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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

4.7 The Relations of Ideas (186–7/235–6)

Brief summary:

For Deleuze, Ideas are varieties including sub-varieties, and there are three possible “dimensions” of these sub-varieties. 1) Vertical: an Idea for some problem/solution can be reformulated in a “higher” (that is, more basic or general) domain; for example, the Ideas of biology can be reformulated in chemistry, which can be reformulated in physics, which can be reformulated in pure mathematics. 2) Horizontal: an Idea for some problem/solution within one domain can be reformulated or re-instantiated within that domain and also within that same problematic, but in so doing produce other Ideas as a result. Thus in Geoffrey’s anatomy, there is one Idea, the transcendental template for the skeletal arrangement of all vertebrates, for example, and each instantiation in actual species is itself a new Idea, since it has different significances to the ways that the parts relate (and thus it has different “singular points” which would define different Ideas). 3) Depth: Two systems which might seem to instantiate different Ideas given their fundamental incompatibilities in fact on a deeper structural level share the same Idea. For example, irrational values make geometry and arithmetic seems fundamentally incommensurable, since geometry can express them as cuts in a line or figure, but in arithmetic the decimals are interminable and thus not so straightforwardly expressible. However, a closer study of the axioms of both systems reveals that on a deeper level they share fundamental structures.

Summary

Previously Deleuze has been characterizing and exemplifying his notion of the Idea. Now we ask, how do Ideas relate with one another? (SH 148). We first note that Deleuze claims Ideas are varieties containing within them sub-varieties (SH 148, citing DR 187/235). SH says this claim comes from two prior observations. 1) Ideas can be further differentiated to produce Ideas of Ideas, just as differential functions can be further differentiated [although I am not sure yet how this works for Ideas. Perhaps what is discussed below are examples of the further differentiation of Ideas into other Ideas], and 2) Ideas can have various sub-varieties depending on the domain in which they are expressed: “**Ideas are related to the domain in which they are solved. This implies that the same Idea can be expressed in different actual situations, depending on what kind of solution we are looking for**” (SH 149). Now, Deleuze says there are three dimensions of variety for Ideas: a) a vertical dimension: the Ideas as expressed in one field can be expressed another way in a more basic field [I assume this would be like biology’s problems/solutions being expressed in chemistry, which can then be expressed in physics, which can then be expressed in just mathematics, but I am not sure. This is what SH writes: “there are three ‘dimensions of variety’, the first of which is the ‘vertical dimension’ (DR 187/235). This depends on what the elements and relations we are concerned with are. Depending on whether we conceive of the elements as atoms, bones or relations of production, the solution we arrive at will be expressed in the fields of physics, biology or social theory respectively. While we have here different ‘orders’, these orders are still interrelated, in that Ideas of physics can be ‘dissolved’ in higher order problems such as those of biology, and likewise social theory will find itself ‘reflected’ in the structure of the individuals that compose it” (149).] b) a horizontal dimension: [I do not quite follow this one. Let me quote it first:] “The second, ‘horizontal’ dimension deals with ‘degrees of a differential relation within a given order’ (DR 187/235). As we saw in 4.2, by repeatedly differentiating an equation, we can find ‘singular’ points along a curve where the nature of the curve changes. Now, the same Idea can give rise to Ideas with different singular points. Deleuze gives the example of conic sections to explain this concept. In geometry, we can generate a curve by cutting a cone with a plane, just as if we cut a cylinder in half, we would find, on the surface of the cut (the section), a circle. Now, if we take a section of a cone, depending on the angle to the cone at which we take the section, we will have a different type of curve:

[Image from wiki. Similar to figure 2, SH 149]

| Each of these curves has different singular points (points where the gradient is 0, null or infinite), despite the fact that all of the curves are created from the same fundamental shape. In a non-mathematical field, we can note that Geoffroy’s comparative anatomy relies on the fact that the same structure is to be found in the relations between the bones of all animals. Nevertheless, the singular points will vary within species, so the same bones that attach the jaw to the skull in fish are found in the inner ear in mammals” (149-150). [Although I am not following so well, the basic idea seems to be that what is essential are singular points. It seems in the case of the motion of objects, that when we look at changes in velocity, we are looking at the change of a change (that is, the change of position in relation to time). But when we look at a higher order of differentiation, we are looking at acceleration, which is a change of a change of a change, that is, the variations in velocity are the moments of acceleration. And there will be singular points where that acceleration is significant for some reason. Then there is the example of conic sections, which I am not quite getting. Perhaps it is just a metaphor or analogy. Let us begin with it as that. One singular procedure of cutting gives us various shapes, each with very different singular points in the mathematical sense just described above. Analogously, one Idea has many different instantiations within one field, each having their own analogous sort of singular points, somehow. Perhaps instead the cone example is meant to be taken literally. In that case, perhaps the notion is the following. In geometry, we can think of a geometrical field of variation, like the way that the points on the surface of a cone vary from place to place and relate differently throughout. We have the problem of, *what shapes do its cuts make?* And we find that different sorts of cuts give different sorts of shapes that each themselves have their own different singular points, meaning that each such cut differentially relates its parts in its own unique way, and thus each is itself an Idea that is a sub-variety of the larger Idea of the cut cone. Then we have the example of Geoffrey’s anatomy. Here we have one Idea, the transcendental structure, but depending on which species it is actualized in, it will have different singular points, which evidently are any given relation between the parts.] c) as dimension of depth [I am not not familiar with the example, so I do not grasp this one in a profound or clear way. The overall idea seems to be that an Idea can have a sub-variety in depth when two fundamentally incompatible systems in fact share deeper structures, and thus the deeper structure would be the Idea of a sub-variety under the dimension of depth. Perhaps the example can be presented in the following way. Consider how geometry seems to be concerned firstly with lines for example, which secondarily can be determined with numerical values assigned to its points. So we have a line, and we arbitrarily assign its left-most point the value zero, and its right-most side the value of 10, and we can further subdivide the line, each time secondarily assigning numerical values corresponding to proportions of the division. So half is 5, half of that 2.5, and so on. However, there are certain divisions which will not produce a numerical determination. For example, if we divide it according to the golden ratio, we will not be able to express it numerically, since the decimals would go on without termination. This tells us there is a fundamental incompatibility between numerical arithmetic and geometry, since geometry works with continuous magnitudes and arithmetic with discrete ones, and the two systems are not commensurable in cases of irrational values. However, the Bourbaki mathematicians also seem to have discovered that underneath the axioms of both systems are certain shared structures, which are like the deeper Ideas that are sub-varieties of the seemingly incompatible Ideas found within each system.]

Deleuze explains the final dimension of variety, that of depth, with an example from the mathematical theory of groups. He gives the example of ‘the addition of real numbers and the composition of displacements’ in this context (DR 187/236). As the structuralist mathematical collective, Bourbaki, noted, the addition of real numbers and the composition of displacements traditionally belong to two very different fields of mathematics, since one involves discrete units, and one continuous measurement [the following up to citation is Bourbaki quotation]:

quite apart from applied mathematics, there has always existed a dualism between the origins of geometry and of arithmetic (certainly in their elementary aspects), since the latter was at the start a science of discrete magnitude, while the former has always been a science of continuous extent; these two aspects have brought about two points of view which have been in opposition to each other since the discovery of irrationals. (Bourbaki 1950: 221–2)

Bourbaki note, however, that underneath the surface structures of the specific axioms of these different branches of mathematics, we can discern structures that occur in both branches, provided the elements of each branch are understood in a sufficiently undefined manner. Thus, certain relations may hold between elements that are obscured by further specifying their nature. Underneath the structures of geometry and arithmetic are deeper structures which they both share.

(150)

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**.

Bourbaki, Nicholas (1950), ‘The Architecture of Mathematics’, trans. Arnold Dresden, in *The American Mathematical Monthly*, 57, 221–32.

Image credits:

Conic Section.

https://en.wikipedia.org/wiki/File:Conic_Sections.svg

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