# Applied Mathematics and Political Crises

**Ages of Discord: A Structural-Demographic Analysis of American History.** *By Peter Turchin. Beresta Books, Chaplin, CT, September 2016. 274 pages, $35.00.*

*Ages of Discord: A Structural-Demographic Analysis of American History.*By Peter Turchin. Courtesy of Beresta Books.

Peter Turchin, an exemplary interdisciplinary professor and scholar, has set out to remedy this scarcity of mathematical history using classic applied mathematics and solid historical data, and none too soon. His latest book, *Ages of Discord: A Structural-Demographic Analysis of American History*, is a warning shot across the bow of the U.S. ship of state – that dangerous shoals and storms lie dead ahead, with huge swells already rocking the boat. Turchin’s “Political Stress Index” has been growing exponentially, so much so that he likens our current situation to that of the U.S. just before the Civil War, or to France prior to the French Revolution.

Turchin is a founder of what he calls “cliodynamics,” or the application of mathematical modeling, especially dynamical systems, to the study of history (Clio is the Greek muse of history). To this he brings a strong background in population biology, as well as historical studies and anthropology. I highly recommend his classic work of popular history, *War and Peace and War: The Rise and Fall of Empires* (2006). This book is based on his more technical *Historical Dynamics: Why States Rise and Fall* (2003), which discusses the strengths and limitations of a variety of mathematical models as they apply to specific historical periods. These periods are drawn from the Roman Empire, Chinese dynasties, and European history, concentrating on agrarian empires and regimes prior to 1900.

Turchin’s newest book adapts and updates these earlier studies, applying them to the history of the U.S. from the Revolutionary War to the present. The cover of the book shows a wonderful chart, with a blue curve displaying the rise and fall of two narrow “Eras of Good Feelings” (1810-1840 and 1940-1970), alternating with a red curve for his Political Stress Index, which rises exponentially as consensus politics gives way to escalating inequality.

Turchin identifies three factors, the key functions of his “structural-demographic theory,” which often lead to civil war or state collapse. The first and foremost is competition and conflict among a rising population of elites, the second is a declining real wage for an expanding population of workers, and the third, though less prevalent, is state financial collapse. We can view these factors as an empirically-verified version of Thomas Robert Malthus’ classical population theory, but one that identifies vital feedback loops that produce approximate cycles of rise and fall over a century or two, rather than a more or less permanent Malthusian nightmare of poverty for the masses. The Political Stress Index is simply the product of these three factors, which Turchin calls “Elite Mobilization Potential,” “Mass Mobilization Potential,” and “State Fiscal Distress.”

Turchin’s elite population equation is given by a standard first order differential equation (think exponential growth), except that there is an added term involving the wage rate, a term which will send the elite rate of growth into negative territory after a time delay and sufficient fall in the wage rate. The wage rate depends primarily on the gross domestic product (GDP) per capita (a measure of productivity) and the historical supply/demand ratio for labor, and uses a nonlinear product and power formula. Turchin is very careful to explain the equations in detail and to minimize the number of parameters. He starts with simplified equations and concepts before moving into the complexities required to match the historical data. The result is some of the best popular exposition of applied mathematics that I have ever encountered.

Turchin is equally good at the “human interest” level, describing all the historical datasets he was able to locate. These datasets include not just population statistics, GDP, average wages, and national debt, but everything from a surprising history of legal salaries to average male heights, labor battles, Senate filibusters, largest fortunes, immigration expansion and contraction, age at first marriage, and much more. In every case, he seeks multiple datasets to estimate each independent variable or parameter.

Though remarkably successful at illuminating societal discord, Turchin’s analysis obscures the underlying drivers of economic growth or contraction that derive from technologies, resources, and ecosystems. Our new technologies for exploiting fossil fuels have propelled the spectacular growth of the last two centuries compared to the modest growth of land-based empires, thus exposing us to the possibility of greater economic and political collapse. Already returns to investment in fossil fuels are falling while the costs of climate change and other environmental damage are rising, threatening economic contraction ahead and accelerating the factors of discord that Turchin identifies.

Ironically, a successful mathematical analysis that includes such broader variables has been around for over 40 years; namely, the famous *Limits to Growth* scenarios of the early 1970s by Dennis Meadows, Donella Meadows, Jørgen Randers, and William Behrens III. At that time, the mathematical technique was called “system dynamics” (“nonlinear dynamical systems” in today’s lingo), and was well beyond the comprehension of economists, let alone historians. It uses five major variables—population, food supply, resources, industrial output, and pollution—and projects graphs of these variables into the future, based on scenarios for impending parameter values. The “business as usual” scenario (using the best current estimates for the parameters) has held up remarkably well according to a recent study by Graham Turner (Melbourne Sustainable Society Institute, University of Melbourne). I would love to see Turchin’s work extended along these lines, or vice versa.

Peter Turchin concludes *Ages of Discord* by asking, “Will we be capable of taking collective action to avoid the worst of the impending structural-demographic crisis?” *Limits to Growth* suggests that the crisis is actually much deeper. Applied mathematicians have much to contribute.