The concept of “multiplicity” is not only one of Deleuze’s most important concepts, it is also one of his most enduring. That said, it is quite difficult to give a formal definition of the concept due to how dependent it is on relatively obscure forms of mathematics, such as group theory, dynamical systems theory and differential geometry. Ultimately, the concept is intended to replace the concept of an “essence.” The “essence” of something, in philosophy, typically defines its identity. For Deleuze, however, things and processes are described in terms of the morphogenetic processes which give rise to it. In the words of Manuel DeLanda,

“Rather than representing timeless ccategories, species are historically constituted entities, the resemblance of their mebers explained by having undergone common processes of natural selection, and the enduring identity o the species itself guaranteed by the fact that it has become reproductively isolated form other species. In short, while an essentialist account of species is basically static, a morphogenetic account is inherently dynamic.”

Furthermore, Deleuze rejects the existence and usefulness of transcendent entities, particularly when it comes to defining the origins of things and processes. Instead, reality is understood in terms of self-organization from materials immanent to the spatio-temporal, material world. Deleuze introduces the concept of the multiplicity precisely in order to prevent anything from being described in terms of essences, including the processes by which things are determined, since even then, essences may be ascribed to processes, if one is not careful. Deleuze wants to prevent this from happening. DeLanda defines multiplicities as “the structures of spsaces of possiblities, spaces which, in turn, explain the explain the regularities exhibited by morphogenetic processes.”

He reminds us that “space” must not be understood in purely geometrical terms. Instead, it is linked to the concept of process. One can advance in one’s understanding of the multiplicity, however, by understanding the concept of the “manifold,” which is itself a concept used in geometry. More specifically, as DeLanda notes, it is used in differential geometry:

“The term “manifold” does not belong to the analytical geometry of Descartes and Fermat, but to the differential geometry of Friedrich Gauss and Bernhard Riemann, but the basic idea was the same: tapping into a new reservoir of problem-solving resources, the reservoir in this case being the differential and integral calculus. In its original application the calculus was used to solve problems involving relations between the changes of two or more quantities. In particular, if these relations were expressed as a rate of change of one quantity relative to another, the calculus allowed finding the instantaneous value for that rate. For example, if the changing quantities were spatial position and time, one could find instantaneous values for the rate of change of one relative to the other, that is, for velocity. Using this idea as a resource in geometry involved the realization that a geometrical object, a curved line or surface, for instance, could also be characterized by the rate at which some of its properties changed, for example, the rate at which its curvature changed between different points. Using the tools of the calculus mathematicians could now find “instantaneous” values for this rate of change, that is, the value of the curvature at a given infinitesimally small point”(DeLanda)

It is not just the differential character of this new form of geometry which provided Deleuze with the conceptual tools required for his new metaphysics, however. Gauss still used the old Cartesian method for studying a curved two-dimensional surface. That is, it was embedded in a three-dimensional space with its “fixed” set of axes. These axes would be assigned to specific points of this surface. Gauss eventually realized, however, that calculus could study specifically local information. This meant that it “allowed the study of the surface without any reference to a global embedding space.” He would soon realize that N-dimensional curved structures could be defined exclusively in terms of their intrinsic features. It is these intrinsic features which were defined as a “manifold.”

Deleuze’s concept of multiplicity “takes as its first defining feature these two traits of a manifold: its variable number of dimensions and…the absence of a supplementary (higher) dimension imposing an extrinsic coordinatization, and hence, an extrinsically defined unity. As Deleuze writes: “Multiplicity must not designate a combination of the many and the one, but rather an organization belonging to the many as such, which has no need whatsoever of unity in order to form a system.”” This allows Deleuze to do away with essences, since essences require a kind of transcendent space which functions as a container for them.

It is by means of dynamical systems theory which the geometric properties of manifolds require in order to properly define morphogenetic properties. Within this theoretical framework, “the dimensions of a manifold are used to represent properties of a particular physical process or system, while the manifold itself becomes the space of possible states which the physical system can have. In other words, in this theory manifolds are connected to material reality by their use as models of physical processes.” Suppose one is mapping the dynamical behavior of a bicycle. One must determine the number of ways in which such a process or object can change. These are known as the process’s “degrees of freedom.” These degrees of freedom can then be related to one another through differential calculus. Each degree of freedom is mapped onto the dimensions of the manifold, as DeLanda notes. Depending upon what one is mapping, one will require different numbers of dimensions. For example, he notes that while the pendulum will need only two dimensions, the bicycle will need a ten-dimensional space.

“After this mapping operation, the state of the object at any given instant of time becomes a single point in the manifold, which is now called a state space. In addition, e can capture in this model an object’s changes of state if we allow the representative point to move in this abstract space,c one tick of the clock at a time, describing a curve or trajectory. A physicist can then study the changing behaviour of an object by studying the behaviour of these representative trajectories.” The important point in all this, DeLanda notes, is not that their static space is captured, but that they are being measured in terms of processes.

Manuel DeLanda, in explaining the use of mathematics in Gilles Deleuze’s metaphysics, notes that dynamical systems theories allows us to take any space on a map as a single point whose pattern can be examined. One can examine either an object’s change of state or the complex of the modelling state. “An object’s instantaneous state, no matter how complex [can become] a single point…but the space in which the object’s state is embedded becomes more complex (e.g. the three-dimensional space of the bicycle becomes a ten-dimensional state).” A complex process can be studied as trajectories in a space of possible states. This means that new resources can be mobilized in the study of physical problems.

“In particular, topological resources may be used to analyse certain features of these spaces, features which determine recurrent or typical behaviour common to many different models, and by extension, common to many physical processes.” It is at this point that DeLanda launches into a discourse about Henri Poincaré, the great 19th century mathematician. His interest was in modelling real physical systems. He use a very simple equation to model these systems which allowed him to examine recurrent traits with any model with two degrees of freedom. This equation was far too simple to be useful, of course. However, it allowed him to discover the “singularity”; a special feature of two-dimensional manifolds. These singularities exercise a great deal of influence over trajectories. Since these trajectories represent actual series of states in a physical system, they provided a great deal of insight concerning the behaviour of the physical system.

Singularities function as attractors. They (singularities) represent a system’s long-term tendencies. That is, “the states which the system will spontaneously tend to adopt in the long run as long as it is not constrained by other forces.” Trajectories which begin at very different places and engaging in very different patterns may end up in the same final state because they exist within a certain sphere of influence known as a basin of attraction. This is the sphere of influence which an attractor exercises over its specific area. Singularities are specific topological points and “the final state they define as a destiny for the trajectories is a steady state.” Poincaré described certain closed loops that act as attractors as “limit cycles.” So DeLanda:

“The final state which trajectories attracted to a limit cycle (or periodic attractor) are bound to adopt is an oscillatory state. But whether we are dealing with steady-state, periodic or other attractors what matters is that they are recurrent topological features, which means that different sets of equations, representing quite diffferent physical systems, may possess a similar distribution of attractors and hence, similar long-term behaviour.”