INTRODUCTION

Emergence in History

The origin of the modern concept of emergence can be traced to the middle of the nineteenth century when realist philosophers first began pondering the deep dissimilarities between causality in the fields of physics and chemistry. The classical example of causality in physics is a collision between two molecules or other rigid objects. Even in the case of several colliding molecules the overall effect is a simple addition. If, for example, one molecule is hit by a second one in one direction and by a third one in a different direction the composite effect will be the same as the sum of the two separate effects: the first molecule will end up in the same final position if the other two hit it simultaneously or if one collision happens before the other. In short, in these causal interactions there are no surprises, nothing is produced over and above what is already there. But when two molecules interact chemically an entirely new entity may emerge, as when hydrogen and oxygen interact to form water. Water has properties that are not possessed by its component parts: oxygen and hydrogen are gases at room temperature while water is liquid. And water has capacities distinct from those of its parts: adding oxygen or hydrogen to a fire fuels it while adding water extinguishes it.¹

The fact that novel properties and capacities emerge from a causal interaction was believed to have important philosophical implications for the nature of scientific explanation. In particular, the absence of novelty in physical interactions meant that explaining their effects could be reduced to deduction from general principles or laws. Because deductive logic simply transfers truth from general sentences

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to particular ones without adding anything new it seemed like an ideal way of modeling the explanation of situations like those involving rigid collisions. But the synthesis of water does produce something new, not new in the absolute sense of something that has never existed before but only in the relative sense that something emerges that was not in the interacting entities acting as causes. This led some philosophers to the erroneous conclusion that emergent effects could not be explained, or what amounts to the same thing, that an effect is emergent only for as long as a law from which it can be deduced has not yet been found.² This line of thought went on to become a full fledged philosophy in the early twentieth century, a philosophy based on the idea that emergence was intrinsically unexplainable. This first wave of “emergentist” philosophers were not mystical thinkers but quite the opposite: they wanted to use the concept of emergence to eliminate from biology mystifying entities like a “life force” or the “élan vital.” But their position toward explanation gave their views an inevitable mystical tone: emergent properties, they said, must be accepted with an attitude of intellectual resignation, that is, they must be treated as brute facts toward which the only honest stance is one of natural piety.³

Expressions like these were bound to make the concept of emergence suspect to future generations of philosophers. It was only the passage of time and the fact that mathematical laws like those of classical physics were not found in chemistry or biology—or for that matter, in the more historical fields of physics, like geology or climatology—that would rescue the concept from intellectual oblivion. Without simple laws acting as self-evident truths (axioms) from which all causal effects could be deduced as theorems the axiomatic dream eventually withered away. Today a scientific explanation is identified not with some logical operation but with the more creative endeavor of elucidating the mechanisms that produce a given effect. The early emergentists dismissed this idea because they could not imagine anything more complex than a linear clockwork mechanism. But there are many other physical mechanisms that are nonlinear. Even in the realm of human technology we have a plurality of exemplars to guide our imagination: steam engines, thermostats, transistors. And outside technology the diversity is even greater as illustrated by all the different mechanisms that have been discovered in chemistry and biology.

EMERGENCE IN HISTORY

Armed with a richer concept of mechanism the emergent properties of a whole can now be explained as an effect of the causal interactions between its component parts. A large portion of this book will be dedicated to describe the wide variety of mechanisms of emergence that have been elucidated in the decades since the original emergentists first wrote.

Thus, what is different today from the early twentieth century views is the epistemological status of emergence: it does not have to be accepted as a brute fact but can be explained without fearing that it will be explained away. What has remained the same is the ontological status of emergence: it still refers to something that is objectively irreducible. But what kinds of entities display this ontological irreducibility? The original examples of irreducible wholes were entities like "Life," "Mind," or even "Deity." But these entities cannot be considered legitimate inhabitants of objective reality because they are nothing but reified generalities. And even if one does not have a problem with an ontological commitment to entities like these it is hard to see how we could specify mechanisms of emergence for life or mind in general, as opposed to accounting for the emergent properties and capacities of concrete wholes like a metabolic circuit or an assembly of neurons. The only problem with focusing on concrete wholes is that this would seem to make philosophers redundant since they do not play any role in the elucidation of the series of events that produce emergent effects. This fear of redundancy may explain the attachment of philosophers to vague entities as a way of carving out a niche for themselves in this enterprise. But realist philosophers need not fear irrelevance because they have plenty of work creating an ontology free of reified generalities within which the concept of emergence can be correctly deployed.

What kinds of concrete emergent wholes can we legitimately believe in? Wholes the identity of which is determined historically by the processes that initiated and sustain the interactions between their parts. The historically contingent identity of these wholes is defined by their emergent properties, capacities, and tendencies. Let's illustrate the distinction between properties and capacities with a simple example. A kitchen knife may be either sharp or not, sharpness being an actual property of the knife. We can identify this property with the shape of the cross section of the knife's blade: if this cross section has

a triangular shape then the knife is sharp else it is blunt. This shape is emergent because the metallic atoms making up the knife must be arranged in a very particular way for it to be triangular. There is, on the other hand, the capacity of the knife to cut things. This is a very different thing because unlike the property of sharpness which is always actual the capacity to cut may never be actual if the knife is never used. In other words, a capacity may remain only potential if it is never actually exercised. This already points to a very different ontological status between properties and capacities. In addition, when the capacity does become actual it is not as a state, like the state of being sharp, but as an event, an event that is always double: to cut-to be cut. The reason for this is that the knife’s capacity to affect is contingent on the existence of other things, cuttable things, that have the capacity to be affected by it. Thus, while properties can be specified without reference to anything else capacities to affect must always be thought in relation to capacities to be affected. Finally, the ontological relation between properties and capacities displays a complex symmetry. On one hand, capacities depend on properties: a knife must be sharp to be able to cut. On the other, the properties of a whole emerge from interactions between its component parts, interactions in which the parts must exercise their own capacities: without metallic atoms exercising their capacity to bond with one another the knife’s sharpness would not exist.

A similar distinction can be made between emergent properties and tendencies. To stick to the same example: a knife has the property of solidity, a property that is stable within a wide range of temperatures. Nevertheless, there are always environments that exceed that range, environments in which the temperature becomes so intense that the knife is forced to manifest the tendency to liquify. At even greater intensities the molten metal may gasify. These tendencies are as emergent as the shape of a knife’s blade: a single metallic atom cannot be said to be solid, liquid, or gas; we need a large enough population of interacting atoms for the tendency to be in any of these states to emerge. Tendencies are similar to capacities in their ontological status, that is, they need not be actual to be real, and when they do become actual is as events: to melt or to solidify. The main difference between tendencies and capacities is that while the former are typically finite the latter need not be. We can enumerate, for example, the possible

states in which a material entity will tend to be (solid, liquid, gas, plasma) or the possible ways in which it may tend to flow (uniformly, periodically, turbulently). But capacities to affect need not be finite because they depend on the capacities to be affected of innumerable other entities: a knife has the capacity to cut when it interacts with something that has the capacity to be cut; but it also has the capacity to kill if it interacts with large organisms with differentiated organs, that is, with entities that have the capacity to be killed.

Since neither tendencies nor capacities must be actual to be real it would be tempting to give them the status of possibilities. But the concept of a possible event is philosophically suspect because it is almost indistinguishable from that of a real event, the only difference being the former's lack of reality. Rather, what is needed is a way of specifying the structure of the space of possibilities that is defined by an entity's tendencies and capacities. A philosopher's ontological commitment should be to the objective existence of this structure and not to the possibilities themselves since the latter exist only when entertained by a mind. Some possibility spaces are continuous having a well-defined spatial structure that can be investigated mathematically, while others are discrete, possessing no inherent spatial order but being nevertheless capable of being studied through the imposition of a certain arrangement. The space of possible regimes of flow (uniform, periodic, turbulent) is an example of a continuous possibility space in which the only discontinuities are the critical points separating the different tendencies. The space of possible genes, the on the other hand, is an example of a discrete space that must be studied by imposing an order on it, such as an arrangement in which every gene has as neighbors other genes differing from it by a single mutation. As we will see in the different chapters of this book the structure of possibility spaces plays as great a role in the explanation of emergence as do mechanisms.

The chapters are deliberately arranged in a way that departs from the ideas of the original emergentists. These philosophers believed that entities like "Space-Time," "Life," "Mind," and "Deity" (not "god" but the sense of the sacred that emerges in some minds) formed a pyramid of progressively ascending grades. Although the levels of this pyramid were not supposed to imply any teleology it is hard not to view each level as leading to the next following a necessary sequence.

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To eliminate this possible interpretation an entirely different image is used here, that of a contingent accumulation of layers or strata that may differ in complexity but that coexist and interact with each other in no particular order: a biological entity may interact with a subatomic one, as when neurons manipulate concentrations of metallic ions, or a psychological entity interact with a chemical one, as when subjective experience is modified by a drug. The book begins with purely physical entities, thunderstorms, that are already complex enough to avoid the idea that their behavior can be deduced from a general law. It then moves on to explore the prebiotic soup, bacterial ecosystems, insect intelligence, mammalian memory, primate social strategies, and the emergence of trade, language, and institutional organizations in human communities. Each of these layers will be discussed in terms of the mechanisms of emergence involved, drawing ideas and insights from the relevant fields of science, as well as in terms of the structure of their possibility spaces, using the results of both mathematical analysis and the outcomes of computer simulations.

Simulations are partly responsible for the restoration of the legitimacy of the concept of emergence because they can stage interactions between virtual entities from which properties, tendencies, and capacities actually emerge. Since this emergence is reproducible in many computers it can be probed and studied by different scientists as if it were a laboratory phenomenon. In other words, simulations can play the role of laboratory experiments in the study of emergence complementing the role of mathematics in deciphering the structure of possibility spaces. And philosophy can be the mechanism through which these insights can be synthesized into an emergent materialist world view that finally does justice to the creative powers of matter and energy.

CHAPTER ONE

The Storm in the Computer

Let’s begin with the simplest emergent properties, properties like temperature or pressure characterizing wholes made out of a large number of identical parts, such as a body of water in a container. Being composed of billions of molecules that are qualitatively the same makes a body of water much simpler than, say, an ecosystem in which hundreds of different species constantly interact. But this simplicity is what makes the mechanism of emergence behind temperature or pressure a promising starting point for philosophical thought. To begin with, in what sense are these properties emergent? Temperature is defined as the average energy that a molecular population has by virtue of the motion of its parts, the more violent the motion the more intense the temperature. Pressure is defined as the average degree to which the population pushes against the walls of the container by virtue of the momentum of its parts, the faster and more massive the molecules the more intense the pressure exerted. These definitions have tempted philosophers in the past to think that temperature and pressure can be reduced to kinetic energy and momentum, that is, that they are not emergent. But temperature and pressure are in fact irreducible because they are the result of an objective averaging process that takes place spontaneously in molecular populations.

To understand how this works let’s imagine two bodies of water at different temperatures. The moment we place these bodies in contact with each other energy will flow from the body with higher temperature to the one with lower temperature, the flow continuing until the

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temperature difference disappears. In other words, the difference in temperature will display a tendency to average itself out. Thus, saying that a body of water possess a certain temperature, and that possession of that property defines an enduring state, implies that departures from that state are constantly being counteracted by an objective tendency. For the same reason defining “sameness of temperature” can be done by placing two bodies of water into contact and verifying that no flow of energy is taking place between them. Thus, in this simple case the irreducible status of a property like temperature is established by elucidating a mechanism through which the property emerges, a mechanism involving the manifestation of tendency. Accounting for this tendency, in turn, demands switching scales and focusing on the interactions between the parts of the whole, interactions in which the parts exercise their capacities to affect and be affected. In particular, for a temperature difference to cancel itself the component molecules must exercise their capacity to collide and redistribute energy in those collisions.

We can visualize the series of events leading to the emergence of an average temperature by comparing the states of two bodies of water before and after the dissipating tendency has been manifested. At the start of the process the existence of a temperature difference means that the water molecules are distributed with a high degree of order, that is, that they are neatly sorted out into two parts, one hot and the other cold. At the end of the process the entire population is uniformly warm and this order has disappeared. A disordered state is characterized by the fact that we can make a large number of changes in the molecular distribution and leave the bulk state basically the same. In other words, a much larger number of combinations of individual kinetic energies will result in the same warm body of water than the number that will yield the state in which one container is hot and the other one is cold. This affects the probability that one or the other state will occur spontaneously. Because in this case the interactions between molecules are entirely random these odds make all the difference in the world: the state characterizing a warm body of water will have a much higher probability of occurring as a result of random collisions than the one in which they are sorted into hot and cold subpopulations. It is this difference in the odds with which

the ordered and disordered states can occur that explains the tendency for one to become the other.¹

The mechanism of emergence just described for temperature is basically the same for pressure, density, and other intensive properties of molecular populations. Despite their relative simplicity these properties are important because the spontaneous flow of energy that takes place as intensive differences cancel themselves can be tapped into to fuel other processes. The whole composed by two containers of water at different temperatures, for example, has the capacity to drive another process partly because the high temperature container stores a lot of energy, much more than the low temperature one, and partly because we can extract that energy by placing the former in contact with the latter.² This capacity is relatively short lived, however, because once the intensive difference disappears the energy left behind becomes much harder to extract. But if the difference is continuously refreshed, by placing the first container on top of a fire, for instance, then the whole formed by the hot and cold molecular populations can become a component part of a larger whole, playing the role that a gasoline tank or an electric battery play in an automobile or an electronic appliance. The capacity of intensive differences to act as energy storage devices will play such a prominent role in the explanation of emergence in many other examples that it will be useful to have a more compact term for them. We will refer to them as gradients.

In addition to serve as energy sources gradients can serve to generate the moving parts of larger wholes. For example, if a gradient is intense enough and if it is prevented from dissipating it can cause a molecular population to self-organize into a circular motion pattern that will persist as long as the gradient persists. The coordinated movement of billions of molecules needed to yield such a pattern is a highly unlikely event and yet it occurs spontaneously in the ocean and the atmosphere every single day. This coherent circulatory flow, referred to as a convection cell, is produced by the gradient as the means to cancel itself even as the imposed constraints prevent it from doing so.³ The mechanism of emergence behind a convection cell can be explained using the same example of a water container: when the container is heated from below it becomes divided into a warm bottom

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and a cool top; as the bottom water warms up it expands and becomes less dense tending to rise, while the high density cold water on top tends to sink; these up and down movements are counteracted by the internal friction generated by the viscosity of the water, but when the temperature difference becomes intense enough this resistance is overcome and the upward and downward flows join together to form a circular pattern.⁴ Because this pattern is very stable it can literally be used as a building block to construct larger emergent entities.

What kind of entities can be built using gradients as fuel tanks and convection cells as moving parts? The most dramatic example is a thunderstorm, a typical storm containing five to eight convection cells each a few kilometers in diameter.⁵ Viewed from the outside a thunderstorm appears as a large complex cloud with a well-defined form. At the center of the storm is a massive column-like structure called the "central pillar." This vertical structure adopts an asymmetric horizontal shape at its top called an "anvil" for its resemblance to the metal block used by blacksmiths. The central pillar often overshoots the anvil creating a dome at its top. Finally, at the bottom of the pillar there are flanking horizontal clouds lined up in the opposite direction to the anvil. This complex form is one of the emergent properties of a thunderstorm, its directly observable property. But behind its observable form there is the internal machinery of the storm. In addition to gradients and convection this machinery uses phase transitions, the transition from gas to liquid or from liquid to solid, as energy amplifiers. One of the differences between a material such as water in its gas, liquid, or solid state is the degree to which its composing molecules move around and therefore the amount of kinetic energy the material contains. In a solid the molecules are relatively confined to fixed positions so their activity is relatively calm. In liquids this confinement is relaxed: molecules still exert some restraining influence over one another but they allow a more energetic movement. In gases the molecules are even more excited since their movement is not constrained at all. A gas therefore contains more energy than a liquid or a solid. When rising vapor becomes rain some of this extra energy becomes available as a surplus that can be exported to the surrounding medium, increasing the amount of energy available to the thunderstorm. This exportable energy is referred to as "latent heat."⁶

To understand the mechanism of emergence behind a thunderstorm we need to explain how these different components are coupled together. First of all, a difference in temperature between the surface of the ocean and that of the atmosphere must exist to get the process started. This vertical gradient causes an upward flow of air and vapor forming one leg of a convection cell. As the warm moist air moves up it becomes cooler eventually reaching the critical point at which vapor becomes liquid water. At first this phase transition produces very small liquid droplets that become suspended in the surrounding air. The concentration of these tiny droplets makes the upward air current visible as a small cauliflower-shaped cloud that becomes the base of the future thunderstorm. Although at this point the air should start turning sideways and head for the downward leg of the convection cell, the latent heat released by the phase transition increases the temperature of the air current adding buoyancy to it and propelling further up. This self-stimulating interaction is repeated several times allowing the updraft to reach great heights eventually becoming the giant cloud described above, with its central pillar, anvil, and dome. The death of a thunderstorm, in turn, is linked to processes that counteract its sustaining gradients: the higher the air reaches the colder it gets, the more saturated it becomes, and the larger the liquid drops and ice crystals that condense from it. When the weight of these drops and crystals reach a tipping point—the point at which the downward force exerted by gravity becomes stronger than that of the updraft—it begins to fall in the form of rain and hail dragging air with it, stealing energy from the updraft and eventually destroying the internal machinery of the storm.

Other features of this emergent meteorological entity are also explained by gradients. A severe thunderstorm is usually accompanied by the production of lightning, either intensely bright flashes created within the cloud or powerful bolts between the cloud and the ground. Lightning is the result of an electrical gradient, a difference in charge between the upper and lower regions of the cloud, or between the cloud and the ground, the brilliant discharge being the form created by the gradient to cancel itself. Thunderstorms are also the birth place of tornadoes, whirling masses of air made visible by the matter (vapor, dust, debris) that they suck into their intensely rapid circulation.

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Tornadoes are born from the same vertical temperature gradient that causes the updraft to which a steep horizontal pressure gradient is added. The latter is caused by the fact that the updraft sucks the air from the center of the tornado greatly reducing the pressure inside of it compared to that of the outside.⁷ As an emergent whole a thunderstorm is characterized by its properties, such as the heights it reaches or the speed with which it moves; by its tendencies, like its tendency to move in a certain direction or its tendency to consume all its energy and die; and by the capacities it exercises when it interacts with other entities. From a human point of view these interactions are mostly destructive: its lightning kills people and starts brush and forest fires; the heavy rain along its downdraft can result in floods; and the tornadoes it spawns can violently flatten entire towns. These capacities can surely inspire awe and respect at the destructive potential of a thunderstorm but they should not lead to an attitude of intellectual resignation or natural piety toward it: we can explain how it is born, how it lives, and how it dies.

Let's pause to consider the argument so far. The objective reality of emergent properties can be established by elucidating the mechanisms that produce them at a one scale and by showing that emergent entities at that scale can become the component parts of a whole at a larger scale. Mechanisms of emergence may, of course, undergo revision or elaboration, and some are better understood than others, but the possibility of improvement or change in the proposed mechanisms should not force us to take emergence at any scale as a brute fact. There are, on the other hand, aspects of the concept of emergence that this argument does not address. In particular, similar emergent effects can be produced by entirely different mechanisms suggesting that there is more to the emergent properties of a whole than the interactions between its parts. Let's return to the case of convection cells. The self-organized rhythmic flow characterizing convection emerges in many kinds of materials. The flows of molten rock that lie underneath the surface of the earth, for example, exhibit the same coherent circular motion as the air and water above it. More importantly, the same self-organization is displayed by other rhythmic patterns that have nothing to do with the movement of matter in space. A good example comes from the world of chemistry. The gradients in this case are differences in the concentration of

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certain substances, that is, they are gradients of matter not of energy. The rhythms are the rates at which new compound molecules are synthesized—the chemical reaction switches spontaneously from the production of one type of molecule to the production of another following a perfect beat—not collective circular motions. Yet despite these differences a convection cell and a chemical clock, as these reactions are called, are qualitatively the same. This implies that a full explanation of these emergent entities must possess a component that is independent of any particular mechanism.

It could be argued that the similarity in rhythmic behavior is superficial and that it does not demand complicating the concept of explanation, but there are other shared characteristics that cannot be so easily dismissed. In particular, the periodic behavior in both cases is stable against perturbations, that is, if a convection cell or a chemical clock are disturbed by an outside shock they will tend to return to their original period and amplitude after a relatively short time. This tendency is referred to as asymptotic stability. Not all oscillating entities possess this kind of stability. A pendulum in which friction has been carefully eliminated, for example, will not react the same way: a small push will permanently change both the duration and intensity of its swing, the pendulum acting as if it "remembered" the perturbation. A convection cell or a chemical clock, on the other hand, quickly "forget" the perturbation acting as if nothing had happened.⁸ When we explained convection by the causal mechanism outlined above—a temperature gradient that creates a density gradient that, in turn, amplifies fluctuations into a circular flow—we were giving only part of the explanation because the causal chain behind the emergence of a convection cell does not account for the fact that its properties are stable against perturbations. And similarly for a chemical clock.

Adding to the explanation of emergence a mechanism-independent component will involve introducing entirely new ideas so it will be useful at this point to justify the need for the extra complexity. So far the concept of emergence has played an ontological role, showing why it is legitimate to believe in the existence of objective properties, tendencies, and capacities. But once we add the mechanism-independent component the concept of emergence leads to two important epistemological consequences: it explains why we can use partial models to learn about reality and it provides an account for the capacity of those

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models to mimic the behavior of the processes they model. The first consequence derives directly from the notion of asymptotic stability. When the emergent properties of a whole are stable they can survive changes in the details of the interactions between its parts. A given degree of temperature in a body of water, for example, may result from a number of different interactions between the kinetic energy of its component molecules. This implies that we can take the existence of temperature for granted when explaining the circulatory pattern in a convection cell, that is, that we can legitimately leave out of the explanation a detailed census of the kinetic energy of each of the molecules in the population. To put this differently, a stable emergent property is "indifferent" to local changes in the interactions that give rise to it, and this objective indifference translates into an objective explanatory irrelevance of the details of the interactions: including the latter in an explanation would be redundant because many different interactions would yield the same outcome.⁹ Being able to take for granted the existence of emergent properties at one scale in order to explain properties at another scale is arguably a basic requirement for scientific research. If scientists had to build models that captured all scales simultaneously no scientific field would ever have succeeded in explaining anything. We would be trapped in a block universe in which every aspect is inextricably related to every other aspect and our incapacity to separate levels of emergence would leave us cognitively powerless.

The second epistemological consequence derives from the very notion of mechanism-independence: if processes as different in detail as a convection cell and a chemical clock can exhibit the same behavior perhaps mathematical equations can also display that behavior. To set the stage for the argument let's first give a simplified account of the relation between mathematical models and laboratory experiments. Let's assume that we want to understand the behavior of the air currents forming the updraft and downdraft of a thunderstorm. We can use a mathematical model of the dynamics of non-viscous fluids that has existed since the eighteenth century: a set of differential equations that relate the properties of density, pressure, internal energy, and velocity to the flow of air. Using these equations we can generate a series of numbers that indicate the course of the modeled

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fluid at discrete intervals of space and time and then give this series of numerical solutions a visual form, such as a plot in a piece of paper. The expression "the behavior of equations" refers to the pattern generated by its numerical solutions as presented graphically by the plot. Next we move to the laboratory and create an experimental situation in which an actual flow of air is affected only by those same properties, using an apparatus that can exclude any other causal factor from affecting the flow. We run the experiment and take measurements of the airflow at different points in space and instants of time and plot the measured values on a piece of paper. To be able to compare the two plots we must make sure that the values of the variables in the equations and the values of the properties in the apparatus are approximately the same at the beginning of the run, that is, that both the equations and the apparatus are set to the same initial conditions. If the mathematical model captured the real dynamics then the two plots should be geometrically similar.¹⁰

This is, of course, a highly idealized picture of the relation between theory and experiment but it points to the crucial question: the similarity between the two graphic plots suggests that the behavior of the numerical solutions to the equations is isomorphic to the behavior of the physical properties inside the apparatus, a highly improbable behavioral isomorphism that cries out for explanation. Moreover, the advent of computer simulations has allowed scientists to tackle not just simple air currents but entire thunderstorms and the fact that the geometric similarity has persisted has made the underlying behavioral isomorphism even more problematic. In recent decades the equations for non-viscous flow used in the previous example were coupled to another set modeling the phase transitions in water and were numerically solved for every point of a three-dimensional grid, each point representing a box one kilometer wide and half a kilometer high. Enough of these boxes were included in the simulation to fit a regular size thunderstorm. To add a temporal dimension the equations were solved recursively, that is, the solutions obtained as outputs at any one instant were used as inputs for the equations to get the solutions for the next time interval. This makes the expression "the behavior of equations" less metaphorical because recursion transforms a static mathematical object into a dynamic computational process.

A set of values to serve as initial conditions was obtained from actual measurements of wind, temperature, and humidity of an area of the ocean where an actual thunderstorm had developed.

After feeding the computer the initial values the recursive procedure took over repeatedly generating populations of solutions for every time interval, the final product rendered using standard computer graphics software. To everyone’s surprise a central pillar, an anvil, a dome, and a flanking line of clouds spontaneously emerged despite the fact that none of those features had been explicitly modeled. The updraft and the downdraft forming the internal machinery of the storm also emerged, made visible by adding purely graphic entities (weightless spheres) that followed the simulated air currents.¹¹ Part of the explanation for the success of the simulation is the decomposability of reality made possible by emergent properties. The microscopic interactions between molecules at the surface of the ocean and those in the air above it, for example, did not have to be modeled in detail. The effect of friction between water and air molecules that starts the process of storm formation was introduced from the outside as a perturbation of the interface between the two fluids. Similarly, macroscopic details like the influence of the earth’s rotation were simply ignored. That the correct geometrical form emerged despite these simplifications shows that natural phenomena exhibit a recurrent part-to-whole relation, in which wholes at one scale become parts at the next larger scale, and that interactions between scales can be either left out of a model or added exogenously.¹² On the other hand, the behavioral isomorphism between the solutions to the equations and the physical flows in a real thunderstorm is not explained by the decomposability of reality. This isomorphism has mystified physicists for as long as there has been evidence of its existence, some of them resigning themselves to accept the unreasonable effectiveness of mathematics as miraculous.¹³

But as argued above an explanation of this “miracle” can be given using the notion of mechanism-independence. Let’s clarify this notion first for the case of material processes. As mentioned in the Introduction the distinction between properties on one hand and tendencies and capacities on the other is that the former are always actual—actual characteristics of the state of a whole at any given point in time—while the latter need not be: the tendency of liquid water to

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solidify at a critical point of temperature may not manifest itself if the temperature always remains above that point; and the capacity of liquid water to acts as a solvent may not be exercised if the water never comes into contact with soluble substances. The ontological status of both tendencies and capacities is therefore different from that of properties. As the simple case of temperature or pressure shows the part of the explanation of emergence that depends on mechanisms involves the actual manifestation of tendencies (the tendency behind the objective averaging process) and the actual exercise of capacities (the capacity of molecules to collide and redistribute energy). The mechanism-independent component of an explanation, on the other hand, demands clarifying the status of tendencies and capacities when they are not actually manifested or exercised. We could, of course, characterize that status as that of a possibility but that would be too vague: an unmanifested tendency and an unexercised capacity are not just possible but define a concrete space of possibilities with a definite structure.

Let’s imagine this abstract space as a set of points each representing a different possibility. The structure of this space can be conceived as the subset of those points that have a much higher probability to become actual. When we described the mechanism of emergence behind the tendency of a gradient to cancel itself we said that it was based on a probabilistic argument: the state in which the gradient is alive is much more ordered than that in which it is dissipated, and in a molecular population in which all interactions are basically random the disordered state is a vastly more probable outcome of those interactions. An alternative way of saying this is that in the space of possibilities for the molecular population there exists a special point, the point of maximum disorder, and that the population is attracted to that state because it is much more probable than the others. A similar idea can be applied to convection cells and chemical clocks. We can imagine that in their space of possible states there is a set of points forming a closed loop that has the highest probability of actually occurring, forcing a physical or chemical process to repeat the same series of states over and over. If the process is subjected to an external shock it will move away from that loop, existing momentarily in less probable states, but then it will tend to return to it. This informal argument points to the solution to our problem: the stability

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of emergent properties is explained by the structure of a possibility space and the fact that this stability can be displayed by entirely different mechanisms is explained by the fact that their possibility spaces share the same structure.

The concept of a possibility space can be made rigorous using the results of several centuries of mathematical investigation on the nature of abstract spaces. In mathematics a basic distinction is made between metric spaces, the best known example of which is Euclidean geometry, and non-metric spaces exemplified by a variety of other geometries: projective, differential, topological. A relatively simple way of distinguishing metric from non-metric spaces is by the way in which the component parts of a space, individual points, are identified. The metric solution is to give each point an "address" by locating the space relative to a set of fixed coordinates and determining the distance that each point has from those axes. But this familiar procedure is not the only way of individuating points. One can, for example, determine the rate at which the curvature of a space changes at a given point and use this instantaneous rate of change to identify it.¹⁴ When we do this a space ceases to be a set of coordinate addresses and becomes a field of rapidities and slownesses, the rapidity or slowness with which curvature varies at each point. The structure of an abstract space, in turn, can be characterized by those properties that remain unchanged when the space is transformed, when it is moved, rotated, folded, stretched. Metric properties like length or area remain invariant under the smallest set of transformations, while those of the least metric spaces stay unchanged under the largest set. For the purpose of understanding in what sense two different mechanisms can share the same structure we need highly invariant structural properties since the metric details of their possibility spaces are bound to be different. A very important example of these invariant properties is the existence and distribution of special or remarkable points (or sets of such points) called singularities. This is the concept that we need to make the remarks in the previous paragraph less metaphorical: the possibilities with the highest probability of occurring are topological singularities acting as attractors.

Let’s now apply this line of thought to mathematical models. To create a mathematical model the first step is to enumerate all the relevant ways in which the process to be modeled is free to change.

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Let’s imagine that we are modeling a simple physical process characterized by two changing properties, such as temperature and pressure. These are called its "degrees of freedom." The process may, of course, also change in an infinite number of irrelevant ways, the art of mathematical modeling being based in part on the ability to judge what changes do, and what changes do not, make a difference. Once the relevant degrees of freedom of a physical process have been identified the model can be given a spatial form by assigning each of them to a dimension of a topological space. Each point in this space will be a combination of values of temperature and pressure representing an instantaneous state of the process being modeled, while the set of points as a whole represents the space of all possible states for the process. For this reason the abstract space is referred to as state space (or "phase space"). Finally, since a given process changing in time follows a sequence of states its behavior appears in state space as a series of points, that is, as a curve or trajectory. It was by observing the tendency of many of these trajectories to converge on specific areas of state space, to converge on singularities, that the existence of asymptotic stability was first established.¹⁵

Using these ideas the explanation for the unreasonable effectiveness of mathematics can be phrased like this: a mathematical model can capture the behavior of a material process because the space of possible solutions overlaps the possibility space associated with the material process. The two possibility spaces need not be identical but merely overlapping because most mathematical models mimic the behavior of a process only within a certain range of values of their control parameters. A sufficient overlap can nevertheless exist because the singularities that structure both spaces are independent of both causal mechanisms in a process and formal relations in an equation. This explanation implies an ontological commitment to the autonomous existence of topological singularities, or more generally, to the structure of possibility spaces. Once this ontological commitment has been made the term "singularity" ceases to be a purely mathematical concept and becomes a properly philosophical one. In particular, once singularities are taken to be as real and efficient as causes the nature of their reality becomes a problem for philosophers. Do they exist, for example, as transcendent entities in a world beyond that of matter and energy? Or are they immanent to the material world? If all the matter

and energy of the universe ceased to exist, would singularities also disappear (immanent) or would they continue to exist (transcendent)? Although these questions are not mathematical but philosophical the practice of mathematicians can still provide insights into the answers.

If singularities are immanent they must be both irreducible to any particular material process while at the same time requiring that some process or another actually exists. These two conditions are reflected in the way singularities are studied. Topologists, for example, do not study the singularities structuring the possibility space of a model free to change in its temperature and its pressure, but of all models with two degrees of freedom whatever these may be. It can be proved, for example, that in a two-dimensional space only certain kinds of singularities exist: four different types of point singularities distinguished from each other by the form of the flow of nearby trajectories (nodes, saddle points, foci, and centers) as well as one type of periodic singularity.¹⁶ In three-dimensional spaces, the four point singularities are still part of the repertoire but now periodic singularities come in three different forms: stable, unstable, and saddle-shaped loops. In addition, a new type of singularity becomes available, one that can be pictured as a loop that has been repeatedly stretched and folded (a so-called chaotic attractor).¹⁷ This implies that topological facts about possibility spaces can be discovered without reference to the nature of the degrees of freedom, only to their number, and without reference to the nature of the gradient (thermal, gravitational, mechanical, chemical) only to its existence.¹⁸ But the fact that the existence of a gradient, any gradient, is necessary confirms the immanent status of singularities.

Singularities are, therefore, perfectly acceptable entities in a materialist philosophy. The main problem confronting us now is the extent to which we can generalize from these ideas. State space is only one kind of possibility space, a space useful to study tendencies but not capacities. Capacities involve a much larger set of possibilities than tendencies because entities can exercise their capacities in interaction with a potentially innumerable variety of other entities. The more complex possibility spaces associated with capacities, and the nature of the singularities that structure them, are not nearly as well understood as those of tendencies. On the other hand, computers can supply the means to explore these other possibility spaces in a rigorous

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way because the interactions in which capacities are exercised can be staged in a simulation and varied in multiple ways until the singular features of the possibility space are made visible. Each of the following chapters will explore how staging a different type of simulated interaction (chemical, biological, social) can tease out the singular structure of their possibility spaces. The first step in this exploration, however, will not address the question of the relation between models and reality. We first need to clarify the concept of emergence in the case of simulations. This is what the following chapter will attempt to do.