The Topology of Deleuze’s Spatium

Abstract

One of Gilles Deleuze's major ontological categories is that of a virtual continuum which, much like Spinoza's substance, presents two sides-pure extension and thought-or, rather, two powers: the power of being and the power of thinking. This virtual continuum receives a variety of designations throughout Deleuze's corpus: "intensive spatium" in Difference and Repetition,1 "ideal or metaphysical surface" in The Logic of Sense,2 "plane of consistency" in A Thousand Plateaus (written with Felix Guattari)3 and "plane of immanence" in What is Philosophy? (equally coauthored with Guattari).4 While these diverse terms may be argued to accentuate different aspects of the continuum so designated, Deleuze's characterization of the latter remains, nevertheless, fundamentally constant-such that, as one commentator puts it, the various "objects" in question (spatium, surface, plane of immanence or, again, hyperspace) are all rigorously homothetic.5 Such a continuum is, accordingly, consistently described as a pre-extensive, non-qualified "milieu" or "space-stratum" enveloping complexes of differential relations, pure intensities and singularities, with Deleuze seeking to determine in this way an impersonal and pre-individual transcendental field that, assembling the conditions of real-and not merely possible-experience, would neither resemble the corresponding empirical fields (with their correlation of a consciousness and its objects) nor amount to an undifferentiated "depth" or groundlessness (sans-fond indifferencie) identified as pure chaos. Aiming in this way to escape the ultimatum, laid down by both metaphysics and Kant's transcendental philosophy, which claims the transcendental field must be defined in terms of a supremely individuated Being or an intensely personalized Form if it is not to end up inevitably confounded with an undifferentiated abyss or pure chaos, Deleuze consistently mobilizes what we might call a "topological model" in order to describe the properties of this real transcendental field. Along lines similar to Michel Serres' positioning of topology as the first objective elaboration of a transcendental field ("field of invariants") or, indeed, to Merleau-Ponty's exhortation to "take topological space as the model of being"-Deleuze views the topological "categories" of position, junction, and connection as the best able to explain the way in which the transcendental plane gives consistency to the elements (multiplicities, pure intensities or singularities) that populate it. One of the most rigorous statements of the "topological" nature of the transcendental field is found in Deleuze's early article on structuralism where, after defining structuralism's "scientific ambition" as being "topological and relational," he notes that, for this reason, "structuralism is inseparable from a new transcendental philosophy where places are more important than what fills them."6 In short, "what is structural is space, but a non-extensive, pre-extensive, pure spatium."7 Such a space is a space of co-existence, articulated by a system of differences that, far from being external relations defined with regard to pre-existing entities, are, on the contrary, relations constitutive of the "reciprocal determination" of the singularities which occur in the space "like topological events" and which compose multiplicities. As Deleuze writes in Difference and Repetition: "These elements must, in effect, be determined, but reciprocally, by reciprocal relations which allow no independence whatsoever to subsist. Such relations are precisely non-localizable ideal connections, whether they characterize the multiplicity globally or proceed by the juxtaposition of neighboring regions. In all cases, the multiplicity is intrinsically defined, without external reference or recourse to a uniform space in which it would be submerged."8 In sum, it is, above all, the category of connection that Deleuze takes from topology in order to determine his transcendental plane in terms of a structurally stable (or better, "metastable") system auto-regulated by a relational and dynamic, non-quantitative, organization of potential values. …