December 13, 2004
Researcher Uses Mathematical Models
To Resolve Historical Questions
By Leslie Virostek

Peter Turchin, a professor of ecology and evolutionary biology.

Photo by Jordan Bender

Sociologists and historians have put forth many theories to explain why states
expand and shrink, why empires rise and fall. But can any of these theories
be substantiated? Peter Turchin thinks so. His suggestion: Do the math.
Turchin, a professor of ecology and evolutionary biology in the College of Liberal
Arts and Sciences, is a pioneer in applying nonlinear dynamical methods to the
analysis of historical data. In his recent book, Historical Dynamics: Why
States Rise and Fall (Princeton University Press, 2003),
he takes sociological theories on the workings of agrarian societies, translates
them into mathematical models, then compares what the models predict will happen
over time with what the empirical data show did happen historically.
After applying his methods to various historical periods and polities in Europe
and China, Turchin concludes, “History is not just a huge number of random
factors interacting in very complex ways – it is not ‘one damned thing
after another.’ There are some strong patterns that come out. There are some
reasonably simple explanations at work for these patterns.”
Data and Methods
Turchin’s mathematical approach to history and sociology begins with quantitative
data relating to various aspects of history. For example, William the Conqueror’s
famous Domesday Book census, as well as church records for the last several
centuries, are some of the elements that make up a solid data set on England’s
preindustrial population numbers. In addition, historical atlases give detailed
information about the size of polities and empires over
the centuries.
The second step is applying the methods. Thanks to advances in computer technology
and in theoretical nonlinear dynamics – the study of systems described by
nonlinear equations of motion – as a discipline, it is now feasible to create
models that reveal the complex and often surprising behaviors of dynamical systems
characterized by nonlinear feedbacks. In the past, biologists have constructed
useful mathematical models that apply to life in the animal kingdom. For example,
the cyclical predatoryprey interaction and its effect on an animal population
over time can be mathematically expressed.
Human history is also rife with dynamical processes – from the boom and bust
of economies to the rise and fall of nations – that can be described mathematically.
The idea, according to Turchin, is “to put modern methods of analysis with
data sets to resolve historical questions.”
In his book, Turchin creates mathematical formulas that reflect such variables
as class structure, collective solidarity, perceived legitimacy of the government,
and population density, depending upon the theory he is examining. The “geopolitical” theory
advanced by Randall Collins and others, for example, posits that success in war
increases territory and provides more taxes for the state and recruits for the
army, which in turn engenders further success in war. However, there is a negative
feedback in terms of the higher cost of waging war as the distance from the state’s
home base increases. This is sometimes referred to as “imperial overstretch.”
There are other factors as well, but in the end, Turchin finds that the geopolitical
model of territorial expansion, when followed to its mathematical conclusion, does
not explain the cycles of expansion and contraction seen in available data. He
does find, however, that the “demographicstructural” theory of Jack
Goldstone, which involves interactions among such variables as population growth,
inflation rate, intraelite competition, and popular discontent, is well supported
by empirical data on the political instability of England from the 16th century
through the 18th century.
Predictable Patterns
Among his broad findings are that population dynamics – the cycles and patterns
of population growth and decline – are significantly and predictably linked
to state breakdown. He also shows how preindustrial human populations typically
follow predictable oscillation patterns that take two or three centuries to
complete.
Neither of these ideas is brand new or counterintuitive, says Turchin, who
collaborates with historians, sociologists, and anthropologists. But what is revolutionary
about his research is that he uses quantitative analysis to test historical and
sociological theories that generally rely on qualitative thinking.
Mathematical methods have greatly advanced understanding of such sciences as physics
and chemistry, he points out in the introduction to his book. They can do the same,
he believes, for history.
