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Complexity and Scientific Modelling

3 A Framework for Analysing Modelling

To frame the discussion of complexity, I will outline a model of modelling.

Firstly it is important to distinguish between the form and predictive meaning of such models. The models themselves are always held in some form. The set of such possible forms can be considered as a language in its broadest sense - frequently it may correspond closely to an actual language, either natural or formal. I will call this the modelling language. Such models are amenable to some form of inference, in that they can be used to predict some property given some other information (even if sometimes some of the necessary information is only available after the predicted event). At least some of the information comes from what is modelled in the form of measurements. The models correspond, loosely, to what they model via these predictions.

Thus we distinguish two aspects of a model: its form and the correspondence between possible information and the predictions that one could infer from it. This set of information along with the respective predictions can be thought of as defining a subspace of the space of all relevant possibilities. I will call this subspace the model's semantics, because one can draw an analogy between a logic's syntax and its semantics in terms of the set of logical models a statement is true for.

The primary way in which these models can be judged is by the degree of correspondence between what is modelled and the predictions of the models - its error. This however does not rule out the default model, that "anything can happen". Such a model is always trivially correct (and thus is typically chosen as a starting point). Thus we also need an additional goal of preferring the more specific (or refutable) model. I will call this the model's specificity. A modeller with infinite resources and time need only use these two measures as guides in its choice of model. In some cases, of course these dual aims might be in conflict. In a given modelling language one might be forced to choose between a vague but accurate model and a specific but more erroneous one. In some accounts the specificity of models are sometimes left out of analyses of modelling because the types of modelling languages considered are inherently precise.

For us more limited beings, with very distinct practical considerations, the complexity of our models become important. As we shall see below we have to balance the complexity, the error and the specificity of out models. Note that here, complexity is a property of the form of the models while the error and specificity are properties of its corresponding model semantics.

Complexity and Scientific Modelling - 04 APR 97
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