‘Chaos’, in the sense that concerns us, is essentially a mathematical concept, and ‘chaos theory’ is a mathematical theory. Fully to grasp the concept or understand the theory means grappling with the relevant mathematics. A few basic ideas are explained in the course of this book, alongside the more conceptual or methodological discussions.

Relatively little background mathematics is presupposed in the main text. However, at various points it is illuminating (or perhaps simply fun) to go just a little further into the details in a way that may presuppose rather more – though familiarity with some calculus and the general idea of a differential equation, plus the ability to follow a moderately abstract mathematical argument, should largely suffice. These more taxing episodes are set in sans serif type and can be skimmed or skipped by the reader.

The same typographical device also serves to mark off other episodes that in a similar way pursue philosophical details rather further than some might want to follow. By picking and choosing among these passages, readers with various interests should be able to find a path through the book to suit.

To avoid further complicating the text, references to the mathematical and philosophical literature are kept to a minimum in the main body of the book; for a modest crop of additional references, see the final section, ‘Notes’.

In the opening chapter, we saw that the chaotically complex behaviour in the paradigm Lorenz case is dictated by a ‘strange’ attractor with an infinitely intricate structure. In the following chapter, we were able to sharpen up that informal talk of infinite intricacy; the attractor, we said, is a fractal. We must now face the question which immediately arises. How can an infinitely intricate structure like this possibly play an essential part in a competent scientific account of some natural phenomenon? For by the lights of our own best physical theories, quantities such as fluid circulation velocity, temperature, the proportional concentration of a chemical in a mixture and so forth – that is, macroscopic quantities of the type dealt with in paradigm chaotic models like Lorenz's – cannot have indefinitely precise real number values. Hence their time evolutions cannot really exemplify infinitely intricate trajectories wrapping round a fractal attractor, any more than a coastline can exemplify a genuinely fractal pattern.

What is being claimed here is something much stronger than the trite epistemological point that there is a limit to the precision with which we can know facts about the values of physical quantities. The claim is that there is no fact of the matter about the exact values of quantities like circulation velocity. We know, for example, that fluids are gappy distributions of molecules in motion.

The argument of the last chapter raises as many questions as it answers. Chaotic models will idealize and misrepresent the facts by involving patterns of dynamical behaviour which have an intricacy that the modelled phenomena must typically lack. Agreed, other theoretical models also misrepresent by idealizing and simplifying: and we argued that chaotic models may not be dramatically worse off in this respect. Still, it might be wondered, just how comforting is the thought that chaos theory is in the same unseaworthy boat as other theories?

Perhaps we can cheerfully live with the fact that chaotic theories won't be strictly true if we can make sense of the natural fall-back position, namely that these theories may yet be approximately true. But is the notion of approximate truth in good enough shape to rely on here (the literature is littered with failed philosophical analyses)?

That's business for Chapter 5. First, however, there's a more immediate issue. We suggested that a dynamical model can represent the stretching and folding of trajectories in a way which meets standard canons of simplicity and yet as a result be chaotic. But chaos involves sensitive dependence on initial conditions, meaning that the inevitable errors in setting initial conditions inflate exponentially, which in turn means (it seems) that the practical predictions of a chaotic model must go spectacularly wrong. The question then seems to arise: what kind of ‘modelling’ can there be when there must be spectacular and pervasive predictive error?

And what, finally, is chaos? So far, we have taken a few paradigm cases of mathematical models with complex behaviour (e.g. the Lorenz system, the logistic map), noted their intricate features, and then asked some key questions about the role that such infinitely structured models can have in representing a messy world and explaining natural phenomena. Answering those key questions – the central concern of this book – does not at all depend on having a sharp official criterion for separating the strictly chaotic from the non-chaotic cases.

A quick glance at the research literature (in journals like Physica D) shows that working applied theorists also proceed without any agreed precise definition of what counts as ‘chaos’. Rather, the term is typically used quite loosely, to advertise the presence of some interesting cluster of the phenomena that we have illustrated – e.g. exponential error explosion, the existence of a fractal attractor, the equivalence to a ‘symbol shift’ dynamics with product-random output, and so forth. Still, there is some interest in reviewing various options for giving a tidy definition (and some philosophical interest in reflecting on the nature of this definitional enterprise).

In this chapter, then, we consider possible definitions of chaos for (the dynamics in) a mathematical model. We have already offered a rough indication of how, given such a definition, we can extend the notion to apply to some real-world dynamical behaviour.

‘Chaos theory’: the very name suggests a paradox. For chaos, in the ordinary sense, is precisely the absence of order; and how can we hope to impose theoretically disciplined order on the essentially disordered?

It is far too late to change the name. So let's make it clear at the very outset that ‘chaos’ here must be taken as a term of art, stripped of most of its ordinary connotations: and ‘chaos theory’ is just the popular label for a body of theory about certain mathematical models and their applications. A first introductory task, then, is to say something about the kinds of mathematical models that are in question.

One mark of ‘chaos’ is sensitive dependence on initial conditions: that is to say, a chaotic system starting off from two very similar initial states can develop in radically divergent ways. Such sensitive dependence is often referred to as ‘the Butterfly Effect’. A small blue butterfly, let's suppose, sits on a cherry tree in a remote province of China. As is the way of butterflies, while it sits it occasionally opens and closes its wings. It could have opened its wings twice just now; but in fact it moved them only once. And – because the weather system exhibits sensitive dependence – the minuscule difference in the resulting eddies of air around the butterfly eventually makes the difference between whether, two months later, a hurricane sweeps across southern England or harmlessly dies out over the Atlantic.

An applied chaotic dynamical theory claims, in essence, ‘the time-evolution of the relevant real-world quantities has this intricate structure’. As we have seen, despite its non-empirical surplus content, such a theory can count as approximately true; and, despite sensitive dependence on initial conditions, it can be richly predictive. It will not be straightforwardly causal-explanatory (§5.1): but that does not stop the theory being explanatory in other ways. And our next major task is to say something about the sorts of explanations that chaotic dynamics can deliver.

As background, we need to take on board rather more of the elementary mathematics of chaos. That's the business for this largely expository chapter. Careful exposition will, in particular, remove the appearance of number-magic that can attach to chaotic dynamics (§6 below). As usual, the headline news is in the main text; the purely mathematical interludes here fill in a few details and sketch some proofs, but can be skipped.

We begin by turning our attention away from the Lorenz system to another standard exemplar of chaos, the so-called ‘logistic map’, which is a discrete map defined over the unit interval. At first sight, this might look like a radical change of tack. So far, we have been discussing the behaviour of phase space trajectories that are solutions to sets of linked first-order differential equations in the standard form (D).

Chaos – of the sort that concerns us – is a feature of certain dynamical models which exhibit sensitive dependence on initial conditions plus ‘confinement’ plus (typical) aperiodicity. Which is to say, roughly, that tiny differences in initial states can exponentially inflate into big differences in later states, but the values of the relevant state variables eventually remain confined within fixed boundaries although typically never exactly repeating.

Consider again, in the most general terms, how we can get sensitive dependence and confinement. Sensitive dependence means that trajectories through nearby points must tend to spread further apart from each other, confinement means that trajectories need to fold back on themselves so they keep within bounds. This kind of continual spreading and folding back is going to lead to a pretty tangled ball of possible trajectories. In fact, if there is typically to be no exact repetition, and also no merging or crossing of trajectories, then the ball must have a kind of infinite complexity. Since any given aperiodic trajectory will (eventually) be confined inside a finite region, it will have to keep revisiting some neighbourhoods of its earlier locations indefinitely often as it unendingly winds round and round. So segments of trajectories will have to be packed into such neighbourhoods infinitely densely.

In the last chapter, we saw how empirically observed period-doubling leading to apparent chaos might be explained (or at least, be partially explained). For we can show why period-doubling is endemic in a wide class of cases where other marks of chaos are also present. The relevant mathematics is intriguing; but on the face of it there seems nothing exceptional about the way that it may be brought to bear in explaining the phenomena. True enough, the modellings we are working with are often very partial and extremely idealized and perhaps (at this stage in the game) rather lacking in detailed physical motivation: but then, that's common enough in frontier science.

What goes for the explanations of period-doubling goes too, I claim, for other explanations in applied chaos theory. The new mathematical models may be distinctive; but the sort of explanations which their application yields is not. However, philosophers writing on chaos have repeatedly argued otherwise – they have claimed that there is something rather special about the explanations delivered by chaotic dynamical theories. Such explanations have variously been described as being characteristically ‘ex post facto’, ‘qualitative’, ‘holistic’ (or ‘anti-reductionist’), and ‘experimentalist’; it has also been suggested that we should recognize here a rather distinctive ‘Q-strategy’ of explanation. But why so?

Suppose it is maintained, first, that prediction and standard modes of explanation go together – roughly, a standard explanation after the event deploys materials that, before the event, could have been used to frame a prediction.

To provide a better sense of my picture and wishes, I shall discuss one recent theory of causation that fits my picture without satisfying my wishes. This is the so-called transference theory of causation presented and defended in the works of Aronson (1971a,b), Braddon-Mitchell (1993), Byerly (1979), Castaneda(1980, 1984), Dowe (1992a,b, 1995, 1996), Fair (1979), and Salmon (1994). In Jerrold Aronson's formulation, causes transfer some quantity (“e.g. velocity, momentum, kinetic energy, heat, etc.”) to their effects through contact (1971a, p. 422). In David Fair's formulation, causation “is a physically-specifiable relation of energy-momentum flow from the objects comprising cause to those comprising effect” (1979, p. 220). According to Hector-Neri Castaneda, something in need of specification, which he calls “causity,” is transferred. More recently Phil Dowe and Wesley Salmon describe causal interactions as intersections of causal processes that involve an exchange of some conserved quantity. Thus, for example, rolling billiard balls are causal processes, and their collisions (which involve an exchange of momentum) are causal interactions. I shall focus on Salmon and Dowe's version of a transfer theory.

Transfer theories fill in my picture of causation. They portray causation as fully objective. “Out there” world lines intersect, and there are exchanges of conserved quantities in some of these intersections. The study of causation, as the study of exchanges of conserved quantities, is a part of physics. In developing the notions of causal processes and causal interactions, Salmon and Dowe modify my naive ontology of substances and events; but they preserve the basic idea that causes and effects are things extended in time.

In the previous two chapters I examined the relations among three theories of causation: the independence theory (CP), Lewis's counterfactual theory (L), and one version of an agency theory (ATg). That discussion left many loose ends, which this chapter takes up. In particular I need to say something about a token-level formulation of agency theory and about how agency theories and counterfactual theories relate to one another. This chapter also addresses a criticism of the independence condition, which one defender of an agency view has formulated as a criticism of Lewis's theory. Examining that criticism will show the centrality of independence to causation. This chapter ends by pulling together the defense of the independence theory scattered through previous chapters and summarizing the relations between time, agency, counterfactuals, and independence.

Agency, Counterfactuals, and Independence

Agency theories can be formulated as counterfactual token-level theories rather than (as in chapter 5) as type-level theories. Furthermore, the notion of an intervention helps counterfactual theories avoid counterexamples. One might wonder whether L might be derivable from some version of agency theory. Agency and counterfactual theorists might profitably join forces.

The following token-level formulation of agency theory is analogous to the type-level formulation of chapter 5:

One might want the results of interventions to provide truth conditions for a nontransitive relation of causal dependence and then define causation, as Lewis does, in terms of chains of causal dependence.

The first two sections consider whether there is any way to strengthen the DN model to deal with the problem cases without relying on the notion of causation or on elements of a nonconditional analysis of causation. The third section provides proofs underlying the conclusions drawn in §8.5.

Nomic Sufficient Conditions and Explanations

In his essay “The Direction of Causation and the Direction of Conditionship” David Sanford offers an account of causal priority in terms of a nonsymmetrical relation among propositions, “is a causal condition of.” If his account succeeds, then he has provided the core of a purely conditional analysis of causation. Sanford also implicitly offers an account of scientific explanation. Sanford does not claim to have done so much, because his account also relies on an unanalyzed notion of “causal connection in a direct line.”

Let us, following Sanford, say that “A is nomically sufficient for B in circumstances C” (or “B is nomically necessary for A in circumstances C”) if and only if “A&∼B&C&L” is logically inconsistent where A, B, C, and L are all propositions, the conjunction is strongly nonredundant, and L is a (conjunction of) law(s) of nature. A and B state that events with properties A and B occur (1976, pp. 200–1). The exposition differs here from Sanford's in four significant ways: (1) Sanford's exposition is more general and applies to other sorts of necessity. (2) Sanford uses the notion of a logical impossibility rather than the narrower one of logical inconsistency.