The author, a well-known mathematical biologist, traces possible methods of extending the mathematics of developing organisms to the development of societies over long spans of time and circumstance.
The author emphasizes that this work is a first step in a new direction, that his guides include analogy and intuition. It is not an attempt to create a fully developed “mathematical history.” Nor is he committed to any one philosophy or mechanism of history. Indeed, he suggests that mathematics might be used to decide among conflicting models. For example, the “postdictions” (deductions of past events from present ones) yielded by a deterministic model analyzed using an appropriate mathematics (such as differential equations), can be compared with the postdictions of an indeterministic model analyzed with the aid of statistics and probability theory, and both can then be judged against known facts.
It is true that much history is nonquantitative – but then so is much of mathematics. In particular, those branches that are grounded under the term “relational mathematics” – including set theory and topology – may be appropriate for describing these aspects of history. Where numerical data are available, the quantitative branches of mathematics may be applied.
This initial effort may prove to be as much a boon to historians as the introduction of carbon-14 and other scientific techniques have been to archeologists. Certainly, mathematics has proved indispensable to cosmologists, who face problems rather similar to those of the historian: the problem of postdicting remote events in realms largely beyond our direct experience.
Most of the mathematical apparatus employed here is within the limits of the undergraduate mathematics curriculum.
The specific studies include mathematical investigations of the kinetics of transition from one equilibrium to another; the role of prejudice in history, and the mechanics of its decay; the individual as a maker of history; and the effects of geographical factors (like specific shoreline) and the extent of trade on the rates of historical development. The book closes with a discussion of history as a development of a social organism.