Since what seems to be the first treatment in the 1882 [M] as an “algebra”, the topic now known as “kinship algebra” has focused on the correct composition of strings of symbols used to form empirical systems of kinship terms, as used in naturally occurring languages. The papers and Comments here, especially [B, R2, WD] have summarized much of the history since that initial treatment. While their principal focus is on a particular set of kinship terminologies which have come to be known as Dravidian, the papers and discussion raise a number of issues on the basic form of, and purpose of, the use of mathematics in developing cultural theory. [B], following the extensive discussions in [T], recognizes the nonassociative context of natural languages and of kinship generally, but refers to associative algebras for kinship, while [R2] seems to simply assume the necessary forms are associative. [B] applies group theory, as have many others summarized in the citations of these papers; [WD] evaluates that use. Finally, the Comments also focus on “careful ethnographic description” [WD].

We examine a theory of cultural rules as mathematical transforms. Certain cultural rules may be represented as set functions (called here “transforms”) between possible structures (called here “configurations” denoted “C”) on generations of an evolutionary sequence. If R is a rule and R its transform, the outcome of R acting of a starting configuration C is a set denoted RC of possible configurations. The smallest fixed point of the transform R of a rule R (called the “minimal structure” of that rule) is the descriptive diagram for illustration of the operation of certain rules traditionally used by ethnographers. A combinatorial density computing certain key population statistics of a cultural system is derivable from the minimal structure of the rule, enabling empirically testable (and successfully tested) predictions of observable population measures on systems using that rule. Therefore we may conclude that cultural structure and the uncertainty inherent in cultural systems are but two parts of one framework. Cultural theory thus has a structure in some ways like that of quantum theory, and is a physically testable physical theory. But quantum theory has been under development for a century. The task for a comparable cultural theory is simply to get started.

Over the last several decades, anthropology has created testable theory comparable to that found in other sciences. This paper summaries one view of that theory, the perspective of rule bound systems theory, or RBS. RBS theory turns out to imply that much of what has been taught and applied as “social statistics” for the least several decades, is based on the wrong combinatorial density function, which in turns helps explain why social forecasting is so often simply wrong. RBS however has made several tested correct forecasts, including in areas that have proven resistant to previous forecasting. RBS theory also has testable implications on broad scale questions of evolutionary biology.