Stahl67D bmp

208 WALTER R. STAHL

are intrinsically partial models, as defined by certain particular similarity criteria. The naturę of these criteria varies within the three principal groups of models, namely physical analogs, numerical representations, and cybemetic-algorithmic models. In all cases, however, an objective computation procedurę (algorithm) can be used to define the specific naturę of similarity between model and prototype (or among related systems).

Considerable success has been achieved in making partial models of a large variety of biological systems, but only in very few instances have they allowed prediction of unknown quantitative data. The available simulations are commonly invariant under only a few of all possible pertinent similarity criteria, which is what makes them “models,” but not under a large enough set of such criteria to assure real physical or operational similarity in the generał sense.

Modeling is a synthetic method, contrasted with the purely analytical approach of conventional mathematics, and is freąuently able to provide insight into situations far morę complex than manageable, e.g., by use of differential calculus. Computer modeling of both physical and cyber-netic systems has added broad new capabilities to the synthetic simu-lation approach and promises to be of much value for biological simula-tion in all its varieties.

A model and prototype, or two similar systems, can be transformed or mapped into each other in some sense. During this transformation process some mathematical properties shall be found to be invariant and these properties are precisely those defining the extent of modeling. The mapping jnay be very “practical,” as in the case of numerical scale-up theory, or entirely abstract, as found in relational, topological, and algorithmfc representations of allegedly similar biological systems. In generał, any two systems which are “similar” to each other share at least one invariant of some sort. This may rangę for a ratio of Iengths of sides of triangles for Euclidean geometrie similarity, Reynolds number for ordinary fłow models, relative heart weight for two mammals, or intejli-gence ąuotient or “control score” for a young and an old human. The invariance is maintained during certain stated transformations of space, time, size, age, physiological state, location of functioning, etc.

A. VaLIDITY OF THE MODEL

In a volume on technological modeling theory Murphy (1950) distin-guishes the following types of models on the basis of invariance of pertinent criteria of similarity and quantitative validity of results obtained from the model.