5 Jan 2013

Pt3.Ch6.Sb5 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘Berkeley and the Foundations of the Calculus’. summary

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[Note: All boldface and underlining is my own. It is intended for skimming purposes. Bracketed comments are also my own explanations or interpretations.]


Henry Somers-Hall


Hegel, Deleuze, and the Critique of Representation.

Dialectics of Negation and Difference


Part 3: Beyond Representation

Chapter 6: Hegel and Deleuze on Ontology and the Calculus

Subdivision 5: Deleuze and the Calculus

Very Brief Summary:

Deleuze avoids conceiving the differential relation using either finite or infinite representation, because he regards its components as subrepresentational.

Brief Summary:

For Deleuze, there are three moments in the differential relation. dx is nothing in relation to y. The relation on its component level is subrepresentational, but the components reciprocal value is determinable through their differential relation, and by doing so we obtain actual sensible determinations. It is undetermined. It is subrepresentational. But, even though they are undetermined, they are still determinable in relation to each other, because together they form a finite value. So they escape representation, but because they are determinable in relation to one another, through their reciprocal interaction, they can generate determinate representations.  But these relations produce determinations, like how the phase portrait can be seen as the Idea or the problematic, the virtual field of differentially generated incompossibilities that can become actualized. So something subrepresentational is responsible for something representational. The differential can be conceived either using finite or infinite representation. Deleuze chooses neither, because both are representational and thus have the problems of representation. Thus his solution is to see it as subrepresentational but still determinable.



Previously we saw how Hegel’s dy/dx expresses the sublation of being and nothingness.

Now we turn to Deleuze on the calculus. The straight line in the eyes of calculus is the limit of a curve. [For Bergson, space is the limit case of duration and vice versa, so] This “mirrors Bergson's own relation of the two categories of space and duration.”  (172) But Bergson thinks that science misses the lessons of differential calculus, and Deleuze picks up on this.

As the method we have outlined shows, as the section of the curve by which we draw the tangent shrinks, the curve and the straight line become interchangeable. Instead of using this fact to open up a kind of thinking in duration, modern science for Bergson instead uses this fact to approximate the curve using a series of straight lines. Instead of, for example, using the idea of the curve to represent the relation between the biological as durational to matter as extensity, therefore, science attempts to reconstruct an approximation of the curve from a series of tangents, thus reducing duration to extensity (the tangent as the atomic element from which, for the scientist, the curve is constructed). This failure of the science of Bergson's time to adequately understand the role that motion plays also opens up the possibility of a new relation to science that Bergson recognizes as a future possibility and that Deleuze will try to actualize through his interpretation of the calculus. (172)


According to Deleuze

“just as we oppose difference in itself to negativity, so we oppose dx to not-A, the symbol of difference to that of contradiction" (DR, 170). [172-173]

But Deleuze’s analysis has similarities to Hegel’s. He wants a metaphysics that takes dx seriously. Like Hegel, he does not want the differentials to be infinitesimals. Deleuze rejects Leibniz’s infinitesimals because

"quanta as objects of intuition always have particular values; and even when they are united in a fractional relation, each maintains a value independently of the relation" (DR, 171). *(173)

For the algebraic equation of the circle,

x2 + y2 - R2 = 0

We can substitute arbitrary values in for the variables. However, for the differential formulation of the circle, we cannot.

ydy + xdx = 0

Thus calculus does not operate using algebraic terms.

Deleuze claims that “ ‘quanta as objects of intuition always have particular values; and even when they are united in a fractional relation, each maintains a value independently of the relation’  (DR, 171).”  (173) So there are three moments, undetermined, determinable, and determination, for each moment Deleuze assigns a principle, and all these principles together expresses a principle of sufficient reason.

Stage 1: The undetermined as such: the differentials themselves (dx, dy).

dx is nothing in relation to x. This is because “it cannot be captured by either (Kantian) intuition or the categories of quantity.” (173) This makes it “continuousness”, because discrete quantities can be intuited and captured by the categories of quantity. Continuousness and continuity are not the same thing. [Continuousness is the feature that things have which cannot be captured by intuition or categories of quantity. Continuity is the unbrokenness of things in the sensible world, like continuous variations such as curves.] Continuous is the ideal cause of continuity found in the sensible realm. For Deleuze, we should not think of the differential relation in terms of variability. Now, differential elements escape determinations as quantities. Through their reciprocal determination, determinate quantities are generated by the differential function. [So because the differential elements are continuousness, and because their relation allows for the constituent differential relations in variability] continuousness “represents the transcendental condition for variability.” (174a) [Because dx cannot be captured by intuition or categories of quantity]

In this sense, the differential , dx, as a symbol of difference, is "completely undetermined" (DR, 171), that is, as the representation of the "closest noumenon" (DR, 286), difference, it escapes the symbolic order. The symbols, dy and dx, and their values of 0 in respect to y and x, therefore represent the annihilation of the quantitative within them in favor of what Deleuze calls the "sub-representational" or "extrapropositional." (174)

Stage 2: Determinable.

Deleuze replaces Leibniz’s infinitesimal with the notion of the limit, but not the set theory one that conventionally replaced the infinitesimal.

Instead, Deleuze introduces the notion of limit, but not the limit of the modern interpretation, whereby the differential is the value an infinite series converges on. Instead, "the limit must be conceived not as the limit of a function but as a genuine cut" (DR, 172). Deleuze's point is that whereas dy and dx are completely undetermined in relation to x and y, they are completely determinable in relation to one another. (174)

For Leibniz dy and dx fall out of the equation because they are two small. This “led to the difficulty that, on the one hand, they relied on this quantitative moment to form a ratio and thereby determine the value of x, but on the other hand, this moment had to be ignored in order that we could determine the value of x itself, rather than x + dx.”  (174) But Deleuze does not characterize dy and dx quantitatively, “so, when we determine the quantitative answer, they fall out of the equation, as they cannot be captured by the categories of quantity. That is to say, the differentials themselves escape representation but, through their reciprocal interaction, are able to generate determinate representations.” (174) For Hegel, dy and dx are only determinate in relation to one another. The same goes for Deleuze, but also for him, they are still determinable prior to their determination. So they are unlike Hegel’s ultimate ratio.

Deleuze uses differential calculus to characterize the Idea. He says it has for its object the differential relation. (175a) For Deleuze, the differential relation is understood in terms of its difference from the function that is differentiated (the primitive function). We used the example of y = x2. Differentiated, it gives dy/dx = 2x. Now, we can substitute values here to obtain the tangent. So that makes the differential a function too. The primitive function gives us the y values for the curve. The differential function gives us the tangent for any point on the curve. So there is a difference between the function and the differential relation, and it is a qualitative difference [because it determines something difference in kind and not number]. [For some reason]

The primitive function, which deals with the relations of actual magnitudes, is tied to representation, whereas the differential function, which specifies values in terms of dy and dx, is instead tied to the Idea. The movement from the differential to the primitive function is therefore seen by Deleuze to be a movement of generation, akin to the movement from virtual to actual. (175)

Stage 3: Complete determination.

Complete determination corresponds to the actual values of dy/dx. The tangent determines ‘singularities’, points where the nature of the curve changes. So consider this animation.


(Thanks wiki)

Points of singularity are where “dy/dx becomes null, infinite, or equal to zero, represent points of transition within the structure of me curve itself, as for instance, when me gradient of a tangent is equal to zero”. (175)

The importance of these singular points is that they capture me specific nature of the curve and correspond to me singularities of cycles and spirals mat we saw to be definitive of me geography of me phase portrait. This means, for Deleuze, that the differential calculus allows us both to specify the general, in the form of the curve as a whole, as well as me singular, in the form of me series mat emerges from the repeated differentiation of the area around a singularity. (175)

The Idea for Deleuze is “ ‘a system of differential relations between reciprocally determined genetic elements’ (DR, 173-174)”. (176) The Idea and the problem for Deleuze both should be understood in terms of calculus.

the problematic field is to be defined as a field of differential relations, its solution can be defined in terms of finding the primitive function that gives rise to those differential relations. Integration is the inverse operation of differentiation, and i n this sense, the primitive function can be said to be contained within the differentiated function (i .e., immanent to it). (176)

The problematic is transcendent to the solution because the primitive function deals with magnitudes but the problematic with continuousness. [I cannot explain the last part of the following]

The movement between the problem and the solution, as we saw in our previous discussion of it, is a fundamentally ontological movement. The way an organism is structured, for instance, is seen by Deleuze to be a solution to a problem. Because of this, the calculus is essential to Deleuze in allowing him to generate the kind of differential transcendental philosophy that we saw was needed to overcome the limitations in the Kantian system thrown up by Sartre . (176)

Hegel saw the calculus as dialectical. Its notion of the ultimate ratio as vanishing can only be understood using infinite thought which thinks sublated contraries. The set theoretical approach tried to continue conceiving calculus using finite representation. The distinction between finite and infinite involves the metaphysics of calculus, for Deleuze. Deleuze gives the differential neither a fictive nor a real status. It is subrepresentational.

We now discuss the relations between the differential, the phase portrait, and the Idea. Bergson explained the concept of light as the organicism of the integration of all the colors of light converging to make white.

In order to move away from an understanding of the Idea built on analogy, Deleuze moved to the concept of phase space, which was essentially a space that allowed the total dynamic possibilities of the system to be specified simultaneously, while still affirming the reality of different possible trajectories. The phase space thus allowed us to see how the general structure of the system could be specified without the specific | states being opposed to one another. (177-178)

The phase portrait is constructed using differential calculus. But for Deleuze, this calculus cannot rely on finite [set theory] or infinite [Hegel?/Leibniz?] representation. This is why for Deleuze the differential is subrepresentational but still determinable. (178)

Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.


  1. Replies
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