# The mathematician's lens

January 25, 2010 in

A beautiful article in the NYTimes contrasts abstract mathematics with the chilling reality of the Mexican drug cartel wars:

I was born in Mexico City, in a world that seems less and less familiar to me. I live now in the opposite corner of the continent. I am training to be a political scientist at Harvard. My passion has remained the afflictions of my homeland, but at Harvard I have found new ways to address them, to use mathematical models — matrices, vectors, equations, regressions — to understand the Mexican drug crisis.

The cartel wars are extremely violent, and the gangs are responsible for reprehensible kidnappings and deaths. They rank among the most deadly periods of organized crime in human history. The author's goal isn't to explain how she can analyze the wars from up in an ivory tower; it's to describe how her mindset and toolkit inform her understanding of the world in any situation.

The article captured me because it never mentions what the author actually models. Instead, it presents her frightened thoughts and her efforts to calm herself by looking at the world through a mathematical lens. But it's not what you think; there are no emotionally-distant mathematicians here. The author communicates her fascination with tying reality to abstract models, expecting and preempting the protest that reality is too complex and math too simple:

In this violent world, with the man in the blue Chevy whispering at me behind the window, math is my shield. Speaking up about drugs is in these parts a dangerous game. But not if you speak in the language of sigma and conditional expectations. Math protects me from the immediacy of the violence, and it protects me from them.

The beauty of my method lies in its simplicity. With mathematics I’m able to codify and simplify reality to make it manageable and, more important, malleable. I represent each possible individual as an equation in which each term symbolizes tastes, goals, profession and abilities. All people get portrayed: Policemen, politicians, citizens and drug cartels start living in this mathematical world as planes and hyperplanes and, as in real life, they interact and affect one another, sometimes colluding, sometimes colliding, sometimes neither.

I then use optimization to predict the form of interaction that will be the most probable to emerge and remain over time. Math starts speaking. It tells me, for example, under what conditions the outcome would be a drug war; when would the government prefer to cooperate with cartels; or when cruel intra-cartel purges will become the norm.

There is a part of every modeler's mind which is constantly teasing out variables from constants. The statisticians among us may take a frequentist view, and wonder what would happen if a scene played itself out a million times; the programmers will deduce the underlying algorithms from the fuzzy result; the pure mathematicians will see manifolds everywhere:

In this abstract microcosmos, reality can be frozen or just slightly changed. I move and look at my hyperplanes from different angles. Let’s change the penalty code. No, let’s increase patrolling. Or reduce wages. Allow less contact between policemen and dealers. Assume the police force is corrupt. Assume it is not. I solve the equations and there it is. My answers come as Greek letters and probabilities.

I know, I know, this is weird.

Ultimately, "free will" becomes the clarion of the independent. At least, it's the best response to this explanation:

It may seem strange to examine this shadowy world with equations. But mathematics is transforming the social sciences. In the same way that physicists can predict the movement of atoms in space, we can use mathematics to model how individuals and groups will make decisions and interact in a society.

But free will has a (somewhat tentative) analogue in Heisenberg's uncertainty principle, and with that philosophy and math (or theology and physics) are combined -- but there's been plenty of pop-sci written on that topic.

I found this brief article remarkable in how it was able to demonstrate the overlay mathematical thought on an extremely "human" subject without ever needing to explain either one.

(Via Drew Conway)

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