Abstract
In this paper, we argue that rather than exclusively focusing on trying to determine if an idealized model fits a particular account of scientific explanation, philosophers of science should also work on directly analyzing various explanatory schemas that reveal the steps and justification involved in scientists’ use of highly idealized models to formulate explanations. We develop our alternative methodology by analyzing historically important cases of idealized statistical modeling that use a three-step explanatory schema involving idealization, mathematical operation, and explanatory interpretation.
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Notes
In this paper, we follow several philosophers in characterizing abstraction as the “leaving out” of details and idealization as the intentional introduction of distortions or falsehoods into a model or theory (Weisberg 2007b). For a more detailed discussion of this relationship see (Rohwer and Rice 2016).
We put things in terms of “formulating an explanation” because we want to be careful about equating the model itself with the explanation. Instead, we think of model-based explanations as explanations that necessarily appeal to a model in the explanation (Bokulich 2011, 2012). In addition, we want to investigate the use of idealized models to discover or reveal explanatory information before scientific modelers have a complete explanation on hand. In these cases, the model is used to discover and construct an explanation, but it may not be an explanation on its own.
A notable exception to this lack of focus is Hughes’ DDI account (Hughes 1997).
We do not have a more general account of justification to offer here—that is beyond the scope of this paper. We are simply using justification to mean having good reasons for performing various steps in the modeling process. Thus, we are focused on justifying various actions and inferences used in scientific modeling rather than justifying particular beliefs. Moreover, we are not discussing justifying the belief in a scientific explanation (i.e. confirmation) warranted by reasoning such as inference to the best explanation.
For additional details on the historical significance of the Galton case see Stephen Stigler (2010).
This isn’t to say that these models might not also be used for other purposes (e.g. prediction), but these modelers are explicitly interested in providing explanations and the cases are widely held to be explanations by philosophers and scientists alike. If one wanted to argue that these cases are nonexplanatory, we think the onus is on philosophers to account for why they are so widely held to be exemplary explanations.
Michael Weisberg has several papers investigating this kind of question in his excellent discussion of what makes a scientist a modeler (Weisberg 2007b).
It is important to note, however, that this statistical independence does not require that the individual-level events also be causally independent. Indeed, the events do not even have to be causal events. For this reason, instead of interpreting Galton as an instance of causal abstraction, or an attempt to isolate particular causal factors, we contend that what Galton is doing is using an idealized model as a proxy for the entire suite of causal processes involved in heredity.
It is important to note, however, that while Galton and Quetelet were interested in similar target explananda, their explanations for that kind of explananda ended up being quite different because of how they interpreted the statistical results (Ariew et al. 2015).
More specifically, “Galton did not strictly deduce it, but rather demonstrated it by the device of his shot-dropping machine, the quincrunx, in which an analogy of this effect could be observed. That led him to the remarkable thought: the phenomenon that puzzled him could be deduced from the fact (or assumption) that traits were distributed according to the standard statistical law, the law of errors” (Hacking 1990, p. 186). For more details of Galton’s demonstrations see (Hacking 1990; Stigler 2010).
This corresponds to what Batterman and Rice (2014) call having the model and the real-world system within the same universality class. In its most general form, universality is just a statement of the fact that drastically different physical systems will display similar patterns of large-scale behavior that are largely independent of the details of their physical components and interactions (Kadanoff 2013).
Of course, several other conditions are also required in order to derive the central limit theorem; e.g. the random variables must all have finite variance. For the sake of simplicity, we only focus on the two conditions discussed above since we think the justification for these other assumptions will parallel the justification discussed above.
This is just one example of the kind of change that could be explained by appealing to the ideal gas law. However, what scientists really want to explain with the ideal gas law are the ensemble-level regularities across many different changes in the pressure and temperature of gases.
This equation is a statistical model because pressure and temperature are ensemble-level averages that are defined within the framework of statistical mechanics.
It is also worth noting that, as was the case with Galton’s use of statistical laws, this application is not an instance of causal abstraction—i.e. abstractly describing an isolated causal process. Instead, the distribution is a proxy for the total suite of underlying events that need not be causal events.
We should also note that while Morrison calls Fisher’s model abstract, she uses mathematical abstraction to mean what other authors have called ineliminable idealizations (Batterman 2002; Batterman and Rice 2014; Rice 2012, 2015). For more details, see (Morrison 2015, p. 21). Indeed, like the other two cases, there is no causal abstraction going on here. Instead, Fisher is using his idealized model population as a proxy for evolving systems.
The law of large numbers states that as the samples size increases, the average of the quantities sampled will be increasingly closer to the expected outcome of those quantities.
Furthermore, as additional applications of the model are found, more and more justification for employing those idealized models to develop explanations of similar statistical phenomena is accumulated
Of course, showing that the model is predictively accurate in the way described in step (2) might be an essential part of providing an explanatory interpretation of the model in step (3). Despite this connection, we think it is useful to separate the predictive accuracy of the model from the identification of the minimal conditions on which the explanandum counterfactually depends. Thanks to an anonymous reviewer for helping us clarify this point.
Indeed, in our first historical case from Galton it is difficult to distinguish the three steps from one another. This is not particularly surprising, however, because Galton was developing novel techniques that would later be analyzed in more detail and applied in a wide range of cases.
The steps involved in developing these explanations are: (1) demonstration that most of the features of a class of systems are irrelevant to their large-scale patterns of behavior, (2) using that information to delimit a universality class of systems that will display that behavior, and (3) showing that the idealized model and the real-world target system(s) are both within that universality class.
Indeed, many of these derivations are mathematically necessary. Therefore, to deny them would be irrational. Thanks to an anonymous reviewer for helping us see this point more clearly.
It is important to note that, while a counterfactual dependence will sometimes hold in virtue of a causal dependence, it does not require a causal dependence; e.g. a phenomenon could counterfactually depend on a statistical feature of the population such as is mean, variance, or population size.
Thanks to an anonymous reviewer for pressing us to be clearer about this point and the challenges with discussing approximation relations.
For example, Hempel’s account allows for shadows, together with some laws of optics, to explain the height of flagpoles (Bromberger 1966).
While there may be multiple ways to satisfy those criteria—e.g. Hempel discussed both deductive nomological and inductive statistical explanations—the Hempelian project is, essentially, to attempt to identify the necessary and sufficient conditions that distinguish explanations from nonexplanations.
Thanks to an anonymous reviewer for helping us clarify this point.
In addition, Lange argues that there are distinctively mathematical explanations (Lange 2012).
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Acknowledgements
We would like to thank James Woodward, Robert Batterman, and two annonymous reviewers for there helpful feedback on previous versions of this article.
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Rice, C., Rohwer, Y. & Ariew, A. Explanatory schema and the process of model building. Synthese 196, 4735–4757 (2019). https://doi.org/10.1007/s11229-018-1686-y
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DOI: https://doi.org/10.1007/s11229-018-1686-y