Modelling in Natural Sciences

In Chap. 1 we demonstrated the ample range of the meaning of the term model. In spite of the diversity of those examples they all share some characteristics; for instance, they are all linked to a modelling process. In some of those examples the models are prototypes and subject to this process, in others this mapping produces the model as a simulacrum.

The examples compiled in Sect. 1.1 show that models are connected with an imaging process, which either results in the model as an image (simulacrum) of an underlying entity or produces an image as a copy of the model in question. In that case the model is the prototype of its copy. These relations are depicted in Fig. 3.1 (resembling Fig. 1.7). The arrow in the diagram represents the mapping or the imaging process. This mapping produces an image; the object of the imaging process, the projected entity, is called the counter — image of the mapping.

The examples in Chap. 1 show that models are images constructed in many different ways and that they are developed to serve various purposes. In Chap. 2 we have given a more detailed definition of the structures involved in the process of modelling. We have described them as systems, and we have concentrated on the dynamic and open systems as most relevant to modelling. In the preceding chapter we have finally studied the fine-structure of modelling processes as well as those of the modelled entities and the models themselves more closely.

When we speak about ‘the art of modelling’ ¹, there is the idea at the backs of our minds that modelling is a creative process (cf. Sect. 2.4). As creativity is a talent and cannot be learned or taught, it seems to be difficult to treat it in a textbook. The artist learns by experimenting, creating works of art, rejecting them if they are not to his satisfaction or trying to improve them so that they come up to his expectations. We for our part use different methods of working; they are more technical than those of an artist, and thus we replace creative experimenting and practice by the description of some select experiments already carried out. These descriptions are precise case studies of models designed to serve various purposes in different fields of research; they are compiled in the appendix to this book.

As we have seen, validating a model mirroring reality means to compare it with its co — model, the counter — image which contains the relevant details reflected by the model in question. The validity of a model thus is a means to evaluate the quality of the correspondence between reality and its image represented by the model. As a measure validity is determined in degrees, and so the quality of a model depends on its degree of validity. This means that the conclusion that a model is accepted as being ‘good’ or that it is rejected as being ‘poor’ depends on this degree. Conclusions are the results of inference based on the available evidence and they are subject to specific laws. There are very rigid and formalistic rules of deducing conclusions from a set of assumptions; this is the way the mathematician proves his propositions and the philosopher founds his statements.

The dependability of a model is determined by its validity, the way it is well-founded. The quality of the foundations of a model becomes obvious in certain degrees depending on the evidence available to the modeller. There may be observations in favour of a model, thus supporting and confirming it, and there may be observations not in concordance with a model. The latter will certainly reduce the belief in the dependability of a model and cast doubt on it; in the worst case its futility will be proved and it will be refuted. This information will help us to induce a belief in the dependability of a model, which can be measured in degrees. This does not mean that we trust to a model blindly, but that our evaluation is founded on evidence and induced from experience gained from experiments. It is thus a degree of rational belief and may be quantified in terms of probability.

As we have seen in the previous chapters, the dependability of a given model is a rather relative measure. At best it depends on intersubjective agreements, at worst it must be based on entirely personal estimations if normative criteria are not at hand. At the same time modelling and the quality of the imaging process leading to the model are influenced by the kernel of the mapping — this is all the information getting lost when the real system under consideration is mapped into the model. Any statement about the dependability of a model is probabilistic and thus affected by uncertainties. To some degree the loss of information due to mapping as well as the stochastic process of the evaluation of a model are founded on the evidence available to the modeller and to the relevant scientific community. We thus have to grant a certain tolerance to the performance of a model when we compare it with its counter — image (cf. Subsect. 2.4.3), and we will even try to characterize it by degrees itself.

As soon as a model has been developed, there is the necessity of testing it. These tests are performed from various points of view. Each of these aspects has been subject of very particular terminologies. Some of the frequently used terms are quite ambiguous, and to some degree the specific interpretations within the various fields of sciences seem to be contradictory; the same also goes for the terminology applied to different stages of testing a model. In order to avoid any confusion based on semantic or pragmatic considerations when testing a model, we give some very formal definitions in order to mark out the scopes of the testing procedures; some of them aim at the objectives of testing and will mostly be derived from the usage of terms in logic. We admit that these definitions will neither be disjunctive nor complete, but they help to avoid some misinterpretations of terms generally applied.

We have already touched on the term validity of a model in various sections of this book, and we have emphasized that it is a means to measure the degree of dependability up to which a model is well-founded. This interpretation is justified by the etymological roots of the word; and if we agree to use the term in that sense it is not an absolute measure, but rather refers to a quality of a model characterized by degrees of validity. This is due to the fact that the foundations of scientific propositions (theories, hypotheses or models) are very subject-specific and vary from one field of research to the next, and even within each of those fields they depend on the stage of scientific progress and on the available evidence.

In addition to the references already quoted in the preceding chapters we would like to mention some further select articles and books dealing with the subject. They facilitate the access to various aspects of modelling and contain thorough bibliographic references.