Thursday, January 31, 2008

In the old days, we lived at the center of the Universe. We had dominion over the fish of the sea and the birds of the air, and every other living thing that moved on Earth.

These living, moving things, however, were not docile or subservient subjects. As likely as not they were dragons or vicious three-headed dogs, to be valiantly slain. The seas were infested with monsters and booby-trapped with whirlpools, and if we did not fight, they would rend our ships to splinters. Paradoxically, we were at once lords and victims of the universe.

Over the ensuing centuries, Earth was shaken loose from its secure spot at the cosmos’s heart. First the usurper was the Sun, once a planet revolving around us, now grown in stature to a star marshalling the planets. Then even the Sun was forced to bow to the Milky Way Galaxy, where far bigger and brighter stars shone by the score. And still the dizzying descent did not stop: the Milky Way was only one very average spiral galaxy, clustered with countless others.

Domestically, however, matters seemed to improve. Despite the ballooning foreign policy disaster and concomitant loss of stature, we managed to drive the giants and minotaurs to extinction. Our cosmological demotion went hand in hand with increased understanding and mastery of the elements, and over time the Enlightenment and Industrial Revolution became inevitabilities.

Now, living on an obscure outpost of rock circling a mediocre star, we are finally the masters of creation. Scylla and Charybdis are gone, choked or buried by the oil from our ships, which sail the seas unmolested. Perhaps emboldened by the consciousness of our triviality, we lay waste with abandon.

Have we not drawn the correct conclusion from the knowledge of our smallness? In the greater scheme of things, we surely cannot have any real or lasting effect. The temperatures may rise a little here, or the storms worsen there, but these effects are temporary. It has all happened before, and what we do cannot seriously change or damage the Earth.

This is fallacious reasoning.From the state of the art in the 19^{th} century, one might have concluded such a thing, but among the most important discoveries of 20^{th} century mathematics has been the sometimes crucial effect even of tiny variations. Popularly known as “the butterfly effect,” this phenomenon is studied in the branch of mathematics called “chaos theory.”

Consider that every process mathematics can describe – physical, biological, or economic – is encoded in an equation involving rates of change over time. Like a riddle, this equation tells us something about how the quantity we are interested in changes. The value of that quantity itself at any given time must be teased out by solving the riddle. Such an equation describing rates of change is known as a “differential equation.”

However, a differential equation alone is not sufficient for complete knowledge of the quantity in question. The world’s most famous differential equation describes the motion of an object in free-fall, and is so simple it can be stated in words: the acceleration of an object in free-fall is 32 feet per second, per second. This means that with each passing second, the object’s downward velocity will increase by another 32 feet per second.

But as it stands, this riddle has no solution. How fast was the object moving to begin with? Was it a baseball thrown high in the air? A champagne cork spurting from a bottle? To know the current velocity, we need to know not only how velocity changed, but also where it started. We need an “initial condition.”

How crucially does the trajectory of an object depend on initial condition? This is the question which leads to chaos theory. For many systems, the dependence is not crucial. If you slightly change the angle of your billiard cue, you will only cause a correspondingly slight change in the ball’s trajectory.

This, at least, is how ordinary billiards works. In 1898, however, the French mathematician Jacques Hadamard invented his own version of billiards, and with it chaos theory.

Picture a billiard table shaped like the inside of a perfectly hemispherical bowl. The billiard table is made of a metal so highly polished that we can assume it has no friction. Since gravity would spoil the game by clumping all the balls at the bottom of the bowl, Hadamard billiards should be played by astronauts in outer space, using magnetic balls which cling to the surface of the billiard table.

Imagine the frustration of the astronauts when they discover just how difficult Hadamard billiards is to play! Every possible trajectory of a Hadamard billiard ball differs exponentially from every other. Adjusting the cue or the position of the ball by even a hair’s breadth results in the ball’s taking a radically different path.

With the invention of this bizarre game, Hadamard opened the sluice-gates for study of systems highly sensitive to initial conditions. As it turns out, these are not limited to annoying astronaut pastimes.

In 1961, American meteorologist and mathematician Edward Lorenz ran a computer simulation using differential equations for the behavior of weather systems. Compared with the complexity of real-world weather, Lorenz’s model was quite basic, using only three quantities to characterize the convection of air-currents. Roughly, these quantities were the strength of the convection currents, the temperature difference between rising and falling air, and a measure of temperature variation. The rate of change for each quantity over time obeyed a differential equation, relating it to a simple combination of all three quantities. Having constructed a solution, the computer printed its results in a chart:for each time-step was a list of the three values of interest.

Noticing an intriguing effect in one solution, Lorenz wished to run the simulation again from the same initial condition. However, only the latter half of the solution was of importance, and to save time he did not start the simulation from its beginning but from a later time, using his printed chart to enter the three values at that time-step as initial conditions.

To the astonishment of Lorenz, the computer printed a solution radically diverging from the one he had wished to reproduce. What could be the difference of starting in the middle rather than at the beginning? Given an initial condition, there is only one solution to a differential equation. Lorenz had copied a number from the previous solution, so the simulation should simply have resumed from that point as before.

The answer lay in round-off error. Though the computer performed its calculations using six decimal places, the print-out was rounded to three decimals. When Lorenz had entered the figures from the print-out as initial conditions, they were in fact not the numbers the computer had used in the first calculation, but very slightly altered values: decimals of .506 rather than .506127, to be precise. Lorenz wrote, “One flap of a seagull's wings could change the course of weather forever.”

Chaos had moved much closer to home. The artificial and idealized game of billiards envisioned by Hadamard was replaced by a pragmatic and basic model of convection currents, exhibiting similar complex behavior. At once the dream of long-term accurate weather forecasting was shattered, while the study of chaos theory burgeoned in magnitude and importance. Today, everything is chaotic, from brainwave frequency to exchange rates in the East European black market.

If a seagull can flap its wings once and change the course of weather forever, surely the 80,987 airline flights taking off worldwide each day can do as much, too? ‘Perhaps,’ the skeptic might reply, ‘but if this system is so fantastically sensitive to initial conditions, how can we predict what effect we will have? How then can we frame any reasonable ground rules? How could we be sure they would work?’

The answer lies in the difference between weather and climate. While weather refers to the immediate conditions of temperature, humidity, etc, climate is determined by averages of these quantities over some appropriate window of time, and region of space. For this reason, small random disturbances to the weather are averaged out when one passes to the level of climate. Butterflies and seagulls can be considered random in their motions; therefore while capable of changing the weather, they have little effect on climate, which appears to be non-chaotic in nature.

Our motions, however, are not random. Unlike the turbulence of butterflies, our effects do not average out in the calculation of climate, because we have uniformly raised weather temperatures by our carbon emissions.

It seems that our position is reversed from that of the Middle Ages:rather than important but impotent, we are now trivial but powerful. Or, in more familiar (and more flattering) words – small but mighty.

The Earth is no longer a hostile barbarian to be ruled with an iron fist. The dragons have left their caves, and the serpents their briny deep. It is time for a new narrative of our position in the world. Perhaps it could begin with the words of the 20^{th} century mathematician and physicist Paul Dirac, “Pick a flower here on Earth, and you move the farthest star.”