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THE ROLE OF MODELS IN THEORETICAL BIOLOGY 175

if, whenever a relation holds between corresponding individuals of M it also holds between individuals of M', and vice versa.”

The approach of Robinson and also that of Suppes or Apostel (cf. Freundenthal, 1961) treats all mathematical expressions as “sentences” in an algebraic language, and presence of a model is determined by isomorphism of the algebraic structures describing the model and proto-type. It should be kept in mind that one can never really compare a model and prototype in an abolute sense. Always interjected into the comparison is some “theory” or rationale for making observations, taking measurements, and writing eąuations. In this sense one really compares the “theories” covering the model and prototype, which may be in the form of eąuations with associated numerical invariants, sets of test “words” acted upon by automata, theoretical abstract eąuations on paper, or Computer programs simulating yarious kinds of systems. On this matter, Suppes comments-(in Freudenthal, 1961, p. 171): “To establish a representation theorem for a theory T is to prove that there is a class of models of T such that every model of T is isomorphic to some member of this class.”

Suffice it to say that modem mathematical theory can encompass all entities commonly called “models” in biology or technology and even the distorted transformations of skuli shapes for various species shown in Thompson (1959). There has been considerable recent interest in the philosophical problem of modeling in the Soviet Union (Glushkov, 1963a,b; Venikov, 1961; Novik, 1964). Soviet biological modelers have emphasized cybernetic models rather than materiał physical ones, but have taken mathematical isomorphism in one form or another as the most basie definifion of a model.

The application of mathematical modeling concepts to biology will be madę elear by a number of examples from the published literaturę dealing with the construction of various systems that simulate animal function.

III. Physical Analogs of Organisms

Physical analogs of biological systems include the following: (1) Animals, or natural models, used as substitutes for humans (see Stahl 1963a,b, 1967), (2) physical models madę of artificial materials, or in some cases using parts of natural organs, e.g., a reał aorta, tested arti-ficially under conditions of physical similarity; (3) electrical analogs which simulate physical analogs and are based on physical similarity eriteria, but substitute electrical parameters for physical ones; (4)