1 Jan 2013

Pt2.Ch4.Sb2 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘The Two Multiplicities.’ summary

Corry Shores
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[Note: All boldface and underlining is my own. It is intended for skimming purposes. Bracketed comments are also my own explanations or interpretations.]


Henry Somers-Hall


Hegel, Deleuze, and the Critique of Representation.

Dialectics of Negation and Difference


Part 2: Responses to Representation

Chapter 4: The Virtual and the Actual

Subdivision 2: The Two Multiplicities

Very Brief Summary:

Riemann space has heterogeneous metrics. Phase space describes the dynamic behavior of systems, and it can be understood topologically. Such topological models are like Deleuze’s notion of the Idea, because they are multidimensional, they are neither sensible nor conceptual, and they are actualized in various spatio-temporal relationships.

Brief Summary:

Bergson’s continuously integrated heterogeneous multiplicity is based on Riemann space. For Deleuze it opens new possibilities of thought. This non-Euclidean geometry is based on a curving space with no parallel lines. The metrics of the space are nowhere the same (it is non-metric continuous space). It does well to describe topological features of surfaces. Here distances between points do not matter, only the relations between them do. So figures can be deformed drastically through stretching so long as there is no cutting and reattaching of parts. This model helps us understand the behavior of dynamic systems, because we can consider phase space topologically. To describe the trajectories a system can undergo, we make a phase portrait diagram that shows the vectors of variation between related variables, which often tend toward a state of equilibrium, dynamic or static, called an attractor state. The phase portraits can be rendered topologically by considering the attractor state to be like indentations in a surface to which water flows from high to low. We can use this to understand Deleuze’s definition of the idea as an n-dimensional, continuous, defined multiplicity. The topological phase diagram is not sensible, but one of its trajectories actualizes in the sensible. [It is not conceptual, because the dynamics are too complex to conceptualize. We can only see it as diverse surface features.] And yet, the phase diagram is not a duplication of sensible actuality, because actuality only produces one trajectory of development, where the phase space is an infinity of them. In fact, not all can be actualized together, only one can, so the topological phase space is a field of incompossibilities. Yet every actuality sensibly expresses implicitly the whole of this virtual field. This is one trait of the Idea. Note also that the attractor state does not mean there is a teleology to the development of the system, because it is not the cause but rather the result of the behavior. Another trait is that the elements determine each other reciprocally; for, there is not a homogeneous space imposed on them but rather the elements determine their own topological space. The third trait of the Idea is that it actualizes in diverse spatio-temporal relationships; we saw how the same phase topology actualizes in many different instances, and also, the pendulum motion is a sine-wave, and this structure actualizes in many different ways in the natural world. Living systems are non-linear, which means slight changes in initial conditions can lead to drastically divergent outcomes. Such systems are deterministic, because each moment follows from the prior, but they are noncomputable; so they are deterministic but unpredictable. The Lorenz attractor shows us a phase space that is ordered but infinitely complex. This is how living systems operate.



Recall Bergson’s two sorts of multiplicities. One regards the structure “provided by set theory. As we saw, this structure relied on the category of negation and worked by analogy with a certain conception of geometrical space.” (93) The second kind is based on non-Euclidean, Riemannian space.

Deleuze thinks non-Euclidean geometry opens new possibilities of thought outside mathematics. Euclid’s fifth axiom is: “through a point not on a given straight line, one and only one line can be drawn that never meets that given line”. (93) [Or, ‘only one line can be drawn through a point parallel to another line.' The basic insight here is that any other line than a parallel one would eventually intersect with the first line.] There are two problems with this axiom. [1] it relies on a concept of infinity, which lies outside our spatial intuition, and [2] this axiom cannot be derived from other axioms. In the nineteenth century, Lobatchevsky and Bolyai try to prove the fifth axiom through reductio ad absurdum, but the rejection of the axiom led to no inconsistencies. In fact it led to a new form of geometry. We will examine here Riemann’s geometry that he developed by rejecting the Euclid’s fundamental axioms. (93)

Note again that the fifth postulate requires we think of parallel lines extending infinitely. Riemann, however, distinguishes the endless from the infinite. A circumference has no end but it is not infinite. Riemann then postulates that all straight lines are finite but endless. Riemann’s geometry no longer corresponds to our intuition of space, and this leads to a distinction between mathematical and physical geometry. “Whereas for Euclidean geometry, there is an intuitive interpretation of its axioms immediately available to us (physical space), it is not immediately clear how we are to understand the meaning of a geometrical space where all lines are both finite and endless.”  (94) But we will work through some of the implications of Riemannian geometry so to obtain a physical interpretation of it. So we begin with Euclid’s parallel lines: "through a point not on a given straight line, one and only one line can be drawn that never meets the given line ." (94) So Somers-Hall draws this diagram.


Euclid's fifth axiom

(Figure 4.1 from page 94)


Only one line is parallel to another. Any other line would intersect the first one eventually. Also, imagine lines from P moving further and further down the series of Q’s. Line K is the limit case of these lines that increasingly tend toward being parallel. But this also means that as we move to infinity along the series of Q’s, there at the limit case will be a line that both intersects P and Q yet is also be parallel like K. “We can therefore say that parallel lines are lines that meet at infinity, which is equivalent to saying that no matter how far we extend them, they will never meet.”  (94)

Now consider another of Somers-Hall’s drawings.


euclid's fifth axiom in non-euclidean space

(Somers-Hall’s diagram from page 95a)


It is Riemann’s equivalent of the previous diagram. [We will explore its theoretical properties, but first take note that in this presentation it is not apparent how to visualize those properties.] For Riemann, we noted, straight lines are endless but not infinite. So L is finite, and we cannot move Q infinitely to the right. “Instead, as L is of finite length, but unbounded, when Q is moved far enough to the right, it will instead coincide with R, just as if we trace a path on the circumference of a circle for long enough, we will end up back at the point from which we started.” (95) Because we cannot move Q infinitely to the right, there cannot be parallel lines in Riemann’s geometry. Consider another of Euclid’s axioms, that a line is the shortest distance between two points. This is at odds with Riemann’s geometry. Two points for him may determine more than one line. And in fact, all perpendicular lines to a straight line meet at a point.


euclid's fifth axiom in curved space

(Somers-Hall’s diagram from page 95d)


Notice also the three sided figure in the center. We see it has two right angles. That’s already 180 degrees, not even counting the third angle at the top. So unlike in Euclidean geometry, the sum of the triangle’s angles is greater than two right angles. And yet, Riemannian geometry can still apply to Euclidean, because it gives us a geometrical interpretation of the surface of a sphere.

If we give Riemannian geometry this interpretation, the meaning of the various axioms now becomes clear. A straight line is the shortest distance between two points, and the shortest distance between two points on the surface of a sphere is the circumference, or great circle, which passes through the two points. Therefore, the notion of a straight line is to be interpreted in terms of a great circle. Just as all great circles intersect, all straight lines intersect in Riemannian geometry, meaning that there are no parallel lines. Bearing this in mind, the following diagram should make it clear how, on this interpretation, the sum of angles in a triangle can be more than two right angles and how the perpendicular lines meet at a point. (96b)


euclid's fifth axiom in spherical geometry

(Somers-Hall’s diagram from page 96d)

So a two dimensional Riemannian plane can be described in terms of three dimensional Euclidean space. Also, Riemann’s geometry is as consistent as Euclid’s is. It also brings about a distinction between mathematical and real space, and it allows for different forms of geometry. Also, Riemannian geometry describes the curvature of its surface within itself and not by means of an added third dimension like in Euclidean geometry. But this spherical space we so far have examined has a constant curvature. Riemann builds from the work of Gauss and develops three other important novel features to his geometry: variable curvature (found with differential calculus),  an unlimited number of dimensions,  and heterogeneous metrics.


[1] Variable curvature:

Using differential geometry Gauss describes a space with a variable curvature.

Riemannian geometry was able to characterize certain features of the surface intrinsically, such as its curvature, whereas Euclidean geometry needed to understand the surface in terms of another dimension, thereby treating the plane as the surface of a body in three-dimensional space. In the geometry that we have looked at so far, the space represented is one of constant curvature, the surface of a sphere. Such a geometry is for obvious reasons called "spherical geometry." In fact, we can generalize from the spherical geometry outlined above to cases in what is known as differential geometry where the space instead has a variable curvature. In order to achieve this, Gauss leveraged the power of the differential calculus, which allows one to consider not only the curvature of a line but also the rate of change of that curvature. As we can use the differential calculus to describe any curve, and not just the constant curvature of a sphere, when this is incorporated into the geometry, we are able to describe a complex surface through geometry without having to add an extra, extrinsic, dimension. Such a surface would not have the fixed curvature of the surface of a sphere but would instead contain peaks and dips more akin to the topology one would find on a map. Just as in the case of spherical geometry, however, this space can be defined intrinsically, not requiring an extra dimension to capture these changes in curvature. In this sense, it is as if the contours on a map were directly present in the paper without the need for them to either be drawn on or for the map | to extend into a third dimension. (97-98)


[2] Unlimited number of dimensions:

If we were dealing with physical geometry, we would be limited to three spatial dimensions. But Riemann’s mathematical geometry does not have this limitation.  “Instead, we can use Riemannian geometry to describe n-dimensional spaces, where n is any integer.” (98)

[3] Heterogeneous metrics. Consider a space with variable curvature. Gauss calls ‘geodesic’ the shortest distance between two points. But because the metrics are never constant in variably curved space, a geodesic will have a different size and shape depending on where it is placed in that space. Somers-Hall’s diagram below shows how the size and shape of a triangle will vary depending on its location in variable space. (98)


triangle topology

(Somers-Hall’s diagram page 98)

Riemann’s space is the sort dealt with in topology. The distance between points cannot define them as it can in the homogeneous metrics of Euclid. So in Riemann space, we can stretch or fold the distances between points however we like, so long as they maintain their relations to one another. “As such spaces are not defined metrically, they are known as nonmetric, or continuous, spaces. This leads us to topology, the branch of mathematics concerned with the relations of figures apart from their specific metric relations”. (99)


Riemann’s geometry allows Einstein to conceive of curved physical space. And even though Riemann’s axioms are counter-intuitive, they led to “very concrete revisions of the foundations of the sciences.” (99)

Just as the surface of a sphere looks like a flat plane over short distances, so Riemannian geometry looks like Euclidean geometry over short distances. This similarity has parallels with the fact that the breakdown of Newtonian physics is only apparent at extreme velocities. The scope of geometry is wider than providing a model for physical space. Fluctuations in share prices, for instance, can be plotted on a graph, which is in effect a two-dimensional Euclidean plane. Such a graph of price to time does not claim to model actual space, but rather the abstract space of | the market. In a similar way, Riemannian geometry can be used to provide a model, without any reference being made to the nature of physical space. (99-100)

For Deleuze what is important is how this geometry is “used to map how complex systems, such as markets, populations, or organisms, vary through time.” (100) And “With some qualifications, we will see that this move to a dynamic characterization of the system will provide the basis for a Deleuzian notion of virtuality which will fulfill something like the role of essence in classical systems.” (100) In modern science, movement is related to any-instant-whatevers, to constantly varying durations.

System theory aims “to provide a model of the dynamics of a system, or in other words, the different states of a particular system, as well as the relation between these states, that is, the way in which the system moves between these states.” (100) The systems it studies are varied, including simply physical systems like pendulums and complex ones like the dynamic relations between predator and prey populations.

We begin to model the system by choosing the variables that will describe the state of the system. In the case of a pendulum, every state of the system can be described by two variables, namely, the angular velocity and the angle of the pendulum itself. For a predator-prey system, the variables we choose would be the number of predators and the number of prey. In each case, these variables represent the degrees of freedom of the system. (100)

So we may describe these two systems using just two values. “These two values together can be used to describe points on a two-dimensional plane, with each axis representing one of these two variables . This plane, known a s a phase space, therefore contains all possible states of the system, in other words, each possible combination of states of the different variables.” (100) What we see with such a depiction is that we are more concerned with the system’s general behavior than with any of its specific states. And also, we are less concerned with the states and more with the movement between them.

We therefore begin by checking the relations of the variables at various points while the system is in motion. As well as registering these points, we are interested in the vectors present at these points, that is, the direction in which the variables are moving.  In the case of the predator-prey model, this would be found by determining the population levels that follow a certain relation of populations. By determining the change in values which follow from a particular point in the phase space, we get what is known as a vector. To determine the general tendencies of the system, we therefore begin by populating this space with vectors. (101)

Below we see “A phase space with two degrees of freedom populated with vectors.”

vector phase space

(Somers-Hall’s diagram from page 102)

So we populate the space with these vectors. When we have enough to adequate describe the system’s dynamics, we obtain a “vector field, which will contain the totality of the experimental data.” (102) [The vectors are like tangents to curving variations in the space, so they are like the differential relations in calculus. Thus we can perform integral calculus to sum them up.] By using integral calculus, we may create a “phase portrait” of the system, which “reintegrates these individual vectors into paths through the phase space, such that if we know the state of the system at a given time, we can trace the path of the system into the future.” (102)

[Let’s examine the phase space of pendulum movement. So first look at this diagram by James D Jones [all the tan ones are by him, many thanks for his excellent site].

[Thanks James D Jones, Copyright James D Jones 2011]

As time moves forward, the pendulum will swing from extreme positions, with the bottom-most being a central axis, such that when it swing from bottom to left, it is moving through increasingly negative numbers. Now consider this diagram.

[Thanks James D Jones, Copyright James D Jones 2011]

As the pendulum moves to the right, while time moves forward, we see that the velocity, the green line, increases too. When the pendulum is at position 0, the velocity is at its peak. But we are talking about velocity, which has a direction. Since the movement to the left is a negative movement in direction, the velocity is reaching a peak in its negative value, but it is still at its top speed. Then as it swings back upward to its most extreme position on the left, it speed decreases until it is at a standstill. Then of course the velocity increases, now in the positive values, as the position nears the 0 point. Now consider this diagram.

pendulum phase space simple airpower.mexwell.af.mil nic2

[Thanks David Nicholls / Todor Tagarev and Airpower Journal]

Here we no longer have an axis for time. Instead it is just position and velocity. So as it nears position B, the pendulum reaches its maximum speed, given either with negative or positive values (the top or bottom parts of the circle). Here is James Jones’ rendition.

[Thanks James D Jones, Copyright James D Jones 2011]

This circle draws the phase space of this frictionless pendulum’s motion. We can see in this animation how the velocity line is longest when it is at the bottom position, and it points backward on its right-to-left swing, which is why it takes the lower circle position of negative velocity values.


[Thanks Ruryk and wikipedia]

Hence we see how it draws a circular phase space.

Pendulum_60deg     Pendulum_170deg

[Thanks Lucas V. Barbosa (Kieff) and wikipedia]

Here is a diagram of pendulum phase space, but exactly what it represents I cannot say, however it resembles the other ones.


[Thanks wikipedia]

See something similar here in the ‘complicated behavior’ applet at this site, and turn the damping constant and driving frequency both down to zero.]

If we therefore imagine a pendulum unaffected by friction, we will find that the paths drawn through the phase space will form circuits, as the motion of the pendulum repeats itself. In fact, a series of paths through the phase space will be shown on the phase portrait, in | the form of a 'dart board' pattern of circles with the same center (although with enough energy, a different path will emerge as the pendulum completes a full swing). This is because the phase portrait shows the behavior of the system given a certain relation of the variables, and as the initial state of the system (the amount of energy in the system) may differ, so the movement of the pendulum will differ (i.e., if there is less energy in the system, the circle on the phase portrait will be smaller). Thus, the system will return to its initial conditions with the frictionless pendulum, as no energy is lost from the system, with the circumference of the path through the phase space determined by the initial amount of energy, since the more energy that is put into the system, the higher the angular velocity the pendulum will attain. (101-102)

We could also create a phase portrait for predator-prey population dynamics. [Consider this diagram.

preditor prey time series Fig3.5

[Thanks vanderbilt.edu]

Assume that the red x line stands for prey, and the blue y line for predators. Now look at a moment when prey are at their greatest. At that same time, there is a modest amount of predators. But with all that food, the predator population increases. But follow the trend of the correlation for each moment in time. As the predator population increases, they eat more prey, so the red line goes down as the blue one goes up. When the predator population reaches its greatest amount, the prey population is tending downward toward its least amount. So the predator’s food is lessoning, which is why the predators become more sparse. But when predators reach their lowest level, prey can increase their population, repeating the cycle. We see this in this phase portrait diagram.

preditor prey phase Fig3.2

[Thanks vanderbilt.edu]

We see here with the arrows that when predators are the greatest, prey population trends down, taking the predator population with it. So the top of this circle corresponds to the peak of a blue line with its decreasing red line. Prey population then increases, brining up predators with it, and we see that the highest prey population corresponds with a moderate amount of predators whose numbers are rising following the rise of the prey.]

We could also, for instance, use a phase portrait to examine the interactions of predators and prey within a system, correlating the population of each to one dimension of the phase portrait. Here, we would find that the trajectories of the system tended toward a stable circle, as too many predators leads to a drop in the population of the prey, which in tum leads to a drop in the population of predators, as they do not have enough food to support themselves, and a consequent rise in prey, as there are now fewer predators around. Regardless of the initial conditions of the system, it will tend to settle down into this state, as the number of predators to prey reaches an optimum value. (102)

[Dynamic systems like this often tend toward some particular state of equilibrium. With our frictionless pendulum, it was a dynamic equilibrium, but still one that is regular.]


Pendulum_60deg Pendulum_170deg

[Thanks Lucas V. Barbosa (Kieff) and wikipedia]

If there were friction, the pendulum swings would gradually wind down, with velocity and speed decreasing until the pendulum rests at the 0 point.

[Thanks James D Jones, Copyright James D Jones 2011]

[Thanks James D Jones, Copyright James D Jones 2011]

James Jones also shows this phase space three dimensionally.

[Thanks James D Jones, Copyright James D Jones 2011]

So there is equilibrium in both the frictionless and friction pendulums, however it is a dynamic equilibrium in the first case and a static one in the second.]

Regardless of the initial conditions of the system, it will tend to settle down into this state, as the number of predators to prey reaches an optimum value. Likewise, if we add friction to the model of the pendulum, we will find that it settles into a particular state-in this case, not a circuit, but instead a point, as the pendulum eventually stops moving: [102]


[Henry Somers-Hall image from page 102]

Somers-Hall now looks at three ways that Riemannian geometry affects our understanding of the phase portraits: [1] it uses peaks and trenches in the surface to represent attractors and repellers, [2] the portraits may be multidimensional, and [3] the phase space is continuous and non-metric.

[1] Surface features for attractors and repellers.

The phase portraits draw the development for a particular trajectory, but to describe the behavioral dynamics on a whole, we would need a line for each trajectory, which is not a practical method. Stephen Smale offers a solution using topology. So we are no longer using Euclidean discrete metric space but rather nonmetric space. So we are not concerned with the trajectory lines but rather with the sort of surface the dynamics describe. Thus for the friction pendulum, we might think of the center point as being like a dip in a cone. It attracts the behavior of the pendulum like how water runs from high to low ground. We might then think of the frictionless pendulum as having a rut that it circles through. So the low grounds are the attractors, and the high grounds the repellers.  (103d)

In diagrams of phase portraits, such as the one above, the tendencies of the system are represented by lines that move toward a central point representing the stopped pendulum. These lines function something like contours of the system, which show the way in which the system flows from state to state. If we consider the various phase portraits that we have described so far, we can see that certain topological features appear within their spaces. Thus, in the case of the entropic pendulum, all trajectories move toward a single point (the pendulum at rest). For the nonentropic pendulum, however, the trajectories instead proved to be cyclical. These features, which describe the tendencies of trajectories, are called singularities. They function like dips and peaks in the phase space, giving the phase space something like the structure of a 'landscape'. (103)

Now consider a diagram for predator-prey populations.


[Somers-Hall image from page 104]

In this case the x axis is the predator population. [As we saw before with this diagram, predator and prey populations tend toward a dynamic equilibrium.

preditor prey time series Fig3.5

[Thanks vanderbilt.edu]

Imagine if we started with conditions not on this chart above. So suppose at time 8 there is a greater number of predators but also a greater number of prey. The same cycle will repeat, but when we return again to peak prey, there will be fewer prey than at the beginning, and relatively slightly fewer predators. That pattern will continue until it reaches it circular offset pattern. Also if the populations begin low, they will increase to this dynamic equilibrium.]


[2] Multidimensionality of the portraits:

The portrait can have multiple dimension.

If we were to model a more complex situation, for example the interaction of two billiard balls, we would instead need four dimensions for each of the billiard balls, as each will have a location specified by two axes and a velocity along those axes. Thus, in an eight-dimensional space, a single point could represent any possible state of the system. Similarly, the action of two atoms in a gas would require a twelve-dimensional space, as each has both a location and a velocity defined in terms of three dimensions. (104)

[3]  The continuous, non-metric topological geometry of the phase space:

In a non-metrical space, the topological features can be understood apart from the distance relations of the points. So the same structure can be deformed to great extents and still be the same structure, so long as we do no cutting and gluing of the parts [so even folding is fine too].

Thus, topologically, a circle and a square are equivalent, as one can be deformed to look like the other. Since a figure of eight has two holes, however, it cannot be deformed to look like a square or circle. These structures are not, therefore, topologically equivalent. If we view the structure of phase space purely in terms of this topological understanding, we can see that different systems will have topologically equivalent phase portraits-that is, they are governed by the same singularities. (105)

So consider when Deleuze writes that “ ‘an Idea is an n-dimensional, continuous, defined multiplicity’ (DR, 182)” (105) We will later see that Deleuze replaces Aristotle’s notion with the Idea.

The n-dimensional Idea represents the various dynamics that together generate the system itself, much as the phase portrait represents the dynamics of the system. The continuity refers to the fact that the space we are dealing with is nonmetric. By "defined," Deleuze means that the n-dimensional space is governed by the interrelations between the different dimensions. It is these interrelations that create the topology of the space. (105)

Deleuze also gives three criteria for how the Idea may apply to the world.

[1] Idea is a multiplicity whose elements cannot have a sensible or conceptual form [so it is like the topology of phase space].

The Idea is a multiplicity, so it has elements. These elements cannot have a sensible form, a conceptual signification, or any assignable function. Recall Deleuze’s transcendental empiricism from the first chapter. [He is looking for a transcedental field, which would explain how it is we are able to make judgments about the world. But this field would be given immanently in our experiences. So as given to our experience, it does not have a conceptual form. And as transcendental, it does not have a sensible form.]

First, "the elements of the multiplicity must have neither sensible form nor conceptual signification, nor, therefore, any assignable function" (DR, 183) . This follows from our conclusions from the first chapter. As we saw in chapter 1, if the transcendental field is to be conceived of as different in kind from the empirical, it will have more than a merely conditioning role. It cannot merely replicate it at another level. (105)

[Recall how Aristotle’s and Russell’s systems regarded entities as atomic and externally related. This meant that homogenous space was the medium for the entities relations. So space and the entities were distinguishable. But because things are externally related, they may enter external relations with themselves. This caused the paradox of the set of all non-self-inclusive sets for Russell. And it caused the problem of explaining change for both of them, because something can relate to itself at discrete time points, but unless these moments are internally related and not externally related, we cannot explain the movement between the moments.]

As a follower of Bergson, Deleuze also does not want to separate the genesis of space from the genesis of the object itself, as this would be to reintroduce the kind of dualism that we saw was at the root of the problems thrown up by the logical systems of Aristotle and Russell. (105)

We now consider the idea in terms of the possible, the real, the virtual and the actual. For Kant, the idea of the possible has the same structure as the idea of the real. The only difference is that the real exists and the possible does not. But the transcendental [for Deleuze?] has a different structure [than the real?]. So we consider again a topological interpretation of phase space. There are many trajectories, depending on the given conditions of the system, that all lead to an attractor state. All the different trajectories are possible for the system, but they all cannot be actualized together. Only one trajectory can, so they are incompossible. The Idea is like this topological space. There are many virtual incompossible tendencies there in its dynamic behavioral structure. So the phase space itself is an inherent part of the system, but it is different in kind from the one actualized trajectory.

This difference is not in terms of reality, as the phase portrait captures the | tendencies, or long-term behavioral trends, of the system and is therefore real, but rather in terms of the distinction between virtuality and actuality. It is able to capture all potential states of the system without these states needing to be understood as opposed to one another, as they are all present in the structure of the space itself. (106-107)

But the fact that the trajectories inevitably move toward the attractor state, this does not imply some kind of teleology. The phase space is not the cause of the development but rather merely a description of the results of the system’s interactions. Also, as virtual topological multiplicities of the Idea move to physical actualities, there is a transition from non-Euclidean to a Euclidean multiplicity. So the transcendental is not a redundant reproduction of the empirical as it is in Kant.

[2] The Idea’s elements must be determined reciprocally.

This means that “the space is not determined from the outside through the imposition of another space” (107) Homogeneous metrically space is like the other space that is imposed on the space of shapes in Euclidean geometry. [In topological space, the geometrical properties of the shapes do not depend on an additional metric to define them.]

[3] The Idea’s constituent multiplicity “ ‘must be actualized in diverse spatio-temporal relationships’ (DR, 183)”. (107)

So the same virtual Idea, like the topological phase space, must be actualized in many different actual cases; it “does not simply describe a singular situation but instead has a kind of generality.” (107) The frictionless pendulum creates a sine-wave through its phase space. And the sine wave form is found in so many places and ways in the natural world. So this topological phase space is like how the Idea is actualized in diverse spatio-temporal relationships.

Dynamic systems show more complex similarities, however. If we take the case of two different systems, such as soap bubbles and crystals, we find that although spatially the two structures differ, the phase portraits of the two systems are identical, in that the dynamics by which both systems attempt to move to the lowest energy state, in the first case by minimizing surface area and in the second bonding energy, are the same .  In this case, a generality in the dynamic process that generates the system leads to systems with two very different material structures. Whereas the essence of the soap bubble might be traditionally considered as a sphere, and that of a crystal, a cube, the dynamic approach, in identifying systems with their process of genesis, allows the same underlying virtual Idea to generate both forms. In that the vector field maps the generation of the system over time, we also have the model of the temporal morphogenesis of the system. (107)


Recall how we noted that the Bergsonian analysis uncovers a tendency other than entropy. Living systems tend toward greater complexity and differentiation, rather than toward a homogenization of structure. So far we have dealt with linear mechanical systems, which means that they do not involve any exponential functions. Thus small changes in initial conditions will lead only to small changes in final conditions. These are deterministic systems, like the pendulum. But as in the case of the three body problem, a linear analysis cannot describe the system’s behavior. Small changes in initial conditions lead to drastically divergent final conditions. “This possibility of large disparities resulting from initial conditions breaks the linearity of the relations between the initial state and the final state, so such systems are called nonlinear.” These systems are still deterministic in the sense that each state follows deterministically from the prior one. However, the dynamics of the system are noncomputable, so we cannot predict the systems behavior over larger time scales. So they are deterministic but unpredictable. But as computers grew in power, it became possible to draw phase portraits for non-linear systems, such as the Lorentz attractor. First consider a layer of fluid. On the bottom it is heated, and on the top it is cooled. When the heat is little, it moves by conduction to the top. It is a stable equilibrium, because the motion of the particles remains in place. Small disturbances to the system dampen and return to the stable equilibrium, so its phase portrait is like the entropic pendulum. However, if we increase the temperature differential (between the bottom and top), the fluid moves cylindrically as hot fluid rises, cools, and falls again, repeating the cycle. Again, small disturbances would cause it to return to this dynamic state of equilibrium, so its phase portrait is like the nonentropic pendulum.

If the cell is heated further, however, then, the motion of the fluid begins to change. At first, the motion becomes faster until, at a particular point, the hot fluid does not have enough time to dissipate its heat before it reaches the top of the cell. It therefore starts to resist the motion of the fluid, leading to a wobbling motion as the cylindrical motion switches direction unpredictably. The equations governing convection are nonlinear in nature, so Lorenz's model remained a nonlinear model, although he reduced the number of variables to three. With three variables, the phase portrait of the system can be constructed within a three-dimensional Riemannian space. Within this system we find that despite the absence of predictability, a definite structure emerges. This structure, called a Lorenz attractor after its discoverer, looks like the wings of a butterfly ( the two "wings" in this case are the two directions of flow of the system), and although the trajectory through the space forms a clear pattern, at no point does this trajectory intersect with itself. (108)

lorenz replicatedtype.com Picture-86

[Thanks Sean Roberts / replicatedtypo.com]


So this Lorenz system phase portrait is ordered, yet it is infinitely complex. This kind of order cannot be found when taking the mechanistic view. In fact, non-linear dynamics are inherent to living systems. But first take note of the slaving principle:

while systems of a high degree of complexity will have an enormous number of degrees of freedom, it is the case that within complex systems, the number of degrees of freedom actually reduces as the amount of nonlinearity within the system increases. This is because the linear functions of the system fall into line with the nonlinear functions. Thus, as the chaos of the system increases, so does the simplicity. (110)

Recall also the example of embryonic development, which was a process of symmetry-breaking. Complex systems like the convection system display this reduction of symmetry too. At different phases, it had different phase portraits, because its attractor states were different. And consider water’s state changes. Both ice and liquid water can have a temperature of 0 degrees, so all it takes is the slightest disturbance in heat to radically change its state and structure. Living systems make use of these far-from equilibrium states in order to function properly. It would actually be unhealthy for linear dynamics to take over, because that would lead to a state of dying. Now recall how in the Aristotelian system we distinguished entities by their properties. Deleuze’s approach will be to define organisms instead by “the dynamic properties that form them.

"There are greater differences between a plow horse or draft horse and a race horse than between an ox and a plow horse" (SPP, 124). This amounts to understanding the organism in terms of its dynamics, or how it is able to interact with the world, rather than according to the actual hierarchy of genus and species that is central for, for instance, Aristotle's approach. (111)


Image credits:

Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.


A B pendulum circle phase space:

David Nicholls / Todor Tagarev and Airpower Journal



Acceleration/velocity pendulum:


Thanks Ruryk


Pendulum circle phase animations:


Thanks Lucas V. Barbosa (Kieff)


Multiple ring pendulum phase space:



Predator prey cycles:



Tan pendulum diagrams:

Copyright James D Jones 2011



Lorenz attractor:

Thanks Sean Roberts


Texts cited:

Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.

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