ABSTRACT
The relation between the successive terms of a power series expansion The Deleuzian concept of intensive quantity is explicated in this chapter as determined according to the logic of differentiation. It is according to this logic that intensive parts are differentiated from one another such that they are each determined as different degrees of power or intensity, each of which thereby constitutes a different intensive quantity. The mechanics of the logic of different/ciation is then examined in relation to the concept of individuality in order to determine the extent to which this logic is actually able to be mapped onto Spinoza’s theory of relations, the implications of which is investigated in relation to the work of other scholars working in the field of contemporary Spinoza studies in the chapters which follow. Intensive and extensive quantities are determined in relation to Spinoza’s theory of relations respectively as the singular modal essences complicated by an attribute, and the finite existing modes explicated by an attribute. Each singular modal essence that is complicated in an attribute is an intensive part of different degree of power or intensity, or a different intensive quantity. To every different intensive part, that is, to every intensive quantity or singular modal essence, there corresponds an infinite collection of extensive parts, which is explicated by an attribute as an extensive quantity or finite existing mode. The mechanics of both the complication of singular modal essence and the explication of finite modal existence, it is argued, are determined according to the same logic of different/ciation. However in order to determine the convergence of the logic of complication and explication, it is necessary to elaborate in detail how intensive parts are differentiated from one another quantitatively, and to distinguish these intensive parts from the infinite collections of extensive parts which correspond to them. According to the logic of differentiation, the infinitely small differentials of Letter XII, which correspond to the most simple bodies of Spinoza’s Ethics, and which Deleuze characterizes as ‘extensive parts’, are only determined by the differential relations which they effect with each other. When considering extensive infinity, Deleuze argues that ‘it is of course divisible into the extrinsic
‘But’, he continues, ‘these extrinsic parts always come in infinite collections’.1 While the principles of determinability and reciprocal determination correspond to the determination of extensive parts, it is the principle of complete determination which corresponds to the effective determination of infinite collections of extensive parts. So, according to Deleuze, extensive parts exist collectively, that is to say they have no distributive existence but rather enter necessarily into infinite collections, as infinite collections of infinitely small extensive parts. What are these ‘infinite collections’ into which the extensive parts necessarily enter? According to the logic of differentiation, an infinite collection of infinitely small extensive parts is expressed by a power series expansion.2 A power series expansion is an infinite series, which, insofar as it converges with a local function according to the principle of complete determination, is constitutive of an actual infinite. Now, because it actually ‘divides into a multitude of parts exceeding any number’,3 a power series is not a sum in numerical terms, its parts being neither additive nor distributive. The parts into which this actual infinite is divisible are rather the successive terms of the power series, each of which is determined by the repeated differentiation of a differential relation. Each part of the series therefore consists of this differential relation, differentiated to various successive degrees, and a corresponding variable, whose exponent is the same, or has been raised to the same degree or power.4 So each part of the power series, each successive term, implicates those extensive parts determined by the included differential relation. What then is the nature of this implication? What is the relation between each successive part of the power series and the extensive parts which they implicate? According to Deleuze, each of these parts, both successive and extensive, are distinguished as different kinds of parts. There are ‘extensive parts’ which are determined in and by the differential relation, which is itself only a component of each ‘successive part’ of the series. If the successive parts of a power series implicate extensive parts but are not themselves extensive parts, what kind of parts are they? Insofar as a power series expansion is actually infinite, it corresponds to the actually infinite quantity of intensive parts into which the intensive quantity of an attribute is divisible. So, the successive terms of a power series expansion correspond to the intensive parts of an attribute. Each successive term of a power series can therefore be understood to be an intensive part. A power series therefore involves an infinity of intensive parts as its
1 Deleuze, Expressionism in Philosophy, p. 203. 2 See the section of chapter 3 entitled ‘Extensive parts: infinite collections of the
infinitely small’, p. 86. 3 Deleuze, Expressionism in Philosophy, p. 203. 4 Recall that each term of a Taylor Series is comprised of a differential relation
differential relation, intensive parts cannot exist independently of an actually infinite power series expansion in which they are complicated as its successive terms. Each intensive part of such a series implicates a differential relation which has been repeatedly differentiated in each successive term of the series. It is this differential relation which is determinative of the extensive parts generated by the series. Therefore, to each intensive part of the series there corresponds those extensive parts generated by the differential relation which it implicates. It would therefore seem that intensive and extensive parts correspond to one another within each successive term of a power series. However, according to Deleuze, ‘extensive parts and … intensive parts … in no way correspond term for term’.5 On the contrary, Deleuze maintains that to every intensive part there corresponds ‘an infinity of extensive parts’.6 The extensive parts determined by the differential relation implicated in each intensive part are infinitely small extensive parts, thus constituting what is for Deleuze a ‘lesser’ infinity of extensive parts. Insofar as a power series expansion converges with a local function, it is actually infinite, and it divides into an infinite quantity of successive terms or intensive parts. The quantity of infinitely small extensive parts corresponding to such an infinite quantity of intensive parts constitutes a ‘greater infinity’ of extensive parts. To each intensive part of a series there corresponds a lesser infinity, and to each power series expansion there corresponds an infinite collection of lesser infinities which together constitute a ‘greater’ infinity. So when Deleuze writes that ‘extensive infinity is thus an infinity necessarily conceived as greater or less’,7 he is arguing that extensive infinity, or an infinity of extensive parts, is always constituted by infinite collections (greater infinities) of the infinitely small (lesser infinities). The actual infinite of a power series expansion, although determinative of greater and lesser infinities of extensive parts, is therefore rather an intensive infinite, composed of, or divisible into, an actually infinite quantity, or collection, of intensive parts. As a power series expands, the exponent, or power, of each successive term increases in degree, and the series increasingly approximates or converges with its function, which in this case would be an attribute. Insofar as each of the terms of the series corresponds to one of the actually infinite intensive parts into which the intensive quantity of an attribute is divisible, there are an actually infinite quantity of terms, the degrees of which range from zero to infinity. Each intensive part, as a successive term of a power series expansion, therefore has a degree, or a degree of power, and the power series expansion can be expressed as the infinite collection of these different degrees of power. So, the intensive parts of a power series expansion are determined collectively as the different degrees of power or
5 Deleuze, Expressionism in Philosophy, p. 207.
