Ralph Abraham : Computational Mathematics and the Social Sciences
MathKnow08 : Mathematics, Applied Sciences, and Real Life
Politecnico di Milano, Milan, Italy, May 22, 2008
Introduction

In past writings I have developed examples of "bolts from the blue" in which artists have made significant contributions to mathematics --- eg, the perspective theory of Leon Battista Alberti of 1436. This is the obverse of the more familiar contributions mathematicians have made to art --- eg, Benoit Mandelbrot's fractals. This latter example is the epitome of the artistic influence of the computer revolution, or more precisely, the computer graphic revolution, as it impacted mathematics and the arts in the 1870s. In this talk I will be interested not by these mathematics and art connections, but rather, by some more recent mathematics and culture connections --- especially: chaos theory, the social sciences, and real life. The computer revolution has begat new branches of mathematics, --- eg, chaos theory and fractal geometry --- and their offspring --- agent based modeling and complex systems. These new methods of math modeling have extended and changed our understanding of the complex systems in which we live. In this talk I will discuss three case studies of the significance of chaos theory for real life --- especially, policy making.

Case study #1: The Atlantic cod stock collapse (Why simple dynamical models are not predictive)

In Useless Arithmetic, Orrin Pilkey and Linda Pilkey-Jarvis present several cases of the misuse of mathematical modeling, including the collapse of Atlantic cod stocks, prediction of stock prices, body counts during the Vietnam war, the safety of nuclear waste storage at Yucca Mountain in Nevada, the rise of sea levels due to global climate warming, shoreline erosion, toxicity of abandoned open-pit mines, and the spread of non-indigenous plants.

In the first case, the models accused are simple dynamical systems derived from population dynamics of a single species, the Atlantic cod. Interacting populations in the ecosystem were ignored. The quantitative predictions of these models overestimated the safe catch, and the collapse of the Grand Banks cod fishery in 1992 was the result. Even if other factors were included in a complex dynamical model, chaos theory implies that even qualitative predictions are unreliable. Chaotic attractors, fractal boundaries, and catastrophic bifurcations all mitigate against credible forecasts. It is not practical to establish that a model is structurally stable.

Case study #2: Global climate warming (Why complex dynamical models are not predictive)

Al Gore and the Intergovernmental Panel on Climate Change (IPCC) shared a Nobel prize for their work on climate prediction. Global climate warming has been, is, and will be, controversial. Skeptics have called it the greatest scientific hoax of all time, and the IPCC has been accused of major deception. Millions of people have seen Al Gore climbing a ladder to show the predicted rise in sea level. James Lovelock, the Gaia Hypothesis guru, expects a rise of 200 feet, while the IPCC expects 2 feet. Many climate models have been  extensively studied, from the simple two-component daisyworld of James Lovelock, to massively complex models including most of the known factors.

Here we must recall the significance of chaos theory for complex dynamical models, also known as system dynamics models. The early history of system dynamics, developed by Jay Forrester in the 1960s, predated chaos theory. Many complex system models behaved chaotically, as we would now expect, but at the time this irregular behavior was considered misbehavior, and the supposedly faulty models were ignored. The advent of chaos theory breathed new life into system dynamics. Nevertheless, chaotic attractors suffer from sensitivity to initial conditions (the butterfly effect) and thus cannot be used for quantitative prediction. The modeling activity is nevertheless crucial to the hermeneutical circle that drives the advance of science. The qualitative behavior of a model provides a cognitive strategy for understanding complex systems. But even the qualitative behavior of a model cannot be trusted as an indicator for the natural system being modeled, due to structural instability of the model.

In short, mathematical modeling is valuable for comprehension, but not for prediction. Ignoring this simple fact is the cause of most of the misuse of mathematics in policy making.

Case study #3: Financial bubbles and crashes

The failure of predictive models of stock market behavior is one of the cases targeted by the Pilkey book, Useless Arithmetic. But this context provides an opportunity to illustrate our claim that modeling is a useful cognitive strategy for complex natural systems. Physical systems may be understood in terms of physical and chemical forces for which we have superb mathematical models. Social systems, on the other hand, are subject to forces that we have yet to identify and model. 

The advent of agent-based modeling (ABM) --- a new paradigm for computer modeling of complex systems --- has occasioned a giant leap forward for the social sciences. Several ABM environments have been created by social science departments at universities around the world in order to take advantage of this new world of opportunity. One of these, NetLogo from Northwestern University, has been used as an exploratory tool to discover candidate rules for the dynamics of a financial system in joint work with Dan Friedman, a mathematical economist at the University of California at Santa Cruz. In an NSF sponsored project, we have build a NetLogo model for a trading community of money market managers. We explored various rules for a trader to adjust the division of her portfolio between risky and riskless investments, e.g., stocks and bonds. Our goal was to discover a rule that created behavior in the model market showing price bubbles and crashes.

Besides providing a rapid development environment for ABM, NetLogo also provides an adjunct feature called HubNet, enabling human traders to interact with the robot traders (agents) of our market model. Using this feature, we have been able to test the newfound rules with human subjects. In this way we found candidates for social forces underlying bubble-and-crash behavior in human societies.

Conclusion

Mathematical physics took off around 1600 AD. Mathematical biology got underway around 1930. And the social sciences are now rushing to catch up, thanks to the advent of agent-based modeling. Math models and computer simulation are ineffective as predictive tools due to the restrictions of chaos theory. But they still function crucially as part of our cognitive strategy to understand the complex social and environmental systems in which we live.

References

Pilkey, Orrin H, and Linda Pilkey-Jarvis.
Useless Arithmetic : Why environmental scientists can't predict the future.
Columbia University Press, 2007

Abraham, Ralph.
Climate warming skeptics page.
http://www.vismath.org/research/gaia/2007/climate-nay.html

Abraham, Ralph, Dan Friedman, and Todd Feldman.
Selected papers.
http://www.vismath.org/research/landscapedyn/articles/

====================