity.  It appears when the focus shifts from a single fold with two sides to the unlimited process of folding that produces endless folds within folds.  This may sounds quite abstract, but one the beauties of Deleuze's approach is that we can keep in mind a very concrete image to guide us.

The single line of infinite variation or inflection, the line of the universe, curves every which way.  It forever and categorically separates one side from the other.  But at the same time, it interweaves these two sides so tightly at so many scales that they become almost indiscernible.  The convexity of one side becomes the concavity of the other, but within each convexity there is concavity and vice versa, endlessly.  And at the un-draw-able limit, the two sides even complete one another, lean on one another like the sticks of a teepee. 

This limit of the closure of the monad we saw last time -- where the world is in the monad, but the monad for the world -- is clearly the most difficult part of the image.  But while we shouldn't forget about this strange twist or "torsion" in the concept of expression, it doesn't seem to be the main focus of this chapter.  Here, the most important criteria of the baroque seem to be that: 1) the fold keeps going -- with the baroque, the fold gains an autonomy, and, if you will, becomes a verb rather than a noun, an "operative act", and 2) while the folds are densely interwoven at every scale, there are only two sides to the fold.  Together these two make up everything, achieving a kind of closure or convergence.  While Deleuze discusses these two facets of the baroque at separate points, they are really two aspects of the same idea.

What is properly Baroque is this distinction between and division into two floors. The distinction between two worlds was well known in the Platonic tradition: it knew the world through innumerable floors, following a movement of descents and ascents that confronted each other at each step of a stairway that lost itself in the eminence of the One and disintegrated in the ocean of the multiple—the stairway universe of the Neo-Platonic tradition. But the Baroque contribution par excellence is a world with only two floors, separated by a fold that reverberates on both sides in accordance with a different regime. It expresses, as we shall see, the transformation of the cosmos into a "mundus." F, 36)

If we want to maintain the operative identity of the Baroque and the fold, we must show that the fold remains limited in the other cases, whereas in the Baroque it enjoys an unlimited freedom, whose conditions are determinable. The folds seem to give up their supports—cloth, granite or cloud—in order to enter into an infinite contest (F, 48)

... the Baroque invents the infinite work or operation of the fold. The problem is not how to finish a fold, but how to continue it, how to make it pass through the ceiling, how to carry it to infinity. (F, 49)

We can think of the main concern of chapter 3 as working out all the corollaries of this definition of the baroque, and illustrating each of these ideas with examples from painting, architecture, and of course Leibniz's philosophy.  I'll explore these conclusion in roughly the order Deleuze gives them.

First, because there are two (and only two) categorically distinct floors of the baroque world, they separate into a pure interior without an exterior and a pure exterior that has no interior.  These are of course the monad, and matter respectively.  Everything is contained within the monad. It is a sealed chamber with no windows, and as a result has a "dark depth (F, 31).  If there is light within it, this must somehow come from an internal source of illumination, one that at this point remains a bit mysterious.  Deleuze mentions a number of architectural examples, the most interesting being Corbusier's final work, the chapel at Sainte Marie de La Tourette.  While the chapel technically has windows, it's clear that the light here is designed to 'just appear', and to illuminate only a portion of the interior in a very particular way by analogy to the way the monad contains everything internally, but only clearly perceives its own body.  This is a particularly interesting example because we don't usually think of Le Corbusier as "baroque", and yet his extreme formal simplicity does end up being kind of ... elaborate.

On the other side of the fold, we find a pure exterior.  Matter is a system of holes within holes, caves within caves, a sort of infinite sponge or Cantor dust.  Architecturally, this pure exterior corresponds to a facade that attains a 'life' of its own, independent of the structure it covers. 

The monad is the autonomy of the interior, an interior without an exterior. But it has, as its correlate, the independence of the façade: an exterior without an interior. The façade, for its part, can have doors and windows—it is full of holes—although there is no void, a hole simply being the place of an ever more subtle matter. The doors and windows of matter open or even close only from the outside and onto the outside. (F, 35)

Yet somehow the pure and separated interior and exterior nevertheless correspond without contact, in a sort of harmony or resonance (though Deleuze has yet to use this word).  We can see this intuitively in our initial image.  The push of one side is the pull of the other.  The labyrinth of the fold means that penetration is met seamlessly with invagination.  Is that what this is a picture of?  The two sides are inverted images of the same line.  The baroque transforms this reciprocity into a spiritual vector of high and low, rise and fall  -- souls rise towards rationality and fall towards the insensibility of pure matter.  But because these two floors are interlaced in a fractal pattern, the movement of rising or falling is never finished.  Souls move endlessly up and down unfolding towards rationality and refolding towards insensible matter upon the death of their body.  Though they are forever separated, the process of folding matter and soul together simultaneously results in the ongoing folding and refolding of each side independently.  The continuation of a single act of folding pushes these categorically distinct sides, "... to the point of their indiscernibility" (F, 52).  

At this point Deleuze introduces a distinction between "actualization" and "realization" which he returns to several times.  Though we've heard some about how the soul actualizes the virtual world or line of the universe, he has yet to make clear how realization works.  The brief mention of the concept at the end of chapter 2 (F, 30) suggests that it must in some sense be the correlate or duplication of actualization, but it's not immediately obvious why this should be needed as part of condition of the closure of the monad / the convergence of the world.  Still, I think our image gives us an intuition for the necessary reciprocity the fold introduces between the two sides -- the fold between matter and soul is always also a folding of matter and soul, respectively; the folding of either matter or soul is simultaneously an unfolding of the other term as well as a new folding between the terms.  In short, we always fold something already folded, which ensures that folding and unfolding cannot be opposites.  Thus the folding of the world which allowed it to be placed once and for all in the monad (actualization) must mean that the unfolding of the monad in time (realization) must correspond to a folding of the world.

Hence, the ideal fold is the Zweifalt, a fold that differentiates and is itself differentiated. When Heidegger invokes the Zweifalt as the differentiator of difference, he means above all that differentiation does not refer to a prior undifferentiated, but to a Difference that never ceases to unfold and refold itself on each of its sides, and which unfolds one side only by refolding the other side, in a coextensivity of the unveiling and veiling of Being, the presence and withdrawal [retrait] of beings. The "duplicity" of the fold is necessarily reproduced on both of the sides that it distinguishes, but which it relates to each other by distinguishing them: a scission in which each term stimulates [relance] the other, a tension in which each fold is pulled [tendu] into the other. (F, 40)

Deleuze illustrates this theme of the two floors by citing a number of paintings, the most amazing of which is El Greco's The Burial of Count Orgaz. 

It seems hard to imagine an image which better illustrates the folding of two radically distinct floors along the line of heads, as well as the repeated folds within each of the floors. 

By this point we can already start to observe that the list of consequences of our definition of the infinite baroque fold is starting to read like a "coincidence of opposites" (F, 44).  Exterior and Interior.  High and Low.  Deleuze now extends this list to include Light and Dark.  The fold is a type of Chiaroscuro (literally light-dark or clair-obscur in French).  We've already pointed out that the monad lacks windows, and thus has a dark background.  Nevertheless, it still possess a sort of internal light that illuminates its quartier or neighborhood, its 'place'.  While it obscurely contains the entire world in its dark depths, it clearly illuminates only a specific region of it, while the remainder trails off into shadow.  

More rigorously, since monads have no such slits [fentes], a luminosity that has been "sealed" within it is lit in each monad when it is raised to the level of reason, producing a whiteness through all the little internal mirrors. It produces whiteness, but it produces shadow as well: it produces a whiteness, which is coextensive with the clear area [quartier] of the monad, yet which becomes obscure or shades off [se dégrade] toward the dark background, the "fuscum," from which things emerge "through well-executed shadows and hues in varying degrees of intensity." (F, 42)

This connects to Deleuze's critique of the Cartesian notion that our ideas should be clear and distinct (D&R, 146).  With the monad, clarity is always 'shading off' into the obscurity of the background.  The micro-perceptions that compose this background are too small and infinitely numerous to be clear.  They fall outside its radius of illumination.  Yet they are an integral part of the infinite series that makes the monad a distinct individual, different from all the other monads.  Each monad expresses the whole world, but it's this ordering of light and shadow that makes it unique, that gives it a distinction without lending it a contour, just as a Chiaroscuro drawing constructs a figure from nothing more than patches of light and shade.

But as always, this light dark alternation happens on both sides of the matter-soul divide.  Rather than being made of solid and indivisible atoms, Leibniz's matter is full of holes.  In a sense, it's nothing but holes, since it's infinitely cavernous or spongy, and even the material parts of the sponge contain yet more holes inside of holes.  There shouldn't be anything here that would reflect light.  And yet there's no void in Leibniz's philosophy, and these folds of nothingness (perhaps more precisely of measure zero) give matter a texture and even a quality.

Hence texture depends not on the parts themselves, but on the strata that determine their "cohesion." The new status of the object, the objectile, is inseparable from different strata that dilate themselves, like so many occasions for detours and folds [replis]. In relation to the folds it is capable of producing, matter becomes a matter of expression. (F, 54)

This is a conception of the object [as objectile, as modulation] that is not only temporal but qualitative, inasmuch as sounds and colors are flexible and taken in modulation. The object is mannerist and longer essentialist: it becomes an event. (F, 22) 

Again like a Chiaroscuro, folding produces solid seeming form out of nothing, as if it were condensing or compressing matter from the air rather than assembling it from component parts.  In this way the fold can become a model for the distinction between emptiness and the void.  Every fold enclose only an 'empty' space, but the infinity of folds within folds prevents us from ever reaching a foundational void.  With texture, matter takes on a 'life' of its own, but without thereby becoming a substance in itself (a role which is reserved for the monad).  In this context Deleuze mentions several contemporary sculptors I'm not familiar with.

The Folds of the Soul

Whereas chapter 1 seemed relatively straightforward, chapter 2 is definitely not.  There's two factors that make this chapter particularly tough.  First, there's the exploration of Leibniz's mathematical ideas of continuity and differentiation.  While in the end I always discover that Deleuze has a fine comprehension of the mathematics he writes about (ie. the problem is not a postemodern pastiche of terms he doesn't really comprehend), I still don't think he is the most lucid expositor of the subject.  This is largely because he's always trying to extract a philosophical point rather than explain the mathematics involved.  Understanding what he's getting at often requires some independent study that serves as background.  So we'll have to take a detour into analysis and calculus.  The second thing that makes this chapter difficult is the way Deleuze employs the ideas in Bernard Cache's Earth Moves.  While he was clearly influenced by Cache, Deleuze doesn't just apply all of Cache's complex scheme (itself partly derived from Simondon), to his study of Leibniz.  Instead, he latches onto certain aspects of Cache's analysis (and not the most obvious ones) and repurposes them to fit with the system he sees in Leibniz.  Shortened into a thesis statement, Deleuze's framework would be:

We can therefore distinguish between the point of inflection, the point of position,and the point of inclusion. (F, 27)

This three point classification only roughly corresponds to Cache's distinction between inflection, vector, and frame.  But since it recurs several times throughout the chapter, it can serve as a map that leads us through the labyrinth.

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The foundation of the chapter is the concept of inflection as articulated by Cache -- the point where the tangent crosses the curve.  Inflection is the fold, the basic unit of Leibniz's metaphysics that replaces the point or atom.  The first important thing about the point of inflection is that it is between two things -- between the concave and convex side of a curve.  So while it's a single specific point, it already refers to the interconnection and distinction of two things.  The second important aspect of an inflection point is that it constitutes an intrinsic singularity of the curve because it will not change position under transformations likes rotation, reflection, projection, or stretching.  This is what makes it fundamental.  The minima and maxima of a curve are non-intrinsic or 'second order' singularities precisely because they depend on what coordinate system we use to embed the curve (ie. they are not invariant under those transformations). 

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Deleuze tells us about three possible transformations that this simple inflection point can undergo.  Reflection, projection, and (roughly) multiplication with scaling.  He attributes this classification to Cache, but it's more accurate to say that this classification is his reading of Cache.  The relationship between these transformations and the full description of folding that Cache articulates in Earth Moves (chapter 9: Oscillation: pgs. 102-117) is pretty complicated, so I'll come back to it at the end.  For now, we can get some intuition for these transformations by looking at some diagrams. 

1) In the first case, inflection can be reflected.  Deleuze makes it clear that he's referring to Cache's discussion of the gothic arch or ogive (EM, 85).  For Cache, the ogive was the sign of a coupling of the inflection point to a vector, in this case the medieval vectors of gravity or spiritual ascension. 

Deleuze, however, is more interested in the way that this reflection transforms the point of inflection into a fold where the curve doubles back on itself.  The singularity of the inflection point is transformed into a new type of singularity -- the cusp point or, as the translator of Earth Moves clumsily has it, the 'backup point' of a curve.  As we'll see in a moment, this connects it to the mathematics of irrational numbers and differential relations.

2) The second case is when the inflection point undergoes a projective transformation.  In this case, we move from the inflection point of the curve, to the points off the curve that lie at the center of the curvature of the concave or convex sides.

For some reason neither Deleuze nor Cache draw the simplest version of this concept, which looks a bit like the symbol for the Dao, turned sideways and with a few spokes.

While the initial explanation of this transformation involves a somewhat obscure reference to catastrophe theory (pg.17), the concept reappears more clearly a few pages later as the point of view.

Starting from a branch of inflection, we determine a point which is no longer the point that traverses [parcourt] the inflection, nor the point of inflection itself, but the point where the perpendicular lines meet the tangents in a state of variation. It is not exactly a point but a place, a position, a site, a "linear focus," a line born of [issue de] line. It can be called a point of view, inasmuch as it represents the variation or the inflection. Such is the foundation [fondement] of perspectivism. (F, 22)

By its nature, inflection always branches in two directions, and if we define one of these as convex, the other becomes concave.  We find the centers of these shapes by examining the point of intersection of lines drawn perpendicular to the curve's tangent lines.  As we pass through the inflection point, this center will flip from one side of the curve to another.  It's almost as if we were projecting a segment of the curve onto this point, or as if we could view this entire convex or concave segment from a single coherent vantage point.  Clearly, it's a bit of a stretch to call this a point, since the perpendiculars will not converge on a single point unless the sides of the curve are perfect semicircles, but we can still see the idea that there is a focusing of the curve in a particular place that summarizes its variable curvature in a unified way.   The 'projective transformation' creates a new type of singularity from the initial singularity of inflection, and, as we'll see later, this one too is connected to mathematics by way of the conic sections.

3) Finally, there's a third type of transformation that the point of inflection can undergo.  Homothetic transformations are essentially scaling plus translation.  Deleuze's initial description of this transformation is quite a bit more elaborate than what he provides in the other two cases, and this time he puts part of the mathematical analogy upfront, by comparing it to the Koch curve.  

Inflection is infinitely multiplied and the curve folded inside itself again and again in the same way, though each time at smaller scale.  But, however historically 'monstrous' the Koch curve was (because it is everywhere continuous and nowhere differentiable) it is just an initial analogy for a much more abstract version of this transformation.  Imagine if the generator we use changed at each stage of the construction.  This would produce a much more complicated curve that would continue to surprise us with new twists and turns as we zoom in through an infinity of scales.  Instead of the multiplication of a single fold through self-similarity, we would have an endless transformation of this first fold, which in a sense plunges to infinity without ever limiting itself.  'The' fold would become the complete infinite set of folds, an actually infinite fold, complete yet unlimited, much like the Mandelbrot set.

Everything changes when a fluctuation is made to intervene, rather than an internal homothesis. This is no longer the possibility of determining an angular point between two other points, no matter how close they may be, but rather the latitude to always add a detour by making every interval the site of a new folding. (F, 18)

With the full version of this transformation, it's as if we've raised inflection to a higher power to discover the inflection of inflection, as it were, so that inflection itself creates every form in the manner of a vortex. 

The transformation of the inflection no longer admits any symmetry or privileged plane of projection. It becomes vortical [tourbillonaire] and is produced through delay or deferral [par retard, par différé], rather than through prolongation or proliferation. In effect, the line is folded into a spiral in order to defer [différer] the inflection in a movement suspended between sky and earth, which indefinitely moves closer to or farther from the center of a curve, and at each instant "takes flight or runs the risk of collapsing." But the vertical spiral does not prevent [retient] or defer the inflection without also promising it and making irresistible, in a transversal dimension: a turbulence that is never produced alone, and its spiral follows a fractal mode of constitution by which new turbulences are inserted [s'intercalent] between the initial ones. Turbulence nourishes itself with turbulences; and it effaces the contour or outline, which becomes little more than a watery froth [un écume] or a flowing mane [une crinière]. It is the inflection itself that becomes vortical [tourbillonaire], at the same time that its variation opens onto a fluctuation, and becomes a fluctuation. (F, 18)

 

The reference here to "delay or deferral" obviously corresponds to the way we plunge into the curve without ever reaching its end; we can always insert qualitatively new vortices into the details of any vortex.  The edge of the Mandelbrot set teems with an infinite novelty.  But Deleuze also wants to connect this transformation to Cache's idea that inflection must always pass through a frame before it becomes visible (see footnote 39 on (F, 18) referring to (EM, 95)) which results in its continual oscillation between the poles of pure crystal and pure chance (EM, chapter 9).  Since Cache's idea owes much to Simondon's theory of 'higher' individuations (biological, psychical, and social) as types of 'neoteny', the details of this can get complicated pretty quick.  So, again, I want to save them for the end.  For now, I think we can just keep in mind that inflection can be transformed into an infinitely varied curve that is nevertheless completely determined and bounded.  It may be perpetually unfinished, but this isn't because we can in principle forever accumulate more and larger inflections as if each were a separate finished substantial unit.  Instead, the new is inside the old, whose production has been suspended, left unfinished like a set of open internal parentheses, and not on top of the old, as if it were closed off like a completed brick.

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At this point Deleuze leads us on a short detour into Baroque mathematics.  The goal is to show us how Leibniz had a precise mathematical conception of the inflection point that explained the surprising complexity in the interior of something usually thought of as infinitely small and simple.  In short, the point is already a fold, and one fold is already multiple, ad infinitum.  

Deleuze gives us two examples from Leibniz's mathematical writings which take a bit of puzzling over before they yield up some interesting philosophical insights.  In the first example, the inflection point can be found hiding in plain sight. 

We think that the line from A to B is continuous and straight -- completely 'rational'.  Leibniz, however, didn't believe that there was any such thing as a perfectly straight line.  To be continuous, even things like the line segment AB have to have tiny fragments of curves mixed in.

... there is no figure that is exact and unmixed, as Leibniz said, no "straight line without curves intermingled," but also "no curves of a certain finite nature unmixed with some other, in small arts as well as in large," such that one "will never be able to fix upon a certain precise surface in a body as one might if there were atoms." (F, 15)

To illustrate this, Leibniz describes a simple geometric construction that can be used to prove that √2 is irrational.  We draw a right isosceles triangle that has our segment AB as hypotenuse.  The ratio between AB and the sides of the triangle (AC and BC) will be √2/1.  Next we draw a circle centered on A with radius AC that cuts through AB at point X.  Now, if we imagined that the line AB is composed of only rational numbers, we're surprised to discover that, since we already know the √2 is irrational, X is not on the line AB.  Using the arc of the circle, we have cut through the line at what turned out to be an 'empty' point.  To create a continuous line, we can imagine adding this point back by including a tiny fragment of the circular arc in the otherwise straight line.  Unfortunately, since our construction has reproduced the proportions of the initial triangle on a smaller scale in the form of XYB, we're going to have to do this an infinity of times.  We get a clear sense of what Leibniz means by the "labyrinth of the continuum" if we imagine the infinity of tiny curves we have to 'fold in' to the seemingly simple straight line just to make it continuous.

The arc of the circle is like a branch of inflection, an element of the labyrinth, which makes the irrational number a point-fold where the curve encounters the line. (F, 21)

The second example shows us how Leibniz thought of the differential relation as another sort of point-fold.  In justifying his infinitesimal calculus, Leibniz provides the following diagram:

Smith provides a nice discussion of the importance of this idea in his essay on Deleuze's reading of Leibniz (ED, 53), so I won't belabor the point.  The idea is that the dashed line ec remains parallel to line ECY as it travels away from that line in the direction of point A.  The whole way though, the ever-shrinking triangle eYc remains similar to the larger triangle EAC, and in particular, the slope of the line remains c/e=CA/EA= tan(angle in C) no matter how close we come to the point A.  So in a sense, we can think that this angle is 'already in' point A, despite the fact that we don't normally think of a point as something able to 'have' an angle. The reasoning seems distinctly odd when we apply it in this geometric context, but in fact it's the exact same reasoning that we use when we say that a curve has a tangent line (ie. a derivative) at a point.  The differential relation remains something determinate 'folded into' the point, even when the terms of its numerator and denominator shrink to zero. 

In both these examples, the requirements of an infinite continuity lead us to discover that the point is a much more complicated entity than it first seems.  It's no longer a variable, but contains within itself a sort of infinite, though entirely well defined, variability.  This is quite a contrast to, say, algebra, where quantities may be unknown but always assume a single determinate value, which is why Deleuze is calling it "Baroque" mathematics.  The next obvious question is what happens when we collect all these point-folds we've been discussing.  In fact, this is exactly what the calculus is all about.  To the question, "how is a particular curve y(x) everywhere folded?" we can reply with the derivative function, dy/dx.  We usually think of the derivative as exactly that -- derived from the initial curve.  But of course we could think of these differential relations as the primary thing given to us, and the curve which corresponds to their integration as merely a particular example of an object satisfying these relations.  Indeed, this is basically a summary of the entire methodology of physics.  As Smith points out in his essay (ED, 52), Leibniz is trying to give us a single variable function that expresses the coherence of all the possible point-folds.  This is the philosophical import of the derivative function f'(x).

Leibniz posits the idea of a family of curves that depend on one or several parameters: "Instead of seeking the unique straight tangent at a unique point for a given curve, we can go about seeking the tangent curve in an infinity of points with an infinity of curves; the curve is not touched, it is touching; the tangent is neither straight, nor unique, nor touching, but now becomes a curve, an infinite, touched family" (the problem of the inverse of tangents). (F, 21) 

The ultimate object of baroque mathematics is then this "surface of variable curvature", this manifold, that specifies a continuity of  variability, rather the identity of a variable.  Deleuze will often call this "the objectively problematic", because it is not an unknown variable that the correct solution will eliminate.  In fact, it's important to see that with integration, there is no single correct solution -- even with a first order differential equation, a solution can only be expressed to within a numerical constant.  Differential equations have whole families of solutions that may look nothing like one another, and require the specification of initial and boundary conditions to distinguish them.  This is what Deleuze has in mind when he compares the distinction between this surface of point-folds and the curve it specifies with Cache's distinction between the objectile and the object (EM, 88) or Simondon's distinction between a continuous temporal modulation and a molding.  All of these ideas refer to a law of variability that acts like a problem -- a recipe for constructing a whole variety of related solutions.  In short, inflection can become a whole differential field.

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In a sense, everything we've said so far has been only preliminary to the main concern of this chapter -- the folds of the soul.  But now that we have a tangible mathematical image of the point of inflection as a field of differential relations that precedes the object, we can quickly see how there will be a corresponding change in the subject.  As we saw earlier, the variable curve (or, in 3D, the surface of variable curvature) has a center of curvature between each of its inflection points.  This is the point of view that represents multiple inflections in a unified way.  Naturally, the simplest version involves only two inflections.  But since we've seen that we can go on nesting inflections in smaller and smaller folds, it's easy to imagine a single point of view encompassing an infinity of inflection points, each nested pair of which will have its own sub-view, so to speak.  Each of these centers of curvature represents a singularity where a subject can 'stand' and 'look at' inflection in a coherent and 'focused' way -- in short, a perspective.  In reality, this second type of singularity is inherently related to the first, in the same way that we cannot separate the singularities that summarize the topology of a differential field from the field itself. 

Perspectivism does not mean a dependency with regard to a subject defined in advance; on the contrary, a subject will be what comes to a point of view, or rather what remains in the point of view. This why the transformation of the object refers to a correlative transformation of the subject: the subject is not a sub-ject, but a "super-ject," as Whitehead says. The subject becomes a superject at the same time that the object becomes an objectile. Between variation and point of view, there is a necessary relation: not simply by reason of the variety of points of view (although there is such a variety, as we shall see), but first of all because every point of view is a point of view on a variation. It is not the point of view that varies with the subject, at least in the first instance; on the contrary, it is the condition under which a possible [eventual] subject grasps a variation (metamorphosis) or a something = x (anamorphosis). (F, 22)

So the point of view is the integration beneath or behind the subject in the same way that the derivative function was the variation beneath of behind the object.  In a way, the dualistic pairing of subject and object (or superject and objectile is we use the terminology of the quote) is analogous to the relationship between differential and integral calculus.  The subject will define the 'domain' over which we carry out the integration of the derivative to recover a coherent figure or object.  The reference to anamorphosis is helpful here because it illustrates the difference between our typical understanding of subjectivity as a variation of truth, and the truth of variation that Deleuze attributes to the point of view.

It is not a variation of truth according to the subject, but the condition under which the truth of a variation appears to the subject. This is the very idea of Baroque perspective. (F, 23)

Obviously, the view of any picture varies depending on where we stand with respect to it.  But anamorphosis reverses this vapid observation by putting the variation back into the picture, so that the picture itself shows us where to stand to make sense of it.  Holbein's Ambassadors is a perfect illustration of this.  The point of view is not just any old place that a subject stands but more like the foothold that the world affords that allows for the act of standing.  It may not be unique, but it is 'singular', because it orders and relates a whole series of variations.  This is why the point of view is connected to the theory of conic sections.  Points, lines, circles, ellipses, hyperbolas, and parabolas don't seem to have anything in common.  They're clearly totally distinct forms.  But they are also all solutions to a single problem -- the relationship between the vertex of a cone and a plane which sections it.  When we project the cone onto the plane from the particular viewpoint of its vertex, we can find all those forms just by varying the angle between the plane and the axis of the cone.  The vertex is like a secret spot where all these shapes make sense in relation to one another, and all appear as variations of one another, solutions to the same problem, without thereby losing their distinction.

While we're not quite there yet, it's clear that the point of view introduces the type of unified center we associated with the soul or monad.  That is, the point of view is not yet the soul or the subject, but the more like the place these will occupy within matter.   Before we take this final step though, Deleuze points out that the relationship between the first and second types of singularity -- between the point of inflection and the point of view -- already allows us to see how continuity is compatible with discreteness and unity compatible with multiplicity.  Consider an extrapolation of the image we've been working with that indicates some of the nesting with a dashed line:

The curve folds smoothly back and forth at every inflection point.  But every time it does this, the center of curvature or point of view abruptly switches sides.  Each point of view indicates a (nearly) closed region or domain, and each of these appear distinct, but they are nevertheless just the flip side of the inflection points that constitute the continuity of the curve.  We could measure things in terms of the radii of the centers (assuming we embed the curve in some coordinate system) or we could measure along the length of the curve.  But because of the many folds, these measures need not be in close correspondence. 

The continuum is made up of distances between points of view, no less than the length [longeur] of an infinity of corresponding curves. Perspectivism is indeed a pluralism, but for this very reason it implies distance and not discontinuity (obviously there is no void between two points of view). Leibniz can define extension (extensio) as the "continuous repetition" of the situs or position, that is, of the point of view: not that extension is thereby the attribute of the point of view; rather, it is the attribute of space (spatium) as the order of distances between the points of view, which renders this repetition possible. (F, 23)

I find this subtle distinction between space (length along the curve) and place (location in the coordinate system the curve has been embedded in) very interesting.  As Deleuze mentions (F, 23) this distinction allows us to reconcile Leibniz's principle of continuity ("nature does not make leaps") with his principle of the identity of indiscernibles (more accurately called the principle of the non-identity of discernibles).  At first these two principles seem like they should be opposed.  On the one hand, every time I can discern a difference, however tiny, between, say, two leaves on a tree, I conclude that I am dealing with two completely distinct individuals.  There doesn't seem to be a third leaf somehow 'between' these two that would allow one to imperceptibly shade into the other and erase their differences.  Each individuals seems to have an 'atomic' unity that sets it apart from all others.  On the other hand, this seems to violate the idea that nature is continuous.  But if the place -- the point of view, or center of identity -- is not the same thing as space -- the curve that both joins and separates these identities -- then we can have both indivisible units and their continuous connection as part of the same scheme.  Place is 'quantum' and irreducible, but space is the "place of places".

... the mathematical point in turn loses exactitude in order to become position, site, focus, place, the place of conjunction for the vectors of curvature—in short, point of view. The latter thus takes on a genetic value: pure extension will be the continuation or diffusion of the point, but in accordance with the relations of distance that define space (between two given points) as the "place of all places." (F, 26)

As Smith points out this is exactly the problem that Deleuze addresses with his own theory of singularities that he derives from Leibniz.  The singularity of the point of view is just the inverse side of the singularity of inflection, or said differently, singularities and the differential field are dual concepts.  There not the same -- the point of view is not on the curve -- but they're not exactly different either -- we can go back and forth between viewing the inflections of the curve and its centers of curvature.

... the continuum is the prolongation of a singularity over an ordinary series of points until it reaches the neighborhood of the following singularity, at which point the differential relation changes sign, and either diverges from or converges with the next singularity. The continuum is thus inseparable from a theory or an activity of prolongation: there is a composition of the continuum because the continuum is a product.

     In this way, the theory of singularities provides Deleuze with a model of individuation or determination: one can say of any determination in general (any "thing") that it is a combination of the singular and the ordinary: that is, it is a "multiplicity" constituted by its singular and ordinary points. (ED, 56)

Singularities (of type 2 -- point of view) are prolonged over the ordinary points (singularities of type 1 -- point of inflection) that are folded within them, until they reach their 'largest' inflection point, the edge of their envelope, so to speak, that signals a transformation to another singularity (of type 2), another basin of attraction or point of focus on the other side of the curve.  This is why when people ask whether Deleuze was a philosopher of the One or the Many, the correct answer is Mu -- he was a non-dual philosopher.

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Finally, we're ready to discuss the soul itself.  We've moved from the point of inflection to the point of view.  But have we really constructed a subject that is in the world?  If we equate the curve with the world, its seems that insofar as the point of view doesn't lie on the curve, we have identified a subject standing outside the world.  Have we then just rediscovered Descartes' mind-body split in different terms?  And didn't we literally condense any internal complexity of the subject into a point in order to create this observer?  Following (I think) some vague hints left by Cache (EM, 105) Deleuze seems to pose this problem as a question of "visibility".  The point of view is obviously the place from where the subjects sees the inflection points which enclose it (and the sub-inflection points which it encloses).  But how did a place become an action?  What is it that 'occupies' or, as Deleuze says, what is it that "remains" in the point of view that can do the seeing?  This is what we need the soul for.

However, we hesitate to say that the visible is in the point of view. We would need a more natural intuition to make us admit this passage to the limit. Now, such an intuition is a very simple one: Why would something be folded, if not in order to be enveloped, placed in something else? It would seem that the envelope here takes on its ultimate sense, or rather its final sense: it is no longer an envelope of coherence or cohesion, like the egg, in the "reciprocal envelopment" of organic parts. But nor is it a mathematical envelope of adherence or adhesion, where it is still a fold that envelops folds, as in the enveloping envelope [l'enveloppante] that touches an infinity of curves in an infinity of points. It is an envelope of inherence or of unilateral "inhesion": inclusion, or inherence, is the final cause of the fold, such that one moves insensibly from the latter to the former. Between the two, a delay [décalage] is produced, which makes the envelope the reason of the fold: what is folded is the included, the inherent. One might say that what is folded is only virtual, and only exists actually in an envelope, in something that envelops it. (F, 25)

This is a very complex idea that the rest of the chapter is devoted to unpacking.  But the intuition is indeed a simple one.  The reason that things are visible to a subject is because the entire world is folded up inside the subject, the monad.  The subject never actually see anything but itself.  This subject is placed in the point of view, but shouldn't be confused with it.  From the point of view, we move to the point of inclusion, which adds an extra metaphysical dimension to our analogy.  Because the entire curve is included in the soul, and not merely the (infinitely many) folds contained within the radius of the point of view, we need to see things from a point that's not merely off the curve, but on a 'higher plane' above it -- the point of inclusion.  It's only a soul that can actually 'possess' a point of view.  And it can only do this because it encloses, or we might say 'surveys', an entire world.

That in which inclusion takes place, and never ceases to take place—or that which includes in the sense of a completed act—is not the site or the place, nor the point of view, but that which remains in the point of view, that which occupies the point of view, and without which the point of view would not be a point of a view. This is necessarily a soul, a subject. It is always a soul that includes what it grasps [saisit] from its point of view, that is to say, inflection. Inflection is an ideality or virtuality that exists actually only in the soul that envelops it. (F, 26)

 

It's clear that Deleuze's point of inclusion is Lebiniz's monad.  Each monad encloses a whole world, including everything that will happen to it in time.  And each monad is closed on itself, or as Leibniz famously said, a monad has no windows.  It's easy to interpret this lack of windows as a problem.  But the monad only lacks windows because it doesn't need them -- it can see the entire world without ever going outside itself.  The whole world, with all of its inflections, is folded up inside the monad. 

This intuition of closure may be simple, but the strange circular relation it introduces between the soul and the world is definitely not.  Several questions immediately arise.  If the whole world is folded into each soul, why is there more than one soul?  Or if there are infinitely many souls, how can they all enclose the same world and yet remain distinct?  The point of view made sense as a distinct relative center of concavity, surrounded by neighboring centers of convexity on the other side of the curve.  But extending the idea to an absolute closure and inclusion seems to introduce insoluble paradoxes in the relation of the One to the Many.  

The problem introduces us to the second great function of the Fold as a metaphor.  The first was the way a single fold already implicates two sides; the fold is a fractal concept.  In a complementary fashion, the second function of the fold is to confuse outside and inside. The inside of one fold can be the outside of another.  And because the fold is fractal, inside and outside can go on exchanging positions ad infinitum as we fold and fold again.  This is precisely what we need to explain how one soul can encompass a whole world while one world can simultaneously be encompassed by many souls.  The world is in the soul, but the soul is in the world, which is in the soul, which is in the world, which is ...  We keep plunging down, folding in as we go, without ever reaching the stopping point of an absolute inside or an absolute outside.

Deleuze claims that it's this process of going to infinity that really distinguishes Leibniz's monad from earlier, Neoplatonic, versions of the concept.  Instead of thinking of the world as a definite finite object, Leibniz turns it into a convergent infinite series that requires some sort of harmony between the souls that compose its terms.  This type of world, though, can never be completely and objectively given, but must be 'comprehended' as a passage to the limit, something that can only happen within a soul.

But if we ask why the name "monad" remains connected with Leibniz, it is because of the two directions Leibniz followed fixing the concept. On the one hand, the mathematics of inflection allowed him to posit the series of the multiple as a convergent infinite series. On the other hand, the metaphysics of inclusion allowed him to posit the enveloping unity as an irreducible individual unity. In effect, as long as the series remained finite or indefinite, individuals risked being relative, called upon to merge [se fondre] into a universal spirit or world soul capable of complicating all the series. But if the world is an infinite series, it thereby constitutes the logical comprehension of a notion or concept that can now only be individual; it is thus enveloped by an infinity of individuated souls, each of which retains its own irreducible point of view. (F, 27)

The simple image of a world within a soul within a world ... may give us the impression that the two sides are completely interchangeable, like a set of identical Russian dolls.  But this quote (and the two previous ones) helps us understand that the two sides are not symmetrical.  If the world is infinite, it cannot be given as actual.  It can only be "expressed" or "logically comprehended" as a concept, that is, as something virtual.  But this moment of logical comprehension is itself something actual.   Thus, each time we go through another cycle of nesting inside within outside within inside, we move from actual to virtual to actual again, so that there's always a virtual between two actuals (and vice versa).  In order to distinguish the way each side is 'in' the other, Deleuze will say that the world is in the soul, but the soul is for the world.  The soul, with the world folded up inside it, is the only actual thing.  But this actuality only exists as it does so that the virtual series of the world converges.

Because the world is in the monad, each monad includes the whole series of the states of the world; but because the monad is for the world, none of them clearly contains the "reason" of the series of which they are all a result, and which remains external to them as the principle of their accord. We thus go from the world to the subject, at the price of a torsion that makes the world exist actually only in subjects, and also makes the subjects all relate to this world as to the virtuality that they actualize. (F, 29)

Here, Deleuze is clearly alluding to Leibniz's idea of God's pre-established harmony between monads.  But the interesting thing is the way he sees this harmony as essentially identical to the condition of closure of each monad.  It's not that all the monads are created first, each closed or sealed up on itself without consideration for the others, and then once those are finished they are brought into harmony through some sort of tuning process.  In fact, there can really be no 'first monad', no single monad, and even no countable set of monads that would have unity and actuality independent of one another.  Monads are only actualized as what they are by the process of tuning them, meaning that they must all be created at the same time, as the precise structures which will all resonate together, all complete themselves as they converge on the world.  The monad can only be closed because the world it expresses is the convergence of an infinite series.  But the series can only converge if all the monads are closed.  The convergence of the world gives a finality to the monad -- in the sense of both its closure or completion, and in the sense of its 'end' or final cause.

Because the world is in the monad, each monad includes the whole series of the states of the world; but because the monad is for the world, none of them clearly contains the "reason" of the series of which they are all a result, and which remains external to them as the principle of their accord.58 We thus go from the world to the subject, at the price of a torsion that makes the world exist actually only in subjects, and also makes the subjects all relate to this world as to the virtuality that they actualize. (F, 29)

This strange mutual resonance of metaphysically distinct sides is of course a very difficult concept, right up there with emptiness is form and form is emptiness.  Deleuze's brief comments on Heidegger are surprisingly helpful here.  Heidegger, never a particularly subtle or generous reader of other philosophers, assumed that Leibniz thought of monads as closed off substances, each created independently.  By contrast, Heidegger conceived his own dasein as completely open to the world, immersed in-the-world, and hence without even a possibility of having windows.

Martin Heidegger, The Basic Problems of Phenomenology: "As a monad, Dasein needs no window to see what is outside, not, as Leibniz believes, because everything is already accessible inside its capsule…but because the monad, Dasein, in its on being (transcendence) is already outside".   Merleau-Ponty has a much better comprehension of Leibniz when he posited, simply, that "our soul has no windows: that means In der Welt Sein" (F, 30, footnote)

But Deleuze suggests that Heidegger simply misread Leibniz -- the monad was already dasein.  The world is only contained within the 'capsule' of the monad because the monad was specifically created to complete the world.  Thus it isn't 'thrown' into the world as Heidegger often puts it, leaving it to confront a world that has nothing to do with it in some moment of existential crisis.  Instead, the monad is for the world.  The closure of the monad and the convergence of the world are the same thing.

Closure is the condition of being for the world. The condition of closure is valid [vaut] for the infinite opening of the finite: it "finitely represents infinity." It gives to the world the possibility of beginning over again in each monad. The world must be placed in the subject in order for the subject to be for the world. This is the torsion that constitutes the fold of the world and the soul. And it is what gives expression its fundamental character: the soul is the expression of the world (actuality), but because the world is the expressed of the soul (virtuality). Thus God creates expressive souls only because he creates the world that they express by including it: from inflection to inclusion. (F, 30)

So the "point of inclusion", the monad, the soul, turns out to be everything, the whole world.

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I've highlighted the word "torsion" in each of these quotes as a way of bringing us back to our final point, the one I've put off a couple of times.  What is the relationship between Deleuze's reading of Leibniz and Cache's reading of Simondon?  The simplest answer is that Deleuze replaces Cache's concept of the frame with Leibniz's concept of the world.  Both serve analogous roles as a sort of overall structure within which individuation can happen and inflection can become form.  But this answer is actually too simple to make sense without some significant unpacking.

I think Deleuze mentions "torsion" by analogy to a Möbius strip or Klein bottle.  These are both shapes that introduce a confusion between inside and outside or one side and the other.  The Möbius strip also makes an appearance in Earth Moves (EM, 72), precisely at the point where Cache begins to discuss how the frame -- that is architecture as a whole -- doesn't merely define an inside, but actually always allows the outside to pass inside and vice versa.  For Cache, the frame can never captures everything within it without simultaneously making us aware of an out-of-frame that escapes it.  Which of course means that we're paradoxically framing the out-of-frame by constructing a building.  This movement of outside to inside is actually the idea that explains the subtitle of the book: The Furnishing of Territories.  In what we might call a 'torsion', Cache considers furniture as a sort of interior geography.  It provides a milieu for our daily existence within the frame.  In the other direction, the facade of the house serves as a projection of our desires towards the world.  The living room is like a landscape designed expressly for the dimensions of the human body, while those hideous double high entryway columns project the internal self-importance of the occupant, or, by contrast, the brutally blank facade of public housing projects the (supposed) characterless indifference of the souls within.  The architectural frame is the space within which a life is created and from which a life is expressed.

But this literal architectural notion of the frame is just the beginning for Cache.  Throughout the book, he periodically discusses the frame in more abstract terms.  There are the social frames of the gift (EM, 60) or of Rawl's veil of ignorance (EM, 61).  There's also the body as a frame (EM, 73, 130, 142).  But most importantly, Cache's whole scheme ultimately revolves around the idea of the process of individuation as a frame (EM, 117) bordered on one side by chance and on the other by the crystal.  Here he mixes Simondon's scheme of crystalline individuation with his own classification of images.  For Cache, the whole point of 'architecture' is to open a space, a frame, for a particular type of life to flourish (EM, 24).  But as we've seen, a frame does not separate an interior space from its external environment without connecting these two sides.  Which is almost to say that a frame which works too well, which is too tight or does not have windows that permit the continuous back and forth between inside and outside, won't support life at all.  That wouldn't be a frame but a "sack". (EM, 130).  Ultimately then, life is not something that happens within the frame, but in the passing back and forth between the frame and the out-of-frame, in a sort of weaving or folding that Cache captures in his diagram of its "oscillation" (EM, 117).  

This weaving back and forth between too solid frame and too chance variation is related to how Simondon conceived of 'higher' individuations as folded inside unfinished 'lower' ones, rather than built on top once these are completed.  Biological organisms are aperiodic crystals, crystals that were never able to complete their perfectly repetitive structure, but that continue the crystallization by other means.  This is the fractal infolding we saw in our initial discussion of the three transformations of inflection.  The crystal plunges into itself, as it were, taking new detours all the time on its way to perfect crystallization, spawning and (sometimes) overcoming new inflections that threaten to stop it entirely.  It's in this context that Deleuze emphasizes Cache's use of the concept of delay or deferral (F, 18, footnote 39).  In fact, "torsion" and "delay" are almost the same thing here.  Each time we go back and forth through the frame, we create a new level inside the old one.  It's almost as if we're caught in one of Zeno's paradoxes or on Buzzati's Tartar Steppe.  Halfway through one individuation, another begins, suspending progress in the first.  But then halfway through the second a third individuation begins, delaying the completion of the second, and so on.  None of the individuals, it seems, can be completed unless they are all completed.  'The' frame is the whole series of nested frames, each of which defines a relative inside and outside, and thus a type of life.

Leibniz 'frame of the world' is the totality that the infinite series of each monad's infinite series actually converges to, and whose converge is the same as the completion of all the nested individuals.  The individuals are all 'delayed' pending the convergence of the world, but the world won't converge without just the right series of completed individuals.  The virtuality of the world functions a frame, or perhaps more accurately a process of framing that allows for the actualization of individual life.  Unlike Leibniz, Cache doesn't necessarily posit any condition of convergence of closure of all the frames.  They function more like the open dasein we discussed earlier.  It's not clear to me exactly how Deleuze would need to modify Leibniz's idea to take this potential for divergence into account.  Though Smith has some useful suggestions in this regard.

On the other hand, Deleuze is not content simply to provide a reading of Leibniz. "These impersonal and pre-individual nomadic singularities," [monads] Deleuze writes, speaking in his own name, "are what constitute the real transcendental field" (LS 109). Difference and Repetition and Logic of Sense are Deleuze's attempt to define the nature of this transcendental field, freed from the limitations of Leibniz's theological presuppositions, and using his own conceptual vocabulary (multiplicity, singularity, virtuality, problematic, event, and so on). In Deleuze, the Ideas of God, World, and Self take on completely different demeanors than they do in Leibniz. God is no longer a Being who chooses the richest compossible world, but has now become a pure Process that makes all virtualities pass into existence, forming an infinite web of divergent and convergent series. The World is no longer a continuous curve defined by its pre-established harmony, but has become a chaotic universe in which divergent series trace endlessly bifurcating paths, giving rise to violent discords. And the Self, rather than being closed on the compossible world it expresses from within, is now torn open by the divergent series and incompossible ensembles that continually pull it outside itself (the monadic subject, as Deleuze puts it, becomes the nomadic subject). (ED, 57)