# Introduction. Why Do We Need. Mathematical Models of Historical Processes

Almanac: History & Mathematics: Historical Dynamics and Development of Complex Societies

Many historical processes are dynamic (a dynamic process is one that changes with time). Populations increase and decline, economies expand and contract, while states grow and collapse. How can we study mechanisms that bring about temporal change and explain the observed trajectories? A very common approach, which has proved its worth in innumerable applications (particularly, but not exclusively, in the natural sciences), consists of taking a holistic phenomenon and mentally splitting it up into separate parts that are assumed to interact with each other. This is the dynamical systems approach, because the whole phenomenon is represented as a system consisting of several interacting elements (or subsystems).

In the dynamical system's approach, we must describe mathematically how different subsystems interact with each other. This mathematical description is the model of the system, and we can use a variety of methods to study the dynamics predicted by the model, as well as attempt to test the model by comparing its predictions with the observed dynamics.

Generally speaking, models are simplified descriptions of reality that strip away all of its complexity except for a few features thought to be critical to the understanding of the phenomenon under study. Mathematical models are such descriptions translated into a very precise language which, unlike natural human languages, does not allow for any double (or triple) meanings. The great strength of mathematics is that, once we have framed a problem in mathematical language, we can deduce precisely what are the consequences of the assumptions we made -- no more, no less. Mathematics, thus, is an indispensable tool in true science; a branch of science can lay a claim to theoretical maturity only after it has developed a body of mathematical theory, which typically consists of an interrelated set of specific, narrowly-focused models.

The conceptual representation of any holistic phenomenon as interacting subsystems is always to some degree artificial. This artificiality, by itself, cannot be an argument against any particular model of the system. All models simplify the reality. The value of any model should be judged only against alternatives, taking into account how well each model predicts data, how parsimonious the model is, and how much violence its assumptions do to reality. It is important to remember that there are many examples of very useful models in natural sciences whose assumptions are known to be wrong. In fact, all models are by definition wrong, and this should not be held against them.

Mathematical models are particularly important in the study of dynamics, because dynamic phenomena are typically characterized by nonlinear feedbacks, often acting with various time lags. Informal verbal models are adequate for generating predictions in cases where assumed mechanisms act in a linear and additive fashion (as in trend extrapolation), but they can be very misleading when we deal with a system characterized by nonlinearities and lags. In general, nonlinear dynamical systems have a much wider spectrum of behaviors than could be imagined by informal reasoning. Thus, a formal mathematical apparatus is indispensable when we wish to rigorously connect the set of assumptions about the system to predictions about its dynamic behavior.

Modeling of any particular empirical system is as much art as science. Models can be used for a variety of purposes: a compact description of the system structure, an investigation into the logical coherence of the proposed explanation, and derivation of specific predictions from theory that can be tested with data. Depending on the purpose, we can develop different models for the same empirical system.

There are several heuristic rules that aid development of useful models. One rule is: do not attempt to encompass in your model more than two hierarchical levels. A model that violates this rule is the one that attempts to model the dynamics of both interacting subsystems within the system and interactions of subsubsystems within each subsystem. For example, using an individual-based simulation to model interstate dynamics violates this rule (unless, perhaps, we model extremely simple societies). From the practical point of view, even powerful computers take a long time to simulate systems with millions of agents. More importantly, from the conceptual point of view it is very difficult to interpret the results of such a multilevel simulation. Practice shows that questions involving multilevel systems should be approached by separating the issues relevant to each level, or rather pair of levels (the lower level provides mechanisms, one level up is where we observe patterns).

The second general rule is to strive for parsimony. Probably the best definition of parsimony was given by Einstein, who said that a model should be as simple as possible, but no simpler than that. It is very tempting to include in the model everything we know about the studied system. Experience shows, again and again, that such an approach is self-defeating.

Model construction, thus, always requires making simplifying assumptions. Surprisingly, however, the resultant models are often quite robust with respect to these initial assumptions. That is, "first-cut" models can be investigated mathematically as to the consequences of relaxing the initial assumptions for theoretical predictions. Repeated applications of this process can extend theory and simultaneously increase confidence in the answers that it provides. The end result is an interlocked set of models, together with data used to estimate model parameters and test model predictions. Once a critical mass of models and data has been accumulated, the scientific discipline can be thought of as having matured (however, it does not mean that all questions have been answered).

The hard part of theory building is choosing the mechanisms that will be modeled, making assumptions about how different subsystems interact, choosing functional forms, and estimating parameters. Once all that work is done, obtaining model predictions is conceptually straightforward, although technical, laborious, and time consuming. For simpler models, we may have analytical solutions available. However, once the model reaches even a medium level of complexity we typically must use a second method: solving it numerically on the computer. A third approach is to use agent-based simulations. These ways of obtaining model predictions should not be considered as strict alternatives. On the contrary, a mature theory employs all three approaches synergistically.

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One of the main causes for the expansion of the application of formal mathematical methods to the study of history and society is the deep changes that have taken place during recent decades in the field of information production, collection, and processing (as well as in the field of information technologies, in general). These changes affect more and more fields of academic research. The enhanced possibilities for the development of databases, the increasing speed of their processing, in conjunction with the growing availability of many forms of digital information, the diffusion of personal computers and more and more sophisticated software provide all the grounds needed to forecast not only the expansion for the formalization of new methods of information processing and presentation, but also the expanding application of formal mathematical methods in such fields that seem to have nothing to do with mathematics. We may note some serious changes in the attitudes of the "humanitarians" toward formal mathematical methods. The application of formal mathematical methods in the humanities is not just a fashion, or the way to make one's research faster and more comfortable. It becomes evident that such methods create necessary conditions for intellectual breakthroughs, for the establishment of new paradigms, for the discovery of new research directions. To a considerable extent this is accounted for by the very character of many historical processes.

The times of "Pure History" when historians were only interested in the deeds of kings and heroes passed long ago. A more and more important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical methods.

This almanac continues a series of edited volumes dedicated to various aspects of the application of mathematical methods to the study of history and society (Grinin, de Munck, and Korotayev 2006; Гринин, Коротаев, Малков 2006; Малков, Гринин, Коротаев 2006; Коротаев, Малков, Гринин 2006). This edited volume considers historical dynamics and development of complex societies. Its constituent articles treat historical processes at very different levels of scale. Some articles study global dynamics during the last millennia covering the formation and development of the World System. Other articles focus on the dynamics of single societies, or even communities. In general, this issue of the almanac constitutes an integrated study of a number of important historical processes through the application of various mathematical methods. In particular, these articles trace the trajectories of political development from the early states to mature statehood. This almanac also traces trajectories of urban development, and important demographic, technological, and sociostructural changes.

The almanac demonstrates that the application of mathematical methods not only facilitates the processing of historical information, but can also give to a historian a deeper understanding of historical processes.