In the previous article, Manuel DeLanda began to explain the revolutions in geometries as a preface to the relevance of these revolutions to the metaphysics of the philosopher Gilles Deleuze. As seen before, these newer geometries demonstrated that the metric space with which we are familiar from high school geometry “emerges from a nonmetric continuum through a cascade of broken symmetries. “Thus, this space is not abstract mathematics, but is a “concrete physical process in which an undifferentiated intensive space (that is, a space defined by continuous intensive properties) progressively differentiates, eventually giving rise to extensive structures (discontinuous structures with definite metric properties).” Indeed, spontaneous symmetry breaking has come to influence increasingly mainstream forms of physics. It may unify the four basic forces of physics, according to DeLanda (gravitational, electromagnetic, strong and weak nuclear forces).

At extremely high temperatures, according to physicists, such forces “lose their individuality and bend into one, highly symmetric, force. The hypothesis is that as the universe expanded and cooled, a series of phase transitions broke the original symmetry and allowed the four forces to differentiate from one another. If we consider that, in relativity theory, gravity is what gives space its metric properties (more exactly, a gravitational field constitutes the metric structure of a four-dimensional manifold), and if we add to this that gravity itself emerges as a distinct force at a specific critical point of an intensive property (temperature), the idea of an intensive space giving birth to extensive ones through progressive differentiation becomes more than just a suggestive metaphor.”

This brief review will act as a preface into how these revolutions in mathematics and physics inform Deleuze’s radical revision of ontology. Unlike other “post-structuralists,” Deleuze is not content to reduce essences to social constructions and nothing more. Instead, he wants to replace them with a different ontological status. This, of course, moves us into the realm of speculative physics, at which point many of the the more analytically inclined philosophers will lose sympathy. It is also at this point that Deleuze’s speculations begin to inform the burgeoning speculative realist movement.

In order to further understand Deleuze’s concept of multiplicity, it will be important to understand the different “operators” involved in the construction. As DeLanda reminds us, “given a relation between the changes in two (or more) degrees of freedom expressed as a rate of change, one operator, differentiation, gives us the instantaneous value for such a rate, such as an instantaneous velocity (also known as a velocity vector. The other operator, integration, performs the opposite but complementary task: from the instantaneous values it reconstructs a full trajectory or series of states.” It is from these two operators that we obtain the concept of the structure of state space.

One chooses a manifold which acts as a state space. From experimental observations of the changes this system undergoes in tie, or “from actual series of states as observed in the laboratory,” it is possible to generate trajectories whose purpose is to populate such a manifold. Such trajectories constitute raw material for repeated application of the differentiation operator to such trajectories, and, as DeLanda points out, each application “generating one velocity vector and in this way we generate a velocity vector field.” Ultimately, the integration operator allows us to generate from this vector field subsequent trajectories which will help predict future observations of the states of the system.

it is at this point that Deleuze distinguishes between trajectories as they appear in one “phase portrait of a system” as opposed to the vector field. On the one hand, a specific trajectory or integral curve models a set of actual states in a system. On the other hand, the vector field represents intrinsic tendencies of these trajectories and systems, allowing us to see how these actual, physical systems tend to behave. In the first states, specific actual states are modelled or represented, and in the latter case, broader trajectories or singularities are measured. “These tendencies are represented by singularities in the vector field, and as Deleuze notes, despite the fact that the precise nature of each singular point is well defined only in the phase portrait (and by the form the trajectories take in its vicinity), the existence and distribution of these singularities is already completely given in the vector (or direction) field.

Singularities and attractors are differentiated into the two distinct mathematical realities, the intensive and the extensive. Such trajectories approach attractors “asymptotically,” always becoming closer while never reaching it. While trajectories constitute actual states of objects, attractors are themselves never quite actualized. Indeed, who trajectory ever quite reaches the attractor. As DeLanda explains, “It is in this sense that singularities represent only the long-term tendencies of a system, never its actual states. Despite their lac of actuality, attractors are nevertheless real and have definite effects on actual entities. In particular, they confer on trajectories a degree of stability, called asymptotic stability.” Tiny shocks can interrupt a trajectory from its general attractor, but shocks that are sufficiently small will not push it out of its basin of attraction. Such a trajectory will re-stabilize and remain a stable state.

By “structural stability” is meant a distribution of trajectories. The stability of specific distribution of attractors may be affected when the vector field is perturbed. This happens when a small vector field is added to the main one. Distributions of attractors are structurally stable and this is why we witness the recurrence of distinct physical systems. Sometimes, however, the perturbation is unusually large. This may cause a distribution of attractors to no longer be structurally stable. Sometimes, they will bifurcate into an entirely new one. “Such a bifurcation event is defined as a continuous deformation of one vector field into another topologically inequivalent one through a structural instability.”

It is at this point that DeLanda felt comfortable to give an ultimate, formal definition of multiplicity: “A multiplicity is a nested set of vector fields related to each other by symmetry-breaking bifurcations, together with the distributions of attractors which define each of its embedded levels.” On the one hand, there is the component of the model in which information about the actual world, such as trajectories as “series of possible states,” is distinguished from the non-actualized component. Multiplicities themselves, as we will see in the next article, are “virtual.” What this is, and how it differs from the apparently similar modal category of possibility, will be covered in the next article.