For Gilles Deleuze, the multiplicity exhibits concrete universality, whereas the essence exhibits abstract universality. An example of such concrete universality is exhibited in the following quote by the biologist Brian Goodwin:
“The point of the description is not to suggest that morphogenetic patterns originate from the hydrodynamic properties of living organisms . . . What I want to emphasize is simply that many pattern-generating processes share with developing organisms the characteristic that spatial detail unfolds progressively simply as a result of the laws of the process. In the hydrodynamic example we see how an initially smooth fluid flow past a barrier goes through a symmetry-breaking event to give a spatially periodic pattern, followed by the elaboration of local nonlinear detail which develops out of the periodicity. Embryonic development follows a similar qualitative course: initially smooth primary axes, themselves the result of spatial bifurcation from a uniform state, bifurcate to spatially periodic patterns such as segments [in an insect body], within which finer detail develops . . . through a progressive expression of nonlinearities and successive bifurcations . . . The role of gene products in such an unfolding is to stabilize a particular morphogenetic pathway by facilitating a sequence of pattern transitions, resulting in a particular morphology.”
Thus, concrete sets of attractors are instantiated in physical processes and linked together by bifurcations, which are themselves linked together in abrupt phase transitions. Universality is thus embodied concretely rather than remaining general and abstract, as in the case of essences. While essences deal in discrete and abstract differences, multiplicities deal in concrete and continuous differences. Indeed, Deleuze distinguishes between “differentiation” vs. “differenciation.” In differentiation, what is observed is the progressive “unfolding of a multiplicity through broken symmetries” whereas “differenciation” refers to “the progressive specification of the continuous space formed by multiplicities as it gives rise to our world of discontinuous spatial structures” Each individual embodiment of the aforementioned non-essential universalities, furthermore, is unique and distinguishable from other examples:
“the different realizations of a multiplicity bear no resemblance whatsoever to it and there is in principle no end to the set of potential divergent forms it may adopt. This lack of resemblance is amplified by the fact that multiplicities give form to processes, not to the final product, so that the end results of processes realizing the same multiplicity may be highly dissimilar from each other, like the spherical soap bubble and the cubic salt crystal which not only do not resemble one another, but bear no similarity to the topological point guiding their production.”
Deleuze thus invites us to think of a continuum of several multiplicities which produce “differenciations” into apparently static, and, by some accounts, “essential,” spatially structured entities. Differentiation is thus continuous, whereas differenciation is discrete. DeLanda invites us to explore this concept of “differenciation,” in which, as noted before, space progresively articulates discontinuous or discrete spaces. DeLanda asks us to keep in mind that space, rather than being a set of points, is constituted in such a way that there is a “neighborhood” of well-defined relations of contiguity or proximity. This is quite different from the Euclidean geometry with which we were raised, according to which there are fifxed distances and lengths, both of which “determine how close points are to each other.” “Length,” indeed, is a “metric concept.” Euclidean geometry is constituted with “metric spaces.” These are not the only spaces with which geometry has to deal, however.
“A topological space…may be stretched without the neighbourhoods which define it changing in nature. To cope with such exotic spaces, mathematicians have devised ways of defining the property of “being nearby” in a way that does not presuppose any metric concept, but only nonmetric concepts like “infinitesimal closeness.” However one characterizes it, the distinction between metric and nonmetric spaces is fundamental in a Deleuzian ontology. Moreover, and this is the crucial point, there are well-defined technical ways of linking metric and non-metric spaces in such a way that the former [metric spaces] become the product of the progressive differentiation [as opposed to “differenciation”] as the latter [non-metric spaces].”
Understanding what DeLanda is saying here requires a brief exploration of the history of 19th century geometry. During the 1800s, the comprehensiveness of traditional Euclidean geometry was radically called into question. It was no longer universally taken for granted that the structure of physical space was exhaustively captured by Euclidean geometry. In the non-Euclidean geometry of Lobatchevsky, for example, metric concepts, although they were used, were not considered fundamental. The differential geometry of Gauss and Riemann has already been explored, and there were a couple other forms as well. Even though Euclidean geometry was the most highly regarded, it became increasingly evident that its metric concepts were not fundamental, but were derived from more fundamental non-metric concepts. Felix Klein, for example, argued that “all geometries…could be categorized by their invariants under groups of transformations, and…the different groups were embedded one into the other. In modern terminology, this is equivalent to saying that the different geometries were related to each other by relations of broken symmetry.”
Euclidean geometry is constituted by “rigid transformations.” Thus, concepts like length, angles and shapes are not altered by rotations, reflections or translations. Such metric properties do not, however, remain invariant in other geometries. These different geometries end up constituting a hierarchy of non-Euclidean geometries, producing progressively more primordial, non-metric concepts:
“There is one geometry, called affine geometry, which adds to the group characterizing Euclidean geometry new transformations, called linear transformations, under which properties like the parallelism or the straightness of lines remain invariant, but not their lengths. Then there is projective geometry, which adds to rigid and linear transformations those of projection, corresponding to shining light on a piece of film, and section, the equivalent of intercepting those light ways on a screen…These transformations do not necessarily leave Euclidean or affine properties unchanged, as can be easily pictured i we imagine a film projector…and a projection screen at an angle to it…If we picture these three geometries as forming the levels of a hierarchy (projective – affine – Euclidean) it is easy to see that the transformation group of each level includes the transformations of the level below it and adds new ones. In other words, each level possesses more symmetry than the level below it. This suggests that, as we move down the hierarchy, a symmetry-breaking cascade should produce progressively more differentiated geometric spaces, and, vice versa, that as we move up we should lose differentiation [and increase in differenciation]…In short, as we move up the hierarchy figures which used to be fully differentiated from one another become progressively blending into a single one, and vice versa, as we move down, what used to be one and the same shape progressively differentiates into a variety of shapes.”
Ultimately, it becomes increasingly evident that the metric space physicists study is derivative of more fundamental, non-metric, topological continuum “as the latter differentiated and acquired structure following a series of symmetry-breaking transitions.” This DeLanda refers to as a “morphogenetic view of the relation between the different geometries.” It is within such a context that DeLanda begins to articulate Deleuze’s distinction between intensive and extensive physical properties. Extensive properties are metric properties such as area, length and volume, as well as concepts such as energy and entropy.
Extensive concepts are intrinsically divisible. Intensive properties, however, “are properties such as temperature or pressure, which cannot be divided. If we take a volume of water at 90 degrees of temperature…and break it up into two equal parts, we do not end up with two volumes at 45 degrees each, but with two volumes at original temperature.” Deleuze argues that the intensive property is not so much one that is indivisible, but one, according to DeLanda, which “cannot be divided without involving a change in kind.” Thus, to divide the intensive is to produce a fundamental change in kind. As DeLanda points out, “The temperature of a given volume of liquid water…can indeed be “divided” by heating the container from underneath creating a temperature difference between the top and bottom portions of the water. Yet, while prior to the heating the system is at equilibrium, once the temperature difference is created the system will be away from equilibrium, that is, we can divide its temperature but in doing so we change the system qualitatively. Indeed…if the temperature is made intense enough the system will undergo a phase transition, losing symmetry and changing its dynamics, developing the periodic pattern of fluid motion…”