Changing something changes everything

Tomorrow is a lot like today, but not always.  Beware the law of unintended consequences.

Our chief correspondent writes:

I’m taking a risk today, trying to explain something using a mathematical metaphor.  Our consultants understand it easily, but I’m writing for people who are not used to thinking about everything mathematically.  I hope it will still be clear; in any case, I’m describing something you already do.

We call something linear (a system, equation, object, whatever) if doubling the input doubles the output.  It does happen: driving along the highway at a steady speed, going twice as long gets you twice as far.  And it’s what you assume almost all the time in predicting the future: two months of bills will add up to twice what one month comes to, or close to it, most of the time.  At the heart of a linear system is the feature that the change you’re making (driving longer, living another month, buying twice as much) doesn’t change anything else.

It’s not always true.  In fact, linear systems in mathematics are very rare.  Most equations, functions, objects in mathematics are not linear.  But linear systems are easy to study, and a lot is known about them.  In one sense, the large sections of textbooks devoted to them form an example of the Lamppost Problem.

[Late at night, a man walking through a city came upon a drunk on his knees, carefully searching a section of pavement directly under a streetlight.  “Lost something?” he asked.  “Yes,” came the answer, “my pocket-watch.”  They both examined the sidewalk and gutter area for several minutes.  “Are you sure you lost it here?” the man finally asked.  “No, I lost it several blocks back.  But the light’s better here.”]

The saving grace of linear mathematics is that almost everything is very close to linear, if you look at small enough parts of it.  An object goes about as far, for a short time, as it would if it were traveling at a constant speed, even if it is actually accelerating strongly.  The trick is to work out just how small a part is small enough, how short a period of time is short enough.  That keeps a number of mathematicians occupied.

Well, we know life is nonlinear.  Every investment prospectus tries to hint that the future will be just like the past, while declaring (in the fine print) that it won’t be.  Every thing you do changes the world around you, changing your change.  This is the Law of Unintended Consequences.  And it can be ruinous.

Noticing that a village in Africa had no access to clean water, an international aid agency arranged for a supply.  The intended consequences were an improvement in everyone’s health, women not having to spend hours each day fetching water from a distance (it was always the women), and similar good things.  What actually happened was the village became a more valuable section of land, and (in that country of casual corruption) a powerful man evicted all the villagers and took possession of the water supply himself.

It is vital to remember that we live in a nonlinear world, and at least to make a guess as how it might react upon the change you contemplate.