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174 WALTER R. STAHL

C. Mathematical Naturę of Modeling Invariants

The different varieties of modeling criteria listed in Table I can be unified from two different mathematical viewpoints. The first of these is the algorithmic-operational one, in accordance with which one may say that a modeling criterion is equivalent to an algorithm (set of rules) for determining when a particular model system is present. For a physical analog of blood flow, this algoiithm might include computation of the value of the Euler number for the model and prototype and checking to see if they are the same. In the case of a Computer model of human medical diagnosis, the comparison algorithm would involve obtaining probability scores or yes-or-no answers for certain inputs of diagnostic information. This approach may appear overly formal, but most papers on biological models do not, in fact, give the algorithm or set of rules which constitutes a definition of the desired model system.

Stating the matter in slightly different form, one would like to define the program for a robot which will determine when two systems (model and prototype or many similar entities) are similar. In a certain sense this approach is similar to a “Turing’s test” (Turing, 1956) by which one attempts to determine if a thinking entity at the end of a teletype-writer is a human or a Computer solely by asking objective ąuestions. Turing concluded that one simply could not distinguish a brain from a computing machinę by such means. This constitutes another example of both the arbitrary naturę of modeling criteria and the need for stating an algorithm by which presence of a model system is to be ascertained.

The second basie mathematical approach to modeling theory is based on abstract algebra and includes numerical mathematics as a special case. Space precludes detailed discussion of “model theory in algebra” as discussed, for example, by Robinson (1963) or in a symposium edited by Freudenthal (1961), but several ąuotations indicate the generał linę of reasoning used in algebraic model studies. Robinson (1963, p. 1) says: “Model theory deals with the relations between the properties of sen-tences or sets of sentences specified in a formal language, on the one hand, and of mathematical structures or sets of structures which satisfy these sentences, on the other.” Later (p. 10): “If all sentences of a set K hołd in a structure M under a correspondence C then we say that M is a model of K (under C)”; and still later (p. 11): “A one-to-one correspondence C between the individuals and relations of a structure M and the individuals and relations of a structure M' will be called an isomor-phism if the relations of any given order correspond to each other, and