Complexity and Scientific Modelling

There is an understandable wish to measure the complexity of real systems rather than just of models of systems, but if natural systems have inherent levels of complexity they are beyond us. In practice there is no practical upper bound on their complexity as one has only to consider them in more detail or by including more aspects. On the other hand, the effective complexity of systems does depend on our models - the exact motions of the planets may be puzzling when you have to describe them in terms of epi-cycles but much simpler in terms of ellipses.

It may be objected that even if such "real complexity" is so intractable, one can still make comparative judgements; i.e. it is natural to judge some natural systems as more complex than others. An example of this is the claim that a cell is simpler than a whole organism. However in defending such judgements one is always forced into relating that which is compared within a common model. It is only once you have abstracted away what you consider to be irrelevant details, that such judgements of relative complexity become evident. It may be argued that some such models and frameworks are privileged but this pre-judges decisions about relevance and so is not helpful to an analysis of complexity and its place in scientific modelling.

In any case complexity is more critically dependent upon the model rather than what is modelled. So we will approach it from this point of view and leave the reader to judge whether this distinction is fruitful in understanding the processes involved.

For our purposes we will define complexity thus:

"The difficulty associated with a model's form when given almost complete information about the data it formulates."This is a special case of the definition given in [5].

The relevant type of "difficulty" depends somewhat upon your goals in modelling, but here it will indicate the difficulty in finding the model in a search starting at the smallest model forms. This could be size, depth or some indication of the computation that is necessary to discover it.

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