The mathematics in A Scientific Model may look intimidating, but it will be trivial for anyone who understands basic calculus and it doesn't even require that (the analysis with calculus is left to an appendix). Graber does a good job of explaining the necessary concepts; if anything I feel he spends too much time working through repetitive mathematics. In a couple of places I did find his use of mathematics disconcerting: it is one thing to try to quantify variables, another to insist that complex entities can usefully be represented by real numbers.
The state of general cultural evolution then must be proportional to the product of population density and mean societal size. (Definition 3, page 11)
Defining population pressure, then, as the average "load" L placed by a population on its resource base B, we have the definition L = P/B. (Page 80)I would expect both "sociocultural state" and "resource base" to be vectors with many terms, and I certainly wouldn't expect the first to be linearly dependent on demographic variables or the second to have a meaningful inverse.
Graber begins by presenting two different views of population: a geographical view where the important variables are area and density; and a political view where they are number of societies and average societal size. He suggests two "principles of inertia" — in the absence of constraints on expansion, both population density and societal size will stay constant despite population growth. The extent to which population growth is expressed by increases in density and societal size can then be used as a quantitative measure of inhibition of expansion and proliferation. This is used to quantify Carneiro's circumscription theory. All of this material seems pretty straightforward. One of the case studies — of Rhesus monkeys — seems to offer only ambiguous support for the inertia principles, but the other — of the westward expansion of the United States — was one of the more convincing in the volume.
I found the "symmetry thesis" of chapter four somewhat less plausible. The inertia principle states that when expansion is uncircumscribed the rate of formation of new societies is proportional to the population growth rate. The symmetry thesis states that when circumscription is complete the rate of decrease in the number of societies is proportional to the growth rate, and, more generally, that between these extremes there is linear variation. I was initially very sceptical of this thesis. Not only does the motivation for it seem obscure, but the three case studies in the chapter seem more an exploration of what can be done assuming its truth than an attempt to provide evidence for it. It does have the merit, however, of producing concrete figures for the time taken for integration of societies under pressure from population growth, something which may be appreciated by archaeologists seeking a theoretical model with which to compare their reconstructed sequences. (Two of the case studies are based on English and Peruvian archaeological data.)
Chapter five considers geographical unevenness in population densities. Graber argues that it is the rate of increase in density (rather than density itself) which tends to equalize between areas and that proportional density increase is a suitable measure of population "pressure". Once again, this is illustrated with data from the expansion of the United States. Chapter six tackles decay (decrease in population). Graber suggests that during decay density decreases and societal size shrinks (so decay does not reverse expansion) and that consequent growth undoes the effects of decay before resuming its normal course (a kind of hysteresis). This I also found rather implausible — while it is very convenient from a modeling point of view, I think it underplays the importance of variations from simple exponential growth.
Chapter seven is about "culture", with the idea that "something" important varies as the square of the population divided by the product of area and number of societies. While I'm certain all those variables influence aspects of culture, I remain entirely unconvinced that this particular product should be reified. The example explored in the chapter — the 20th century — is particularly unhappy, with a simple count of states being used as the "number of societies". Not only is the existence of African states in their current number and form obviously a function of 19th century European history rather than of recent demographics, but I feel that any perspective on sociocultural evolution which ignores the integration of the entire world into one economic system must be missing something important. (While it attempts to incorporate technological innovation in a couple of places, Graber's model doesn't deal with economics at all.)
Most of my criticisms are recognised by Graber himself, however. In his conclusion he writes
Chapter 4 simply presents ambitious speculations recommended more by parsimony than by evidence, and chapters 5, 6, and 7 are no more than tentative explorations along lines suggested by previous chapters.and his whole approach is more "let's see what happens if we try this" than "this is how it is". Even allowing for its massive simplification and abstraction, A Scientific Model is likely to be wrong in many ways, but it is thought-provoking and likely to be of great heuristic value. It is worth having a look at if you are at all interested in the relationship between demographics and political evolution.
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