Appendix: Axiomatics

Summary

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell, “Recent work on the principles of mathematics”

The order of presentation in this book has been historical rather than rigorously logical. It is only now, after we have introduced the particular types of models, that we define their foundations and discuss the general concept of a mathematical model. In the course of this discussion we will develop the axiomatic basis of the concepts relation, network, and group, and define the equivalence of two models, called isomorphism, and the simplification of one model by another, called homomorphism.

Every utilization of mathematics in the preceding chapters as a theoretical framework for topics in anthropology has been an instance of a mathematical model, which is now defined precisely. For this purpose, we must develop the notions of an axiom system and of a model of an axiom system.

Axiom systems

The quote from Bertrand Russell that begins this chapter is explained by the following definition of an axiom system. In order to avoid circular definitions, in which ultimately a concept is defined in terms of itself, certain basic terms called primitives must be deliberately left undefined, and then all other concepts are constructed on this definitional base. Similarly, in order to avoid circular reasoning in which a statement is utilized as a reason for its own validity, certain basic assertions called synonymously axioms or postulates are assumed to be true without proofs.