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 1: The problem of Representation
Chapter 2: Difference and Identity
Subdivision 7: Preliminary Conclusions
Very brief summary:
Aristotle and Russell construct hierarchies whose problems derive from the fact that their systems omit a form of difference which would explain cases where the principle of excluded middle does not hold, of example (a) in transitional phases when something has mutually exclusive states (like something being both wood and fire in the transitional phase of its ignition) and (b) ring species where creature A can interbreed with B, and B with C, but not C with A.
Brief Summary:
So far in this chapter we have seen the problems with Aristotle’s and Russell’s hierarchies. For example, it is difficult to speak of the highest category. For Deleuze, these problems arise from the four shackles of mediation: identity (what allows classificational grouping), analogy (comparable relations between concepts), opposition (maximal differences allowing differentiation between species), and resemblance (variational similarities allowing differentiation of individuals in a species). These shackles result from common sense (which calls for partitions between concepts) and good sense (which calls for hierarchical subsumptions). For Deleuze, all four cases miss his notion of a deeper sort of difference. Common sense wants general concepts that are distinguishable from others. This requires clear demarcations, or maximal difference, following the principle of excluded middle, ‘either P or not-P but not both’ (P v –P). This means there is no ambiguity between concepts. However, there are two important cases where this principle does not hold, namely, transition (when something is between states and thus has two mutually exclusive determinations together) and ring species (where a first species can interbred with a middle species, which can interbred with a third species, and yet the first and third cannot interbred. So how do we classify their species?)
Summary
Previously we examined Russell’s solution to the paradox of the class of all non-self-inclusive classes. We saw that he distinguishes levels of classification, and one level can only refer to the level below it, and not to itself.
Now Somers-Hall makes some preliminary conclusions. In Aristotle’s system, we cannot define the highest genus, being, because definition requires giving the genus and species, but the highest genus has no greater genus for its definition. [Yet we call both the highest genus ‘being’ and all its members ‘beings.’ Does this mean that ‘being’ and ‘beings’ are synonymous, on account of them sharing both the same name and the same definition? Aristotle instead says they share related meanings and a similar name. They are paronymous.] But we found in our examination of Russell’s and Whitehead’s Principia Mathematica that the problem runs even deeper. For in this case, in order to make a system of classification consistent, it needs to rectify the paradox of the class of all non-self-inclusive sets. Russell’s solution is to say that one level can refer to the one below it, but not to itself. But this makes it impossible to designate a highest class and also to make a universal statement regarding all classes. And when we say we are speaking of all properties of a class, we are implicitly making just as many statements as there are levels under that class.
While Aristotle's logic may prevent the determination of the highest genus, our discussion of the Principia showed that this difficulty is in fact far more serious than it appears in Aristotle's work, as if this highest genus could be specified, then the concept of the totality of the system could only be achieved at the cost of consistency. Thus, within representational logic from Aristotle to Russell and Whitehead, totality and consistency remain mutually exclusive. (61)
[So for Russell, as we noted, when we speak of all the properties of a class, we are implicitly making a statement for every class below it. This means that there is an ambiguity between the levels that allows for one statement to refer to all of them indirectly. In Aquinas, we are not able to define God, or being, the highest category; however, we can specify the relation of beings to being, using analogy. Just as some species is to some genius, so to are beings (or Man) to God.]
As both Aquinas and Russell make clear, and without Aquinas having any direct influence on Russell, the concept of a totalized thinking of being remains a necessity, so in both cases the notion of analogy is brought in to provide the same kind of pseudototalizing effect on the systems. For Aquinas, this allows us to talk of God and man in the same terms, regardless of their differences in being. For Russell, systematic ambiguity allowed us to refer to a series of statements of different types as if they were one universal statement through structural analogies between them. (61)
We also saw that both Aristotle and Russell were unable to explain change, because there were only able to say something about states at some moment, and not about the transition between states.
For Deleuze, these issues spring from what he calls the “four shackles of mediation (DR 29)” (61): identity, analogy, opposition, and resemblance.
Deleuze’s Four Shackles of Mediation:
1) Identity: the undetermined concept that unifies the system and is presupposed in the form of the highest genus [perhaps it is something like the principle that allows anything to be a class, like saying, things can be unified into groups as if there is a higher identity for all of them, even though there can be no explicit specific highest identity unifying them all.] "This is present for Russell as the universal statement that cannot be more than a verbal pronouncement, one that is comprehensible but cannot be explained through the system itself.” (61)
2) Analogy: the “relation between the ultimate determinable concepts, that is, the categories for Aristotle or the isomorphic statements of variable type for Russell” (61).
3) Opposition: [what distinguishes things from one another] “Opposition deals with the relations between determinations within concepts and thus refers to the differentiae, which relate the determinations of the genus to each other through a process of exclusion, and to the clear specification of the predicate for Russell, which delimits a class from other classes through purely bivalent criteria.” (61)
4) Resemblance: Resemblance allows for the differentiation of individuals sharing the same species. For Aristotle, essence is what allows for this sort of resemblance, and we see something similar in Russell’s theory of change, which explains how the same thing differs at different moments of time. (61d)
Deleuze says that the source of these problems is common sense and good sense, which we discussed earlier when discussing Kant. Common sense again refers to the partitions of concepts. In the context of Kant, good sense referred to subsumption of the predicate under the subject, but now in this context it refers to the construction of hierarchy itself (62). The two functions are both present in Aristotle and Russell. But in both cases there are problems with what does not fit explicitly in their systems.
What is at issue in both Russell and Aristotle is what is absent from their systems. Identity provides a concept that is inexplicable, analogy provides an isomorphism between elements within a totality that cannot be defined, opposition provides a stability that is contradicted by phenomena such as ring distributions of species, and resemblance merely shows the gap between differing temporal individuals rather than explaining it. (62)
For Deleuze, the element getting overlooked in all four cases is the concept of difference. Common sense wants a general concepts. This requires a clear demarcation between concepts. In order for the demarcation between concepts to be clear, we need the law of excluded middle (Either P or not-P, but not both) P v –P, which makes use of maximal difference [because they are mutually exclusive, the terms could not be more clearly different, for otherwise they would be ambiguous]. [But change involves the ambiguity of states. In the transition from one state to the next, something is predicated simultaneously with two mutually exclusive states. Also, there are cases like the ring species that also make it unclear whether or not one animal is clearly the same species as another. For, A is a fellow species with B, and B is a fellow species with C, but A and C cannot interbred. So if A and C are in separate species, then what species is B in? But how can two types of animals A and C be in the same species but be unable to interbreed? So there does seem to be ambiguous cases of classification where something is both P and not-P at the same time. There is a sort of middle that must be included. There is the transitional state between P and not-P when something is both at the same time, and there is the intermediary links in ring species that are both in and not in the species on either end of the chain.]
This makes difference essentially oppositional and prevents the understanding of temporality, as well as the diversity present in indeterminate species ( the issue of ring distributions applies just as much to Russell's system as to Aristotle's) . Static states form the entirety of the system, leaving no space for either generation or transition. The points where true difference shows itself are in the places where representational logic breaks down, in the movement between these oppositional points, in the chasms that appear within the system . "Difference ceases to be reflexive and recovers an effectively real concept only to the extent that it designates catastrophes: either breaks of continuity of the series of resemblances or impassable fissures between analogical structures" (DR, 35). (62)
Deleuze’s difference is not oppositional or maximally different as with the difference in the law of the excluded middle. We will discuss Deleuze’s difference in chapter 6. Now we turn to Hegel’s possible response to the systems of identity in Aristotle and Russell. (62)
Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.
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