7 Assemblages as Solutions to Problems

Singularity is beyond particular propositions no less than universality is beyond general propositions. Problematic Ideas are not simple essences, but multiplicities or complexes of relations and corresponding singularities. From the point of view of thought, the problematic distinction between the ordinary and the singular, and the nonsenses which result from a bad distribution among the conditions of the problem, are undoubtedly more important than the hypothetical or categorical duality of truth and falsehood along with the ‘errors’ which only arise from their confusion in cases of solution . . . In this manner the distribution of singularities belongs entirely to the conditions of the problem, while their specification already refers to solutions constructed under these conditions.

Deleuze, Difference and Repetition1

When the term ‘singularity’ was introduced in the eighteenth century, its referent was considered of the greatest importance. The term was coined to refer to special (non-ordinary) solutions to physical problems, when the latter were framed using differential equations. Posing these problems involved selecting among the properties characterising a physical phenomenon those that were the most relevant, and discovering dependencies in the ways in which the properties changed. These problems could be expressed as a question and its presuppositions: given that the properties of this laboratory phenomenon exhibit these dependencies, why does the phenomenon tend to be in this particular state instead of another possible state? In a formal model the properties are represented by variables, dependencies between variables matching dependencies between properties (at least, for ideal phenomena). But even if we manage to correctly capture these dependencies, we still have to determine which particular combination of values for the properties, as represented in the model by solutions to the equations, defines the state that the phenomenon tends to be in. Already in the previous century mathematicians and scientists suspected that the states in which a phenomenon tends to be, the states a phenomenon ‘prefers’, can be identified with those that have either a minimum or a maximum value for a particular variable.

Thus, in 1662 Pierre de Fermat proposed that light propagates between two points so as to minimise travel time. His basic insight can be explained this way: if we knew the start and end points of a light ray, and if we could form the set of all possible paths joining these two points (straight paths, crooked paths, wavy paths), we could find out which of these possibilities is the one that light ‘prefers’ by selecting the one that takes the least amount of time. Given this insight, and some laboratory evidence that these ‘preferences’ were real, what was needed was the creation of a mechanical procedure (an algorithm) that took as an input a model based on a differential equation and yielded as an output the solutions that were maximal or minimal. This need was met by the calculus of variations created by the mathematician Leonard Euler in 1733. Before Euler the main problem was to find a way to specify the set of possible paths so that it was maximally inclusive, that is, so that it contained all possibilities. This was achieved by parametrising the paths, that is, by generating them through the variation of a single parameter.2 But there are many physical problems in which the possibilities cannot be parametrised by a discrete set of variables. Euler’s method solved this problem by tapping into the resources of the differential calculus. Without going into technical details, these resources allowed him to rigorously specify the space of possibilities and to locate the minimum, maximum, and inflection points of the functions that join the start and end points.3 These points became the original singularities. The ontological impact of variational thinking is perfectly captured in this comment by Euler himself:

Since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves . . . Therefore, two methods for studying effects in nature are open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes . . . One ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus is not only one solution greatly strengthened by the other, but, more than that, from the agreement of the two solutions we secure the highest satisfaction.4

Some contemporaries of Euler went as far as claiming that singularities were proof of the existence of God, since a supremely rational deity would surely create a world in which the preferred states were optimal, whether such an optimum state was achieved by minimising or maximising. If we replaced a transcendent deity with an immanent cosmic plane, this assessment of the ontological importance of singularities would come close to the ideas we discussed in Chapter 5. So why is it that philosophers took so long to incorporate this concept and correctly assess its metaphysical import? There are several factors that contributed to this neglect. By the first half of the nineteenth century all the different ideal phenomena of classical physics had been given a variational form, unifying its different branches under a single master equation, the Hamiltonian, and its preferred solution: the one that minimised the difference between potential and kinetic energy. But the older form of this body of knowledge, the one in which problems were framed in terms of forces, was still available, and a new form (framed in terms of fields) would be introduced before the end of the century. So there were competing alternatives. For empiricist physicists, that is, for those who believed that only directly observable entities exist independently of our minds, a decision over the alternatives had to be made on the basis of the observable predictions of each, since neither forces, nor fields, and certainly not singularities were directly observable. The difficulty was that the three versions made the same predictions. Hence, physicists concluded that it was useless to speculate whether there really existed forces that acted as effective causes, or whether reality really contained gravitational fields, or finally, whether gravitational gradients and their singularities, the latter acting as final causes, were the correct explanation.5

This attitude was the first obstacle to the incorporation of singularities in a realist ontology. The second was the belief, shared by many philosophers in the first half of the twentieth century, that mathematics had been reduced to logic at the end of the previous century. In this erroneous version of history, the differential calculus had been reduced to arithmetic when the concept of infinitesimals was replaced by the notion of limit (and the latter shown to depend only on the concept of number), while arithmetic itself had been reduced to set theory, and hence, to logic. After that, philosophers felt justified in their neglect of the actual mathematics used by scientists and attempted to reconstruct scientific practice in purely logical terms. Thus, a model of the way in which theoretical knowledge interacts with evidence from the laboratory was created which was based exclusively on linguistic entities and their logical relations. The former were divided into two classes, theoretical and observational statements, and their relations were specified like this: the set of theoretical statements (which must include some exceptionless laws) is used to derive predictions purely deductively; these predictions are then compared with statements recording observations made in the laboratory; if there is a match the theoretical statements are confirmed, if there is not, they are disconfirmed. This became the standard model of scientific theories.6 But soon counter-examples were found showing that the set of observational statements underdetermines the choice of the theoretical statement considered to be confirmed, and many empiricist philosophers adopted the view that the choice was therefore made on the basis of arbitrary conventions. (This is referred to as the Quine/Duhem thesis.)7 As it turned out, whether one believed in logically necessary or merely conventional relations among statements, it was the very idea that mathematical models can be reduced to statements and their logical relations that eliminated the possibility of taking a serious look at the ontological status of singularities.

More recently, however, a new school of thought has rejected the standard model, urging philosophers to return to the mathematics used in actual scientific practice. Among the leaders of this movement is the philosopher Bas Van Fraassen, who defines the new approach to scientific theories as one that

makes language largely irrelevant to the subject. Of course, to present a theory, we must present it in and by language. That is a trivial point . . . In addition, both because of our own history – the history of philosophy of science which became intensely language-oriented during the first half of [the twentieth] century – and because of its intrinsic importance, we cannot ignore the language of science. But in a discussion of the structure of theories it can largely be ignored.8

Van Fraassen pioneered the study of state space in modern philosophy of science, reintroducing it as a serious subject after a century of neglect since its creation in the 1880s by Henri Poincaré. As we saw in Chapter 5, Poincaré’s innovation consisted in the creation of a geometric representation of the space of possible solutions to differential equations. Each of the dimensions of this space represented one way in which a phenomenon was free to change (its defining properties or ‘degrees of freedom’), while the dependencies in the way that the properties changed were represented by relations between those dimensions. The state in which a phenomenon found itself at any given moment became a point in the space, while the history of the phenomenon as its properties changed became a trajectory (a series of points) generated in accordance with the dependencies captured by the equation. Using the resources associated with his novel approach, Poincaré greatly extended the achievements of Euler, discovering and classifying new kinds of singularities: nodes, saddle points, foci, centres, all of them zero-dimensional, as well as a new kind of one-dimensional singularity, limit cycles.9

In this approach, the set of trajectories is what replaces the set of theoretical statements in the standard model. A similar replacement can be made of the set of observation statements, while the evidential relation between the two sets can be established without using either deductive logic or conventions. Let’s imagine that we have a laboratory where a phenomenon’s degrees of freedom can be restricted (by screening out other factors) and where we can place it in a given initial state and then let it run spontaneously through a sequence of states. Let’s also imagine that we can measure with precision the values of the degrees of freedom (say, temperature, pressure, and volume) at each of those states. We run the experiment several times, starting with different initial conditions, and generate data about the phenomenon’s possible histories. The data will consist, basically, of sequences of numbers giving the values of temperature, pressure, and volume that the phenomenon takes as it evolves. We then plot these numerical series on a piece of paper, turning them into graphic curves. These curves are the replacements for the observation statements. Finally, we run our mathematical model, giving it the same values for initial conditions as our laboratory runs, and generate a set of state space curves. Finally, we compare the two sets. If the mathematical and experimental trajectories are geometrically similar (in suitable respects) this will count as evidence that the model actually works.10 Replacing the standard model with an approach that is more faithful to actual scientific practice is clearly a great improvement, but from a metaphysical point of view this replacement is only the beginning. The next step is to explain why the models work, an explanation that will vary depending on our ontological commitments. In particular, whether we can give singularities an explanatory role will depend on removing the obstacle presented by empiricist commitments.

The first candidates for ontological evaluation are the trajectories themselves, and since these represent possible histories, their metaphysical status will depend on how possibilities are treated in a philosophy. Empiricist philosophers are mostly sceptical about possible entities. Quine, in particular, likes ridiculing them: ‘Take, for instance, the possible fat man in the doorway; and again, the possible bald man in the doorway. Are they the same possible man, or two possible men? How do we decide? How many possible men there are in that doorway? . . . How many of them are alike? Or would their being alike make them one?’11 What Quine is arguing here is that we do not possess the means to individuate possible entities, that is, to identify them in the midst of all their variations. The target of his sarcasm is modal logic, the branch of logic concerned with the analysis of counterfactual sentences such as ‘If JFK had not been assassinated the Vietnam War would have ended much sooner’. We all understand the meaning of contrary-to-fact statements like these, but they are notoriously difficult to handle rigorously. In particular, it is not clear how to exclude variations that do not make a difference, that is, possibilities that are insignificant: while the possible world in which JFK survived is significantly different from the one in which he died, we cannot say the same thing about the possible world in which he has a different haircut, or wears different clothes. But as realist philosophers like Ronald Giere have argued, while Quine’s sceptical remarks are valid for linguistically specified possible worlds, they are not so for state space trajectories:

As Quine delights in pointing out, it is often difficult to individuate possibilities . . . [But] models in which the system laws are expressed as differential equations provide an unambiguous criterion to individuate the possible histories of the model. They are the trajectories in state space corresponding to all possible initial conditions. Threatened ambiguities in the set of possible initial conditions can be eliminated by explicitly restricting the set in the definition of the theoretical model.12

This reply, however, is not enough to satisfy an empiricist. Van Fraassen, for example, can still deny the need to be ontologically committed to possible histories given that, for him, the goal of science is not to explain unobservable traits of reality but merely to achieve empirical adequacy, that is, to increase our ability to make predictions and to increase the level of control of outcomes in the laboratory. For this limited purpose all that matters is that we generate a single trajectory and then match it to a series of measurements of the actual states of a phenomenon. The rest of the set of possible trajectories is merely a useful fiction. This ontological stance towards modalities is referred to as ‘actualism’.13 The realist reply is to argue that understanding a phenomenon involves not just knowing how it actually behaves in this or that specific situation, but also knowing how it would behave in conditions that may not in fact occur. But realists can disagree about what components of state space must be considered to represent mind-independent entities. One alternative is to believe that all possible trajectories consistent with the dependencies between variables should be considered real. The other option is to reject a commitment to possible states and possible histories, and to include in a realist ontology only the structure of the space of possibilities, a structure specified by singularities. As we saw above, Euler thought of the influence of minima and maxima as equivalent to that of final causes. More recently, the reference to Aristotle is downplayed, and instead the singularities structuring state space are said to represent the long-term tendencies of the phenomenon being modelled, that is, its dispositions. In previous chapters we used the term ‘virtual’ to define the ontological status of dispositions that happen not to be currently manifested. The modality of being of something that is virtual should be carefully distinguished from that of a possibility. As Deleuze argues:

The only danger in all this is that the virtual could be confused with the possible. The possible is opposed to the real; the process undergone by the possible is therefore a ‘realization’. By contrast, the virtual is not opposed to the real; it possesses a full reality by itself. The process it undergoes is that of actualization . . . [To] the extent that the possible is open to ‘realization’, it is understood as an image of the real, while the real is supposed to resemble the possible. That is why it is difficult to understand what existence adds to the concept when all it does is double like with like. Such is the defect of the possible: a defect which serves to condemn it as produced after the fact, as retroactively fabricated in the image of what resembles it. The actualization of the virtual, on the contrary, always takes place by difference, divergence or differenciation. Actualization breaks with resemblance as a process no less than it does with identity as a principle. Actual terms never resemble the singularities they incarnate. In this sense, actualization or differenciation is always a genuine creation.14

To make a rigorous formal analysis of the virtual structure of state space we need to take into account a component that most analytical philosophers routinely ignore: the velocity vector field. To understand what a vector field is we need to understand the kind of spaces used in the geometric approach. In particular, these are not metric spaces but rather differential manifolds. The difference between the two can be boiled down to the way in which the points composing a space are specified. In a metric space the points can be specified by their relation to a set of global coordinates, so the space is basically a set of X, Y, and Z coordinates. A differential manifold, on the other hand, is composed of points that can be defined using only local information: the instantaneous rate of change of curvature at each point. This makes the manifold a field of rapidities and slownesses, the rapidity or slowness with which curvature changes at each point. Moreover, each point is not just a speed but also a velocity, since a direction may be assigned to it. If we use a vector to represent a velocity, the space becomes a field of velocity vectors. This matters because the distribution of singularities is not given by the trajectories (or integral curves) but by the vector field itself. More precisely, while the nature of a singularity is established by using the shape of nearby trajectories – whether a point singularity is a focus or a node, for instance, is determined by observing whether the integral curves in its vicinity approach it as a spiral or a straight line – the existence and distribution of the singularities does not need any trajectories to be determined. As Deleuze writes:

Already Leibniz had shown that the calculus . . . expressed problems which could not hitherto be solved or, indeed, even posed . . . One thinks in particular of the role of the regular and the singular points which enter into the complete determination of the species of a curve. No doubt the specification of the singular points (for example, dips, nodes, focal points, centers) is undertaken by means of the form of integral curves, which refers back to the solutions of the differential equations. There is nevertheless a complete determination with respect to the existence and distribution of these points which depends upon a completely different instance, namely, the field of vectors defined by the equation itself . . . Moreover, if the specification of the points already shows the . . . immanence of the problem in the solution, its involvement in the solution which covers it, along with the existence and distribution of points, testifies to the transcendence of the problem and its directive role in relation to the organization of the solutions themselves.15

For Deleuze, the mathematical distinction between a vector field and its virtual singularities, on the one hand, and the possible trajectories, on the other, becomes the philosophical distinction between the defining conditions of a problem and its possible solutions. The conditions of a problem are given by a distribution of the significant and the insignificant. Let’s first illustrate this with a linguistically specified problem, that is, the kind of problem expressed by a ‘Why’ question like this:

Given that the properties of this laboratory phenomenon exhibit these dependencies, then why does the phenomenon tend to be in this particular state instead of another possible state?

In these kinds of cognitive questions we must distinguish the presuppositions of the problem, the aspects of a phenomenon that we are not trying to explain, as given by the part following the ‘given that’ clause. The question itself determines what needs an explanation, but to be fully specified it also needs a contrast space, the part following the ‘instead of’ clause.16 For such a question to be well posed, the presuppositions must state everything that is significant to an understanding of the problem, while the contrast space must specify the relevant alternatives. And similarly for problems posed using differential equations: we must first discover the significant ways in which a phenomenon is free to change and eliminate all the trivial ones; then we must select only those dependencies between variables that make a difference to the outcome, and reject all others; finally, once the equations are given a geometric form, we must distinguish those points that are noteworthy or remarkable, that is, the singularities, from the rest of the points that are merely ordinary. From these considerations, Deleuze derives several consequences. First, the conditions of the problem precede the finding of solutions, and the solutions are only as good as the problem they are supposed to solve: if trivial degrees of freedom are included, or insignificant dependencies selected, the solutions will also be trivial. Second, the conditions of the problem not only precede (and define the adequacy of) a solution, they also survive it: a problem has an autonomous existence, as a virtual entity, and continues to exist even after actual solutions are found. And third, problems are not only independent of their solutions, but have a genetic relationship with them: a problem engenders its own solutions as its conditions become progressively better specified.

The last point can be illustrated with an example from the history of algebraic equations. There are two kinds of solutions to equations, numerical and analytical. Numerical solutions are given by numbers that, when used to replace an equation’s unknowns, make the equation true. For example, an equation like x2 + 2x – 8 = 0 has as its numerical solution x = 2. Analytical or exact solutions, on the other hand, do not yield any specific value or set of values but rather the structure of the space of possible solutions, a structure expressed by another formula. If we remove the numerical constants from the above equation, we get x2 + ax – b = 0, an equation with the analytical solution:

x = √((a /2)2+b) – (a/2)

By the sixteenth century mathematicians knew the exact solutions to algebraic equations in which the unknown variable was raised up to the fourth power, that is, those including x2, x3, and x4. But then a crisis ensued. Equations raised to the fifth power refused to yield to the previously successful method. But two centuries later it was noticed that there was a pattern to the solutions of the first four cases which might hold the key to understanding the recalcitrance of the fifth. The mathematicians Neils Abel and Evariste Galois found a way to approach the study of this pattern using resources that today we recognise as belonging to group theory.17 The notion of a group of transformations was introduced in Chapter 3, but we need a more detailed description now. The term ‘group’ refers to a set of mathematical entities and a rule of combination for those entities. The set must meet certain conditions, one of which is to possess the property of closure, which means that when we use the rule to combine any two entities in the group, the result must be also an entity in the group. Of all the entities that may form groups the most important for our purposes is transformations, in which case the rule is a consecutive application of the transformations. For example, the set consisting of rotations by 90 degrees (that is a set containing rotations by 90, 180, 270, and 360 degrees) forms a group, since any two consecutive rotations produce a rotation also in the group. A group, in turn, can be used to determine the properties of an entity that remain invariant under the transformations, properties that can therefore be thought to have special significance. The geometrical properties of a cube, for example, remain unaltered under the above group of rotations, so the group captures a significant aspect of its identity: its capacity to be unaffected by certain transformations, or its indifference to them.

Galois used certain transformations (permutations of an equation’s solutions) that, as a group, revealed the invariances in the relations between solutions. More specifically, he saw that when a permutation of one solution by another left the equation valid, the two solutions were indistinguishable from one another. In other words, the permutation made no difference to the validity of the equation. Galois also showed that as the original group gave rise to subgroups that progressively limited the substitutions that left relations invariant, the successive transformations narrowed the set of possible solutions until all of them had been found. Deleuze argues that what Abel and Galois showed was that the original group revealed not what we know about the solutions, but the objectivity of what we do not know about them, that is, the objectivity of the problem itself, and that as successive subgroups were applied this objectivity became further specified.18 Hence, the solutions are generated by the progressive differentiation of the problem. As he writes:

We cannot suppose that, from a technical point of view, the differential calculus is the only mathematical expression of problems as such . . . More recently other procedures have fulfilled this role better. Recall the circle in which the theory of problems was caught: a problem is solvable only to the extent that is is ‘true’ but we always tend to define the truth of a problem by its solvability . . . Abel was perhaps the first to break this circle: he elaborated a whole method according to which solvability must follow from the form of a problem. Instead of seeking to find out by trial and error whether a given equation is solvable in general we must determine the conditions of the problem which progressively specify the fields of solvability in such a way that the statement contains the seed of the solution.19

To link this genetic concept of problems to our previous discussion we need to combine the resources of group theory with those of dynamical systems theory, as the study of state space is known today. The relevant transformations in state space are perturbations: adding a small additional vector field to the one defining the structure of the space. Some perturbations leave the topological properties of state space – its distribution of singularities, its dimensionality, its connectivity – invariant, but others do not. In particular, a perturbation can cause the occurrence of a bifurcation that changes the number of singularities in the space, yielding a topologically inequivalent one. In the calculus of variations, singularities are always points (corresponding to steady-states), but in state space a singularity can also be a line, shaped into a closed loop. These are referred to as ‘limit cycles’ or ‘periodic attractors’, and correspond to a phenomenon’s tendency to oscillate or pulsate in a stable way. More recently a third variety of singularity has been found, the result of repeatedly stretching and folding a closed loop. These are ‘chaotic’ or ‘strange’ attractors. The existence of different types implies that the symmetry of state space can also be broken by changing the form of the singularities. For example, the Hopf bifurcation can change a steady-state singularity into a periodic one, while the Feigenbaum bifurcation can turn a periodic singularity into a chaotic one. And just as a series of groups and subgroups progressively specifies the conditions of a problem expressed with algebraic equations, so can a sequence of bifurcations progressively unfold the solutions to a problem posed by a differential equation: steady-state solutions, periodic solutions, chaotic solutions.

How can we connect these ideas to the theory of assemblages that we have developed in previous chapters? So far we have considered problems as they are posed by the human mind. But the objectivity of problems, their autonomy from their solutions, implies that what is problematic is not just what strikes our minds as being in need of explanation. In addition we need to consider the rich variety of physical, chemical, and biological problems confronted by assemblages when they are first born and as they operate throughout their lives. The simplest possibility space, one structured by a single minimum or maximum, defines an objective optimisation problem, a problem that a variety of actual entities (bubbles, crystals, light rays) must solve in different ways. But the optimisation problem survives its solutions, ready to be confronted again when a flat piece of soap film must wrap itself into a sphere; or a set of atoms must conform itself to the polyhedral shape of a crystal; or a light ray must discover the quickest path between two points. Similarly, a symmetry-breaking cascade like the one Galois used to solve the quintic equation structures a possibility space that also defines problems for fluids in motion. Thus, a moving fluid is presented with an objective problem as its speed increases, and it solves it by adopting a different manner of moving adequate to each range of speeds. At very slow speeds the solution to the problem is simple: stick to steady or uniform flow. But after accelerating to a critical threshold that solution becomes insufficient and the moving fluid must switch to a convective or wavy flow just to keep up. Finally, after crossing yet another critical threshold, the faster speeds present the flow with a problem that it cannot solve with a rhythmic movement and it is forced to become turbulent, distributing energy into a structure of vortices within vortices. As these different manners of moving unfold one after another, they display all the physical solutions to the problem of flow. Or better, the problem defined by a symmetry-breaking cascade is not a problem of flow but, depending on the material substratum in which it is to be solved, it is a problem of progressive differentiation.

We have encountered this virtual problem throughout this book in a variety of contexts. In Chapter 2 we argued that the earliest forms of language consisted of monolithic symbolic artifacts that lacked the combinatorial productivity of modern words. The latter can be combined into an infinite number of sentences because each word has different probabilities of co-occurrence with other words, whereas the monolithic artifacts were at first all equiprobable. This original condition can be characterised by its invariants relative to the transformation permutation: if two of the ancient symbolic artifacts were substituted for each other their probabilities of co-occurrence would be left invariant. A process of successive departures from equiprobability, a process that broke that original symmetry, would later lead to the differentiation of words into nouns and verbs, adjectives and adverbs, articles and prepositions. In Chapter 6 we discussed a very different case, the progressive differentiation of chemical species, a process that involved successive departures from rotational symmetry on the part of electron orbitals. And in Chapter 5 we discussed the differentiation of geometric figures that is caused by successively eliminating transformations from the group to leave the features of the least differentiated geometry, topology, invariant, the group containing displacements, rotations, inversions, scalings, projections, bendings, foldings, and stretchings. By eliminating stretching and folding (a bending that creates a crease), we obtain the group that leaves the features of differential geometry invariant; by getting rid of bending we get the group for projective geometry; and so on until we reach the group containing only those transformations that leave rigid features invariant, the group that characterises metric geometries.

We characterised this relationship between a virtual problem and its multiple solutions as one of divergent actualisation. We can refine that characterisation by using Euler’s suggestions regarding the original singularities: a given phenomenon in classical physics can be explained by listing its efficient causes or by displaying its final causes. The latter, such as the preference for a minimum or maximum as a final state, are shared by many different phenomena, but the former vary from one phenomenon to the next. We can also express this point without Aristotelian terms: singularities define a tendency in mechanism-independent terms, but we also need to specify the causal mechanisms that implement those tendencies in actual cases, and these will vary from case to case. This divergent relation between problems and their solutions also has epistemological consequences: if the same singularity can be actualised as a tendency in two very different phenomena, then it may also become actual as a tendency in the behaviour of solutions to an equation. Thus, the cognitive relation between a mathematical model and the phenomena it models would be one of co-actualisation. And more generally, for a cognitive problem to be well posed there must be an isomorphism between the distribution of the trivial and the important, of that which leaves us indifferent and that which merits our notice, in the humanly posed problem and the problem of which the phenomenon itself is a solution.

To conclude this chapter we need to address an important limitation of this approach. Euler’s and Poincaré’s singularities can help us explain the ontological status of tendencies when they are not being actually manifested. But we also need to explain the status of capacities when they are not being exercised. As we saw, capacities differ from tendencies in that we must always consider the coupling of a capacity to affect to a capacity to be affected. Whereas tendencies are like habits, repetitive and limited in their variation, capacities are more like skills, flexible and adaptive. And whereas the manifestation of tendencies can be checked by the mere occurrence of the final state that a phenomenon prefers, capacities involve staging interactions in which an ability to affect is tested against various abilities to be affected. The possibility spaces associated with all the interactions that a given phenomenon is capable of are not nearly as well studied as those for tendencies, and we do not have a well-defined procedure to sort out the points composing those spaces into those that are significant and those that are insignificant. Nevertheless, candidates for these possibility spaces do exist and they must be philosophically investigated. Let’s then finish the chapter by examining one example that may indicate the direction of future research in this area. In the twentieth century a new field of mathematics opened up, discrete mathematics, which is concerned with problems that cannot be posed in terms of continuous quantities, like those framed using the differential calculus.20 Nevertheless, correspondences can be found between the new and the old fields: although possibility spaces can be studied using operators (differentiation) that generate infinitely small differences, they can also be explored using difference equations that use only finite differences between adjacent terms.

For our purposes here we can limit our examination to one particular sub-field of discrete mathematics: cellular automata. Two of the great mathematicians of the twentieth century, Stanislav Ulam and John Von Neumann, pioneered this field in the 1940s.21 Unlike the continuous manifolds used to construct state spaces, the space of a two-dimensional cellular automaton is segmented into discrete shapes: squares, triangles, hexagons, or any figure that completely tiles the plane. Another difference is that while a point in the former represents a state, each of the cells of the latter contain an automaton that can exist in a given state (not just represent it), a state that is the result of its interactions with neighbouring automata. In both cases an exploration of the actual component of the space (the actual trajectories or the actual interactions) involves the use of recursion, the repeated use of the output of an operation as its next input. But while in the former, recursive solutions to the equation are used to generate a continuous series of states, in the latter recursive interactions – or better, recursive application of the rules defining the interactions – generate emergent patterns of states. This has the consequence that in the former what needs to be explained is why the continuous trajectories take the form that they take – why they tend to converge on a particular point in state space, for example – while in the latter what is problematic is differences in the complexity of the emergent patterns. Finally, the two cases differ in the virtual component of the space: the former demands a study of the vector field and the distribution of attractors it determines, while the latter forces us to explore the space of possible rules defining the interactions.

An extremely simple cellular automaton can be used to make these ideas more tangible. Its space is tiled by square-shaped cells, each capable of being in only two states: on or off. Each cell can interact only with those with which it shares an edge or a vertex, so each cell has eight neighbours. And finally, the rules of interaction are very simple: if only one of its neighbours is on, a cell will change from on to off (or stay off); if two or three neighbours are on, the cell will change from off to on (or stay on); finally, if more than three neighbours are on, the cell will be turned off. When we initiate a simulation with a random distribution of on and off states, and then let the recursive application of rules take its course, simple patterns covering several cells spontaneously appear: steady-state patterns that remain stable under many perturbations; periodic patterns that oscillate between states; and periodic patterns that move diagonally across the space. The latter are intriguing because as the pattern moves, its composing states belong to very different cells, showing that the pattern is independent of any individual automaton. These mobile patterns are referred to as ‘gliders’.22 In addition to these spontaneous patterns, a large variety of engineered patterns can be created in this space, some of which have even more remarkable properties. A complex pattern can be designed in which two mobile patterns (shuttles) clash and generate a glider as a by-product; then they reverse direction and clash again, generating another glider. A particularly stable steady-state pattern (a block) must be placed on both sides of the mobile patterns to ‘eat’ the debris produced by their collisions. This designer pattern is referred to as a ‘glider gun’.23

The sheer variety of both spontaneous and engineered patterns is already a phenomenon that demands explanation. But there is more. As just mentioned, each cell contains an automaton the state of which is capable of affecting the states of its neighbours and of being affected by them. These finite state automata are the simplest of all automata. They can carry out many computations as long as the latter do not involve storing intermediate results, like carrying a number when performing a multiplication. In other words, finite state automata operate without any memory. More complex automata can be created by relaxing this constraint: if we allow the automaton to store a single item in memory it becomes a push-down automaton; if we give it a large memory but limited access to it, it becomes a linear-bounded automaton; and finally, if we give it an infinite memory and unlimited access to it, it becomes a Turing machine.24 Modern computers, with their ever-increasing amounts of memory, are approximations to a Turing machine. If we peek at the central processing unit of a digital computer with a microscope, we can see that it is made out of very simple building blocks (gates) that perform the simplest logical operations: and, or, not. In other words, a computer is built from the bottom up out of millions of And-gates, Or-gates, and Not-gates. These gates are used to build more complex components, such as a Flip-Flop, which can act as a simple form of memory, and these, in turn, can be used to build more complex parts. Because this building procedure is so well known, in order to show that a particular medium can be used to carry on complex computations, all one has to show is that the simplest logical gates can be built in such a medium. As it happens, it can be shown that glider guns (and other engineered patterns) can be used to build logical gates, and this leads to the striking conclusion that an interacting population of the simplest automata can be used as a medium to build the most complex one: a Turing machine.25

It was this remarkable result, the unexpected jump from finite-state automata to Turing machines, skipping all intermediate steps, that posed an urgent problem. Why does the particular set of rules just described make this jump possible? Or in the terms we are using, given that finite-state automata can affect (and be affected by) one another in the way specified by the rules, why does the exercise of these capacities allows such a dramatic increase in computational capacity? Answering this question demands an exploration of the space of possible rules, because the interactions are defined by rules. Strictly speaking, we would need to investigate two spaces, the space of possible rules and the space of possible ways defining neighbourhoods, because we must also take into account that only neighbours can affect and be affected by one another. Moreover, to be exhaustive, we would need to explore not only the rules for two-dimensional cellular automata but also cellular spaces of any number of dimensions. For our purposes here we can aim at a less ambitious goal: to discover whether the space of possible rules has a structure that displays the distinction between the ordinary and the singular. For this limited goal, studying one-dimensional cellular automata is good enough. This greatly simplifies the task because in this case the cells become discrete points in a line, allowing us to ignore the geometry of the tiles and to concentrate on the rules.

As was the case with the spaces of possible genes or proteins, the space of possible rules is a discrete combinatorial space that has no intrinsic spatial order: all the possible rules simply lie next to each other without forming neighbourhoods or other spatial arrangements. But as we did with spaces of molecular sequences, we can begin by calculating the size of the space. Because rules determine only changes of state relative to neighbouring states, the size of the possibility space can be calculated from the number of combinations of two variables: the number of states in which an automaton can be and the number of neighbours (including itself) with which it can interact. The number of states raised to the number of neighbours yields the combination of possible states in which each neighbourhood can be. The number of possible rules is given by the number of states raised to the number of possible configurations of states for a neighbourhood. A one-dimensional cellular automaton in which each cell can be in two states and interacts only with its two immediate neighbours has a total of 256 possible rules. By contrast, the number of possible rules for the two-dimensional cellular automaton we just described, one with two states and nine neighbours (eight neighbours plus the reference cell) is the number 10 followed by 154 zeros.

This huge size explains why the first serious investigation of the possibility space for interaction rules was performed in the one-dimensional case. To explore its structure the following method can be used: select a rule from the set of 256 possibilities, give it a starting pattern, and then follow its evolution; as the process unfolds, check whether the number of patterns in which the states can be tends to decrease; if it does, then identify the limiting pattern the evolution approaches in the long run. Using this method, four different limiting patterns were identified, each associated with a different class of rules: the first class of rules leads to fixed, homogeneous patterns; the second class gives rise to periodic or oscillating patterns; the third class leads to random configurations; and finally, the fourth class of rules produces the full repertoire of patterns, including the patterns capable of motion that can serve as raw materials to build a Turing machine.26 For this reason, the rules corresponding to the fourth class of patterns can be considered singular or remarkable, while the other three types are ordinary. This conclusion is strengthened by the fact that rules of the fourth class are relatively rare compared to the other three types.27

At the start of this chapter we pointed out that when singularities were first discovered, their ontological significance was readily appreciated, but an empiricist ontology and a tendency to logical reductionism made singularities invisible to philosophers. Today, these two obstacles have been mostly removed, although as we saw, the debate over what components of state space should be considered to correspond to something real still rages. The conjunction of discrete mathematics and digital computers has increased the repertoire of formal resources that can be used to explore possibility spaces, and should therefore contribute towards a trend for a greater appreciation of virtual structure: even the earliest simulations, like Monte Carlo simulations, allowed us to follow a process until it reached a singularity.28 Moreover, as we just saw, the ease with which interactions can be staged using computers provides us with the means to extend our understanding from the simplest dispositions, tendencies, to the most complex ones, capacities, and to further develop the concept of a singularity to include not just what is traditionally understood by this term but all the formal entities that can sustain distributions of the significant and the insignificant. For assemblage theory this more adequate understanding of dispositions is crucial. If the actual components of an assemblage, as well as its actual emergent properties, show that the assemblage is a solution to a physical, chemical, biological, or social problem, its virtual dispositions reveal what is problematic about it, the objectivity of what we do not know about it: What tendencies would be manifested in novel conditions? What capacities to affect and be affected would be exercised when interacting with other assemblages that it has never interacted with?

Notes