Differential Fields

I still don't really understand Leibniz.  But I can see he's lurking in the background of this section at least as much as Hegel.  In fact, Deleuze seems to see Leibniz as having only just missed the mark:

In this sense, too, Leibniz goes further or deeper than Hegel when he distributes the distinctive points and the differential elements of a multiplicity throughout the ground, and when he discovers a play in the creation of the world. It seems, therefore, as though the first dimension, that of the limit, despite all its imperfection, remains closest to the original depth. Leibniz's only error was to have linked difference to the negative of limitation, because he maintained the dominance of the old principle, because he linked the series to a principle of convergence, without seeing that divergence itself was an object of affirmation, or that the incompossibles belonged to the same world and were affirmed as the greatest crime and the greatest virtue of the one and only world, that of the eternal return. (my emphasis)

How does this distribution of distinctive points and differential elements work though?  All I really know about Leibniz is the thumbnail sketch that he thought of the world as composed of monads, which I've always imagined along the lines of spiritual rather than physical atoms.  In other words, I would have called Leibniz an "ideal atomist", for lack of a better term.  Which is actually a kinda interesting combination, now that I think about it.  I'd long ago given up on the naive version of Materialism that dominates our current scientized popular worldview.  So Idealism as a concept doesn't seem that shocking to me.  It's hard to make headway with it though, when ideas are usually presented as being subjective entities that only exist in the skulls of hairless chimps.  What cosmic arrogance!  However, if ideas are like tiny minds distributed all over the place, and even come in atomic form, then the theory starts to become a lot more interesting.  That doesn't get us over the hurdle that Deleuze is raising in this chapter, namely that monads as ideal atoms are still conceived on a model of identity, and have no internal structure one can investigate.  But it's still a thought provoking change of context.  

Or at least that's what I would have said about Leibniz, until I reviewed the video tape.  It turns out monads do have a sort of internal principle, so aren't quite atoms after all.  This is all from The Monadology:

9. It is also necessary that each monad be different from each other. For there are never two beings in nature that are perfectly alike, two beings in which it is not possible to discover an internal difference, that is, one founded on an intrinsic denomination.

 

10. I also take for granted that every created being, and consequently the created monad as well, is subject to change, and even that this change is continual in each thing.

 

11. It follows from what we have just said that the monad's natural changes come from an internal principle, since no external cause can influence it internally (sec. 396, 400).

 

12. But, besides the principle of change, there must be diversity [un détail] in that which changes, which produces, so to speak, the specification and variety of simple substances.

 

13. This diversity must involve a multitude in the unity or in the simple. For, since all natural change is produced by degrees, something changes and something remains. As a result, there must be a plurality of properties [affections] and relations in the simple substance, although it has no parts.

 

14. The passing state which involves and represents a multitude in the unity or in the simple substance is nothing other than what one calls perception, which should be distinguished from apperception, or consciousness, as will be evident in what follows. This is where the Cartesians have failed badly, since they took no account of the perceptions that we do not apperceive. This is also what made them believe that minds alone are monads and that there are no animal souls or other entelechies. 

 Hoocoodanode!?  Somehow monads contain or represent or are composed by a diverse multitude that in a passing moment of "perception" are unified into a simple substance with no parts.  Isn't this exactly the type of structure we're looking for with a concept of difference in itself producing identity?  Leibniz kinda cruises right past this point; The Monadology goes on to talk about how the soul is connected to the body, how God is the reason for everything, how this is the best of all possible worlds, etc ... He only really comes back to make any use of his passing observation that monads express a multiplicity when he concludes that each monad actually expresses the entire world from a certain perspective.  Each monad is a mirror of all infinity, though it expresses some part of its relationship to that infinite clearly (namely the part that represents its body) and some part obscurely (everything else). It seems like the infinite in question here is basically all the other monads, though this gets a bit confusing because later on he says that monads grow and develop somewhat like multiplying cells, producing higher level monads which are the souls of animals and people.  Which suggests that not only are there an infinite number of monads, but they are sort of infinitely nested or folded into one another as well.  I don't want to get totally lost in the details of his philosophy now, especially the way this relates to God and compossibility and the notorious "best of all possible worlds".

However, I do want to pick up on this thread of a "diversity involving a multiplicity in the unity".  It seems that for the co-inventor of the calculus this wasn't just a philosophical, but also a mathematical concept, or at least that's the way Deleuze interprets it.  In other words, diversity = the differential element (dx or dy).  There are various formulations of calculus, but the one that both Leibniz and Newton used is the one I was still taught in high school -- the differential is an infinitely small difference, and two infinitesimals form a clear relationship that defines the slope of the curve (dy/dx) at the limit of the difference becoming infinitely small.  Once you know the derivative -- formed by these differential relations at each point -- you can integrate them to find the curve.  One of the first steps in imagining that integral is to figure out where the underlying curve is vertical and where it is flat by finding the distinctive points where the derivative is either zero (flat) or infinite (vertical).  Either of these can be referred to as singularities of the derivative, though I feel like you hear that a lot more in the case of the derivative going infinite.  I take this to be the thrust of the comments Deleuze made about Leibniz in the previous section:

In reality, the expression 'infinitely small difference' does indeed indicate that the difference vanishes so far as intuition is concerned. Once it finds its concept, however, it is rather intuition itself which disappears in favour of the differential relation,
as is shown by saying that dx is minimal in relation to x, as dy is in relation to y, but that dy/dx is the internal qualitative relation, expressing the universal of a function independently of its particular numerical values. However, if this relation has no numerical determinations, it does have degrees of variation corresponding to diverse forms and equations. These degrees are themselves like the relations of the universal, and the differential relations, in this sense, are caught up in a process of reciprocal determination which translates the interdependence of the variable coefficients.  But once again, reciprocal determination expresses only the first aspect of a veritable principle of reason; the second aspect is complete determination. For each degree or relation, regarded as the universal of a given function, determines the existence and distribution of distinctive points on the corresponding curve. We must take great care here not to confuse 'complete' with 'completed'. The difference is that, for the equation of a curve, for example, the differential relation refers only to straight lines determined by the nature of the curve. It is already a complete determination of the object, yet it expresses only a part of the entire object, namely the part regarded as 'derived' (the other part, which is expressed by the so-called primitive function, can be found only by integration, which is not simply the inverse of differentiation. Similarly, it is integration which defines the nature of the previously determined distinctive points). That is why an object can be completely determined - ens omni modo determinatum - without, for all that, possessing the integrity which alone constitutes its actual existence. Under the double aspect of reciprocal determination and complete determination, however it appears already as if the limit coincides with the power itself. The limit is defined by convergence. The numerical values of a function find their limit in the differential relation; the differential relations find their limit in the degrees of variation; and at each degree the distinctive points are the limits of series which are analytically continued one into the other.

So by now you can probably see what I'm driving at.  I think Deleuze is borrowing this blurring of philosophy and mathematics that occurs with Leibniz to give us an image of what the world of pure difference is like.  It's a differential field along with the singularities that this field defines.  This is most fun to visualize in two dimensions as a vector field.  I am not the first guy to interpret Deleuze this way.  In fact, most of Manuel DeLanda's reading of him is an elaboration of this basic image.  I don't remember whether DeLanda talks about this or not, but it seems Deleuze himself is borrowing the basic idea from Leibniz.  Basically, each monad is a singularity in a differential field created by the relations of all the other monads.  And a principle of limit and convergence and continuity reigns at every level.  You can define the derivative, even though it is only a relationship of infinitesimals, because of the properties of limits on a continuous number line.  You can integrate all of these differentials into a singular monad because God assures you that all the monads play nice together, and converge on one compossible world -- the best of all possible worlds.  Deleuze is borrowing the image, and removing the converge part.