Model explanation

by pointing out that “any partition of the plane into regions of equal area has a perimeter at least that of the regular hexagonal honeycomb tiling” (Hales, 2001, 1), or “the existence of a honeycomb tiling depends on the shape and size of the lattice”

(Hales, 2001, 7). Thus, the hexagonal tiling is optimal with respect to dividing the plane into equal areas and minimizing the perimeter.

This fact, known as the ‘honeycomb conjecture,’ shows how mathematical ex-planation of physical phenomena and classic scientific exex-planation depend on each other. It also motivates the claim that a successful theory of explanation for com-puter simulation should account for both of these kinds of explanation, given the very nature of computer simulations themselves. Here lies another reason why I be-lieve that the unificationist account is the most suitable candidate for simulations.

According to this latter account, scientific explanation is treated in the same logical way as mathematical explanation of physical phenomena, merging into one account two allegedly different kinds of explanation (recall my first objection to Hempel’s de-ductive nomological model). In this vein, simulated phenomena could be explained regardless of their mathematical nature or their physical characteristics. I discuss the unificationist account in more detail in Chapter 5.

Let me turn to another interesting account of explanation: model explanation.

Given the model-like nature of computer simulations, it is also natural to envisage it as a possibility.

elab-orates on an additional distinction between ‘how-possible models’ and ‘how-actually models’ as a way to identify models that onlypurport to explain and those that ac-tually explain. The difference is mostly grounded in the metaphysical assumptions built into the model. A ‘how-possible model’ would loosely describe the behavior of the underlying mechanism that produces the phenomenon, whereas ‘how-actually models’ would describe (and correspond to) real components, relations, and so forth in the mechanism that produces the phenomenon.⁶⁰ A genuine model explanation, therefore, is one where the model correctly and accurately reproduces the actual mechanism of the target system (i.e., a ‘how-actually model’). To Craver’s mind: “a mechanistic explanation must begin with an accurate and complete characterization of the phenomenon to be explained” (Craver, 2006, 368).

Bokulich objects that these requirements for a model to be explanatory are too strong, for one rarely has a complete and accurate description of the phenomenon to explain.⁶¹ This is a fair objection that concerns not only all the theories of ex-planation for models, but also a theory of exex-planation for computer simulations, where a complete and accurate representation of the target system is rarely, if ever, achieved (or at least to the degree demanded by Craver). In addition, Craver can be interpreted in the context of Kim’s internalist/externalist distinction as taking an externalist relation grounding the explanation. Indeed, to Craver’s mind, in order to have a successful explanation, a “complete characterization of the phenomenon”

(Craver, 2006, 368) must be provided; that is, there must exist an objective corre-spondence relation grounding the explanation. The mechanistic model explanation is, by definition, explanatory externalist and, therefore, of no use for computer sim-ulations.

The covering law-model explanation, as elaborated by Mehmet Elgin and Elliott Sober, is restricted to models in evolutionary biology known as ‘optimality models’;

that is, models that describe the value of a trait that maximizes fitness, given a set of constraints.⁶² The most significant drawback of the covering law-model explanation is its time dependency on actual models of evolutionary biology. Computer simula-tions require a more adaptable theory of explanation, one that allows for time- and science-independence; that is, one whose explanatory relation does not depend on the way science conceptualizes the empirical world in one moment in history, but on general descriptions of the relations holding between explanans and explanandum.

Hence, the covering law-model fails to endure as a good candidate for computer simulations.

The hypothetico-structural explanation, as elaborated by Ernan McMullin, ex-plains the properties of a complex entity by postulating an underlying structural model, whose features are causally responsible for the properties to be explained.⁶³ In an explanation, the original model must successfully capture the real structure of the object of interest. For asuccessful explanation, however, one must first jus-tify the model by de-idealizing it, that is, ‘adding back in’ those features that have been omitted by the process of idealization.⁶⁴ Now, it is precisely this process of de-idealization that makes the hypothetico-structural explanation unappealing for computer simulations. One interesting use of simulations is their capacity to rep-resent systems about which we might only have some theoretical knowledge, but no real empirical content to ‘add back in.’ As if that were not enough, McMullin’s model of explanation would beg the question of how to ‘add back in’ features to intricate and complex systems such as computer simulations. I then conclude that the hypothetico-structural explanation presents unattainable demands for computer simulations.

Finally, Bokulich’sstructural model explanation is a modification of James Wood-ward’s theory ofcounterfactual dependence, although applied to models. According to Bokulich, for a model to be successfully explanatory, three conditions must hold:

First, the explanans makes reference to an idealized or fictional model; Sec-ond, that model explains the explanandum by showing that the counterfactual structure of the model is isomorphic (in the relevant respects) to the counter-factual structure of the phenomenon. This means that the model is able to answer a wide range of “what-if-things-had-been-different” questions. And third, there is a justificatory step specifying what the domain of applicability of the model is and that the model is an adequate guide to that domain of phenomena (Bokulich, 2011, 43).

It is the second condition that seems to contribute to a new view on model expla-nation. However, it entails two problems that Bokulich needs to give an answer to:

first, the nature of the isomorphic relation between the model and the target system is contentious. Although Bokulich is careful enough to say that this notion is being used in a loose sense,⁶⁵ it is not clear how isomorphism can be handled in such a way that it does not run into the typical problems of isomorphic representation.

Indeed, isomorphic relations are objected to on grounds of not being sufficient (i.e., representation might fail even if isomorphism holds) nor necessary for representa-tion (i.e., representarepresenta-tion is possible even if isomorphism fails), and for not having the right formal properties (i.e., isomorphism is symmetric, reflexive, and transitive while representation is not), among others. Speaking of isomorphism in the context

Secondly, the ‘what-if-things-had-been-different’ kind of questions that Bokulich needs to be answered are inspired by Woodward’scounterfactual dependence, which can be interpreted in the following way: the explanation must allow us to see what would have been different for the explanandum if the factors cited in the explanans had been different.⁶⁶ It is not clear, however, that such an account of explanation is viable for computer simulations. Allow me to elaborate.

Let me begin by illuminating the notion of counterfactual dependence using Woodward’s example. Consider an explanation of why the magnitude of the electric intensity (force per unit charge) at a perpendicular distance,r, from a very long fine wire with a positive charge is given by the expression E = _2⇡"¹

0r (Coulomb’s law), where is the charge per unit length on the wire and E is at right angles to the wire.⁶⁷ Woodward’s proposal is that explanation is a matter of exhibiting systematic patterns of counterfactual dependence, that is, what would have happened if the initial and boundary conditions had been different.⁶⁸ To illustrate this last point consider that the above expressionE shows how, altering the charge per unit , one obtains either a repulsive or an attractive system. For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while for a negative charge the direction is the opposite. Therefore,E above shows how the field would change if the charge along the wire, the distance, or any other initial or boundary condition were increased or decreased (among other interventions that one can perform on the system). “We learn from [this example]”, says Woodward, “that despite the fact that Coulomb’s law is an inverse square law, the field produced by a long straight wire falls off as the reciprocal rather than the square reciprocal of the distance from the wire” (Woodward, 2003, 191).

The problem with this approach is simply a matter of practicality. It is humanly impossible to explain a simulated phenomenon by exhibiting what would have hap-pened if the initial and boundary conditions of the computer simulation had been different. Scientists run simulations precisely because they are unable to anticipate the results, and are therefore unable to know what would have happened if condi-tions had been different. Believing otherwise is to misinterpret the uses of computer simulations in the sciences. I conclude that neither Bokulich’s model of explana-tion nor Woodwards’ counterfactual dependence are suitable accounts of scientific explanations for computer simulations.