Somers-Hall, (4.3), Deleuze’s Difference and Repetition, ‘4.3 Ideas and the Wider Calculus (178–84/226–32)’, summary

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.]

Summary of

Henry Somers-Hall

Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide

Part 1
A Guide to the Text

Chapter 4. Ideas and the Synthesis of Difference

 

4.3 Ideas and the Wider Calculus (178–84/226–32)

 

 

Brief summary:
Deleuze has been discussing how differential calculus allows us to understand certain structural features of his notion of the Idea. However, other fields (including physics, biology, psychology, and sociology) as well generate Ideas in response to problems. The important difference is that differential calculus also deals with the issue of the grounds for a problem to be determinable, namely, with the differential relations between undetermined, reciprocally related parts. Since Ideas are composed of many such differential relations, they are multiplicities. There are three criteria for the emergence of an Idea: 1) the parts are determined only through their differential reciprocal relations, and thus they are not determined prior to those relations, 2) when the differential elements are reciprocally related, they lose any sort of independence they may previously have had, 3) the Idea must apply to a variety of spatio-temporal differential relations in the world.

 

 

Summary

Recall from section 3.5 how for Plato, mathematics was the second highest form of knowledge, right below knowledge of the Ideas. [You might recall as well the diagram that we displayed there, which is taken gratefully from the Thesis Eleven website.]

Likewise, “Deleuze wants to provide an account of the world, not just of the field of mathematics” (SH 141). We noted before that problems are on a different order than solutions are, and so mathematics is merely a way of representing solutions:

what is mathematical (or physical, biological, psychical or sociological) are the solutions’ (DR 179/227). These domains do not apply to problems themselves, but only | to problems as expressed in relation to (and within) solutions. The calculus is itself a way of providing symbols of difference, and as such, it is still propositional, and tied to a specific domain. Since, and as such, it is still propositional, and tied to a specific domain. Since what these symbols refer to cannot be represented, however, the calculus points beyond itself to the problem itself.
(SH 142)

[It seems that the idea is the following. Difference itself and the problem exist on a sub-representational level. There are many fields that offer solutions in representational ways to these problems and offer a symbolic expression of difference. But the calculus, unlike other fields, both recognizes that the differential is undetermined while at the same time it explains how that undetermined can become determined (its determinability). Perhaps these other fields like biology and so on only try to generate their own determinations rather than deal with the conditions of determinability itself.]

What is important about the calculus is that it presents an account of how undetermined elements can become determinate through entering into reciprocal relations. As relations exist in domains outside of mathematics, the differential calculus ‘has a wider universal sense in which it designates the composite universal whole that includes Problems or dialectical Ideas, the Scientific expression of problems, and the Establishment of fields of solution’ (DR 181/229).

Recall how for Deleuze, “Ideas are formed from the differential relations of their elements” [and since there are many such relations of elements] “In this sense, Deleuze claims that ‘Ideas are multiplicities’ (DR 182/230); and “They are the reciprocal relationships of elements that in themselves are indeterminate.” (SH 142). [If we are dealing with for example multiple physical things in space, they are determinate on their own, putting aside their combined membership in larger groupings. The differentials, we said however, do not have determinate value on their own, but only in their reciprocal relations. See for example Deleuze’s explanation from his course lecture of 1981/03/10. The next idea is that Multiplicity in this case is substantive. Perhaps the idea is that the parts on their own are not substantive, since they are not determinable, but only as a group in reciprocal relation are they determinable and thus substantial.]

As we have seen, Ideas are formed from the differential relations of their elements. In this sense, Deleuze claims that ‘Ideas are multiplicities’ (DR 182/230). They are the reciprocal relationships of elements that in themselves are indeterminate. Now, when we are dealing with a spatial multiplicity, we talk about multiplicity in terms of a structure possessing many elements. In this sense, we can call it an adjectival notion of multiplicity. The ‘many’ in this case is a way of describing elements that can be in a sense indifferent to being given the classification, ‘many’. They are determinate before they form a group. On the contrary, with differentials, they become determinate precisely by being reciprocally determined. Rather than multiplicity being an adjective that describes a group of substances, Deleuze claims that ‘“Multiplicity”, which replaces the one no less than the multiple, is the true substantive, substance itself’ (DR 182/230).
(SH 142)

We also saw how for Kant, we do in fact experience self-standing elements, and so Kant’s manifold is not Multiplicity in Deleuze’s sense of the term (142). Now, “Deleuze gives three criteria for the emergence of Ideas”: 1) the elements of the multiplicity involved in the [production of the] Idea “must be  determined through their relationships with one another, rather than prior to it” (142). This is because, as Deleuze puts it, “‘the elements of the multiplicity must have neither sensible form nor conceptual signification, nor, therefore, any assignable function’ (DR 183/231)” (SH 142). [The idea seems to be that on their own, the differentials involved in our thinking and experience cannot be sensible or conceivable until they somehow enter into reciprocal relation]. 2) when the differential elements are reciprocally related, they lose their independence [I do not follow this point so well. Let me quote it first:]

Second, ‘the elements must in effect be determined, but reciprocally, by reciprocal relations which allow no independence to subsist’ (DR 183/231). As Deleuze notes, | ‘spatio-temporal relations no doubt retain multiplicity, but lose interiority’. That is, the elements are not intrinsically related to one another, but are simply related by occupying a certain space together. On the other hand, ‘concepts of the understanding retain interiority, but lose multiplicity’ (DR 183/231). When we determine a concept (man is a rational animal, for instance), we do so by subsuming it under another. As such, while they are intrinsically connected, they form a unity, rather than a multiplicity.
(SH 142-143)

[I am not sure what to do with the distinction between spatio-temporal relations and conceptual relations, in how they are dealing with this notion that reciprocal relations of the differentials make it so there is no longer independence of the parts. Perhaps the idea is that neither intuitions nor concepts qualify for ideas. For, intuitions are spatio-temporal, and since they are extensive, they have a multiplicity of parts. However, the parts are related partes extra partes and thus are not intrinsically related. And concepts, because one concept subsumes another within it, are intrinsically related, are still not multiple, since they are unified. Perhaps, then, the Idea would be intensive somehow, since that would allow it to be both multiple (as a degree of variation) while also being intrinsic, since degrees of variation are not made of extensive parts. See for example Deleuze’s discussion of Spinoza’s modes in Expressionism in Philosophy, chapter 12 and chapter 13.] 3) [I am not sure I follow this last point, but it seems that in order for an Idea to emerge, there as well must be both differential relations on the level of the Idea that can be found in various ways and in various domains between spatial-temporal terms. ]

Finally, ‘a differential relation, must be actualised in diverse spatio-temporal relationships, at the same time as its elements are actually incarnated in a variety of terms and forms’ (DR 183/231). That is, if the Idea is to provide some kind of explanation of the structure of the world, it must be applicable to more than one situation; it must capture relations in more than one domain.” (143)

SH notes that these three features of the Idea are found in differential calculus, “but to explain how this account functions more generally, Deleuze provides three examples of Ideas in non-mathematical fields: atomism as a physical Idea, the organism as a biological Idea, and social Ideas” (143). We turn to these in the following sections.

 

Citations from:

Somers-Hall, Henry. Deleuze’s Difference and Repetition. An Edinburgh Philosophical Guide. Edinburgh: Edinburgh University, 2013.

Or if otherwise noted:

DR:
Deleuze, Gilles. Difference and Repetition, trans. Paul Patton, New York: Columbia University Press, 1994/London: Continuum, 2004.

Image credits:

Thesis Eleven.
<https://thesiseleven.wordpress.com/philosophy/platos-republic/simile-of-the-divided-line/ >

<https://osopher.files.wordpress.com/2011/02/plato-divided-line.gif?w=504&h=360>.