Stahl67C bmp

THE ROLE OF MODELS IN THEORETICAL BIOLOGY 207

(theories) imply the same invariants, e.g., a group invariant, they are said to be “models” of group theory, i.e., the particular symbolic con-struct is isomorphic with the generał one describing group properties.

Biological Axiomatics

Abstract symbolic modeling of biological systems generally commences with an attempt to axiomatize their behavior or to construct a “theory” relating various symboLs representing certain properties of the system. This allows one to obtain seąuences of symbols (sentences) characterizing systems of a related type. The form of these seąuences or sentences is then compared from the viewpoint of abstract mathematical properties.

Pioneering efforts to axiomatize certain properties of biological systems were madę by Woodger (1937, 1962) in respect to Mendelian genetics and other aspects of heredity. This same problem was recently analyzed by Lyapunov and Malenkov (1962). A dedication volume to Woodger, edited by Gregg and Harris (1964), deals with many basie ąuestions of biological axiomatic models.

In a long series of works Rashevsky (1965) and Rosen (1958, 1963a,b) have attempted to apply abstract relational algebra and topology to elementary biological problems. The concept of abstract mathematical mapping has been specifically related to self-organizing systems by Leibovic (1963). The author (Stahl, 1965b) discusses algorithmically unsolvable problems in the context of a celi model based on. Turing automata.

A book is now available on the application of modem algebra to biol-ogy (Nahikian, 1964), and it may be expected that further kinds of formal “algebraic models” shall be found to describe the orderly properties of biological systems. A good deal of the work of Thompson on biological symmetries can be subsumed under group theory, as can the whole ąuestion of integrated physical scaling of many physiological functions. Thus, abstract algebra has already proven to be a very important tool for biological modeling theory.

XII. The Role of Biological Modeling

A very large number of reports referring to biological modeling have been published and the annual publication ratę of such reports is in-creasing rapidly. Nearly all bibliographic citations in this chapter (about 285 items) contain the words model or simulation in their title; they represent not over one-third of the available pertinent reports.

There is no “perfect model” except an identical copy, and all models