Why is Leibniz so important to Deleuze?

I came to reading The Fold rather late, and had noted lots of references to Leibniz, especially in Logic of Sense, and Difference and Repetition.  There were discussions in particular of notions such as incompossibility instead of contradiction, but, more generally, some admiration for the calculus.  Other discussions were Leibnizian even though I did not realize it at the time, especially concepts like the singularity, and the stuff about how multiplicities offer a series of ordinary and singular points stretching to the neighbourhood of the next multiplicity.  I think I can now see why Leibniz in general, and the working out of the calculus in particular, are so important.

Let's take the calculus first.  I'm not mathematician enough to understand the precise arguments involved in Leibniz's work, but I can see some general points.  In the first place, the calculus helps us to analyze the shape of, and the areas under, curved lines, whereas classical mathematics had operated with straight lines.  Not only is the curved line an interesting mathematical problem, but analyzing curved lines helps us grasp real problems, in nature, which is riddled with curved lines.  Leibniz proceeds, according to the notes I read on the calculus, to consider infinitesimal differences and the series of them, and then projects this notion forward, as it were, to explain the slope of the lines of tangent, even very very small or infinitesimal ones.  We can see this in terms of the practical procedures of rendering the curved line as a series of straight line tangents, but to get an accurate description, we need an infinite number of tangents.  What Leibniz did was to manage this problem by saying that we can describe infinitesimal differences as ratios, without needing to actually put in real numbers or concrete measures, and this is dead handy if we are discussing infinity, and also very handy if we are discussing the forces that produce curves, which might not necessarily work on the same numerical base.  We also need to talk in abstract terms like lines segments rather than lengths.  The particular ratio that interested Leibniz was the one between rates of change in the Y and X variables.  Those variables can be seen as offering two sides of a right triangle with the tangent as the third side, and if we know the values of X and Y, we can calculate the value of the tangent.  Moving away from concrete values to ratios, we can talk about the abstracted changes in X and Y and their ratio—giving us the famous differential equation of DY/DX, where D stands for the rate of change.  This is a constant ratio, at least between singular points on a curve, where the slope is constant.  At singular points, the curve slope changes.

Although this particular approach needed modification, and we had to ignore some issues for the time being, including intrusive philosophical notions of the infinitely small, we made progress in explaining the slope of the curve, and, in turn, this helped us calculate the area under a curve, or quadrature, and we can do this even for curves that extend even to the infinitely large.  Well done, Leibniz.  But in the process, he had developed mathematics away from concrete values, away even from normal algebra, to the study of relations, something that was to lead to the notion of mathematical functions.  Deleuze often cites the example which makes this point, saying that even when the actual values of X and Y are zero, the relation between DX and DY persists [if I have to followed this, it is because the relation is derived from actual triangles drawn with positive values, which are not equal, whereas if X and Y are reduced to zero, they would be made equal, using conventional rules.  So stuff conventional rules, let's give calculus new rules].

This exercise in the calculus is part of a far more general approach to develop a mathematical understanding of philosophy and of real problems, again by developing a particularly abstract grasp of them.  We can see complex real objects, for example, as a series of curves.  We can explain continuity in a new way, as regular movement from one singularity to another.  This is clearly going to assist Deleuze in his own project to account for chaosmos in terms of mathematical concepts like multiplicity, singularity, and force, an underlying process that produces fixed points.

Equally important is the connection between mathematical processes and reasoning like this, and the more general 17th century notion of sufficient reason.  One of the main implications here, it seems to me, is that if we invoke the principle of sufficient reason, we proceed by rational analysis to unpack the predicates of a particular concept as subject.  Predicates can be events. Mathematics seems to proceed in this way, to nonmathematicians like me, by taking a concept such as the function, and unpacking more and more implications from it, developing implications in the form of a whole series of equations, then exploring the implications, and so on.  Philosophical analysis in general was to proceed in this way, via a number of ingenious discussions in Leibniz to extend the principle of sufficient reason in order to make it consistent with the other philosophies and other concepts like identity.

It strikes me that the sort of philosophy embraced by Deleuze follows exactly this pattern.  When he talks about what the concept is, he means it in the same sense as Leibniz,  a subject with predicates, something that can be inferred by working back from its predicates, although this does not necessarily give us the entire concept, and often means that we have only 'fuzzy' definitions.  And when he talks about how forces turn into realities, via actualizations, he means something like the concept unpacking itself, producing a chain going the other way, so to speak, offering more and more limited definitions, and more and more empirical accounts of the relations between them.  [I can also see some links with Spinoza here].  This also explains his disdain for empirical analysis, of the kind you might find in sociology, which operates on the surface, to change the metaphor a bit, too far down the chain towards the empirical end.  As we get closer to actualities, we can define them empirically and also use relations between them such as causes: but there are deeper definitions and deeper forces and relations further back up the chain, and it is those that philosophy needs to investigate [I am probably using terms developed by DeLanda here].

There are some ore similarities. The Leibnizian method of vice-diction, for example, where
one examines concrete cases, including the accidental ones, in order to trace the operation of the virtual. This is the sort of procedure developed in Logic of Sense ,where it is described as 'counteractualization', going from the actual to the multiplicity,  and in Difference and Repetition. Deleuze's admiration for it is repeated in a lecture he gave on the 'method of dramatization', where he said it 'consists in traversing the Idea as a multiplicity', which produces all cases, the accidental and the apparently essential alike.

Wayne Brooks (Facebook 23/01/2017) In differential calculus, D(x) means let x be any infinitesimal difference in x. Let D(y) be any infinitesimal difference in y. An infinitesimal difference in either x or y, while infinitesimal in itself is significant in the relationship of x to y formulated. The relationship of x is altered by any even infinitesimal difference in x or y [the butterfly effect]. If x is a straight line and y is a straight line, any infinitesimal difference in either one of them will cause them to intersect. What this looks like in a formula is D(x)/D(y)=change in x/change in y, and when x and y are given values (actual, determined), the differential (virtual, determinable) can be calculated. What gets calculated is then actual, however the trajectory of the the calculus, the curve between variables has infinite points along the curve, calculated between 0 and infinity which are clear limits of reality expressed mathmatically.

For example, Velocity = distance divided by time, meaning that we can use this relationship in reality (velocity in this case) to see how differences (variable in relationship to each other) apply (distance and time in this case). Why this matters is because it is an Actual expression of how Virtual Ideas operate for Deleuze—they vary in relationship to one another as any differences/variables vary in the real world. From start to finish, the faster I go and the further I go in my car, the greater my velocity (miles per hour).

These values vary from start to finish, and from distance a, to distance b, to distance c, etc. At the same time there would be time a, time b, time c, etc., so we could calculate the velocity at any point a, b, c, etc. by plugging the values in to the formula: velocity = distance divided by time, =d/t. So how many points a, b, c are there from start to finish? Actually, there are an infinite number of points which we could calculate.

These infinite number of points are actually calculated by differential calculus, a method to cover all the possible points. Well how can that be done? In order to calculate target point x on the start to finish line, Leibniz who had not yet invented differential calculus) chose to start with the average velocity of the car over a time (t) and slowly decrease the slice of time we use to divide the distance (d) travelled so that d/t would approximate (infinitely approach) the target point x. This had the effect of making t differ from d in incrementally smaller time periods until arriving at the target point x. Keep this difference, this derivative, this differential calculus in mind as Deleuze’s concept of how Ideas function in Virtuality. He is merely using mathematics here as a model of the Virtual.

We have a limited faculty of thought, so all we can see, all our brains can calculate are the given, the mere appearance of things, like looking through a darkened glass. Instead of thinking that we fully synthesize an object of perception, we need to realize that our thought is a synthesis in process, an ‘arising’ or ‘flowing.’ Therefore we become subject to the transcendental illusion, or representations of objects as if they were complete and original because we dont automatically do differential calculus of all variables at all times (and not given to intuition).

The differential calculus relates the three moments of the Idea intrinsically. It is:

1) Undetermined in that the differential cannot be given in intuition
2) Determinable when it is put into a relation dy/dx
3) Determined when given specific values

The calculus provides symbols of difference (individual values) which although propositional, points beyond itself to the problem itself. Thus the calculus accounts for how undetermined elements can become determinate through reciprocal relations, not only in mathematics, but in the world at large

The differential calculus “has a wider universal sense in which it designates the composite universal whole that includes Problems or dialectical Ideas, the Scientific expression of problems and the Establishment of fields of solution.” (p. 181)
See also my thoughts on folds

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