NOTES ON: Delanda, M.  (2002) Intensive Science and Virtual Philosophy, London: Continuum

[This is Deleuze for scientists, and also for those readers who want to understand some of the mathematical and scientific background [spattered throughout Difference and Repetition]—explained very well on Delanda’s videos on Deleuze.  It has also been hailed as a philosophical account of complexity theory.  It is a challenging read for sociologists, but I much prefer it to the literary and subjective interpretations, especially those found in educational commentaries. Underlinings are mine, designed to help me look through quickly]

This is to be a ‘reconstruction of [Deleuze's] philosophy, using entirely different theoretical resources and lines of argument’ (4).  This is because the general issue of changing theoretical assumptions and developing a realist ontology needs to be addressed—and arguments are found often in a very condensed way in Deleuze.  It is necessary to discuss not ‘Deleuze’s words [but]…  Deleuze’s world’ (5).  It is not a comprehensive reconstruction.  The main theme will be to replace essentialism with a notion of ‘dynamical processes’ of various kinds.  Deleuze himself often offers only a ‘compressed’ account of these issues, one which ‘assumes so much on the part of the reader, that it is bound to be misinterpreted’ (5).

Deleuze conceives of difference as a positive productive force which drives a process, especially intensive differences such as differences ‘in temperature, pressure, speed, chemical concentration’ (6).  Epistemological/ontological issues ensue, but the most important thing for Deleuze is to correctly pose relevant problems, especially to focus on ‘the singular and the ordinary’ (7).

Although this reconstruction must be adequate, ‘There is a certain violence which Deleuze’s texts must endure in order to be reconstructed for an audience they were not intended for’ (8).  In particular, it is necessary to avoid premature solidification to what is intended to be a fluid and open text.

Chapter one.  The mathematics of the virtual: manifolds, vector fields and transformation groups

[There is an excellent summary of the themes of this chapter in the beginning of chapter 2:

Realist ontology describes ‘a relatively undifferentiated and continuous topological space undergoing discontinuous transitions and progressively acquiring detail until it condenses into the measurable and divisible metric space which we inhabit’ (56).]

The key concept is multiplicity, but this has highly technical definitions based on different sorts of mathematics.  It is best seen as a radical replacement for the concept of an essence.  One example of an essence would be to suggest that what makes humans alike is that they are rational.  Deleuze wants to replace the notion of essence with a ‘morphogenetic process’ (10).  Species are historically constituted rather than representing timeless categories.  All ‘transcendent factors’ are replaced by ‘form –generating resources which are immanent to the material world’ (10).  It is necessary to avoid essentialism at a deeper level, suggesting similarities of process.  ‘Multiplicity’ attempts to do this, by specifying ‘the structure of spaces of possibilities’ (10).

Multiplicity is a concept related to 'manifold', a way of describing geometrical spaces with certain properties.  Understanding of manifolds has developed in geometry, beginning with Cartesian space, a way of locating curves on a two dimensional space.  The location of each point could be expressed as a relation between [X and Y values] numbers.  Algebra could then be used to describe processes and shapes.

Differential geometry (Gauss and Riemann) developed from the use of calculus.  Originally used to calculate rates of change in quantities relative to each other, this led to the idea that geometrical objects such as a curve, could also be described at each point.  The surface can be studied itself, ‘without any reference to a global embedding space’ (12).  The surface was seen as a space in itself.  Riemann went on to develop a geometry of N dimensional space, and the structures in this space ‘were originally referred to by the term “manifold”’ (12).  This led to completely new ways of understanding space, which were to be taken up by Einstein and others.  Deleuzian multiplicities are like manifolds with a variable number of dimensions and with no external unity imposing coordination, quite unlike essences.

It still necessary to explain how multiplicities relate to physical processes.  Here we need some theory of dynamic systems.  The ‘dimensions of the manifold are used to represent properties of a particular physical process’ (13), as a kind of model.  One way to model an actual object is to consider the number of ways in which it can change—its ‘degrees of freedom’ (13).  These changes can then be related to each other using differential calculus [somehow].  If an object like a pendulum can change only position and momentum, it has 2 degrees of freedom, whereas a bicycle has 10—five parts such as handlebars, wheels, crank and 2 pedals, and the possibility of changing both position and momentum.  Each degree of freedom becomes one dimension of a manifold, so that the pendulum needs a two-dimensional plane, and the bicycle a 10-dimensional one.  Each concrete possibility is called a state space, and changes of state can also be described accurately.  This model is a way of capturing processes, so that we can map all the possibilities (although there is a loss of information since each state space becomes a single point in a trajectory).

We can now use topological maths to analyse some other features of these spaces, especially those which described ‘recurrent or typical behaviour common to many different models’ (14).  These special features are singularities, and they influence trajectories and thus physical systems.  They can become attractors for trajectories, for example, influencing any within their ‘basin of attraction’ (14), and producing the long term tendencies of the system.  Some singularities are points, whereas others take the form of closed loops or oscillations.  The example given [described as an abstract machine in the book on war] refer to the classic shapes taken by different physical objects as they attempt to minimize bonding energy.  The minimal bonding energy is the attractor, the objects will actually vary according to their own properties—taking a spherical form for soap bubbles, and a cubic form for salt crystals.  Such abstract machines are mechanism independent, and thus ‘perfect candidates to replace essences’ (15). 

Yet multiplicities have more dynamism, and this helps avoid some essential understanding of process.  Multiplicities are by no means fixed structures, but unfolding ones, ‘following recurrent sequences’ (16).  The example here is the fertilised egg which, as it turns into an embryo, is not following a preformed path, but differentiating progressively as it develops—there is no ‘clear and distinct blueprint of the final organism’ (16).

There is a more precise mathematical way to describe progressive differentiation—the theory of groups.  Elements of groups combine in rule governed ways, such as regular transformations – you can rotate an object by steps of 90°.  To do this with a cube would produce a certain invariance in appearance, but not if we rotated it by steps of 45°.  However a sphere would remain invariant—it is described as having ‘more symmetry than the cube relative to the rotation transformation’ (17).  Classifying objects by the degrees of symmetry offers a way to classify them in terms of process, and responses to events: it is a relative property not an intrinsic one.  It is also possible to change one object into another by altering its symmetry.  The sphere can become a cube by losing symmetry, or undergoing a ‘symmetry breaking transition’ (18).

Symmetry breaking can be seen as a way to explain phase transitions.  In some physical systems, these occur when some parameter changes—water turns to ice or steam at particular temperature points.  Gas has more symmetry than solids.  In the example of the fertilised egg, development occurs as a result of a ‘complex cascade of symmetry – breaking phase transitions’ (18).

When a singularity undergoes such a transition, it can be converted into another one, as a bifurcation.  Again a particular critical value of a parameter is required, a ‘threshold of intensity’ (18).  Bifurcations may take place in a regular sequence, as when point attractors become loops.  The sequence in flows of liquid offer a familiar example, as they go from steady state to cyclic and then turbulent flows, or as heat produces conduction, then convection currents, then turbulence.  Apparently, when studied, these observable transitions take a more complex form.

There are differences between these physical processes and mathematical understandings, since the latter is ‘mechanism independent’ (19).  Physical systems have much more specific sequences of events, although there are some similarities too.  Mechanism independence is what replaces essences in Deleuze—‘multiplicities are concrete universals’ (21).  These mechanisms can result in quite different concrete outcomes, as with the examples of the soap bubble and the salt cube above, since the similarities higher the level of processes not product, hence the ‘obscure yet distinct nature of multiplicities’ (21).  Further, multiplicities can be seen as meshed together, ‘creating zones of indiscernibility…  Forming a continuous immanent space very different from a reservoir of eternal archetypes’ (21).

In this way, multiplicities differentiate (unfold through broken symmetries) and specify a series of ‘discontinuous spatial structures’ (‘differenciation’ [with a 'c'] , and nothing to do with Derrida at all).  One product of differenciation is the normal three dimensional space (22).  We need to understand what space actually is—the set of points grouped into neighbourhoods, which vary in terms of their proximity or contiguity with each other.  Normally, we think of metric ways to measure proximity, but in other spaces, such as topological ones, they do not remained fixed.  This requires non metric definitions of properties like distance.  However, metric spaces can be seen as arising from a progressive differentiation of non metric ones, as a symmetry breaking cascade.

This was realised through developments in geometry, especially non Euclidean ones.  The example of a manifold has been discussed, but there are other forms of geometry [such as affine geometry , and others, ending in topology].  Euclidean geometry can be derived from these later forms, since they are ‘related to each other by relations of broken symmetry’ (23).  [Different forms of transformations are responsible, and metrics do not remain variant with these.  Examples include where some properties such as ‘the straightness of lines remain invariant, but not their lengths’ (23).  Projective geometry is even more mysterious, with transformations called projectivities].  Each form offers more symmetry than the level below—‘as we ascend from Euclidean geometry more and more figures become equivalent to one another, forming a lesser number of distinct classes.  Thus, while in Euclidean geometry two triangles are equivalent only if their sides have the same length, in affine geometry all triangles are the same (regardless of lengths)’ (23). At the highest level, topology, geometric figures can remain invariant since only bending, stretching and deforming transformations are permitted it [the demonstration on the YouTube video is particularly helpful here, showing how a doughnut might be bent and pinched until it becomes a cup].  This is the least differentiated geometry.

In this way, symmetry-breaking cascades produce more differentiated geometric spaces, or more structure, [as you go down the hierarchy].  This could be seen as a metaphor ‘the birth of real space’ (24).  In mathematics, these are purely logical relations, but there is an ontological dimension as well.  This is where we need a distinction between intensive and extensive physical properties.  The latter include metric properties and quantities, which are intrinsically divisible, while intensive properties cannot be divided [we can divide volumes of water but the temperature of the water does not split into halves].  When you do change intensive properties you change qualitatively or in kind, as with rising temperature inducing phase transitions in water.

Objects therefore arise when an intensive space differentiates itself to produce extensive structures [there is an example with quantum physics, 25—the four fundamental forces arose from a series of phase transitions].

Essences assume that physical objects somehow receive external forms, but multiplicities explain how patterns can be developed ‘without external intervention’(26).  The mathematical models can be seen as describing real physical processes producing metric space.  There is no need to see change as only arising in the social constructivist sense [discussed further in Delanda here] More work is still needed to replace even the mathematical metaphors and analogies [next chapters].

Deleuze uses multiplicities to achieve some other goals as well, especially in displacing ‘modal logic…[more in Lof S]   The branch of philosophy which deals with relations between the possible and the actual’ (27).  If this process involves speculation, Deleuze proposes certain constraints to guide such speculation—avoiding essentialism for example.

Deleuze analyses state space in an original way, in specifically analysing the relation between the features of state space and the trajectories which are determined by them, and arguing that there is an ontological difference involved [ie we can explain reality that way].  It is possible to use differential calculus to calculate a specific value for a rate of change ‘such as instantaneous velocity (also known as a velocity vector)’, but integral calculus takes these instantaneous values and ‘reconstructs a full trajectory or series of states’ (28).  Together, the processes ‘generate the structure of state space’.  The process begins with experimental observations of empirical changes, actual series of states, and trajectories are then created from them.  These in turn are used to predict actual velocity vectors, constructing ‘a ‘velocity vector field’.  We then use integral calculus to move beyond empirical observations to generate still further trajectories as predictions.  Together, these trajectories produce ‘”the phase portrait” of the state space’ (28).  For Deleuze, there is an ontological distinction between these empirical trajectories and the possible ones [the former have to be actualised?], or between the vector field and the phase portrait.  It is in the vector field that singularities are distributed in a known manner.

In state space, trajectories always approach attractors infinitely closely, but without reaching them.  Thus we never achieve a state of actualisation [it needs additional features?].  Attractors therefore represent the long term tendencies of the system, and not its actual states.  Nevertheless, attractors are real and have real effects, especially in stabilising trajectories.  Attractors also stabilize entire vector fields [by being distributed among them in a known way—this is apparently tested by adding a small vector field and seeing what happens to the distribution of attractors].  Structural stability is typical, but serious disturbances will lead to structural change such as bifurcation.

It is now possible to offer a ‘final definition of a multiplicity.  A multiplicity is a nested set of vector fields related to each other by symmetry–breaking bifurcations, together with the distribution of attractors which define each of its embedded levels’ (30).  In this sense it both offers actual states and possible ones which are never actualised.  Each multiplicity is not realised when it produces concrete events, because they are real already—they actualise.  Deleuze has to then introduce the term virtuality to describe the status of multiplicities.  Virtualities are also real, since they have  real effects in the objective world and must be considered as part of objects.

We also need to deal with the notion of modality, the ‘ontological discussion of possibilities’ (31).  Classic modal philosophy runs into problems because it focuses excessively on sentences expressing what might have been, but this is entirely speculative, lacking any sort of structure and thus offering ambiguity.  Perhaps analyses of state space can overcome these limitations, because they provide a knowledge of actual trajectories in state space and how they are individuate, and they are able to limit possibilities.  Thus the emergence of individual actualizations ‘is defined by laws…  As well as by initial conditions’ (32), such that only one individuated trajectory becomes possible.  Typically, though, there will be many individual trajectories, ‘one for each possible initial condition’, although particular combinations may be ruled out (32).  There is still a philosophical debate about the status of mathematical possibilities like these, and empirical adequacy might be a test—those producing actual sequences are held to be real.  However, it might not be possible to isolate individual trajectories from the global system, which might contain information about individual trajectories [I think, 33].

For Deleuze, it is important to focus on the realist possibilities, but he does think it necessary to apply some deeper understanding about how trajectories become individuated, which means examining the regularity generated by singularities in the whole state space [I think -- individuation is a first step in actualisation,although only some individuations get actualised?].  Nevertheless, it is the actual vector field which produces actualizations, and so it cannot just be seen as a mathematical possibility.

Philosophers like to find examples in classical physics, but these tend to select simple vector fields relating to linear systems, where there is only one possible attractor.  Here, influences on the chosen vector field are minimal.  However in more typical complex examples, there is distribution of singularities, several attractors of different types and their 'basins of attraction'.  Is it necessary to describe the possibilities in terms of virtualities?  [Remembering that this is held to be a real potential, not just a linguistic possibility].  We could define state space as a set of possible points instead.  However, the point is to explain how actualizations take place [remembering that trajectories never actualize because they never converge on attractors]— something else is needed beyond mere possibility.

Perhaps the old category of 'necessity' can be applied?  This category belongs to classical physics with classic determinism with clear initial conditions and general laws affecting a trajectory at each point.  The new picture adds singularities in a more complex way.  Now, as many initial conditions can end in the same end state, the states that trajectories go through are less relevant and indeed may fluctuate considerably.  Similarly, the attractor has a stronger role compared to the initial conditions  (35).  It is important not to take the simplest example of single attractors whose basins take up the entire space [the classic linear deterministic example].  Only then does determinism specify a single outcome.  More complex space, with multiple attractors, ‘breaks the link between necessity and determinism, giving the system a “choice” between different destinies, and making the particular end state a system occupies combination of determinism and chance’ (35).  There can be contingencies, accidental disturbances and shocks, altering the power of individual attractors.  Bifurcations can alter the distribution of attractors.  [This is where Prigogine, and Coulis, are cited—apparently, ‘chance fluctuations in the environment’ decide which fork of the bifurcation appears, 35-36].

This is not Deleuze’s actual argument, but ‘it follows directly from his ontological analysis’ (36).  His discussion is more general and philosophical.  He has already argued that we must not see virtual multiplicities as essences, and therefore that much modal logic has to be rethought, because specifying possibilities nearly always involves an essence—and this applies ‘also to those physicists who seriously believe in the existence of alternative parallel universes’ (36).  For example, possible universes still assume the existence of fully formed individuals with the same underlying identity.

For Deleuze, individuals have to be explained, and not taken for granted, by referring to the process of individuation.  There is thus a connection between the boundaries of individuals and the ‘objective production of the spatio temporal structure’ (37).  The conventional notion of possibilities does not explain this process, and thus runs into difficulties about whether the possibles share the same essence or not.  Deleuze thinks that he has explained detailed differences in reality instead.

This also helps him avoid typological thinking [like avoiding essences, another proscription guiding his speculations].  Some of these involve essences, but not all of them [Aristotle’s natural states].  Botanical taxonomy is the best example. Similarities and differences were tabulated, and higher order relations of analogy and opposition used to generate new categories.  This was to be a timeless classification. For Deleuze, however, these apparently natural relationships, like resemblance, should be seen as ‘the mere results of deeper physical processes, and not as fundamental categories on which to base an ontology’ (39).  It is necessary to explain judgments, but not by referring to subjective categories or conventions [as in social constructivism], ‘but [as] a story about the world...the objective individuation processes’ (39).

More follows in the next chapter about species and individuals.  Basically, species sort themselves by natural selection and then they consolidate through reproductive isolation.  Together, these factors affect the identity of a species and how clear cut it is, or whether it can be hybridised.  Again, the degree of resemblance is a matter of actual historical details and developments, not a matter of fundamental concepts.  This argument can be extended to all natural events, including the ways in which human beings have become historically constituted.  This even includes apparently natural elements like metals—the example is gold, whose properties actually vary according to the scale of the sample [individual atoms of gold to do not melt, and there are various intermediary structures such as crystals and grains which also have different properties and these change at particular critical sizes (40)]

The negative constraints, avoiding essences and typologies, are accompanied by positive resources, as in the following chapter.  Generally, the task is to follow the traces of the virtual in individual events or entities, and reconstruction can show how multiplicities can 'form a virtual continuum’.  We also have to develop the notion of virtual space and virtual time, and connect with the empirical laws of physics.  Virtuality will be seen to be a useful construction, replacing laws and essences, and 'leading to an ultimately leaner ontology' (41).

Chapter two.  The actualisation of the virtual in space

The process of discontinuous transition and condensation [already summarised above] on page 56 leads to a discussion of examples that are less metaphorical than mathematics.  The key here is the discussion of intensive and extensive properties— the first one is ‘like temperature or pressure…  Continuous and relatively indivisible’, and can produce change through phase transitions, while extensive properties are like lengths or volumes ‘divisible in a simple way’ (56).  Again, the idea is to replace essences and typologies.  It will be necessary also to discuss qualities, as distinct features of empirical objects, again requiring a departure from mathematical analogy.

Species and individuals.  Species used to be thought of as categories expressing an essence or natural state, eternal archetypes.  Darwin replaced this idea by showing that species have histories.  More recently, species have been seen as individuals, not kinds, not having a higher ontological status than actual individuals. Instead of members of species exemplifying species-level qualities, the relation is more one of wholes and parts.  Interactions among individuals produce the characteristics of the whole, as when new breeding patterns among individuals divide a species into two sub populations.  Species do differ from individuals in terms of scale, both geographically and temporally. This is an example of a ‘flat ontology, one made exclusively of unique, singular individuals, or differing in spatio-temporal scale but not in ontological status’ (58).

It becomes important to specify the processes through which the whole emerges, which Deleuze argues is intensive.  This is because there are two basic ideas of ‘population and heterogeneity’, and this implies that population characteristics are only statistical averages, quite unlike the essentialist approach.  In essentialism, individual variation is unimportant and accidental.  For ‘population thinkers’ by contrast, individual variation is ‘the fuel of the evolution’, with homogeneity an unusual event (59).  There is no need for archetypes or ideal forms exemplified in individuals.  Instead, another key concept of Darwinism emerges—‘the norm of reaction’ (59).  This presupposes flexibility between genes and bodily traits producing different sub populations—different rates of sunlight or nutrient, for example, will produce subpopulations of different sizes.  There is no need to insist on one underlying ideal phenotype.  Instead of rating individual specimens according to some degree of perfection, we can operate simply with an idea of ‘relations between rates of change’ (60).  For Deleuze, Darwin broke with ideas of essences in this double way, substituting populations for types, and ‘rates of differential relations for degrees’ (Thou Plats, 60).

There are also structures between species and organisms—‘demes: concrete reproductive communities’ (60). They also feature intensive properties expressed as rates, such as rate of growth of the deme, which in turn depends on things like the rate of availability of environmental resources.  Demes also can have stable states as attractors, and transitions like bifurcations.  They can also have cyclic properties introduced by adding factors to growth rates.

Like multiplicities, with the abstract qualities of differential relations and singularities, physical examples have ‘counterparts’ [but not resemblances, Delanda insists] in these rates of birth, death, migration and resource availability.  The correspondence arises because a ‘given intensive process of individuation embodies a multiplicity’ [in the sense of taking a bodily form?].  The lack of resemblance is explained by ‘the fact that several different processes may embody the same multiplicity’.  In this way, multiplicities replace essences, and intensive individuations that embody them replace general classes (61).

We therefore have ‘three ontological dimensions which constitute the Deleuzian world: the virtual, the intensive and the actual’ (61).  Concrete individuals in actual worlds are the equivalent of the metric structures which condense out of the virtual.  They can exist in different spatial scales, providing the familiar objects in the actual world.  However, actual empirical objects process qualities as well—such as individual organisms ‘playing a particular role in a food chain or having a particular reproductive strategy’ (62).  Thus the intensive has to describe both extensive properties and qualities.

This can be seen in embryological processes, which produce not only definite spatial structures, but qualitative differentiation of cells into specialist cell types like muscle or blood.  First, eggs have to produce spaces, initially as non metric neighbourhoods, defined by ‘chemical gradients and polarities’ and with fuzzy boundaries.  In these neighbourhoods, cells begin to cluster together, but the numbers and location of each cell is immaterial.  It is local interactions between cells which matters.  Aggregates of cells produce either sheets or migratory groups as stable states.  These two states are connected through a phase transition.  The outcome is either migration or folding of cells.  The latter produces three dimensional structures.  The processes are subject to changes in rates such as the birth and death rates of cells.  There is no detailed genetic control, but ‘rather nonlinear feedback relations between birth and death rates and the processes of migration and folding’ (63).  This means that no quantitative precision is available to biologists, [in their explanations] but this is actually a strength indicating ‘the presence of a more sophisticated topological style of thought’ (63).  This renders an understanding of the processes ‘anexact yet rigorous’ [a phrase in Deleuze] (64).

The outcome of migration and folding is the production of definite spatial structures, such as bones.  The materials produced also have qualities, however, such as the ability to bear particular kinds of loads.  These also arise from intensive processes, this time involving the production of specialist kinds of cells, this time from an original set of ‘pluripotent’ cells.  As they circulate, these cells exchange chemical signals which affect their differentiation, in a process called induction controlled by regulatory genes, which seem to work in patterns which act as attractors, each of which represents a particular type of cell.  What actually happens to a [germ cell] depends on which attractors exist nearby, a kind of local trigger, and the degree of  ‘stimulus independence’ [and there is a link back to ‘mechanism independence’] acting at the virtual level.  This is an important ‘part of what defines the traces which the virtual leaves in the intensive’ (65)].  There must however be lots of possible stable states to act as attractors, and this will depend on the connectivity of the network of genes.  At particular ‘critical values of connectivity a phase transition occurs leading to the crystallization of large circuits of genes, each displaying multiple attractors’ (65) [this account is based on some recent biological work cited in the references]. 

Mathematical and biological models do not literally correspond.  In some of the biological models, physical processes replace some of the stages in symmetry breaking cascades.  Eventually, it would be ideal to replace all the mathematical stages with such physical processes.  In another example, it might be possible to talk about assembly processes of organisms.  Manufacturing assembly lines are classically metric and rigid, but biological assemblies are connected by their typology—for example the specific length of a muscle is less important than its attachment points, so the length can grow according to the length of the bones.  Components are transported as diffusion through a fluid.  They randomly collide, and locate each other through 'a lock and key mechanism' rather than exact positioning (66).  This form of assembly also permits random mutations to arise without harming the organism [because there are many normal combinations also available], increasing ‘evolutionary experimentation’ (67). The resulting complexity combines extensive, metric structures, but also qualities.  These are indivisible as well.

Using the word ‘intensive’ involves an application away from thermodynamics, and there is a need to incorporate additional Deleuzian terms to explain how the intensive gets hidden underneath the extensive and qualitative properties of an actual product, producing 'the objective illusion fostered by this concealment' (69).

The issue is not just one of divisibility, since this would mean the intensive and qualitative are similar.  There is also the issue of 'subjectively experienced intensities, such as pleasure' and the difference from 'objective intensive properties' (69).  One way to distinguish the intensive and extensive, apart from divisibility, is to remember that intensive properties can average out rather than add up, as when volumes of water with different temperatures are mixed.  However, average values like this produce certain dynamic aspects as well, driving some process of equilibration, or 'fluxes'.  This is an example of a positive or productive difference, unlike those of extensive properties which produce relations of similarity and difference.  Thus, as Deleuze says 'difference is not diversity…  Difference is that by which the given is given [or produced]’ (70).

Intensities can produce change if differences are large enough, producing a steep gradient, through a phase transition.  It is this positive result that matters rather than the formal property of not being divisible in metric terms.  In biological terms, intensive flows take the form of migratory movements or movements of energy through a population.  Genetic differences are also equivalent [‘an extension of the original notion of intensive gradients, but…  nevertheless related’ (71)].  However, in biological populations, there are much more opportunities to interact than in thermodynamic ones.  In particular, biological organisms have capacities which have no equivalent.

Capacities refer to the potential ‘to affect and be affected by other individuals’ (71).  In chemistry, carbon has a much greater capacity to combine with other elements than inert gases.  Biological components are assembled very flexibly, and have even more ‘combinatorial spaces’ (71).  This notion of greater possibilities alludes to the virtual.  Deleuze actually refers to the virtual and the intensive as possessing singularities and ‘affects (unactualised capacities to affect and be affected)’ (72).

Singularities have been well studied, but affects less well so [so DeLanda pursues some parallel work].  One approach involves the idea of capacities to form novel assemblages, and it is possible that there may well be universal recurrent patterns [the example refers to ‘random grammars’ and ‘algorithmic chemistry’ (72)].  Certainly, adding the idea of capacity takes us away from classic thermodynamics definitions of intensive.  Forming assemblages is one way to think of capacity [and the example is the walking dog, forming an assemblage with the ground and a gravitational field].  Capacities cannot be reduced to the properties of the interacting individuals, and seem to emerge unpredictably. A similar notion is ‘affordance’, which also stresses the relational nature of capacities, being released only when dogs relate to ground.  Spatial scale affects the interaction.  Affordances are also symmetric [reciprocal]—dogs may hide in holes in the ground, but also dig holes of their own.

The ability to ‘articulate heterogeneous elements’ can also be seen as a part of intensivity (73).  The notion ‘extensive’ can be extended to include the articulation of homogenous components.  This helps us explore the crucial notion of difference.  Intensive processes preserve positive differences, ones which generate further differences, as in evolution.

Objective illusion.  It is common to find the intensive concealed ‘under’ the extensive, and the ‘concrete universals (singularities and affects)’ which drive the intensive.  This is easier to see in assemblages where intensive differences remain, where homogenisation does not take place.  Scientists often themselves systematically homogenise processes, however, or study systems in equilibrium there are differences cancel themselves out.  This explains the persistence of the objective illusion.  The problem is exacerbated if physicists only study final states [and linear systems].  There is both an objective and subjective impulse towards this objective illusion [this looks like classic Marxist notion of ideology, where both subjective interests of economists and the misleading surface appearance of economic activity both contribute to misunderstanding] (74).

It is important to study systems which preserve intensive differences.  An example here is work focusing on ‘the field of far–from–equilibrium thermodynamics’ (75).  Such systems maintained flows of matter and energy.  It is also useful to study non-linear systems with multiple attractors—since these are easy to push away from equilibrium [and the work of Prigogine and Nicolis is cited, 75].  Such systems continue to display virtuality, in the form of potential alternative states, which can be produced by shocking the system.  [Here, and later,  a distinction seems to be appearing between complexity and mere diversity.  The former relates to the virtual and its potentials, while the latter seems to relate to actualizations and their combinations?  If this is so, the politics of complexity might lead to clarify whether it is talking about complexity as such or mere diversity—diversity would certainly be easier to manage and change?  In any event, it is not enough just to cite an analogy with a complex system, as Osberg does with Prigogine—we need a much more detailed analysis, as DeLanda  argues at the end of this chapter].

However, the virtual also appears best in systems with high intensities [low intensities are associated with equilibrium].  Again, physicists often prefer to study systems at low intensity values, and thus help to promote the objective illusion again.

Deleuze says we must penetrate beneath the objective illusion using a philosophical method, focusing on constituting processes responsible for consistency, ‘to go back up the path that science descends’ [What is Philosophy, 76].  We need to trace back qualities and extensivities to the intensive processes that produce them, and then back to the virtual.  DeLanda’s example turns on biological classifications again.  We should study objects like the tetrapod limb not by classifying it according to the common properties of limbs, but examining the process whereby they get produced, a matter of ‘asymmetric branching and segmenting’ (77).  In other words, a virtual limb is unfolded through particular intensive sequences, including bifurcations and blocked by vacations.  This will take us back to the intensive, but we need also to get to the virtual: the mathematical analogy of the relation between topological and other spaces helps here.

Extensive structures can be seen as occupying the bottom (metric) level, intensive processes the intermediate [eg affinal] level, and a virtual at the topological level.  No hierarchy is being claimed here—‘a nested set of spaces’ is a better description (78).  Each space in this case needs to be defined by its affects, whether it affects or is affected by specific operations such as rotating, folding and so on.

The virtual as a continuum.  Discontinuous individuals in the actual world are differentiations of this continuum.  This is no simple topological space, however, but a ‘heterogeneous space made out of a population of multiplicities, each of which is a topological space on its own…  a space of spaces, with each of its component spaces having the capacity of progressive differentiation’ (78).  What then would mesh these different spaces together into a ‘plane of consistency’ [nothing to do with logical consistency, DeLanda says, but rather defined as ‘the synthesis of heterogeneities as such’ (78) [apparently explained in What is Philosophy.  It starts to look a bit circular here, since the need to be consistent is rooted in the very definition of the plane, but the explanation of consistency assumes that there is such a plane that can synthesise?].

We can progress by considering the characteristics of the objects of populate the virtual, especially multiplicities.  The mathematical definition takes us only so far, and we have to abstract from the mathematics to get philosophical concepts [that are content free, or pre-individual].  If each singularity is extended into an infinite series, multiplicities can be meshed, but we have to spell this out a bit more detail.

The need to extend singularities arises from considering physical systems with multiple attractors, or biological assembly processes with many combinations.  Both of these cases allude to the virtual, since neither can be precisely described numerically, and novel assemblages are always possible.  How can we explain these unactualised capacities?

Abstracting from mathematical notions such as a function is one step.  Functions normally model systems in terms of input and output variables, the latter indicating a particular state in a state space described by the former.  However, we do not wish to ‘presuppose individuality’ (80).  [We want to maintain the idea of the virtual].  We therefore need some idea of a formless function, without values as such, but referring only to rates of change [this reminds me of the importance of the calculus for Deleuze, where the relation is preserved even when the actual values are zero in both cases].  In this way, virtual relations can be thought of which only determine each other, with no outside determination, and no actual content.  ‘Virtual singularities should be distinguished from individuated states’ (80).  Singularities like this have pre-individual characteristics [it is a mistake to see attractors as special points of state space].

We have to think of singularities as defined by vectors.  Vectors are not individuated states, but ‘instantaneous values for rates of change’ (80).  Occasionally, vectors are stationary, as a ‘topological accident’, and these have the status of events.  But not even these are actual events for Deleuze, but ideal events—‘turning points and points of inflection…  pre-individual, not personal, and a- conceptual.  Singularity is neutral’ [that is lacking content?] (81).

[The quote above comes from Logic of Sense, and DeLanda has an interesting note on it.  The actual events, compared to an ideal one  appear 'as a more fleeting and changing individuation.  Deleuze argues that events have the individuality of a haecceity…  The unique singularity of a moment.  There is a quote from Thousand Plateaus: “individuality [can be] different from that of a thing or subject…  [and]…  consist entirely of relations of movement and rest between molecules or particles, capacities to affect and be affected”'.  DeLanda goes on to say that the walking dog assemblage can be one of these, a concrete event, involving specific animals and specific conditions, rendered by Deleuze and Guattari as “This should be read without a pause: the animal–stalks–at–five– o’clock”.  This event can be described in terms of ‘relations of rapidity and slowness: the ground affords the animal a solid surface only because relative to the speed or temporal scale of change of the animal, the ground changes too slowly’.  By comparison, at the virtual level, singularities are also haecceities, annoyingly enough, but speeds and affects are different—producing an accidental moment in a field of velocity vectors which provides some sort of insulation from the transformations going on in the rest of the field.  So is a haecceity in the first sense a moment of fixity that arises from some sort of empirical combination, whereas at the virtual level, it is a unique interaction of vector fields?].

So, getting back to the notion of extension, a series of ideal events stretch out from each multiplicity.  To take another analogy, there are phase transitions in water at temperatures of zero and 100°, but a series of ordinary events in between them, which only produce limited linear effects.  At the virtual level, singularities in multiplicities produce a series of ‘ordinary [but still] ideal events extending up to the vicinity of other singularities belonging to other multiplicities’ (81).

Naturally, they must not be separated by some sort of metric scale.  To detour into mathematics again, there are clearly cardinal series and ordinal series of numbers.  Ordinal scales are not metric but relational.  However, ordinal scales are related to numerical ones, [as topology is related to geometry?].  Another quality of ordinal scales is that they cannot be added together to cancel their differences.  Deleuze thinks that this is an ontological matter, where the ordinal actually constitutes the numeric ‘through a symmetry breaking discontinuity’ (82).  Thus we can say that the ordinary events between singularities are only minimally actualised, separated from each other by ordinal scales.  The strings of events can be woven into a continuum, again avoiding individuation and  anything concrete,, producing communication between them, and also a proliferation.  As usual, we wish to avoid all those relations like similarity and analogy.

Multiplicities do not actively interact with each other because they are conceived of as essentially impassive, neutral or sterile.  They are independent of any particular mechanisms, and may be affected by several causal mechanisms.  There has to be some causal mechanism, or they would float off into the transcendent, and Deleuze wants to insist they are immanent.  As usual, we have to think of an unusual possibility, that they are incorporeal themselves, but affected by corporeal causes, ‘historical result of actual causes possessing no causal powers of their own’ (83).  They are in quasi causal relationships.  [This gets very close to the delirious and obsessive detail of Anti Oedipus, where one thought has to be immediately justified or qualified by further and further refinements and details in order to preserve the implication of the first term and fight off rivals.  Deleuze seems to have his own rigorous philosophical rules to guide this obsessional pursuit of implications].  We have to work with the dubious notion of quasi cause to account for the invariant properties of multiplicities, which must be produced ‘by at least one operator’ (84).

Naturally, this quasi causal operator cannot be seen as working in the usual way, which would involve too much individuation.  It can only have ‘the most ethereal or least corporeal of relations’, defined as ‘”resonances or echoes”’ (84).  We can begin to understand this further by looking at abstract communication theory [which also avoids content or individuation by referring only to a possible link between two series of events with different probabilities—if one change in one series affects the probability distribution of the other, information is said to have been transferred].  Even this is not abstract enough, however, and Deleuze specifies that the connection cannot be numerical, it must be a matter of ordinal distance, and communication should take place only via ‘the difference between the singular and the ordinary, the rare and so common, without further specification’ (85).  This will be discussed in subsequent chapters.

Has Deleuze gone too far?  Other phenomena can be studied empirically.  Even symmetry breaking cascades can be checked against empirical findings.  But purely intensive processes and their effects cannot be.  Nevertheless, we are moving away from the usual argument that eternal essences can be grasped a priori.  Deleuze’s scheme features ‘concrete empirico–ideal notions, not abstract categories’ (86).  There are hints at least of a quasi causal operator in a new field studying ‘emergent computation’ [discussed further on page 86].  The simplest example [!] refers to what happens in materials near phase transitions—apparently, separated events in the system can communicate, by fluctuating around a given state.  Near phase transitions these fluctuations begin to correlate, and thus transmit information in the very abstract sense.  Again, this seems to occur in a variety of materials, displaying the desirable ‘divergent universality’ [or mechanism independence] (87).  It may be that living organisms have acquired this capacity, and as a result, evolution has kept them at the edge of phase transitions rather than at equilibrium.  There is a notion of populations of cells as ‘poised systems’ (87) [compare with the notion of a trembling organisation?].  All this is still controversial and contested, but there is enough, Delanda thinks, to justify ‘postulating such an entity as a quasi causal operator’ (88).

Although Deleuze’s ontology is unfamiliar, it is at least very detailed—and speculative.  He is prepared to offer an alternative explanation to the notion of essences, and one that does not just refer to social conventions.  He has given a detailed description of the process of individuation, and offered this in the form of a discussion of mechanisms of immanence, not transcendental concepts.  Overall, he has attempted ‘to explain how the virtual is produced out of the actual’ (88).  [Actually, this is a good point.  Essentialism does not really explain how essences are connected to embodiments.  Come to that, social constructivism doesn’t really explain how construction occurs—they need at least some mechanism like habitus? How about critical realism or structuration approaches which also operate with a virtual level -- their mechanism is only human agency?]

Chapter three.  The actualisation of the virtual in time

There seem to be two notions of time in physics, turning on whether or not there is a fundamental asymmetry between past and future—in thermodynamics there is, but in classical physics, including relativist physics, there is an invariance.  The issue turns on whether it makes a difference to the values concerned.  In thermodynamics, running the system one way leads to diffusion then equilibrium, but running it the other way leads to an increasing amplification.  Behind these differences are differences in the ways in which the laws governing the processes vary.

The significance of laws appears in the next chapter, but there has been a tendency to keep the laws intact in a way which allows for symmetry, but which requires irreversibility to be explained [an interesting note give some examples of how this works, including seeing the directionality of time as a subjective effect, or a contingent effect, page 137].  However, nonlinear processes and far from equilibrium states have brought irreversibility back to prominence [and Prigogine has been important here, citing Bergson, and wanting to maintain the positive power of time in becoming, 105].

Deleuze also wants to insist on becoming without being, where individuals arise from an irreversible process of individuation.  Bergson has influenced both.  Generally though, Deleuze wants to extend the discussion into the same terms as used when discussing space, how a ‘temporal continuum…  through a symmetry breaking process yields the familiar, divisible and measurable time of everyday experience’ (106).  We need a discussion of extensive and intensive time, the former divisible into instants, especially sequences of cycles of different kinds.  The sequence is are traceable to underlying processes.  These are complex, with oscillations of various lengths nested within each other.  Intensive characteristics will explain how metric temporality emerges—this will draw on work that shows how external shocks can or induce various oscillations.  Generally, sequences of oscillations will show both singular and ordinary events, which reveals intensive factors at work.

Extensive time.  There is a nested set of cycles in a flat ontology.  The spatial scales involved have already been discussed, but there are temporal scales too, arranged as before in a variety of structures ranging from individuals to the virtual—‘an individual typically displaying a spectrum of timescales’ (107) (daily monthly and yearly cycles, or reproductive cycles and so on).  These cycles can overlap, but it is conventional to assign ‘a particularly prominent timescales’ to each individual level’ (such as our lifetime for an individual member of a species).

Can these cycles be seen as a result of symmetry breaking processes?  There happens to be a particularly suitable bifurcation (the Hopf bifurcation) explaining how steady state attractors become periodic ones.  The example to illustrate this turns on a spatial analogy involving transition from a gas to a crystalline state [it is complex, page 108.  In spatial terms, there is a loss of invariance following this process.  The same goes for time distributions, apparently, and a displacement can produce ‘a sequence of cycles that is out of phase with the original one’.  Again Prigogine and Nicolis are cited in support.  They seem to argue that instead states, we can ignore time, but in periodic motions time becomes important, and this can be referred to as a breaking of temporal symmetry].

Linear oscillators can be explained by looking at the details of their initial conditions, and they are typically regular.  However nonlinear oscillators do not depend on extrinsic constraints, and are better seen as pulses of action, each emerging from its past [difficult stuff again, page 109].  Their cycles can be seen as occupying a nested set as before, implying that time unfolds ‘pulse by pulse’ rather than offering some universal scale.  The normal metric notion of time would be one possibility in this unfolding she if, arising from a particular oscillation or a nest of them.  Deleuze talks about particular syntheses of time as lived presents, where the past and the future are contracted—he calls this ‘Chronos’, and notes that only the present is important, with the past and the future related to it.  There is apparently a vast extended present involved [the reference is to Logic of Sense].  Delanda explains this by saying that because the different scales involved, past and future in a quick oscillation is still only the present in the longer one—geological time for example.  Even biological organisms ‘have many past and future events for oscillate as operating at atomic and subatomic scales’ (110).  It follows that normal metric extensive time also contains shorter cycle oscillations, and thus it is ‘”composed only of interlocking presents”’, quoting Deleuze as above, page 110.  Although Deleuze sometimes refers to ‘lived presents’, he intends no psychological element—‘this is simply a matter of convenience of presentation and not fundamental to his account’ (110). 

A classic example from relativity theory reveals the objective dimension.  It is not just that the space travelling twin looks less old to the earth -bound observer —it is the atomic oscillators in the cells of each person which are ‘objectively affected’ in the travelling twin (111).

‘Lived present’ can be better understood by a relation between timescales and capacities.  We know that spatial scales affect capacities—that small insects may walk on water, but not larger mammals.  Timescales are similar, affecting the perception of change—so extremely slow cycles appear to be not moving at all [so everything really is becoming, including things like mountains or planets that appear to be fixed to us].  Extremely fast oscillations can mean that phenomena are irrelevant.  Subjective experience simply interprets these objective relations.

There is also relaxation time—the time taken to settle into a stable periodic state, to be recaptured by an attractor.  This will vary between phenomena, ‘and in each case they display characteristic timescales’ (111).  Relaxation time can also affect affordances, seen best in the curious materials called glasses which are really liquids flowing extremely slowly.  Observers will be able to detect a flow in glass given ‘sufficiently long observational times’.  Again no psychological or subjective qualities are implied, simply a relation between observation and relaxation times, an interaction.  In this sense, we can replace the observer with some other material and refer to ‘how the glass “appears” to it’ (112).  The materials can interact as with affordances—liquids can flow around glass and also erode glass.  Thus these capacities are affected by relative timescales, especially relaxation times. This helps understand ‘lived presents’, how individuals perceive their own timescales relative to the capacities of others.  Even inorganic things can have a lived present, however. The present therefore is a product of oscillations, movements and the type of matter involved.

Intensive aspects of temporality.  Intensive properties are involved in the production of individual oscillations.  Some work in biology is cited, page 113, on how the internal clocks of various organisms can be shocked, producing a halt in the oscillation.  The shock itself triggers the death of an oscillation, depending on its internal and intensive structure.  Some shocks will completely annihilate them selection if there is already a stable steady state attractor [which kind of replaces the oscillation].  The shock can produce ambiguous behaviour, arrhythmic patterns if the attractor is close to a phase singularity.  It is also possible to shock systems into creating oscillations around a phase singularity.  The results seem to indicate some intensive mechanism independent tendency, applying to biological organisms and even inorganic chemical reactions (113).

Oscillators can also synchronise or entrain temporal behaviour (114). This is similar to the capacities to form assemblages between heterogeneous individuals.  Such assemblage presupposes certain qualities of an individual, but also an adequate ‘outside’ [or environment?].  Entraining appears when for example organisms synchronise their sleep cycles with cycles outside themselves, such as the day cycle of the planet.  [However, apparently purely physical oscillators can also entrain]  Isolating animals from the external cycles can reveal an autonomous internal cycle—25 hour one for human beings, apparently (114).  Planets' rotational periods synchronise these internal cycles and can do so flexibly for different organisms.  Again this is typical of an intensive process, both stimulus and mechanism independent.  There seem to be weak coupling signals involved, but they must operate at a particular level of intensity or strength.

Temporality is therefore sequential, a sequence of oscillations.  But within these are singular and ordinary moments [in the biological work being cited, singular moments are those where phase transitions can occur?  Or are there other ‘sensitive points’ as well, where oscillations can be seriously affected?].  There is also the notion of parallel structures in time, already implied by the notion of entrainment, where several oscillations act in unison.  The spectacular example turns on the characteristics of a slime mould, which can take the form of individual amoebae, then aggregate into a single field, and eventually into a single organism, the process being controlled by a critical levels of the availability of nutrients (115).

So far, then, we have seen an abstract symmetry breaking event, a Hopf bifurcation, and some experimental results from biology.  This is the same sort of process as in the previous chapter, where abstract models have to be made physically plausible, and, as a result, complexity [in the ordinary sense] must be introduced.  It is also necessary to define a virtual continuum.  Something similar has now to be done to explain the birth of metric time.

We know from the biological work how important it is to develop critical timing and parallelism, but biological evolution might also benefit from the explanation of novelty—parallel developments and complex relations between, including relations of timing, can lead to different processes of acceleration in parallel developments, and this can produce novelty.  Rates of change and couplings are important in embryology, and time is involved in many of these notions of important rates.  Different rates of change can affect each other as in the example with affordances and relaxation times.  Even processes operating at similar scales can affect each other according to their rate of change [the example given is processes that produce patterned skin, which depends both on concentration of chemical substances and how quickly they react as they diffuse through an embryo, 117].  There are important rate-independent phenomena too, such as the information contained in the genes, which is not itself decoded at different rates, but produces enzymes with different controlling rates [this leads to discussion about whether or not genetic action can be seen as some kind of computer program—if so, it is a parallel processing network—117-8]. 

This replaces the idea that a novelty can only be added at the end of the sequence.  Instead, ‘new designs may arise from disengaging bundles, or more precisely, from altering the duration of one process relative to another, or the relative timing of the start or end of the process’ (118).  This is apparently called heterochrony, and one biological result is neotony, where sexual maturity exceeds the rate of development of the rest of the body. 

The idea of parallal disengagements further leads to the argument that some evolutionary change involves simplicity not additional complexity, which helps remove the accusation of teleology from Darwinism—Deleuze noted that progress can occur from simplification, from the loss of components and so on (118).  At the end of embryological development, the relatively fixed anatomy conceals these intensive processes, although some remain, as in the ability to self-repair.  And even the most finished individual can take part in other intensive processes, such as those in ecosystems.

Ecosystems are assemblages of heterogeneous species.  The population density of interacting species can vary, and this is another intensive property featuring phase transitions.  An environmental shock, for example, can produce a relaxation time until equilibrium is regained—the resilience of a population.  This sort of intensive property can individuate a species.  There are also different timescales operating simultaneously, such as the birth and death rates of a population and the interaction with those of others, as in a food chain.  Amplification effects can arise affecting relaxation time, according to the degree of connectivity between the species.  Environmental changes can produce longer oscillations.  The whole can be seen as a network of parallel processes. 

However, an ecosystem can also develop temporal cycles associated with evolution.  Evolutionary rates are no longer thought of as uniform or linear, but feature accelerations and decelerations.  This can also produce novel designs, as when a species becomes extinct and new niches are available.  Another affect can be symbiosis, meaning not just a beneficiary relation between two partners, but ‘an assemblage of heterogeneous species which persists for long periods…  and which typically leads to the emergence of novel metabolic capabilities in at least one of the partners’ (121).  This can be a form of coevolution, where the partners exert selection pressures on each other.  Symbiosis occurs at different scales: the cellular level, where cells have combined to generate photosynthesis (121); or where micro organisms cooperate with animals or plants in digestion or nitrogen fixing.  Both of these examples feature accelerated processes which show ‘meshing of the capabilities of two or more heterogeneous populations’ (121).

Deleuze normally refers to singularities and affects, but sometimes he refers to speeds and affects, ‘speeds of becoming and capacities to become’ (122).  Parallel processes can be defined in terms of relative speeds and rates of acceleration or deceleration.  This can become an evolutionary strategy, allowing an individual ‘an escape route from an overspecialised design’ (122).  [A kind of biological version of becoming and escape].  Coevolution can be rendered as a composition of speeds and affects.  Symbiosis similarly can enable ‘a fully formed being [to] cease to be what it is to become something else, in association with something heterogeneous on the outside’ (122).

This helps flesh out the abstract notions of timing and duration, as promised.  A further complication arises when we consider that time is always combined with space to produce ‘spatio- temporal phenomena’ (122).  This helps see that the emergence of metric properties occurs simultaneously with space and time in a single process, preserving the flat ontology.  Virtual space time involves the same elements in Deleuze—a non metric continuum, changing populations of virtual multiplicities and a quasi causal operator which assembles a plane of consistency.  This is speculative, but it does preserve ‘an empiricism of the virtual, even if it does not (and should not) resemble the empirical study of the actual’ (123).  Deleuze has at least posed the problem of how to derive a virtual ontology while avoiding essentialism and typologies.  He has at least pointed to the need to provide a mechanism of immanence.

The quasi causal operator is such a mechanism.  It provides ‘a minimum of actualisation’ for virtual multiplicities by prolonging them into a series of ordinary [but still ideal] events, and suggesting a series of convergence and divergence between them, using abstract communication theory as before.  (123).  Such communication can arise spontaneously in poised systems, even inorganic ones.  In parallel processing networks, other critical levels may produce communication ‘in the neighbourhood of a critical point of conductivity’ (124) Both embryos and ecosystems may also need to be poised to maximize communication.

However, ‘changing distributions of the singular and ordinary’ can also be produced by information transmission.  Virtual series can only be based on an ordinal scale, and statistical distributions cannot be conceived as fixed [if we are to preserve Deleuze’s interest in the virtual as minimally actual or individuated].  Instead there must be ‘mobile and ever changing (“nomad”) distributions in the virtual series, establishing both convergent and divergent relations between them’ (124), produced by the quasi causal operator.

This operator condenses singularities by producing communications between the series emanating from every singularity, linking them, differentiating the series, ‘ensuring they are linked together only by their differences’ (125) [the note on page 145 expands this view and says that Deleuze also thinks in terms of entrainments arising from initially weak forces.  The homely example considers two pendulum clocks initally transmitting weak signals to each other, through vibrations in the floor.  Once resonance is achieved a much stronger connection emerges.  This notion of resonance is used to explain the action of a quasi causal operator {and this also explains why Deleuze is always banging on about vibrations, multiplicities as vibrations, and how they connect with each other avoiding resemblances and analogies and all the other forbidden terms}].

This explains the spatial characteristics of the virtual as a mesh, but there is a temporal dimension too, which Deleuze calls ‘Aion’ [the prat].  This refers to a [mysterious] time of decomposition where spaces become sub spaces.  Spaces condense singularities, but time is needed to complete this event [or something!  125].  Deleuze uses the term ‘adjunction’ here, and this is borrowed from group theory—the ‘” adjunction of fields” is an abstract operation very closely related to the idea of the progressive differentiation of space through a cascade of symmetry breaking transitions’ (125).  The point seems to be to differentiate this process from others such as bifurcation, which feature a sequence of events and stable states, only one alternative of which is actualised [the example is the convection cycle again and how either clockwise or anticlockwise rotation is produced, but not both].  There may well be unstable options produced, but they don’t last long.  In virtual unfoldings, however events can coexist rather than following each other, and ‘each broken symmetry produces all the alternative simultaneously, regardless of whether they are physically stable or not’ (126).

Thus virtual forms of time involve absolute simultaneity.  In relativist physics, two events cease to be simultaneous if they become separated in space, but Deleuze's conception goes further.  In virtual space there are no metric distances, so that the idea of a stretch of time becomes meaningless [you further need the argument that ‘ordinal distances…  join rather than separate events’ (126)].

At the virtual level, not even the laws of relativity apply.  Instead, the very ‘temporality of the laws themselves’ are produced (126).  Although physics does not normally worry about the ontological status of its fundamental laws, philosophers commonly understand them as universal essences and thus as timeless.  But Deleuze’s notion of the virtual sets out to replace this philosophical understanding.

What does a non-metric form of time look like?  It would be unlike any present time, even the longest cycle.  It cannot be entirely timeless, since that would involve essentialism.  One step involves seeing the virtual as ‘populated exclusively by pure becomings without being’, which avoids timeless being (127). A pure becoming, unlike an actual one, must lose any trace of sequentiality or  directionality.  The analogy is with phase transitions, especially unactualized ones.  At the critical event, such as 0° C, water neither melts nor freezes, and both states are actual becomings.  By contrast, ‘a pure becoming…  would involve both directions at once, a melting – freezing event which never actually occurs but is “always forthcoming and already past”’ (quoting Logic of Sense and other references, note 58, 148) (127).  All events in virtual space, including the unfolding of multiplicities and their prolongation into singularities are pure becomings.  Thus, time itself can be seen as unfolding, as a pure order.  It has no present, since having a present would be to stop becoming.  It is instead ‘an ordinal continuum unfolding into past and future, a time when nothing ever occurs but where everything is endlessly becoming’ into the past and into the future (127).  Pure becoming is symmetric, with the normal direction of time only appearing when symmetry is broken in the process of actualisation.

Just as multiplicities are neutral or ‘causally sterile’, so is pure becoming as a temporal dimension (127).  However, just as the quasi causal operator has capacities to affect multiplicities, ‘acting in parallel with physical causality’ (127), and producing a mesh of interwoven multiplicities, so it has a temporal aspect as well.  This goes on at a virtual level, so no normal passage of time is involved—‘this other time must indeed be conceived as instantaneous’ (128).  [This seems to be a parallel here with the notion of pure becoming, pure instant].  Normal time always features of limited duration [a cycle], while the series of cycles can go on to infinity.  However virtual time is unlimited in its duration, but finite [instantaneous].  What the quasi causal operator does is to bring about an event of zero duration [with an incomprehensible quote from Deleuze Logic of Sense, 128, possibly meaning that there must be some form of minimal actualisation again?  The quote mentions Aion].

Delanda admits that this conception of time needs work, unlike the parallel discussion of space.  Even the discussion of space seems unnecessarily speculative—why not just use nonlinear mathematics with its notion of attractors and bifurcations?  However to do so would invoke essentialism again, this time in the notion that platonic ideas were being described by mathematics [a problem for Osberg who just uses Prigogine’s biology as a model for complex organisations].  A mathematician is cited in support of this view that nonlinear mathematics should not be seen as a self organizing system theory for everything (129).  Deleuze would be against any attempt to develop some eternal taxonomy, which leaves him with no choice but to push on to speculate about complex mechanisms for meshing at the most abstract level.

It could be argued that it would be simpler to go for nonlinear mathematics, but this would be ‘an illegitimate use of simplicity’ (130).  Platonism might be more familiar, and there are no other well known attempts to specify mechanisms of immanence, but that is no reason not to proceed.  The complexity is not over yet anyway.  We have seen how a continuum could be built from a population of multiplicities, but where do those multiplicities come from?  They must have been produced, otherwise they would look like essences.  We need yet another immanence mechanism to explain them, another task for the quasi causal operator, to extract multiplicities [from singularities ?] (130).  Deleuze uses the term section or slice here.  This is a mathematical operation which reduces the dimensions of the object [the bit about the circle being a slice through a sphere, a sphere being a slice through some four dimensional figure and so on].  Mathematics already uses this term to analyse attractors, by slicing a complex topological shape in order to simplify for study.

Deleuze has different notions, though.  Taking the example of flow patterns in liquids, empirical studies would see the attractors as the effects of actual causes such as temperature gradients.  However, it would be possible to sample or slice through a system like this in order to get the entire set of attractors defining each flow pattern, and all the bifurcations which mediate the patterns.  What this does in effect is to strip away all the empirical detail, leaving only the topological invariants, ‘the distribution of its singularities as well as the full dimensionality of its state space’ (131)

This is based on the idea of N dimensional manifolds discussed in chapter one.  These manifolds have different dimensions defining relevant degrees of freedom or ways of change.  Each actual multiplicity sampled empirically offers a specific value for the dimensions, since empirically only a finite number of ways of changing is possible.  Each multiplicity would have a different value for the number of dimensions.  When multiplicities are meshed together on a plane, a dimensionally diverse population emerges, ‘a space of variable dimensionality’ (131) [the idea of a plane is a bit misleading, since it implies only two dimensions] [the Deleuze quote seems to go back on this by talking about multiplicities being flattened when joined in the plane of consistency.  The quote is from 1000 Plateaus, so it might mean anything --but see below].  To add to the confusion, sometimes the quasi causal operator does this slicing, and sometimes the plane of consistency itself: ‘the difference between the two formulations is, I believe, unimportant’ (132).  [I’m seriously confused here, I confess.  Do multiplicities have to be sliced before they can be joined?].

For Delanda, the point is that when multiplicities are joined they are still heterogeneous.  However, the quasi cause ‘would operate at N-1 dimensions’ [which seems to imply the slicing process?].  This at least avoids the idea of the transcendental source of unity which would require N+1 dimensions (132).  [The quote from Deleuze (What is Philosophy) puts this in different terms, saying that a multiplicity can never be overcoded with a supplementary dimension, but that they fill all their dimensions already.  Any change seems to have to come from the outside, involving connection with the other multiplicities, and this is described as ‘the abstract line, the line of flight’.  It is possible that this common filling of all available dimensions, and lack of overcoding is what makes multiplicities capable of being flattened on the plane of consistency, regardless of the actual number of dimensions they possess?].

To summarise [!] There are two immanence mechanisms in the quasi causal operator.  There is pre-actualisation, where multiplicities are assembled together, or rather their ordinal series are, with relations of convergence and divergence.  This would poroduce minimal actuality and the first broken symmetry that will lead to full actuality.  Secondly, there is a counter actualisation, following from the actual extensive and qualitative back to the virtual.  This unflattens multiplicities, allowing them to unfold and differentiate again.  This involves sampling all actual events, and unfolding them into the past and future, redistributing the singularities of the past and future which account for the different levels [very confusing], (133).

Preactualisation begins the process of actualisation by giving multiplicities limited autonomy from the intensive, and a basic power to actualise.  At this stage, singularities could exist as a potential alternative state.  It would begin the process which leads ‘down the symmetry breaking cascade’.  In this operation, the quasi causal operator becomes known as the ‘”dark precursor”’, quoting Difference and Repetition, 133.  With counter actualisation, the process works up the cascade from the intensive towards the virtual.  Sometimes this is a spontaneous process revealing the virtual underneath the extensive.  It can be seen as something that ‘accelerates an escape from actuality which is already present in some intensive processes, the quasi causal operator is referred to as a “line of flight”’, quoting 1000 Plateaus.  [Note 77 on page 151 says the lines of flight may be relative or absolute.  Relative ones are found in actual assemblages, as in the examples from embryology and ecosystems, reflecting affects’ and relations of speed and slowness, and allowing an escape from rigid morphologies.  This is only a relative escape though, and absolute lines of flight require these relative escapes to be boosted so that they leave the intensive altogether and head for the plane of consistency.  Indeed, a further Deleuze quote says that these lines of flight actually create the virtual continuum].

Delanda ends by saying that even though these proposals might be speculative, at least Deleuze has asked the right questions and discussed the constraints if we want to abandon essentialism and discuss immanence mechanisms.  There are apparently several different accounts in Deleuze, showing that he was not satisfied with the solutions he gave.  It does help us understand what Deleuze thinks philosophy should do—‘creating virtual events (multiplicities) by extracting them from actual processes and laying them out in a plane of consistency’ (134). This is what the quasi causal operator is supposed to do [so philosophers are merely conscious element of it?  An interesting note 78, 152, discusses what Deleuze means by a concept—not a matter of understanding but referring to virtual multiplicities, itself a concrete universal].  This methodology distinguishes philosophy from science—science is interested in the actualized, while philosophy wants to ‘”extract consistent events from the states of affairs”’, quoting What is Philosophy, 134.  Science and philosophy can therefore be seen as two separate operations or as a single one, but Deleuze does see that there are objective movements which philosophers must grasp.  Indeed, philosophers must ‘become “the quasi cause of what is produced within us, the Operator”’, quoting Logic of Sense, 134.  This leads to a connection between ontology and epistemology—understanding means ‘correctly grasping the objective distribution of the singular and the ordinary defining a well posed problem’.  Spelling out consistent problems implies that they have an objective existence, somehow behind the [empirical?] solutions, ‘just like virtual multiplicities do not disappear behind actualized individuals’ (135).

Chapter four.  Virtuality and the laws of physics.

In a flat ontology, we can avoid the idea of reified or abstract totalities such as institutions, or even nation states: these all become concrete individuals operating at different scales, produced by concrete historical processes.  As with organisms and species, human individuals’ relations with these are best seen in terms of parts and wholes.  [See his book on social assemblages].  Any homogeneity arises from concrete historical practices too.

The term ‘science’ can be misleading if it refers to a totality, especially one defined by an essence.  Instead, there are individual scientific fields, emerging from populations as above [shades of Bourdieu!] .  These populations can include mathematical models and techniques, laboratory phenomena, machines and instruments, experimental skills, ‘theoretical concepts and institutional practices’ (154).  [Quite like ANT here then?].  These fields are affected by historical processes, which produce resemblance or separation, transfer of techniques [‘translations’ for ANT].  As a result, ‘as a matter of empirical fact, science displace a deep and characteristic disunity’ (154), although this is often concealed by philosophical effort, which attempts to develop essentialist and topological thinking.

In classical mechanics, fundamental laws such as Newton’s, are often viewed as general truths from which specific propositions follow simply as a matter of logical deduction.  This typically ignores the productive effect of processes themselves, especially that of causal connections.  In the strict sense, this argues that causal processes literally produce effects as a mechanism.  However, much philosophy assumes that there are merely constant regularities instead of active causes.  This followed from a notion that causality is inherently linear, simple and separated: this made causality look simple and law like.  However more complex forms of causality exist ‘nonlinear and statistical causality, for instance’ and we need some account of intensive production processes to include them (155).

By framing causality as a series of laws, linguistic statements have dominated accounts of causality.  The specificity of mathematical models needs to be restored, however, especially those which involve attractors and state space as before.  For example, we can then see why some solutions to an equation behave in a particular way—they ‘approach an attractor’ (155).  [More excellent examples of possibilities missed by linguistic reduction follow].  Minimising causes, and rephrasing causes in linguistic terms both lead to essentialist thinking.

According to Hacking, Newton’s theory of gravitation did much to avoid the specifics of causality by avoiding specifying the actual mechanisms involved [and Hume is also important].  This is encouraged by the general avoidance of experimental procedures in the philosophy of science—these are much more complex and productive than is suggested by logical deductions from laws.  [Not a bad description of the relation between educational theory and practice as well].

The ‘deductive – nomological approach’ sees explanation simply as logical arguments following from general laws, in the form of propositions ‘declarative sentences…  what two sentences in different languages, expressing the same state of affairs, have in common’ (156).  Usually propositions describe initial and other conditions.  If the prediction generated matches behaviour, then behaviour has been explained, not by tracing specific causes, but by ‘a typological approach: subsuming a particular case under a general category’ (157). The whole process starts with axioms—‘a few true statements of general regularities’, from which we can deduce general theorems, and then use observations in the laboratory to check for truth or falsity.  However, any truth content has ‘already been contained in the axioms’ (157).  Axioms are therefore like essences. 

A new approach requires mathematical models in explanation, including statistical models of ‘raw data’.  Practising science involves drawing upon a population of these models.  Models are sometimes constructed by combinations of fundamental laws and ‘various force functions’.  The link between these models and laws is not a simple one of deduction, but a modelling process, including ‘many judicious approximations and idealisations, guided by prior achievement serving as exemplars’ (158).  In ontological terms, the models involve emerge from concrete historical processes, as an open set, despite occasional closures [unlike the misleading textbook view according to Prigogine, 158].  Cartwright begins her account by saying that axiomatic laws are actually false—that is they achieve generality by compromising accuracy [as when the laws of physics assume frictionless wheels and so on, and have to assume that all other things are equal].  Additional modifications have to be made to increase accuracy, but that loses generality.  Instead, physics operates by deploying ‘causal models tailored to specific situations’ (159).  Truth content does not lie entirely with fundamental laws, since models are not simply deduced from them.  Indeed, causal models themselves vary in terms of whether they focus on specific situations or general laws, with the former much more frequent.

There are also statistical models of data, never raw data as in positivism.  Statistical models were used originally to calculate measurement errors, for example.  Laboratory testing involves ‘a complex practice of data gathering, involving not passive observations but active causal interventions’ (160) [good examples in Latour].  General laws attempt to unify these models, and again this effort is the result of an historical process featuring concrete individuals such as Euler or Hamilton (160).  In the process of unification, however, the notion of singularities became important, rather than abstract forces [discussed 160f.  One idea from Hamilton is the ‘minimum principle’ which apparently unifies a number of accounts—light travels along the path that minimises distance, for example.  This was originally connected with theological notions that God works with an economy of effort.] Eventually, a ‘calculus of variations’ mathematicised this principle, ‘the first technology ever to deal directly with singularities’ (161).

The calculus involves trying to pin down the actual processes, among all the possibilities, that have changed physical systems—‘for example a set of possible paths which a light ray might follow’.  Possibilities can then be sorted into ordinary and singular cases, and ‘the results of experiments show that the singular cases (a minimum or a maximum) are the ones that are in fact actualised’ (161).  The singularities are not proper attractors, but act in a similar way. Attractors define the long term state of a system, an equilibrium.  This helped Euler replace the awkward Aristotleian combination of final and efficient causes with just final causes, which led to the idea of more unified conceptions. [But doesn't this lose detail? Worth it to break with Aristotle?].( For Deleuze, final causes have to avoid any teleological connotations, and so can only be quasi causes).

This abstract approach avoiding details meant that classical mechanics could become some unifying approach—it had discovered a mechanism-independent process.  Other more detailed models were still required however to replace the details.  Nevertheless, this combination could not be reduced to linguistic general laws, and mathematical models proliferated.  Again, some related to the actual world, but others to the virtual world ‘by virtue of being well posed problems’ (162).  For Deleuze, a problem is well posed if it gets right ‘the distribution of the singular and the ordinary, the important and the unimportant, the relevant and the irrelevant’ (162). 

This leads to a problematic approach [one focused on problems] to replace fundamental laws and axioms.  Nevertheless the search for a single law which everything follows still persists.  Again it will be necessary to pursue non linguistic and more specific explanations of things like the distribution of the important and the unimportant.  This can begin by considering explanatory problems.  Traditional explanations downplay the productive mechanisms involved, but remain at the level of explaining regularities instead of looking at why specific processes occur.  Why questions will require specific models that exceeds the linguistic formulation.

One example arises from a philosopher called Garfinkel [!] who points out that requests for explanations can imply different ‘contrast spaces’ [implied alternatives, which may often not be shared in a conversation.  The example is the question why did you rob a bank.  The robber replied because that’s where the money is].  Answers implying different contrasts can be true, but still irrelevant—relevance and validity implies a specific contrast, and this is not apparent in linguistic formulations.  Instead, different contrasts should be seen as possibilities, or even state spaces (165).  Using the vocabulary of state spaces and mathematical models offers a more precise account than the usual linguistic formulations.  Possibilities will then depend on the distribution of singularities and their basins of attraction, or in Garfinkel’s terms ‘“basins of irrelevant differences, separated by ridge lines of critical points”’ (166). 

In Deleuze’s terms, problems may be rendered false [irrelevant?] because they are either under- or overdetermined—vaguely defined so that it is impossible to see if an event confirms one alternative rather than the other, or too sharply defined, so that only specific events will help decide alternatives [wouldn’t this be a fair test, the only kind of help us falsify?].  The examples turn on trying to explain changes in populations of predator and prey.  One explanation might be overdetermined in the sense that it would require us to account for each individual rabbit being eaten by each individual fox.  [So what would be an under determined one?  One that considered irrelevant relations between predator and prey?] What we need instead is an explanation operating at a suitable level of specificity, which would produce a stable relation.  This might vary according to the scale of which we are operating—population level, or individuals.  The trick will be to study the causal capacities at each level.

For Deleuzians, we also need to trace quasi-causes.  In the example of a population these would refer to the long term duration of the cycle, ‘a mechanism-independent aspect which still demands explanation’ (168).  In the case of biology, this has led to the search for attractors governing stable cycles.  However, it is not just biological mechanisms which can be studied like this.  Convection flows and turbulence also pose the problem of a suitable level to study the process – in this case, descriptions at the molecular level are irrelevant, ‘many collision histories be incompatible with the same macro level effect’ (168).  Macro factors such as temperature gradients are required, together with quasi-causal factors such as bifurcations and attractors.

[There is a useful summary of the argument so far 168-9].  The only thing added is that contrast spaces can have the complex structure of a cascade of bifurcations.  In this way, ‘a problem may gradually specify itself as the different contrast spaces’ it contains reveal themselves, one bifurcation at a time’ (169).  A connection with ontology is starting to appear: the relation between problems and solutions ‘is the epistemological counterpart of the ontological relation between the virtual and the actual.  Explanatory problems would be the counterpart of virtual multiplicities’ (169).  Actual explanations would be individuated.  For Deleuze, this means that actual organs in an organism can be seen as solutions to a problem [and presumably this runs the other way round as well, that puzzling out an explanation means recovering the ontological issues involved?].  In Delanda’s  example, soap bubbles and salt crystals are solutions achieved by the molecules involved to the problem of attaining minimal points of energy: ‘It is as if an ontological problem, whose conditions are defined by a unique singularity, “explicated” itself as it gave rise to a variety of geometric solutions’ (169-70).

So problems posed by humans (epistemology) are intimately related to ‘self posed virtual problems’ (ontology), and this is ‘characteristic of Deleuze’ (170).  The two are ‘isomorphic’.  Experimenters who individuate problems’ in the laboratory are acting isomorphically with intensive processes of individuation in reality.  This is counter to the usual view of realism, where a description is produced intending to mirror reality by developing a relation of similarity.  Naturally, resemblance or similarity cannot be accepted by Deleuze [so he weasels with the idea of isomorphism?].  Philosophers by contrast ‘must become isomorphic with the quasi causal operator, extracting problems from law – expressing propositions and meshing the problems together to endow them that minimum of autonomy which ensures their irreducibility to their solutions’ (170).  These isomorphic processes go on the experimental and the theoretical level.

It is important to realise that the material itself in the laboratory behaves, as much as the mathematical models—for example matter can self organise and self assemble, and this is lost by a focus on linear causality.  There are still ‘nonlinear and problematic’ relations between materials, experimental situations and causal models even after simplifying causals to linear ones (170).  Laboratories produce heterogeneous assemblages which are ‘ isomorphic with real intensive individuation processes’ (171).  Theoretical problems also correspond to Deleuze’s analysis of state space, involving trajectories and singularities: ‘ the singularities defining a problem in physics are isomorphic with those defining the conditions of a virtual multiplicity’ (171).  We only see these links once we focus on problems and not solutions as prior.

The unwarranted emphasis on solutions is found in classical physics and its residual view of matter as passive, a mere receptacle of forms, and where all the activity is produced by the experimenter.  Delanda applies this to social constructivists as well (171).  By contrast, a flat ontology assumes that properties emerge from causal interactions rather than being simply a sum.  Stable emergent properties and new causalities are responsible for larger scale individuals.

The usual approach isolates and separates causes in order to study them, simply by ignoring complications.  ‘As Bunge notes, this procedure may be ontologically objectionable but is in many cases methodologically indispensable’ (172).  [This is scientism, where the pursuit of a true method will deliver true results]

It is not the initial simplification into objectivity of causes the problems, but its subsequent reification into a principle.

Classical notions of causality also include assumptions of ‘uniqueness, necessity, unidirectionality and proportionality’ (172).  Emphasizing these has produced an impoverished notion of materiality, ‘clockwork world views’.

Uniqueness.  In practice, several different causes can produce the same effect, and the same cause can produce different effects—heat can be produced by several causes, and hormones like auxin can produce both growth and inhibition of growth in plants, depending on where it is located.  An additive conception of cause would not be able to detect multiple causes, and it would not be able to account for different effects do not add up.

Necessity.  A better conception is to talk about enhanced probability, as when smoking enhances the possibility of developing cancer.  Again effects are distributed probabilistically, and do not just add up for the whole population.

Unidirectionality and proportionality.  In the classic conception, effects do not react back on causes, but we know that every action involves a reaction.  If this is a large enough reaction, proportionality also fails [since it assumes that ‘small causes always produce small effects’ (173)].  In reality, a variety of options are available, including ones where effects amplify causes (positive feedback).  Given this variety of causes and effects and interactions, simple addition becomes rather unlikely.  Criticism of the notion of externality is also implied [another Aristotelian concepts related to efficient cause, apparently].  External causes are supposed to affect passive targets, producing all the affects, but this breaks down if the target ceases to be passive and can react back as above.

In a flat ontology, linear causes are special cases, and most causal relations are statistical probabilistic.  There are no passive receptacles, since internal structures play a part in determining the effect, for example if they feature alternative attractors.  As a result, the domesticating impact of linear causality ceases to apply, and the idea of problems emerges again: it is no longer enough to specify a single external cause or additive effects.  Objects can self organise and self assemble, producing emergent and unpredictable effects from intensive processes.  [The example from 1000 plateaus is the one about artisans needing to work with wooden material rather than attempting to totally dominate it].

In a similar way, experimenting in physics also has a productivity of its own.  It involves individuating stable laboratory phenomena, often involving novel products.  These phenomena can relate to several theories, and persist even if paradigms change, or remain in a problematic state lacking a full explanation.  Individual entities also have to be produced in this way, ‘connecting operations to a materiality instead of deducing the form of the entities in question from a theoretical law’ (176).  So measuring things like the mass of electrons or their charge is a matter of ‘intervening causally in the world’ (176).  Once individuated, physicists learned from electrons by making them part of heterogeneous assemblages and observing their affects.  It was not until the sort of practice developed, that electrons were seen to be real.

This heterogeneous assemblage can include machines, models, phenomena like electrons, and the experimentalists themselves.  These different components are meshed together in a complex process, and models are refined and skills developed.  ‘The whole’ [scientific knowledge itself] gets stabilised by this assemblage as well.

Again, this is an epistemological counterpart of intensive processes in ontology [experimenting is seen as an intensive process, gradually defining the problem by considering what makes a difference, what is relevant and so on].  This is an emergent process, to be compared with the immediate intuition of an essence.  The process delivers extensive products too, like individual bits of data particular solutions.  Thus, for Deleuze, [Difference and Repetition] ‘”Learning is the appropriate name for the subjective acts carried out when one is confronted with the objectivity of a problem…  whereas knowledge designates only the generality of concepts or the calm possession of a rule enabling solutions”’ (177).

So problems can be subordinated to solutions by simplifying complexity into a homogenous linear system, or studying low intensities or equilibria.  This limits the capacity of the material to form new assemblages, and is acceptable [as long as it is not reified].  Subordination also arises when processes such as experimental processes are neglected in favour of formalised statements.  These are abstracted from practice, and they only become important by inserting them into some theoretical framework.

So far we have been discussing models that interact with the empirical.  Those which interact with the virtual tend to be much simpler, as in the case of formulating general laws.  To analyse these models, requires Deleuze on state space.  State space analyses cannot be used for causal analysis because they are too simple [‘typically valid only for models with a few degrees of freedom’ (178)].  They do not refer specifically to causal processes either, although in some cases, it looks as if the successive states can be linked causally, with the initial state as the first cause, and the final state as the effect.  But this is ‘a mathematical expression positivist reduction of productive or genetic aspects of causes to a process of uniform succession’ (179).  Instead, each state is produced from the same determinants, rather than one causing the next one.

There is information about quasi-causal relations, however.  First of all, you have to see that vectors generate the series of states in a trajectory, by producing singularities.  [bits I do not understand here, apparently, singularities define the independent conditions of the problem, while the vector field produces solution curves—179.  The quote from Difference and Repetition is hardly helpful, and involve some notion of complete determination as opposed to the specific determination of singularities]. 

For conventional analytic philosophy, trajectories are used as predictions—measured values are transformed into a curve which is then projected.  Laboratory systems can be produced with similar initial conditions, and thus similar projections.  A perfect match means that the model is true to the system, but normally, we settle for approximate truths.  However, the whole argument is based on the geometric similarity between curves, but there may be deeper topological invariants producing an isomorphic relation between model and reality—the assumption is that the model identifies the singularities of motion correctly.  The implication is that this happens when the model and the system are ‘coactualizations of the same virtual multiplicity’, at least over a given range (181).  So the geometric resemblance must be explained by these common topological properties instead of being left is sufficient—that would be a proper explanation of similarity.

In analytic philosophy, laws are related to trajectories as generals are to particulars [that is a logical relation again?] Particular trajectories reflect different initial conditions.  For Deleuze, however it is the distribution of singularities that matters—the particular state of trajectories is irrelevant because many different ones can ‘end up in the same place, the attractor’.  The distribution of singularities determines which initial conditions are relevant in the first place.  The generality of the law is really produced by ‘the universality of virtual multiplicities of which both model and real system are divergent actualizations’ (181).

This inability to focus on problems rather than solutions has a long history, and is associated with linguistic formations.  However, there is a more modern and specific mathematical process too, occurring whenever problems are judged by their solvability alone.  There have been breakthroughs too, where solvability was itself seen as a result of a well posed problem [the examples are discussed 182 F.  A gloss follows].

Fail to break equations have particular and general solutions, the first involving the substitution of an algebraic term with a number, the second describing ‘the global pattern for particular solutions’, usually as another equation.  [So   ( x squared) plus (3x) minus 4 =0) can be solved in the usual way to give x= 1, but there is also a general form [I'll have to spell it because formulae will not copy over well to HTML] x = the square root of  (A squared over 2) plus B minus (A over 2), where A and B take the place of 3 and 4 in the equation above.  [ I think I can see this].  For mathematicians, such a general solution indicates a well posed problem.  However, this worked to the fourth power of X, but not for values above.  The solution only emerged by thinking of equations as occupying particular groups, following the development of group theory [more detail, 183]. The point is not to take general solvability as the criterion of a well posed problem, but to invert the process, so that general solvability is itself explained in more universal terms from group theory.  These universal terms involve permutations of different solutions, providing groups of equations, and this helps understand what is not known.  For Deleuze, it is important to include what is not known in the very objectivity of the problem.  This particular approach also produced increasingly specific sub groups of equations, so that ‘problem itself becomes progressively better specified’ (184).

In another example, space and time were rethought.  In classical conceptions, changing spaces [choosing different locations for the laboratory] made no difference to physical processes.  The same goes for time.  In this sense, absolute values for space and time are indifferent to the operation of the law, that is they have become irrelevant.  This led to the theory of relativity (183).

This is led to a more general approach to mathematics, moving away from trial and error processes to achieve solvability, and revolutionary step to link problems and solutions in a new way.  The same idea can be seen in modern solutions to the problem of specifying groups of differential equations—a computer generates a population of solutions so that the general pattern can be discovered.  Classical physics had to simplify because it could not generate this kind of solution, and so it tended to neglect models that produced an exact solutions.  This produces an additional bias ‘toward a clockwork picture of reality’, and the solvable equations happen to be the linear ones (185).  It so happens that linear equations can be solved economically, because of ‘the superposition principle, which states that given two different solutions of a linear equation, their sum is also a valid solution’ (185).  This was a considerable bonus at the time.  Linear equations can apply to complex systems, as long as the relevant variables operate at lower levels of intensity.  For all these reasons, linear approaches tended to dominate.

A similar approach was followed by Poincare with differential equations.  The problem was to model the interactions of three solar system bodies, not just by looking at the empirical singularities, but trying to generate  a whole picture of their existence and distribution, or to generate a space for all solutions.  He thus defined the problem itself in spatial terms. 

These models clearly conforms to Deleuze’s idea of the virtual as well.  Hiding problems behind immediate solutions also hides the virtual and ‘promotes the illusion that the actual world is all that must be explained’ (187) [another example of positivism really?  Also a problem for activists?].  The hope was that a super law of everything would emerge from this study.  However, with multiple attractors and non linear models, there will always be emergence, surprise, novelty, no end to problems.  The lack of surprise in classical physics depends on effects already being present in causes, with only quantitative novelties, a world without productive history.

The alternative ontology is fully historical.  Each individual is produced by ‘a definite historical process of individuation’, and, through interaction, each individual can itself drive historical and causal processes.  Even the quasi causal is to list oracle, but with a different form of temporality.  So, ‘in the Deleuze Ian ontology there exist two histories, one actual one virtual, having complex interactions with one another’ (188).  There is an historical process of actual events, and one of ideal events ‘defining an objective realm of virtual problems of which each actualised individual is but a specific solution’ (188)

[Note that there is also a very useful appendix explaining the connection between terms used in this reconstruction and Deleuze’s own terms, written in the usual clear ways.  This clears up some mysteries at last—for me, for example, supporting my hunch that the body without organs was some sort of virtual continuum, and that the desiring machines are really individuating processes.  There is also a really clear discussion of the wretched section on syntheses in AntiOedipus.  Delanda!  What a hero!]

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