Six Senses in which Models Represent

In the context of our discussion, there are two main axes along which the question mentioned above can be developed. Firstly, this question can receive different answers depending on what is meant by ‘good.’ A natural and widely accepted view among philosophers is that a model is good if it is true. Another view, which is more in line with the actual practice of applied mathematicians and engineers, is that a model is good to the extent that it is close enough to truth (i.e., sufficiently accurate). Many philosophers object to the concept of approximate truth; for example, Laudan (1981) claims that it is “just so much mumbo-jumbo.” However, to the extent that approximate truth is to be understood in terms of accuracy, it should not be considered objectionable. Finally, an even weaker criterion—that could be called ‘selective accuracy’— would further restrict the requirement of accuracy to only a few properties of the system that we find ourselves interested with; I consider this to be weaker because it allows for a model to be considered good given a set of questions, and not good given another set of questions.

Secondly, this question can receive two answers that map onto the distinc- tion that we made above between two senses of facticity of models. The first sense of facticity concerns the accuracy of the set of modelling assumptions, and the second sense concerns the accuracy of the model equations. Thus, one can say that a model is good if it is derived from factual (or sufficiently accurate) modelling assumptions, or one can say that a model is good if its model equations are accurate. As a result, we obtain at least six different ways

_

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Criteria

Target

Modelling assumptions Model equations

Truth T-MA T-ME

Accuracy A-MA A-ME

Selective Accuracy SA-MA SA-ME

Table 2.1: Six senses of ‘goodness’ of a model (i.e., a representation).

of answering whether a model is good (see table 2.1). To my knowledge, they do not have names that academics agree on, and so I will merely use acronyms to support the discussion below.

Which sense of ‘good’ will allow us to properly understand the logic of modelling? To begin with, it is important to realize that each of them has a role to play in our understanding of the role and success of mathematics in science. However, to the extent that we are trying to understand when models are good, the discussions that focus on modelling assumptions are not hitting the right target. The reason for which we construct a model is that our interest is not in the modelling assumptionsper se, but rather in deriving model equations that represent the system well. In other words, our interest is not primarily set on what the system is, but rather on what happens in it. In turn, the model equations will be analyzed so that we can extract information to answer questions concerning the behaviour of interest. However, as we noticed above, the two sense of facticity are relatively independent, so unless one already has reasons to believe that the set of modelling assumptions contains the right assumptions, their accuracy will not guarantee the accuracy of the model equations. Consequently, given that we are attempting to understand when models are good, we should focus on the views in the right-hand column of table 2.1.

So, let our subsequent discussion of what a good model is target the model equations. Is truth an appropriate way to understand what a good model is? Firstly, we should emphasize that it is certainly not a bad one, in the following

sense: if a model equation exactly describes what happens in a system,i.e., if it is literally true, then the model provides a good representation of the system. Thus, truth seems to be asufficient condition for a model to be good; however, it is by no means necessary. One the one hand, if truth were required of a good model, then there would not be a widely usable criterion to determine when a model is good. On the other hand, a model is meant to help us answer questions about the behaviour of interest; if it is close enough to the truth to fulfill this function, then it must be considered good. Moreover, as argued, seeking exact truth might impede our ability to know what the model says. Thus, by elimination, the relevant senses in which a model is good is A-ME and SA-ME in table 2.1.

This provides support for the claim made in chapter 1, namely, that an ele- mentary model-theoretic machinery is not sufficient to capture the semantical aspects of model construction and model evaluation. Accordingly, the relevant notion of adequacy of mathematical representations should not be defined only in terms of satisfaction. In the following subsections, I argue that it should rather be understood in terms of perturbation. More precisely, I argue that the semantic evaluation of our models should be characterized in reference to the effects of perturbations on the quantitative and qualitative behaviour of the system. Along this line, I will provide a perturbation-theoretic account of comparative accuracy of mathematical representations that will be seen to be the cornerstone of the justification of the methodological gambits employed in applied mathematics and, as a result, as explaining successes of mathematics in the natural sciences.