Archive for December, 2012

Arms in America

December 30, 2012

On December 21, 2012, the Executive Vice President of the National Rifle Association, Wayne LaPierre, announced, “I call on Congress today to act immediately, to appropriate whatever is necessary to put armed police officers in every school,” prompting this blog.


On Friday, December 14, 2012, a man in black fatigues walked into the Sandy Hook Elementary School in Newtown, CT, with a Bushmaster assault rifle, and killed 20 first grade children and six adults. Within hours, the lapsed discussion on gun control in the US was rekindled and raging in the media on several fronts: availability of assault weapons, tracking of the mentally ill, security in schools, and so on. This blog is about the proliferation of arms in America, not about this particular shooting.

These arguments have been ongoing, especially since the shooting death of James Brady in 1981. Increasingly we have since a differentiation into two camps, the hawks and the doves. The hawks, including some members of the National Rifle Association, believe that the number of deaths by gunshot in the US (the highest by far in the develped world) will be decreased by increasing the number of guns among the public. The doves advocate the opposite. This dichotomy is typical of arms races between nations, which tend to escalate exponentially until war is triggered.

Arms races have been the target of mathematical modeling for a century, and in this blog I will outline this long history. The dream of this kind of mathematics is to serve history by increasing our level of understanding of massively complex systems in which we live. In my book, Chaos, Gaia, Eros, this branch of mathematics is called erodynamics. We begin with some extracts from that work.

Richardson, 1919

Lewis Fry Richardson was an English physicist, meteorologist, and Quaker. A conscientious objector in World War I, he served as an ambulance driver on the front lines in France and saw a great deal of death and suffering. After deciding to devote his life to the elimination of war, he developed a linear model for the arms race between two nations, in which a spiral of increasing armaments in each nation resulted from mathematical laws.

Richardson felt that individual nations caught in this kind of dynamic were innocent victims of an out-of-control global system. He submitted a paper on his model to a scientific journal, fully confident that another war could be averted. The paper was rejected, and World War II began. After this rejection Richardson continued his work, trying to justify the model on the basis of actual armament statistics. In these efforts he founded the field of politicometrics. Richardson’s life work was published posthumously in 1960.

Bateson, 1935

Gregory Bateson adapted the Richardson model to the process of the division of a culture into subcultures, analogous to differentiation in biological systems. He called this universal dynamical process for the development of a schism a Richardsonian process of schismogenesis. In fact, schismogenesis, a social form of bifurcation, was one of Bateson’s main themes.

Thom, 1972

In the 1960s René Thom developed catastrophe theory. He published the theory in 1972, along with a number of ideas for its application in the sciences, linguistics, philosophy, and so on. The final chapter of his book sets out the modern formulation of Erodynamics, in the context of proposed applications to sociology and psychology.

Isnard & Zeeman, 1976

Mathematicians C. A. Isnard and Christopher Zeeman replaced the linear model of Richardson and Bateson with a nonlinear model: the cusp catastrophe of Thom’s theory. They applied their model to the original arms race context of Richardson’s work, showing how the model fit a situation of schismogenesis, in which the voting population of a democratic nation split into two populations, hawks and doves.


The hope of this model is due to its geometry. It is complex enough to capture the essential features of the social dynamic unrolling today in the US, and simple enough to be understood by most educated people. The two dimensions of control variables provide enough wiggle room to empower a process of negotiation and reconciliation of both sides of the debate. The understanding of the dynamics of the argument, too complex to be easily understood without the model, may become clear through a study or simulation of the model. It may even help to know that we are enmeshed in a classical process that frequently ends in tragedy: an arms race.

For the hawks — who believe that if they have more guns, then we have more fear, and thus need more guns, and more hawks — the mathematics (and the historical record) suggest otherwise. For the doves — who believe that fewer guns incline towards fewer deaths, less fear, thus less need for guns, and more doves — the mathematics (and history) is supportive. The unwitting role of the media is to promote more fear. As the hawks are heavily armed and the doves are not, this fear turns doves into hawks.

The only force turning hawks into doves is the force of understanding — for example, mathematical understanding. Our mathematical models provide strategies to escape from an arms race, strategies which may be counter-intuitive, yet effective, like swimming parallel to the beach to escape from a riptide and safely reach the beach.


It is a pleasure to acknowledge the inspiration from Gregory Bateson, Rene Thom, Chris Zeeman, and Gottfried Mayer-Kress for my work on arms races over the years. Also, many thanks to my partner Ray Gwyn Smith for the riptide metaphor and web searches.


Ralph Abraham, Chaos, Gaia, Eros, 1994; Ch. 18.
E. Christopher Zeeman, 1977. Catastrophe Theory: Selected Papers, 1972-1977; Ch. 10.
Gun-related murder rates in the developed world.


December 30, 2012