Deleuze on Multiplicity: A Primer
I've been cobbling together, mostly for my own benefit, a bit of a glossary of Deleuzian terms. This one's on 'multiplicity'. Most definitions of the term start with the distinction between the 'continuous' and the 'discrete', but I'm not convinced this is the best way to go about it. Moreover, the connection between multiplicities and manifolds - which is so vitally important! - sometimes doesn't get as much stress as I'd like it to. So I've tried to foreground that here. Two things that are missing here are a critique of Delanda's influential recasting of multiplicity as 'state space' - something I think is deeply misleading - and a comparison to Badiou's use of 'multiplicity', but I'll leave those for a possible future post. In the meantime, here we go!:
Part I: What is Multiplicity?
"Multiplicity" is easily one of the most important concepts in Deleuze, whose appearance ranges right across his oeuvre, from his early book on Bergson, all the way to his late work with Guattari, What Is Philosophy? The question is then, what is multiplicity? At its most basic, multiplicity is a concept, although it is a concept such that everything that is, is a multiplicity. One of the tough things about grasping the idea of a multiplicity is that it is an adverb treated as a substantive. That is, multiplicity defines less "what" is than how everything is. It defines a kind of manner or way in which things are. It is in this sense that one can say that everything from race, to gender, to states, to people, 'are' multiplicities. It is to say that they all partake in a certain way of being. As for what that 'way' is, Jon Roffe's simple definition works nicely as a start:
"A multiplicity is, in the most basic sense, a complex structure that does not reference a prior unity. Multiplicities are not parts of a greater whole that have been fragmented, and they cannot be considered manifold expressions of a single concept or transcendent unity. On these grounds, Deleuze opposes the dyad One/Many, in all of its forms, with multiplicity. Further, he insists that the crucial point is to consider multiplicity in its substantive form – a multiplicity – rather than as an adjective – as multiplicity of something. Everything for Deleuze is a multiplicity in this fashion." (Roffe, "Multiplicity", The Deleuze Dictionary). This is a good start, but we want to expand on this, insofar as this is better at telling us what a multiplicity is not, rather than what it is.
Before that however, it's worth attending to the strange grammar of multiplicity: why treat an adverb as a substantive? Well, one can think of it as a way of challenging the primacy of the 'is' question as the defining question of ontology ("what is?"). Deleuze's effort is to displace the question of 'what is'? with a series of other questions that get closer, as it were, to the essence of things. As he puts it directly: "contraries may be combined, contradictions established, but at no point as the essential been raised: 'how many', 'how', 'in which cases' ... 'Multiplicity' ... is the true substantive, substance itself. The variable multiplicity is the how many, the how and each of the cases. Everything is a multiplicity..." (D&R, 240). By saying that 'the essential' resides in the 'how?', the 'how many?', and the 'in which cases?', what is at stake is the transformation of the 'is' question into these others: to understand what something 'is', we need to understand how it is: the 'is' and 'how' become indistinct, mirrored in the way in which the adverb is treated as a substantive.
Part II: Multiplicities and Manifolds
One of the quirks of the history of the word 'multiplicity' is its potted translation from German, into French, and then into English. While Deleuze's direct reference is primarily to Bergson, Bergson's own use of the word 'multiplicité' is itself a reference to work of the German mathematician Bernhard Riemann. Riemann's work, in turn, speaks not of 'multiplicity', but of 'mannigfaltigkeit' - manifolds. There's a case to be made then, that all of Deleuze's references to multiplicities should in fact be read as references to manifolds! Without going quite that far, this linguistic clue is enough to at least establish how close the idea of a multiplicity is to that of a manifold. And in fact, it's through a study of manifolds that many of the features of multiplicities will come to light.
So what then, is a manifold? It is, first of all, a way of conceptualizing space. More specifically, it is an incredibly general way of conceptualizing space. To understand this, let's consider a specific way of understanding space. In the history of math, one of the more well-known 'discoveries' was the discovery of 'non-Euclidean geometry'. Non-Euclidean geometry differed from the Euclidean geometry which preceded it by relaxing the constraint on parallel lines. Whereas in Euclidean geometry parallel lines could never meet, non-Euclidean geometry allowed, by means of curving space itself, precisely this constraint to be broken, and for parallel lines to meet. In this sense non-Euclidean geometry can be understood as a more general way of conceiving space than Euclidean geometry, insofar as it operates with one less limit to what it can do.
If this line of thought from the specific to the general can be followed, then to understand manifolds, it must be understood that manifolds are a way of conceiving space in even more general terms than non-Euclidean geometry. How much more general? In a few ways. We will mention three that Deleuze specifically draws attention to. The first is that manifold space is not limited to 3 dimensions. In fact, manifold space - actually better called 'topological space' - can be 'n-dimensional': it can have as many dimensions as need be. The second, and perhaps most important feature, is that topological space - at its most general - is not even measurable in any precise sense. It is space that is, as Deleuze puts it, 'non-metric'. Rather than any 'measure', what defines such topological space is what it can do. In topology certain shapes can be transformed into others, and in fact can be conceived of as identical (or rather, 'equivalent') to others, so long as certain constraints are kept equal. As such, capacity, rather than measure, defines such spaces. We will come back to this below.
Finally, another important 'general' feature of topological space is that it is in fact 'built-up' from a series of 'local spaces'. It is by stringing a series of local spaces together, as it were, that a 'global space' is constituted. To get a handle on this, recall that non-Euclidean geometry allowed parallel lines to meet by curving space itself. Well, in topological space there can be not just one curved space, but many curved spaces, each with its own degree of curvature. By 'adjoining' these locally curved spaces, one arrives as a global space, put together in any which way. Here again we come across the idea that there is no 'prior unity' to multiplicities. They are, as Deleuze puts it, 'pure patchwork'. Putting these three features of topological space together (n-dimensional, non-metric, patchwork), we have a characterization of multiplicities (or manifolds) as space understood in as general terms as can be:
"Riemann space at its most general thus presents itself as an amorphous collection of pieces that are juxtaposed but not attached to each other. It is possible to define this multiplicity without any reference to a metrical system, in terms of the conditions of frequency, or rather accumulation, of a set of neighbourhoods; these conditions are entirely different from those determining metric spaces and their breaks ... Riemannian space is pure patchwork... It has connections, or tactile relations. It has rhythmic values not found elsewhere, even though they can be translated into a metric space. Heterogeneous, in continuous variation, it is a smooth space, insofar as smooth space is amorphous and not homogeneous" (ATP, 485). It is these characteristics that above all, define the manner in which all things are.
Part III: Two Types of Multiplicities
That said, in the quote just presented, one line stands out: that Riemannian space "can be translated into a metric space". This, despite the fact that "it is possible to define this multiplicity without any reference to a metrical system". Here we come to the feature of multiplicity most commented upon by those who discuss the term, including Deleuze himself: the fact that multiplicity comes in two flavors, as it were - continuous, and discrete. This will need a little unpacking, but the first thing to note is that this complicates what we said above about multiplicities characterizing 'how' things are. In fact, insofar as there are two kinds of multiplicities, there are correspondingly two kinds of ways in which things themselves can be. The 'first' way, is the way in which we have been discussing - multiplicities as "pure patchwork", n-dimensional, and non-metric. However, in addition to this, multiplicities can also be made to be homogeneous, of fixed dimensionality, and 'metricized'.
What needs to be emphasized however, is not merely the duality of multiplicities, but the fact that one kind of multiplicity underlies the other; specifically, that continuous multiplicities are the basis out of which discrete multiplicities are derived. To quote Manuel Delanda on this, "the metric space which we inhabit and that physicists study and measure was born from a nonmetric, topological continuum" (Delanda, Intensive Science, p.17). To see how this is the case, we need to think back to how the generality of topological space is made specific by 'adding constraints'. Just as (the more general) non-Euclidean geometry can be 'made' into (the more specific) Euclidean geometry by constraining space such that parallel lines never meet, so is it the case that topological space, when given constraints, can itself be transformed into metric, homogeneous space. While the details of this procedure is too much to get into here, the point to take away is that topological space is primary and constitutive of metric space. Or in other Deleuzian terms: smooth space is primary and generative of striated space.
These two kinds of multiplicities - continuous and discrete - in fact form the root of some of Deleuze's most famous other distinctions. In particular, that between the virtual and the actual on the one hand, and the intensive and the extensive on the other. Here is Deleuze in Bergsonism, making clear the former alignment: "[Discrete multiplicity] is a multiplicity of exteriority ... of difference in degree; it is a numerical multiplicity, discontinuous and actual. The other type of multiplicity, [continuous multiplicity] is an internal multiplicity ... of difference in kind; it is a virtual and continuous multiplicity that cannot be reduced to numbers" (Bergsonism, 38, italics in the original). This distinction between a 'difference in kind' and 'difference in degree' is key here, which brings us, finally, to:
Part IV: Singularities
We have already seen how a topological space is in fact a very 'general' notion of space (the most general!), characterized by the shedding of many of the constraints usually associated with the 'space' we are familiar with. However, such shedding of constraints does not render topological space featureless and bland (like a homogeneous blob). Rather, changes in topological space are 'measured' in a different way from changes in metric space. Changes in metric space are easy to understand: take a square, and double it's size by increasing the length of its sizes. Ta-da! You've made a change in metric space. However, the change that has occurred is merely one of degree. Quite literally, you've taken a (given) measurement, and scaled it up. The square is still the square, only now, double the size.
From a topological POV however, these two squares, metrically distinguished by size, are the exactly the same. To effect a change in topological space a different kind of difference (from metrical difference) is going to have to be in play. The differences that are relevant for topological space are instead differences in kind. Arkady Plotnitsky has perhaps put it best: "Geometry (geo-metry) has to do with measurement, while topology (topo-logy) disregards measurement and scale, and deals only with the structure of space qua space and with the essential shapes of figures. ... Insofar as one deforms a given figure continuously (i.e. insofar as one does not separate points previously connected and, conversely, does not connect points previously separated) the resulting figure is considered the same. Thus, all spheres, of whatever size and however deformed, are topologically equivalent. They are, however, topologically distinct from tori. Spheres and tori cannot be converted into each other without disjoining their connected points or joining the disconnected ones. The holes in tori make this impossible" (Plotnitksy, "Space in Riemann and Deleuze", Virtual Mathematics).
These 'breaking points', points past which one figure cannot be transformed into another without changing in kind, are given a name by Deleuze - 'singularities', or simply 'distinctive points'. Unlike metrical space then, multiplicities are defined by the 'distribution of ordinary and singular points' in topological space. It is singularities which give 'structure' to topological space, and what prevent such 'general' spaces from simply being conceived of in terms of homogeneous, indifferentiated flatness. As Deleuze writes of Bergson in his book dedicated to him, in conceptualizing multiplicity, "Bergson has no difficulty in reconciling... continuity and heterogeneity". Continuous multiplicities, despite not being defined by metrical measure, nevertheless have their own logic, defined in terms of changes in kind, out from which changes in degree ultimately derive.
Such are the logic of multiplicities which define all that is.
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