Gilles Deleuze famously replaces the traditional modal distinction between actuality and possibility with a distinction between actuality and virtuality. This is introduced to block the claim that reality can be fully or adequately represented. If the actual is just the instantiation of the possible, then an actual thing resembles the thing *as conceived* or *as represented* in every respect other than with regard to existence or actuality (Deleuze 1994, p. 211). Whereas the actual thing qualitatively ‘resembles’ the possible thing (like the subject of a hyper-refined bit-map) the virtual is the part of the thing that corresponds to its tendencies. Tendencies are not (so the story goes) the actualization of possible states of the thing but the processes by which the thing self-differentiates. As Daniel Smith points out, this not only short-circuits representation at a fundamental level, but allows for the production of deep metaphysical newness:

*Deleuze will substitute for the possible-real opposition what he calls virtual-actual complementarity: the virtual is constituted through and through by difference (and not identity); and when it is actualised, it therefore differs from itself, such that every process of actualisation is, by its very nature, the production of the new, that is, the production of a new differenc*e (Smith 2007, p. 6).

It’s not clear to me whether this entails global anti-representationalism or whether this position is compatible with the view that that the actual (but not the virtual) *is *representable in some fashion. The latter position seems implicit in those passages where Deleuze equates the actual with the phenomenal (experienceable) and the virtual with those noumenal motors (‘the noumenon closest to the phenomenon’) that bring extensive differences and objects into our phenomenal purview (Ibid., p. 222). Moreover, global anti-representationalism seems to run up against the obvious objection that that an entirely unrepresentable world would be intractable and unknowable, whereas our world seems tractable and knowable in part.

Be this as it may, this does seem to entail that we cannot represent a Deleuzean becoming (a virtuality) as the realization of some possible state of a thing even if (assuming that global anti-representationalism is rejected) we can represent successive actualizations in this way. If this is right, then it raises some interesting questions about the way concepts drawn from the mathematics of dynamical systems have been employed by contemporary Deleuzeans: Manuel de Delanda being the most prominent figure here.

Dynamical systems theory (DST) is all about trajectories in mathematical spaces the points of which describe the possible states of a system (a state space). To quote Robert Devaney it asks ‘where do points go and what do they do when they get there’ (Devaney 1986, 17). Where a differential function f’ describing a dynamical system can be solved it is possible to show how its integral f generates a trajectory with respect to its variables. Where these cannot be solved (which, mathematicians inform us, is true in the majority of cases) it may still be possible to give a qualitative account of the tendencies of the system. Thus the differential equations of a system that describe how its rates of change alter can tell us about attracting sets (attractors) towards which its orbits (trajectories through state space) will approach asymptotically (that is, orbits tend to approach these sets by successive iterations without ever arriving in them).

Now, Delanda has used this geometrical conception of an attractor or ‘singularity’ to explicate Deleuze’s conception of the virtual and to explain why the virtual/actual distinction is metaphysically preferable to the possible/actual distinction. We can, for sure, represent the possible states of a system by a state space. For example, the state space of a 3 layer neural net with 8 inputs inputs + a 4 neuron hidden layer + a 2 unit output layer can be represented in a space of 8 + 4 + 2 = 14 dimensions. Any possible behaviour of this network can be thought of as a point in this 14 dimensional space. If the net can be ‘trained up’ in some discrimination task – like distinguishing round from jagged shapes – the singularities will be points within those partitions of the 4 neuron subspace towards which patterns evoked by ‘jaggedish’ or ’roundish’ stimuli on the input layer will tend to converge.

So far, we have not had recourse to the virtual/actual distinction to describe the behaviour of this system. To be sure, we’ve talked loosely in terms of tendencies: e.g. as stimuli at the input become increasingly jagged the state of the hidden layer in the trained network should tend to approach the prototype ‘jaggedness’ state. But this is really just another way of describing the system’s dispositions – specifying how it would perform given certain kinds of input. So why is DST supposed to help in understanding the virtual? Delanda thinks that the asymptotic nature of singularities is key here. While singularities can be said to specify the behaviour of a system, they do so in terms of states that the system could never ‘actually’ assume:

*A clue to the modal status of these invariants is the fact **that, as is well known, trajectories in phase space always **approach an attractor asymptotically , that is, they approach it **indefinitely close but never reach it . Although the sphere of **influence of an attractor, it’s basin of attraction, is a subset of **points of phase space, and therefore a set of possible states, the **attractor itself is not a possible state since it can never become **actual *(Delanda 2010, 149)

Thus a singularity represents the tendencies of a system but not one of its possible states:

* In other words, unlike trajectories representing possible histories that may or may not be actualized, attractors can never be actualized since no point of a trajectory can ever reach them. Despite their lack of actuality attractors are nevertheless real since they have definite effects. In particular, they confer on trajectories a strong form of stability, called “asymptotic stabili*ty” ..*. It is in this sense that singularities represent only the long term tendencies of a system but never a possible state. Thus, it seems, that we need a new form of physical modality, distinct from possibility and necessity, to account for this double status of singularities: real in their effects but incapable of ever being actual. This is what the notion of virtuality is supposed to achieve* (Ibid., 150).

If this account works, then it appears we can unpack Deleuze’s conception of the virtual without the highly speculative metaphysics used in Smith’s gloss above. But can we do this satisfactorily? My worry here is while an attractor may not lie on an orbit within the dynamical system itself, it does belong to its state space. Moreover, its status *qua singularity* depends on those features of the system which determine the possible trajectories of the orbits. For example, if a singularity is a single point attractor s and the orbits are defined by a mapping of a point F(p), then successive iterations of F (p), F(F(p)), etc. will approach s as the number of repetitions approaches infinity. So this is a property which can be defined in terms of the actual properties of a set: namely the region or ‘basin of attraction’ within which every iteration is a subset of the set generated by previous iterations. The properties which define the singularity thus seem to be structural. They may be very exotic (as we are told is the case with ‘strange’ or chaotic attractors) but their specification does not seem to require any new logical concepts – certainly, no new modal concepts. Maybe I’m missing something vital – I can’t claim a confident grasp of the mathematics of dynamical systems – so I’ll leave it to those better qualified than myself to correct any misunderstandings here.

References:

Deleuze, G. (1994), *Difference and Repetition*, Paul Patton (trans.). London: Athlone Press.

Devaney, Robert L.(1986). *An Introduction to Chaotic Dynamical Systems*, Menlo Park, Ca.:Benjamin Cummings.

Delanda, Manuel (2010). *Deleuze: History and Scienc*e, Atropos Press.

Smith, Daniel (2007), ‘The Condition of the New’, Deleuze Studies, Vol 1, pp. 1-21.

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