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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
Very Brief Summary:
Deleuze’s and Hegel’s responses to classical representationalist philosophy can be compared on the basis of their different interpretations of differential calculus and Kant’s antinomies. We find that for Deleuze we have an unconditioned ground of sensible and intelligible things that is subrepresentational, and for Hegel it is representable, using infinite thought.
Brief Summary:
The calculus differential determines the varying relation between variables that vary with respect to one another. Leibniz saw it as the relation between infinitesimal magnitudes but there are formal problems with this. Newton saw it in terms of vanishing values. Hegel regards the vanishing values as being determinate values that combine finite and infinite, and being and nothingness. This contradiction is only thinkable with infinite thought. For Deleuze the terms of the differential relation are undetermined and subrepresentational, but they are determinable in relation to one another and are the unconditioned condition of conditioned actual determinations. Kant thinks we arrive at antinomous theories regarding whether there is a temporal beginning to the world because our understanding is unable to grasp the unconditioned, the thing in itself, with its categories. Hegel thinks the antinomies go together. Together they express the genuine infinite, because they affirm both that there is a limit and also that it is surpassed. For Hegel the unconditioned, the dialectical contradiction, is representable with infinite thought. Deleuze thinks that the unconditioned is thinkable but not using representational thought but rather using the logic of incompossibility.
Summary
In the previous chapter we examined and evaluated Hegel’s dialectical system. We saw how it generates the categories of our thinking. It involves contrary concepts being thought together and producing third new concepts. Finite thought cannot think this productive self-contradiction, but infinite thought can. Kant derived the categories on the basis of finite thought, which limits him to only be able to talk about the structure of the understanding, but Hegel’s infinite thought allows him to examine the structures of reality as well. Aristotle (and Russell) had a hierarchical system with being at the top. All things are beings because they all fall under the highest genera, being. But that genera as the highest cannot be defined, so his system is incomplete. Because Hegel’s categories are genetically linked, he can have both various categories that are linked but without leading to paradoxes arising when we disallow self-reference and self-contradiction. This also allows him to explain change, which was a problem in Aristotle’s system.
Now in this chapter we compare Deleuze’s and Hegel’s interpretations of the differential calculus [a] to see how they both overcome limitations of classical representational systems in philosophy and [b] to see that Hegel does so with a modified sort of representational system, an infinite sort, while Deleuze does so on the basis of subrepresentational elements.
First we examined the differential relation in calculus. We saw how differential calculus analyzes the varying differential relation that holds between quantities that vary with respect to one another. Leibniz manner of performing this operation involves us seeing the differential as the ratio between [a] an infinitely small amount of one term’s variation, to [b] an infinitely small amount of another term’s variation, at some point along their continuous co-variance.
But the problem with Leibniz concept of the infinitesimal is that it does not hold to the rigorous standards required for mathematical proofs, and also it is treated in a suspicious way, first supposing it, then later removing it when its presence is inconvenient to the rules of mathematics. Newton’s method of fluxions is not really based on an infinitesimal quantity. Instead we are to think of a line as a point in motion. We then think of the dy/dx as expressing and instantaneous velocity at some place along the line’s movement. So first we take note of that place where we want to find the tangent (the derivative). We consider the point moving toward that line, moving so fast along the x axis per how fast it moves along the y axis [so if its x value changes slowly with respect to its y value, then the gradient is turning sharply upward. Think of distance (y) to time (x), if something moves very far in little time, it is going fast.] The moving point then comes to location whose value we want to find. As the moving point reaches that stable-point, the distance between the two points’ x values and y values nears zero. There will be a moment when that difference between them is in the act of vanishing to zero. The x value will be vanishing to zero with a certain degree of intensity with respect to how intensely the y value is vanishing to zero. So here we have eliminated the problematic concept of the infinitesimal. Newton’s fluxion will help Hegel articulate his concept of infinity. He distinguishes the spurious infinite from the genuine infinite. Consider if we want to think about an infinitely great numeral. We might think of a highest, then say, now add one. This makes the previously seemingly infinite value now finite, and it has a limit, on the other side of which is an infinite value. But then we say to that infinite, add another, thus making it finite and creating a new infinite. Hegel’s example is 2/7 = 0.285 7 142 …. The … means that we keep needing to add another and another decimal in order to represent the value that is complete in fractional form as 2/7. But Hegel wants a representation of the genuine infinite with terms that are not external to the concept. In Newton’s dy/dx, we do not have independent values for each term. We only have values for both terms together. Here we have a finite value that is made of vanishing terms, which means this finite value is made of non-finite (infinite) values. Also, its infinite values are nothing unless they differentially relate expressing a finite value. So inherent to the infinity is the finite. Also, because they are vanishing, they are straddling both being and nothingness. So this is like Hegel’s true infinite where the finite and infinite contain one another and are sublated.
Put another way, the fluxions in dy/dx are vanishing, but are still determinate. That means they are still being and not yet nothing. But this is being at the brink of being nothing, so the dy/dx expresses a sublation of being and nothingness.
For Deleuze, the dy and dx are not determinate, however he is also not falling back on the infinitesimals. There are three moments in the differential relation: undertermined, determinable, and determination. Either term alone has no value with relation to its respective variable. So they do not have determinate values on their own, they are undetermined. This also means they are not representable. We cannot give either one a finite determinate value like we can the terms in fractions. But they are determinable in relation to one another, meaning that when they are differentially compared, they produce a determinate finite value. Their determinability is important, we will see, because it means that they are actual like determinate values but they cannot be conceived using representational thinking. That is only possible when they together actualize some finite value, and it is only this finite value that is representable. So value relationships in our world of intellibility and sensiblility are actualized. But they are just one actualization from a set of other virtual incompossible ways that a problematic situation can be actualized. The subrepresentational level conditions the actualized determinations, making it the unconditioned. But note that the unconditioned and the conditioned are ontologically distinct, one being virtual and the other actual.
So Hegel’s concept of the differential is representational, because the terms are determinate. Although, it is infinite representation, because the differential relation expresses the sublation of being and nothingness. The sublation, as what is responsible for the terms, is the unconditioned that conditions the terms, which are the conditioned. They are both representable for Hegel. For Deleuze, however, the components of the differential relation are subrepresentational. So the basis for representable determinations is not itself representational and it is not of the same order.
We will further bring out the difference in Hegel’s and Deleuze’s conceptions by seeing their interpretations of Kant’s antinomy regarding whether the world has a temporal beginning or not. The dogmatic view says there is a beginning, because were there not, there would be an endless series of moments prior to the present one, but without there being a first on which to gain a foothold, they never could have progressed to the present moment. Thus there must be a beginning. The empiricist view says that if there were a beginning, then it is a limit, which means there is something temporal before it. It must be identical moments, but as such they cannot explain how suddenly change appears. So there must be no limit to the world’s temporality. Kant has his own diagnosis of the cause of this irresolvable antinomy. In the first case of there being a beginning, each moment of time is conditioned by its prior, except the first one, which has no prior one to condition it, and thus the first moment is unconditioned. That means the unconditioned is a moment like all the rest, they are all the same thing, moments in time, and as appearings of the world they can be understood by the categories of our understanding. In the second theory where the world has no beginning, also each moment is conditioned by its prior one. However, the series as a whole is assumed to always have been and always will be there. This means that there is nothing outside it that conditions it, for if there were such an external condition, that it would be considered the whole series’ beginning. Thus the series as a whole is the unconditioned. But again, because we are considering the series as a whole the unconditioned, but it is made up solely of parts that are conditioned, then we again are placing the conditioned and the unconditioned on the same level and we are understanding both using the categories of the understanding. Kant says that this is the problem. Instead for him, the unconditioned is on the level of the thing-in-itself, the noumenal level and not the phenomenal, thus it cannot be understood using the categories of the understanding. One view thinks that the world’s temporality is finite, the other infinite. But there is a third option, which is that the thing-in-itself is neither finite nor infinite, just like if we say, ‘bodies have either a good or a bad smell,’ then add that there is a third option, that bodies do not have smells. So for Kant, we encounter the antinomies when we only use our understanding at the urging of our reason which seeks the unconditioned of the conditioned. Hegel, however, thinks that we can know more than just the categories for understanding things, we can also understand the categories of the thing-in-itself. He thinks there is a form of thinking, infinite thought, that is able to conceive of contraries that sublate to produce a new category. In his analysis of the calculus differential, Hegel notes how two non-finite values are in an oppositional relation that produces a finite value. They are non-finite because they are in the act of vanishing, which means they are straddling both a finite value and zero, or being and nothingness. They can only have a value when they are in a differential relation, so the differential relation as something infinite contains in it something finite. But that also means that the finite value of the differential relation is composed of infinite values. This genuine infinite is a sublation of the concepts of infinite and finite and of being and nothingness. Using merely a representational system of logic, like the one Kant bases his metaphysical deduction on when determining the categories of our understanding, leads us to not be able to see how finite and infinite can include one another, how such a contradiction can not only hold but also be generative of new categories. Hegel will say that Kant’s antimony really affirms this infinite thought. So first he notes that each argument in the antinomy is circular. The argument that there is a beginning says that if there were not, we could not have arrived at the present moment. But this present moment that we are taking for granted is a limit, a beginning all its own, because it is the beginning for all moments happening after it. The argument for the limitlessness of time says that the first moment must have an antecedent of some kind, that every moment must have an antecedent, but that is what they were trying to prove. For Hegel, both sides of the antinomy are the parts of an infinite thought of the concept of infinity, because the one side affirms there must be a limit and the other affirms it must always be overcome. For Hegel, the unconditioned is the dialectical sublation, which is found in both reality and thought, and thus for Hegel the conditioned and the unconditioned are the same sort of thing, they and are contradictory, but they can be thought together. In Deleuze’s analysis of the antinomies, he will also say like Kant that the conditioned and the unconditioned are different in kind, but unlike Kant and like Hegel, the unconditioned can be thought. However unlike Hegel, the unconditioned cannot be thought representationally, because its terms, as we saw with the differential’s terms, are subrepresentational. Yet even though they are subrepresentational, that does not mean they are indeterminate or unthinkable; for, like the calculus differentials, the terms are determinable, meaning that when they are related, they produce a determination of finite value, and also they are thinkable, but only either in terms of a virtual multiplicity or in terms of an actual determination that implies the virtual multiplicity that it expresses.
Hegel’s and Deleuze’s interpretations of differential calculus indicate their different responses to classical representational systems in philosophy.
Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.
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