Some results in graph theory and their application to abstract relational biology

Abstract

It has been shown in earlier work that one approach to what Rashevsky has called “abstract biology” is through the study of the class of ($\mathfrak{M}, \Re$)-systems that can be formed in an arbitrary subcategory of the category of sets. The concept of the ($\mathfrak{M}, \Re$)-system, however, depends on the availability of mappings that contain other mappings in their range. It is shown that, by introducing an appropriate measure for this property, the problem of characterizing those categories suitable for a rich theory of ($\mathfrak{M}, \Re$)-systems reduces to a problem familiar from the general theory of graphs. Some new results in these directions are obtained, and it is then shown that any category with mappings that possess properties we might expect to hold in the physical world will also admit a rich theory of ($\mathfrak{M}, \Re$)-systems. In particular, it is shown that a sufficiently large family of mappings drawn at random from such a category will with overwhelming probability contain an ($\mathfrak{M}, \Re$)-system.