Paths of Cultural Systems

Abstract

A theory of cultural structures predicts the objects observed by anthropologists. We here define those which use kinship relationships to define systems. A finite structure we call a partially defined quasigroup (or pdq, as stated by Definition 1 below) on a dictionary (called a natural language) allows prediction of certain anthropological descriptions, using homomorphisms of pdqs onto finite groups. A viable history (defined using pdqs) states how an individual in a population following such history may perform culturally allowed associations, which allows a viable history to continue to survive. The vector states on sets of viable histories identify demographic observables on descent sequences. Paths of vector states on sets of viable histories may determine which histories can exist empirically.Keywords: quantum logic; groups; partially defined algebras; quasigroups; viable cultures Ethnographic FoundationThe structures described here and their consequences imply much of what may be predicted about empirical cultures. Anthropologists very often draw illustrations of structures using methods discussed here, but based on intuition, thus have little notion of what their commonly used diagrams might predict. While [1,2] defined mathematical means to describe the current and future demographic organization of lineage organizations, with empirical examples, we here specify the demography of kinship-based systems [3][4][5][6][7][8] with some related definitions in our Appendix A; and empirical examples in [9][10][11][12][13]. We follow the inspiration of [14]. In an empirical culture many other relations may also occur; we note some of those in our Part 7, Discussion at the end.Our notion of studying viable minimal structures-which are the smallest minimal cultural structures that can "reproduce" the ascribed social relations in one generation-follows from [15]. Our history describes how sustaining those relations allow the culture to reproduce the rules. Cultural rules describing histories may be stated in natural languages, which label the individuals in a descent sequence with a subset called a kinship terminology. Our term viable embodies what anthropologist Radcliff-Brown called "persistent cultural systems" ([10], p. 124). Radcliff-Brown and others often described histories using discrete generations, as do we. Empirical cultures are almost ubiquitously described by many anthropologists using viable histories, typically represented in an ethnography by its minimal structure. The nearly ubiquitous presence in ethnographies of viable histories implies they may be the only observed histories.An example is Figure 1 (whose source is [9]) which is an actual viable minimal structure of a history (that is, a persistent cultural system). The triangles in the illustration are males, the circles are females, descent moves in the downward direction, the labels are names used in the kinship terminology. The sign "=" means "marriage" between the two individuals attached to it; the "=" on the far right in the illu...

Inheritance is a Surjection: Description and Consequence

“…Thus, the first work showing that kinship rules are actually non-associative algebras, but still consistent with the mathematics cited here, was reference [60]. A non-associative algebra can have an associative sub-set (the kinship rules for example), which are allowed by reference [60] and required by reference [48]. Reference [61] is the first paper which derives mathematical kinship groups entirely from purely mathematical assumptions.…”

“…Ballonoff [47,48] recently reviewed the literature of mathematical anthropological description, in relation to the ability of a system to survive. The earliest paper for mathematical description of kinship is from 1882: [49] modeled strings of symbols to represent semantic concepts, such as "MBD" (mother's brother's daughter) for an instance of part of the English term "first cousin."…”

“…The mathematical groups for each minimal set of kinship terms described by Ruheman (or Weil as cited in the next paragraph) is also now called the "minimal structure" [47] of that system of kinship description. It is also described [47,48] as the "history" of that system since it describes how possible parental pairs of that culture may be created. The minimal structure is not a literal description of how all marriages work, though in some cases some empirical relations can also look like the Ruheman minimal descriptions in some ethnographies.…”

“…It can also be called "amplification" of local survivability, or, by analogy to borrowing of capital in the world of finance, "leverage." The reader can look at references [47,48,84,85,87,88] for more details and citations of numerous examples.…”

“…We do not claim here that our statistics tell use exactly what changes, or that what we call "Protestant" were the only choices. However, reference [48] may include that the possible changes among the available sets of histories at time t, might also limit what choices are available, and thus may limit what alternate forms of histories might be possible. The reader might also look at examples such as references [87,88], which discuss not only at the values of ns and ps but also how the distributions of choices (the mechanics of the application of the d −1 devices in our diagram) move parts of the certain other cultural practices when a village might be subdividing.…”