Cellular automata, agent-based simulations, and complex sys-

ing, and justifying a model that involves analytically intractable mathematics (e.g., Winsberg 2001, p. 443; 2003, p. 105; Humphreys 1991, p. 501; 2004, p. 107). Following Humphreys (2004, p. 102-104), we call such a model a

‘computational model’. (Frigg and Reiss, 2009, 596)

Both categories are certainly meritorious and illuminating. Both capture the two senses in which philosophers define the notion of computer simulation. While the narrow sense focuses on the heuristic capacity of computer simulations, the broad sense emphasizes the methodological, epistemological, and pragmatic aspects of computer simulations.

The challenge now is to find a way to address the philosophical study of com-puter simulations as a whole, that is, as the combination of the narrow and broad sense as given above. Such a study would emphasize, on the one hand, the represen-tational capacity of computer simulations and their power to enhance our cognitive abilities, while on the other provide an epistemic assessment of their centrality (and uniqueness) in the scientific enterprise. In order to achieve this, I must first set apart the class of computer simulations that are of no interest from those relevant for this study. Let me now begin with this task.

3.2.2 Cellular automata, agent-based simulations, and

Neumann was working on the problem of self-replicating systems, experiencing great difficulty in finding good results. The story goes that Ulam suggested von Neumann use the same kind of lattice network as his, creating in this way a two-dimensional, self-replicator algorithm. These were the humble beginnings of cellular automata.

Cellular automata are extremely simple: they are abstract mathematical systems in which space and time are considered to be discrete. They consist of a regular grid of cells, each of which can be in any number of states at any given time. Typically, all the cells are governed by the same rule, which describes how the state of a cell at a given time is determined by the states of itself and its neighbors at the preceding moment. Stephen Wolfram defines them as:

Mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighborhood of sites around it. (Wolfram, 1984b, 1)

Although this is a general definition, it already suggests the possibility of complex natural systems modeled by cellular automata, instead of the traditional PDE. To Wolfram’s mind, cellular automata are more adaptable and structurally similar to empirical phenomena than PDE.³¹ In a similar vein, Gérard Vichniac believes that cellular automata not only seek numerical agreement with a physical system, but also they attempt to match the simulated system’s own structure, its topology, its symmetries and its ‘deep’ properties.³² Similarly, Tommaso Toffoli entitled a paper:

Cellular automata as an alternative to (rather than an approximation of) differential equations in modeling physics (Toffoli, 1984), highlighting cellular automata as the natural replacement of differential equations in physics.³³ Unfortunately, the meta-physical assumptions behind cellular automata are far from settled. It is not clear, for instance, that the natural world is actually a discretized system, as opposed to a continuous one, like today’s physics describes it, and therefore whether discrete cells accurately represent it.³⁴

On less speculative grounds, it is a fact that cellular automata have little presence in the natural sciences today.³⁵ The reason for their absence is mostly cultural since the physical sciences are still the accepted models for the natural world as we know it (the widespread use of PDE and ODE is the proof of this). Moreover, even for systems of equations that model the empirical world as discrete, cellular automata have little, if any, presence. Despite Wolfram’s efforts to show that the world might be more adequately represented by a discrete point of view, the natural sciences

Another reason why I exclude cellular automata from this study is because they differ from equation-based simulations in terms of solving the underlying model.

While cellular automata provide exact results of the models implemented,³⁶ equation-based simulations experience all sorts of errors in their results: round-off errors, truncation errors, transformations errors, and so forth (see my study on errors in Section 3.3.3). Moreover, since approximations are almost nonexistent for cellu-lar automata, any disagreement between the model and the empirical data can be ascribed directly to the model which realized the theory, instead of the cellular au-tomata itself. This is clearly not the case for equation-based simulations (see my study on verification and validation in Section 3.3.2). As Evelyn Fox-Keller points out,

[cellular automata are] employed to model phenomena that lack a theoretical underpinning in any sense of the term familiar to physics (phenomena for which no equation, either exact or approximate, exists [...] or for which the equations that do exist simply fall short) [...] Here, what is to be simulated is neither a well-established set of differential equations [...] nor the fundamental physical constituents (or particles) of the system [...] but rather the phenomenon itself.

(Keller, 2003, 208)

The above discussion suggests that cellular automata require a methodology and an epistemology of their own, different from the ones needed for equation-based simulations. For these reasons, and for keeping this study as focused as possible on one class of simulation, I exclude cellular automata from the scope of this work.

However, it must be clear that I am not denying the possibility that this work on explanation is also applicable to cellular automata, rather I am only noticing that the existing methodological and epistemological differences from equation-based simulations call for a proper analysis, one that I will not pursue here.

In a similar vein, agent-based simulations and complex systems will be set outside the scope of this study. Here, agent-based simulations are understood as models for simulating the actions and interactions of multiple autonomous programmed agents that re-create and predict the behavior of large systems such as societies, biological systems, and the like. The notion of ‘complex system,’ on the other hand, covers different activities, such as game theory, evolution and adaptation, systems biology, etc. Both classes of simulation share the property of investigating how the total behavior of a system emerges from the collective interaction of the parts of the simulation. Also, the total behavior of these simulations is sensitively determined by their initial and boundary conditions, leading to the representation of unpredictable behaviors of natural systems. To deconstruct these simulations to their constituent

elements would remove the added value that has been provided in the first place by the computation of the set of rules and equations. It is a fundamental characteristic of these simulations, then, that the interplay of the various elements brings about a unique behavior of the entire system. Thus understood, emergence presupposes epistemic opacity for it is impossible to predict the futures states of the system.³⁷ In this way, agent-based simulations and complex systems also require a methodology and an epistemology of their own, just like cellular automata do.

Another reason for rejecting agent-based simulations, complex systems, and cel-lular automata altogether is that, despite isolated cases, different kinds of computer simulations are used in different scientific disciplines. Traditional physics, for in-stance, still relies strongly on continuous equations (PDE and ODE, for instance), which are mechanistic in essence; so do physics-related disciplines such as astronomy, chemistry, and engineering. Cellular automata, agent-based and complex systems are more prominent in social and biological sciences, where the behavior of agents is better described and analyzed. Naturally, there are overlapping uses of computer simulations in the sciences: cellular automata have proven successful in traditional physics on phase transitions, replacing the use of PDE.³⁸Conversely, the implemen-tation of cellular automata by PDE is also under study.³⁹ More examples can be found in the use of PDE in population biology, as the Lotka-Volterra predator-prey model shows.⁴⁰

Perhaps the best reason for excluding cellular automata, agent-based, and com-plex system from this study stems from the minimal requirements needed for a successful explanation by computer simulations. In other words, in equation-based simulations the explanans can be reconstructed directly from the simulation model since its computation does not add value to the results. In any of the other classes of computer simulations, the interplay of the various elements during the computation must be considered as part of the explanans, for they are part of the success of an explanation. To analyze exactly the leverage that such an interplay (emergent behavior, etc.) has in the explanation of data, and how this affects and influences the construction of the explanans, is a matter that exceeds the limits of this work.

All that it is possible to say is that one could not expect a successful explanation when epistemic and methodological specificities of the simulations are left out. This exclusion does not entail, however, that a proper understanding of these computer simulations could bring together all classes of simulations under the same study on scientific explanation.

Let me now discuss equation-based simulations in more detail.