by Corry Shores
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[The following is summary. All boldface, underlining, and bracketed commentary are my own. Proofreading is incomplete, so please forgive my typos and other distracting mistakes. Somers-Hall is abbreviated SH and Difference and Repetition as DR.]
Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide
Chapter 4. Ideas and the Synthesis of Difference
Very brief summary:
Chapter 4 characterizes the sorts thinking that deal with the fundamental intensive level of reality and also the Ideas that are involved in this thinking. An Idea has two important features: 1) it is made of indeterminate parts that are determined by means of a binding differential relation, and 2) it can find many various spatio-temporal actualizations. The differential in calculus is determinate only through reciprocal relations, and it describes the basic structure of Ideas. There are three other examples that more or less fulfill the two criteria of Ideas: 1) Epicurus’ and Lucretius’ atomism, since atoms become determinate only when in vibratory collision and many combinations are possible, 2) Geoffroy’s homological anatomy, where there is a transcendental template whose parts gain sense only in their combined network and many instantiations can be actualized from it, and 3) the Marxist notion that there are invisible networks of relations that underlie and determine the social, political, and economic surface conditions. Learning happens when first we encounter a problematic situation, then on the basis of a question, we gather fragments to form an Idea, implicated within which are many actualizable solutions. The discordant exercise of our faculties is itself a solution to the problem of unrepresentable Difference that we encounter. Also, negation is not a part of the fundamental levels of reality and thinking. It comes about either when we formulate propositions, which are negatable, or in actualization, which generates determinate properties that can oppose one another. Actualization is dramatized, meaning that it begins with intensive variations in the speeds and distributions of development, then secondly these form the extensive and qualitative properties, but all this occurs dramatically in an unpredictable way.
Now that we know what thinking should not be like, we look at how Ideas and thinking should be understood. Deleuze thinks that Ideas should not be formulated propositionally, in which case it would take on empirical determinations. Kant’s Ideas at first seem to fulfill this requirement, but in the end they do not. Deleuze characterizes the Idea as having three intrinsically related moments, which are expressed in an unorthodox tradition in calculus. Bordas-Demoulin sees the differential as undetermined, since in the differential formulation for the circle we get the essence of a circle without needing reference to particular circles, as with Descartes’ algebraic formulation. Maimon thinks that the differentials involved in intuitive givenness by themselves are not sensibly determined but gain it when they are differentially related. In this way he describes the conditions for determinability. And Wronski is concerned with notable moments in a variation that are numerically determinable, thus he shows the role of determination. Other fields might have Ideas with this structure, but only calculus looks at the structure itself. Ideas emerge under three conditions: the parts gain determination only through their differential relation, they then lose their independence, and the Idea applies to a variety of spatio-temporal differential relational situations in the world. Deleuze offers three examples. The first is Epicurus’ and Lucretius’ atomism. Here the atoms are insensible until colliding, and they can take many various arrangements of combination. The second is Geoffroy’s homological model of anatomy. Here there is a template of sorts that is really just made of differential relations of parts that take a wide variety of instantiations in the different species. The third is Marxist social ideas. Here we are to think of an invisible network of underlying relations that are responsible for the surface conditions in the political, economic, and social situation. Ideas also have three dimensions: vertical (Ideas can be rearticulated in more basic fields), horizontal (one Idea can find different variations within one area), and depth (one Idea can underly seemingly different Ideas in otherwise incommensurable domains). An Idea is virtual, and it explicates into actuality. It is not possible, since the possible is a deprivation of reality, but the Idea has no such deprivations. There are three stages in the formulation and actualization of Ideas, and this is learning. First we encounter a problematic situation. Secondly we gather little Idea fragments, which are differential relations, that we combine to form the Idea. We do this in part by means of the question, which creates the framework for how the Idea should be composed and what sorts of actualizations it implicates. Thirdly this Idea is like a map or graph implicated many actualizable solutions, which may find determinate actualization. Our body itself is a solution that evolution has found to certain problems. Our faculties also, insofar as they operate discordantly, are a solution to the problem of the unrepresentable fundamental Difference, which can only be thought about through this discordant exercise. Negation is not found at the levels of the problem or the Idea. It comes about in two ways. The first is when we formulate the problem propositionally, which allows us to negate the sentences. The second is in actualization, which creates determine properties that oppose one another. This actualization however happens by means of dramatization, since development is unpredictable and involves a complicated network of “actors”. Actualization is first spatio-temporally intensive, as it involves intensive variations in developmental speeds and distributions. Then only secondly is it extensive and qualitative.
(4.1 Introduction: Kant and Ideas) Deleuze seeks a model of thinking where thought does not consist of propositional solutions that relate to empirical objects. So Ideas for Deleuze should not have an extrinsic relation to the empirical world. Instead a) they should be intrinsically related, while b) maintain a difference in kind such that thinking on the level of Ideas is not done using empirical determinations. Kant’s Ideas are problematic since they have no empirical object, but this also means they have no propositional solutions. So they would seem to involve a sort of thinking that deals with problems that are not understood in terms of their possible solutions as articulated in propositions using empirical determinations. However, in the end, the Ideas for Kant receive empirical determinations and thereby extrinsically relate to the empirical world. (4.2 Ideas and the Differential Calculus) Deleuze thinks that the Idea has three intrinsically related moments, namely, indetermination, determinability, and determination. One place where they obtain such a relation is in an unorthodox tradition in the history of calculus that grants the differential its proper metaphysical status. The three main thinkers in this tradition are Bordas-Demoulin, Maimon, and Wronski. They will relate the three moments intrinsically in this way: the differentials dx and dy are by themselves undetermined, but they obtain their determinability when brought into differential relation, which can then be determined numerically. Bordas-Demoulin shows how the differential formulation for the circle gives us its essence, since ydy + xdx = 0 tells us that its points tend back steadily to its point of origin. Descartes’ alternative formulation x2 + y2 – R2 = 0 only tells us what specific values of a circle are, and thus it requires determination. But since the differential formula tells us what circles are without needing to deal with specific ones, it sees the differential as being undetermined. For Maimon, the differential terms dx and dy by themselves cannot receive sensible determination, but they can when they are differentially related. In this way he gives the conditions for determinability. Wronski is concerned with moments in a variation that are drastic while also being numerically determined, thus he supplied the notion of determination. Again, the differentials begin undetermined on their own, but they are determinable when differentially related, which creates a value that may be numerically determined. (4.3 Ideas and the Wider Calculus) The Idea for Deleuze has these structural features, and differential calculus brings that structure to light. Other fields generate Ideas, but unlike calculus they might not explore the structure of Ideas themselves. An Idea is a multiplicity since it is composed of many differential relations. An Idea emerges under three criteria: 1) the parts gain their determination through their differential relation, but they are undetermined prior to this relation, 2) after differentially relating, they lose any independence they might previously have had, and 3) the Idea that emerges applies to a variety of spatio-temporal differential relations in the world. (4.4 First Example: Atomism as a Physical Idea) Deleuze gives three examples in the history of philosophy that seem to fulfill these three criteria for the emergence of the idea. 1) Epicurus’ and Lucretius’ atomism. The atoms that make the world are falling downward too quickly to be perceived, thus on their own the parts are undetermined. They only become sensibly determined when they collide repeatedly and vibrate in a general region. Thus the parts gain their determination through their differential relation. And since they may enter into a variety of relations, they apply to a variety of spatio-temporal configurations. Deleuze is not completely satisfied, since this model understands determination too much as sensible determination. (4.5 Second Example: The Organism as Biological Idea) 2) The second example is Geoffroy Saint-Hilaire’s homological model of evolutionary anatomy. The alternative model by Cuvier, comparative anatomy, gives the same name to different species’ parts that are similar in form and function. Thus a fish fin and a human arm have these different names. But as species evolve there might be evolutionary connections between these parts, even though their form and function mutated. Cuvier’s model will miss these connections. For Geoffroy, however, there is a transcendental structure that is merely a set of relations between parts. All species, then, have these relations, even though their parts differ. Thus it is the same relational juncture in the transcendental structure that in fish takes the form of a fin and in humans an arm. So, the parts in the transcendental model on their own have no sense or conceptual content were they taken on their own, but they gain some conceptual determination when they are understood as occupying their place in the whole total network of differentially related parts, and there are many possible evolutionary instantiations of this transcendental structure. It would seem then that Geoffroy’s model fulfills Deleuze’s requirements for an Idea, but Deleuze thinks that it is still too much tied to actual instantiations. (4.6 Third Example: Are there Social Ideas, in a Marxist Sense?) 3) The third example is Marxist social ideas. There are certain visible surface structures like laborers and means of production. Althusser thinks that Marx is not so concerned with these but more so with deeper, invisible, unspecifiable political and ideological relations that are responsible for the generation of the visible surface structures. These deeper structures also fulfill the three requirements for an Idea, since the parts of these deeper structures have no sense taken independently, but they gain it as a network of such relations, which can take on many different manifestations under different historical circumstances. (4.7 The Relations of Ideas) There are three dimensions of varieties of Ideas. 1) Vertical: an Idea for a problem or solution can be reformulated in a more basic or general domain. So biological ideas might be reformulated using chemistry concepts, which can be reformulated into physics notions, which finally might be reformulated in mere mathematics. 2) Horizontal: here an Idea finds variations of itself within one domain. For example, Geoffrey’s transcendental structure finds many different arrangements (perhaps for creatures with different basic structures, like plants and vertebrates), each being its own Idea. 3) Depth: two systems might seem to be very different and to have very different Ideas might on a deeper level share the same Idea. Geometry and arithmetic for example seem very different, since the one can represent irrational values simply and the other cannot. But on a deeper level they share fundamental structures and thus the same Idea underlies the seemingly different Ideas in each system. (4.8 Essence, Possibility and Virtuality) Ideas for Deleuze become actualized, but we are not to understand them as if there were essences in the traditional sense of the term. For Aristotle the essence is obtained by removing everything that is inessential. The essence for Bergson however is something like the combined network of all actualizable manifestations of the Idea, which are folded in together in a complicated way. His analogy is color. White light is like the essence of color, since it can be broken down into any actualizable color, all of which when recombined form white light. This notion of the Idea involves the concepts of virtual, actual, and possible. The Idea is virtual, and it explicates into actualities. The virtual is not the possible, since the possible is the real deprived of actuality. However the virtual is fully real and lacks nothing, since it implies its infinity of actualizable instantiations. The virtual has three important properties: 1) it is real without being actual, since it is responsible for the genesis of actualities 2) it is complete without being entire, since it contains all actualizable instantiations (making it complete), but since they are infinitely many and various, it can never be exhausted by its actualizations (making it not entire). 3) it is differentiated without being differenciated, since it is composed of a network of differential relations (making it differentiated), but these relations and its terms cannot be ascribed predicates (and so it is not differentiated). (4.9 Learning and the Discord of the Faculties) Learning for Deleuze does not involve drawing inferences from propositions. Instead, we learn when we engage with the problematic situation at hand, then take from many sources little Idea fragments in the form of significant relations in the situation, and combine them into a map or graph of sorts that implies a wide variety of directions to pursue, which is the Idea, then lastly to develop solutions that can change the conditions of the original situation in such a way as to solve its problematic elements. Evolution has already done such a thing, since each of our organs is like a solution to certain problems. Likewise, each faculty is like such a solution. Now, the world is composed fundamentally not of distinct recognizable entities but rather with intensive differences. This means that were our faculties to operate concordantly to recognize a common object, they are not thinking about this fundamental layer of reality. Instead, their discordant exercises, whereby each has its own object of a different kind, enables such a thinking of intensive difference, and thus the discordant exercise of the faculties is like a solution to the problem of Difference itself. (4.10 The Origin of Ideas) So the problem is found in our encounter with an intensive field of differential relations that we cannot process using our given resources. We then formulate an Idea by putting fragments together, and on its basis we find solutions to the problematic situation we originally encountered. What relates the original problematic solution to the Idea we come to form is the question, which sets a particular framework for the construction of the Idea. Deleuze uses the metaphor of the dice throw to illustrate this. The problem is like being in a situation that calls for the throw of the dice, which will be the solution. The dice themselves are the Idea, since they contain the actualizable outcomes, only one of which at a time is in fact actualized. Also, the dice faces can be redesigned, just as the Idea can be reformulated. And the continual reformulation of the Idea is like the repetition of difference. (4.11 The Origin of Negation) For Deleuze, negation is not a part of how things are generated. However, there can be negation after they are generated. The problematic situations do not have negation in them, nor do the Ideas and the actualized solutions. We get negation however when we formulate the problems propositionally, which allows “it is not the case that…” to be affixed to them. This propositional denial is one origin of negation. But it misleads us into thinking that there is negation in the fundamental levels of reality. There is another origin of negation, which is the process of differentiation by which distinct actual states of affairs are generated. But this negation is secondary to the non-representable fundamental structures where there can be no negation. (4.12 Actualisation) Implicated in the Idea are many actualizable instantiations of the Idea, which serve as solutions to problems. But there is dramatization in actualization. It is like the development of egg, where new stages are not predictable from prior ones, and also there are many interrelated parts that interact complicatedly like dramatic actors do. Actualization in this form of dramatized development is initially spatio-temporally intensive, because it involves intensive variations in the accelerations of development and in the distributions of elements. Then, as a result of the intensive spatio-temporal dynamics, extensive and qualitative features come about secondarily.
Somers-Hall, Henry. Deleuze’s Difference and Repetition. An Edinburgh Philosophical Guide. Edinburgh: Edinburgh University, 2013.
Or if otherwise noted:
Deleuze, Gilles. Difference and Repetition, trans. Paul Patton, New York: Columbia University Press, 1994/London: Continuum, 2004.