Metaphysical and Epistemological Issues in Complex Systems

METAPHYSICAL AND EPISTEMOLOGICAL ISSUES IN COMPLEX SYSTEMS Robert C. Bishop 1 INTRODUCTION There are multiple senses of complexity. For instance, something can be considered complex because it is complicated or intricate (think of an engine or a watch with many moving parts and interrelated systems). However, systems that have a complicated set of interacting parts may actually exhibit relatively simple behavior (this is the case for engines). In contrast, the notion of complexity of interest in “complexity studies” centers on systems whose behavior is nonlinear and typically exhibits self-organization and collective effects. Such behavior appears to be any- thing but simple (see sec. 2.5), yet it is the case that systems with relatively few interacting components can exhibit this kind of intricate behavior. For instance, phase changes in Ising models and other systems with large numbers of components are examples of systems exhibiting complex self-organizing behavior. But even three-particle billiard-ball-like systems can exhibit the requisite complex behavior. On the other hand, many n-body systems, where n is large, do not exhibit complexity (e.g., Brownian motion of molecules) because the interactions among the constituents are not of the right type. An important feature of the systems of interest in complexity studies is that the interactions of the system components be nonlinear. Still, characterizing the resulting behavior, and the complex systems which exhibit it, is one of the major challenges facing scientists studying complexity and calls for the development of new concepts and techniques (e.g., [Badii and Politi, 1997]). A number of metaphysical and epistemological issues are raised by the investi- gation and behavior of complex systems. Before treating some of these issues, a characterization of nonlinear dynamics and complexity is given. Along with this background, some folklore about chaos and complexity will be discussed. Although some claim that chaos is ubiquitous and many take the signal feature of chaos to be exponential growth in uncertainty (parameterized by Lyapunov exponents, see sec. 2.4), these examples of folklore turn out to be misleading. They give rise to rather surprising further folklore that chaos and complexity spell the end of predictability and determinism. But when we see that Lyapunov exponents, at least in their global form, and measures for exponential divergence of trajectories only apply to infinitesimal quantities in the infinite time limit, this further folklore also Handbook of the Philosophy of Science. Volume 10: Philosophy of Complex Systems. Volume editor: Cliff Hooker. General editors: Dov M. Gabbay, Paul Thagard and John Woods. c 2011 Elsevier BV. All rights reserved. 106 Robert C. Bishop turns out to be misleading. Instead, the loss of linear superposition in nonlinear systems is one of the crucial features of complex systems. This latter feature is related to the fact that complex behavior is not limited to large multi-component systems, but can arise in fairly simple systems as well. The impact of nonlinearity on predictability and determinism will be discussed including, briefly, the potential impact of quantum mechanics. Some have argued that chaos and complexity lead to radical revisions in our conception of determinism, namely that determinism is a layered concept (e.g., [Kellert, 1993]), but such arguments turn on misunderstandings of determinism and predictability and their subtle relations in the context of nonlinear dynamics. When the previously men- tioned folklore is cleared away, the relationship among determinism, predictability and nonlinearity can be seen more clearly, but still contains some subtle features. Moreover, the lack of linear superposition in complex systems also has implica- tions for confirmation, causation, reduction and emergence, and natural laws in nonlinear dynamics all of which raise important questions for the application of complex nonlinear models to actual-world problems. 2 NONLINEAR DYNAMICS: FOLKLORE AND SUBTLETIES I will begin with a distinction that is immediately relevant to physical descriptions of states and properties known as the ontic/epistemic distinction tracing back at least to Erhard Scheibe [1964] and subsequently elaborated by others [Primas, 1990; 1994; Atmanspacher, 1994; 2002; d’Espagnat, 1994; Bishop, 2002). Roughly, ontic states and properties are features of physical systems as they are “when nobody is looking,” whereas epistemic states and properties refer to features of physical systems as accessed empirically. An important special case of ontic states and properties are those that are deterministic and describable in terms of points in an appropriate state space (see secs. 2.3 and 3.2 below); whereas an important special case of epistemic states and properties are those that are describable in terms of probability distributions (or density operators) on some appropriate state space. The ontic/epistemic distinction helps eliminate of confusions which arise in the discussions of nonlinear dynamics and complexity as we will see. 2.1 Dynamical systems Complexity and chaos are primarily understood as mathematical behaviors of dynamical systems. Dynamical systems are deterministic mathematical models, where time can be either a continuous or a discrete variable (a simple example would be the equation describing a pendulum swinging in a grandfather clock). Such models may be studied as purely mathematical objects or may be used to describe a target system (some kind of physical, ecological or financial system, say). Both qualitative and quantitative properties of such models are of interest to scientists. Metaphysical and Epistemological Issues in Complex Systems 107 The equations of a dynamical system are often referred to as dynamical or evolution equations describing the change in time of variables taken to adequately describe the target system. A complete specification of the initial state of such equations is referred to as the initial conditions for the model, while a characterization of the boundaries for the model domain are known as the boundary conditions. A simple example of a dynamical system would be the equations modeling a partic- ular chemical reaction, where a set of equations relates the temperature, pressure, amounts of the various compounds and their reaction rates. The boundary con- dition might be that the container walls are maintained at a fixed temperature. The initial conditions would be the starting concentrations of the chemical compounds. The dynamical system would then be taken to describe the behavior of the chemical mixture over time. 2.2 Nonlinear dynamics and linear superposition The dynamical systems of interest in complexity studies are nonlinear. A dynamical system is characterized as linear or nonlinear depending on the nature of the dynamical equations describing the target system. Consider a differential equation system dx/dt = Fx, where the set of variables x = x1, x2, ..., xn might represent positions, momenta, chemical concentration or other key features of the target system. Suppose that x 1(t)andx 2(t) are solutions of our equation system. If the system of equations is linear, it is easy to show that x 3(t)=ax 1(t)+bx 2(t)is also a solution, where a and b are constants. This is known as the principle of linear superposition. If the principle of linear superposition holds, then, roughly, a system behaves such that any multiplicative change in a variable, by a factor α say, implies a multiplicative or proportional change of its output by α. For example, if you start with your television at low volume and turn the volume control up one unit, the volume increases one unit. If you now turn the control up two units, the volume increases two units. These are examples of linear responses. In a nonlinear system, linear superposition fails and a system need not change proportionally to the change in a variable. If you turn your volume control up two units and the volume increases tenfold, this would be an example of a nonlinear response. 2.3 State space and the faithful model assumption Dynamical systems involve a state space, an abstract mathematical space of points where each point represents a possible state of the system. An instantaneous state is taken to be characterized by the instantaneous values of the variables considered crucial for a complete description of the state. When the state of the system is fully characterized by position and momentum variables (often symbolized as q and p, respectively), the resulting space is often called a phase space. A model can be studied in state space by following its trajectory, which is a history of the model’s behavior in terms of its state transitions from the initial state to some 108 Robert C. Bishop p • (qf , pf ) • (qo , po ) q Figure 1. chosen final state (Figure. 1). The evolution equations govern the path — the history of state transitions — of the system in state space. There are some little noticed yet crucial assumptions being made in this account of dynamical systems and state spaces. Namely, that the actual state of a target system is accurately characterized by the values of the crucial state space variables and that a physical state corresponds via these values to a point in state space. These assumptions allow us to develop mathematical models for the evolution of these points in state space and to consider such models as representing the target systems of interest (perhaps through an isomorphism or some more complicated relation). In other words, we assume that our mathematical models are faithful representations of target systems and that the state spaces employed faithfully represent the space of actual possibilities of target systems. This package of assumptions is known as the faithful model assumption (e.g., [Bishop, 2005a; 2006]). In its idealized limit — the perfect model scenario [Judd and Smith, 2001]— it can license the (perhaps sloppy) slide between model talk and system talk (i.e., whatever is true of the model is also true of the target system and vice versa).

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