A model is not a ‘thing in itself’ but is ‘about’ something – otherwise it is just a computer program or a set of equations. A formal model is a system of inference (logical or computational) plus a relationship between parts of the model and what is being modelled. Some models are ‘analogical’ – they are used to support discursive understanding of the world. In these, the mapping is informal and up to each listener to construct. Analogies (of all kinds) are essential for thinking about the world and what to do, especially when dealing with situations or phenomena we do not fully understand. Others are more empirical, where the mapping is well-defined from the model to specified kinds of data.
With empirical models, the mapping between the model and the relevant observation-derived data is crucial. When using such models there are three stages; (a) one uses the mapping to get from some data to the model (b) use the model for some inference and (c) use the mapping again back to data (Hesse 1963). This includes both predictive and explanatory uses of ABMs (although the sequence of use is different in each of these). These are distinguishable from abstract uses because there is a precisely described relationship between the model and the data, this is not flexible on a case-by-case basis. In these cases, the reliability comes from the composite (a-b-c) mapping, so that simplifying step (b) can be counterproductive if that means weakening steps (a) and (c) because it is the strength of the overall chain that is important for these uses (Edmonds 2010).
This contrasts to using a model for analogical purposes since the human mind is very creative at inventing connections from even simple models onto new and complex phenomena. This might be why we like simple and abstract models, because our minds will create the mappings automatically and sometimes unconsciously, obscuring the complexity of this relationship and easing our conscious cognitive load. Thus, analogical models can give an illusion of empirical generality that either falls apart when one tries to make it precise or requires substantial changes in order to make the model applicable to each case.
The example of the development of models in quantum mechanics illustrates the difference between analogical and empirical models. Although the mental models and stories about quantum mechanics have changed over the years (e.g. from the Copenhagen interpretation to a realist or many-worlds interpretation), the formal equation-based models have developed around a stable core. This stability comes from their extraordinary empirical success, a success that remains despite the considerable “cognitive dissonance” that these theories cause. This example shows that, at least sometimes, the development of the formal models, driven by empirical adequacy, can be more important than the attendant abstract models used to get a feel for what is happening (ABMs have not been around long enough for a clear example of this to emerge) Formal models have the right characteristic for a evolutionary ‘replicator’. That is, a pattern that (a) is reliably and exactly replicable, (b) can be copied and varied (e.g. by a community of researchers) and (c) can be selected according to some criteria (e.g. their success at intended purpose). Thus, unlike ideas or “theories” whose composition may not be precise, models can provide the substrate for a process with the necessary properties for evolution (Hull 1988). This could form the backbone for a useful science, developing a menu of models that work in practice, regardless of their intellectual appeal or accessibility. Around this core may be other models whose purpose is to help think about the empirical models (and by extension the phenomena) in a looser, analogical manner. These are necessary since we do need ways of thinking about what we are doing and why, but these are one step removed from the core empirical processes, just described, whose nature and purpose are not analogical.
Under this evolutionary view (of models rather than knowledge), formal models are essential to science, not because their kind of representation is innately suited to what it represents, but because it is supports an evolutionary process of non-mental (formal) representations. ABMs are more expressive than analytic mathematical models because they do not have to be simple enough to be solvable in terms of closed form solutions. Rather, like the invention of multi-cellular organisms in the Cambrian explosion, allow a wider universe of distributed and complicated structures that are difficult to summarise with equations.
Such an evolutionary picture of models depends critically on a number of aspects. If what is replicated is not a formal object but something more fluid that is re-interpreted in slightly different ways by each recipient, then the ‘noise’ of variation between versions can easily overwhelm any progress by evolution. More crucially, how the models are selected drives what results. If models tend to be copied due to empirical success then the process will tend to produce models with this property, but if models are copied because they are interesting and simple then there is no reason to suppose that the result will be any increase in empirical relevance over time.