by Corry Shores
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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 6: Hegel and Deleuze
Very Brief Summary:
Hegel’s interpretation of the differential involves a representational view, Deleuze’s does not.
Brief Summary:
Hegel’s concern with the dy/dx of calculus is with the relation between its finite value and its infinite (insofar as they are vanishing and thus non-finite) constituent values. But for Hegel, the differential still is a matter of representation, even if it is infinite representation [it is infinite because it involves the combination of contraries.] For Deleuze, the components are subrepresentational. This is like how the virtual is not yet explicated. The value of the differential for Deleuze is that it can explain the basis for determination without resorting to representation.
Summary
Previously we saw Deleuze views the components of the differential relation as subrepresentational, and this allowed him to conceiving the differential relation using neither finite or infinite representation.
Now we will compare Deleuze and Hegel on the calculus. In sum:
Hegel takes a broadly Newtonian line in his interpretation of the calculus, whereas Deleuze takes his inspiration from the work of Leibniz, albeit with a nonintuitional interpretation of the differential. For Deleuze, the difficulty of differentials appearing in the resultant formulae is resolved through the belief that we are dealing with two different ontological planes, for Hegel once again through recognizing that the status of the nascent ratio differs from that of normal numbers. Aside from this difference, we should note that what Hegel discovers in the calculus is "the infinity of 'relation"' (DR, 310, n.9), meaning that what is important in the calculus is the relation between the two fluxions, to such a degree that this relation not only defines the determination of them but also their existence. (178)
For Deleuze, the relation between differential values is one of determinability, meaning that the are undetermined with respect to their variable, but together determine a value. So their existence is not just a matter of their being differentially related. In the fourth remark to the dialectic of becoming, Hegel refers to the differential calculus when explaining vanishing. On this basis Deleuze characterizes Hegel’s whole philosophy as being one of infinite representation. Hegel speaks of the differential as vanishing and being on the point of disappearing, but not having vanished yet. It is not that he renounces finite representation. Rather, he wants it integrated inside the infinite [the finite value of the differential relation is based on the non-finite values of dx and dy. So in the infinite is the finite. Here the finite representation of quantities is integrated inside the infinite representation, when those infinite representations are differentially related so to constitute the finite representation/value.] Somers-Hall explains:
In this sense, for Deleuze, Hegel's criticism of those who went before him is that they had not taken difference to the level of absolute difference, the contradiction. In making this final move, Hegel goes from finite, organic representation to what Deleuze calls "infinite, orgiastic representation." Organic representation is given content by participating in orgiastic representation, just as both being and the differential are maintained as being just on the point of vanishing. (178d)
Hegel’s emphasis is on the structural elements of the differential ratio dy/dx. Deleuze’s concern instead is “the difference in kind between the primitive function and the differential and the fact that by integration, the primitive function can be generated from the differential”. (179) Recall how for Bergson disorder is just a different sort of ordering. The primitive function and the differential are different in kind, but it might be said that they are two different kinds of distributions involved, “sedentary (representational) distributions and nomadic (differential) distributions.” (179) For Deleuze, negation and opposition appear in the world to the degree that they are “ ‘cut off from their virtuality which they actualise, and from the movement of their actualisation’ (DR, 207)” (179) Deleuze provides an account of the genesis of determination. But he does not reduce the absence of determination to indifference. He is able to do this, because differential relations for him have a structure of their own, and because the differential has a structure that escapes representation. [Determination is generated through differential relations which are undetermined yet not indifferent; they are subrepresentational rather than being determinate; but they are determinable in relation to one another, producing determinations.] “In doing so, Deleuze walks a tightrope by treating the virtual as both real and not actual.” (179)
Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.
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