In explaining how Gilles Deleuze’s metaphysics gives us a new way of understanding reality, Manuel DeLanda seeks to provide an example of how singularities produce an entirely new way of thinking about the genesis of physical forms.

“There are a large number of different physical structures which form spontaneously as their components try to meet certain energetic requirements. These components may be constrained, for example, to seek a point of minimal free energy, like a soap bubble, which acquires its spherical form by minimizing surface tension, or a common salt crystal, which adopts the form of a cube by minimizing bonding energy. We can imagine the state space of the process which leads to these forms as structured by a single point attractor (representing a point of minimal energy). One way of describing the situation would be to say that a topological form (a singular point in a manifold) guides a process which results in many different physical forms, including spheres and cubes, each one with different geometric properties.”

Basically, this is what Deleuze is referring to when he says that singularities are “implicit forms that are topological rather than geometric.” This is to be contrasted, DeLanda notes, with the essentialist approach in determining the genesis of entities and processes, according to which “the explanation for the spherical form of soap bubbles…would be framed in terms of the essence of sphericity, that is, of geometrically characterized essences acting as ideal forms.” In short, Deleuze is interested in determining the genesis of reality from within reality itself rather than constituted from on high.

The singularity, in determining long-term behavior patterns, constitute possibilities which themselves constitute state space. These state spaces structure the possibilities “open to the physical process modelled by a state space.” Keep in mind that singularities repeat themselves habitually over a long period of time (indeed, that is their very definition). Singularities, or the multiplicities which they define, are good candidates for replacing essences because of their mechanism-independence. The singularity, or long-term behavior, is not the equivalent of the “essence of a process,” however, as DeLanda cautions. There are other formal properties of multiplicities, he notes, which distinguish them from essences.

Essences, on the one hand, he notes, have clear and distinct natures. Multiplicities, on the other hand, are “obscure and distinct” :”the singularities which define a multiplicity come in sets, and these sets are not given all at once but are structured in such a way that they progressively specify the nature of a multiplicity as they unfold following recurrent sequences.” DeLanda asks us to think of a fertilized egg prior to its development into an organism. For the essentialist, the entirety of the essence exists prior to its development. Biologists today reject such a notion, and argue that “differentiated structures emerge progressively as the egg develops.” Even though the egg possesses requisite chemical and genetic material for the eventual development of a human, provided adequate conditions are met, these materials lack a “clear and distinct blueprint of the final organism,” and cannot therefore be said to possess a metaphysical “essence.”

DeLanda goes on to explain how this notion of progressive differentiation can be articulated through the field of mathematics known as “group theory.” This form of mathematics developed during the 19th century; a form of mathematics which came to be very important in 20th century physics. It is known as “group ” theory because it involves a set of entities given special rules for combination. DeLanda notes that the most important property these objects have is that of “closure,” according to which “when we use the rule to combine any two entities in the set, the result is an entity also belonging to the set.” DeLanda’s example is that of the positive integer. Adding any two positive integer yields another positive integer. DeLanda explains:

“Although sets of numbers (or many other mathematical objects) may be used as illustrations of groups, for the purpose of defining progressive differentiation we need to consider groups whose members are not objects but transformation (and the combination rule, a consecutive application of those transformation). For example, the set consisting of rotations by ninety degrees (that is, a set containing rotations by 0, 90, 10, 270 degrees) forms a group, since any two consecutive rotations produce a rotation also in the group, provided 360 degrees is taken as zero. The importance of groups of transformations is that they can be used to classify geometric figures by their invariants: if we performed one o this group’s rotations on a cube, an observer who did not witness the transformation would not be able to notice that any change had actually occurred (that is, the visual appearance o th cube would remaininvariant relative to this observer). On the other hand, the cube would not remain invariant under rotations by, say, 45 degrees, but a sphere would. Indeed, a sphere remains visually unchanged under rotations by any amount of degrees. Mathematically this is expressed by saying that the sphere has more symmetry than the cube relative to the rotation transformation. That is, degree of symmetry is measured by the number of transformations in a group that leave a property invariant, and relations between figures may be established if the group of one is included in (or is a subgroup of) th group of the other.”

But what does this mathematical revolution have to do with Deleuze? It represents an important break from essentialism in mathematics insofar as “Classifying geometrical objects by their degrees of symmtry represent a sharp departure from the traditional classification of geometrical figures by their essences.” It used to be that the mathematician would think of a cube in terms of what it shared with al other cubes. The same is true of spheres. No longer are these figures classified in terms of the static properties they share with other figures. Instead, the figures are classified in terms o how they are affected by active transformations; that is, they are classified in terms of “their response to events that occur to them.” As DeLanda explains:

“Another way of putting this is that even though in this new approach we are still classifying entities by a property (their degree of symmetry), this property is never an intrinsic property of the entity being classified but always a property relative to a specific transformation (or group of transformations). Additionally, the symmetry approach allows dynamic relations to enter into the classification in a different way. When two or more entities are related as the cube and the sphere…when the group of transformations of onfe is a subgroup of the other, it becomes possible to envision a process which converts one of the entities into the other by losing or regaining symmetry. For example, a sphere can “become a cube” by loosing [sic] invariance to some transformations, or to use the technical term, by undergoing a symmetry-breaking transition. While in the realm of pure geometry this transmutation may seem somewhat abstract, and irrelevant to what goes on in the worlds of physics and biology, there are many illustrations of symmetry-breaking transitions in these more concrete domains.”

Phase transitions occur by means of broken symmetry, for example. A phase transition is an event which takes place when critical values, such as temperature or size, are reached. This switches a physical system from one state to another, “like the critical points of temperature at which water changes from ice to liquid, or from liquid to steam. The broken symmetry aspect here can be clearly seen if we compare the gas sand solid states of a material, and if for simplicity we assume perfectly uniform gases and perfect crystal arrangements.” Indeed, in the case of the progressive differentiation which a fertilized egg undergoes, determinations subsequent to the initial spherical egg can be articulated in terms of a complex cascade of symmetry-breaking phase transitions.

Manuel DeLanda goes on to demonstrate how the concept of progressive differentiation can be articulated in state-space terms. Keep in mind that DeLanda has previously noted that defining an entity in such a way that replaces essences entails that state space was articulated in terms of its singularities, or long-term patterns. One singularity or group of singularities can undergo symmetry-breaking transitions and be transformed into another one. These transitions are known as “bifurcations” in non-linear dynamics. One state can be added to another by “control knobs” which “determine the strength of external shocks or perturbations to which the system being modelled may be subject.

These control parameters tend to display critical values, thresholds of intensity at which a particular bifurcation takes place breaking the prior symmetry of the system. A state space structured by one point attractor, for example, may bifurcate into another with two such attractors, or a point attractor may bifurcate into a periodic one, losing some of its original symmetry.” Indeed, just as attractors are found in recurrent forms, bifurcations may articulate recurrent sequences of sch forms. In one sequence, there is a point attractor which becomes a control parameter at a certain critical value, destabilizes and bifurcates into a “periodic attractor.” As DeLanda points out, “This cyclic singularity…can become unstable at another critical value and undergo a sequence of instabilities (several period-doubling bifurcations) which transform it into a chaotic attractor.”

Such a symmetry-breaking cascade of bifurcations is indeed found in real physical processes, such as hydrodynamic flow patterns. In each recurrent flow pattern, there is one after the other at very specific critical thresholds of speed or temperature. Each phase transition can be initiated by heating an underlying water container. When temperatures are low, heat flows from top to bottom. This is known as thermal conduction. It is steady and simple and exhibits a pattern that is pretty bland, since it possesses the symmetry degree of a gas. However, at a certain critical point of temperature, convection takes place, causing coherent rolls of water which rotate either clockwise or counter-clockwise. This structuring of the water container has caused it to lose some of its symmetry. When the water gets even hotter, and approaches another critical threshold, yet another new pattern emerges. This is known as turbulence, which is more complicated. Indeed, there are at least seven distinct flow patterns revealed by the Couette-Taylor apparatus when the liquid movement reaches a certain speed (rather than heat). “…thanks to the simple cylindrical shape of the apparatus, each phase transition may be directly related to a broken symmetry in the group of transformations of the cylinder.”