The Butterfly That Started a Revolution

Chapter 1: The Butterfly That Started a Revolution

A Small Mistake, a Vast Consequence

It began with a number. To be precise, it began with a number that Edward Lorenz typed, almost carelessly, on a Tuesday morning in the winter of 1961, and then refrained from typing. The numbers he had entered the day before had been 0.506127; the number he entered now was 0.506. A rounding error so ordinary that anyone, in any other context, would have dismissed it as the kind of clerical liberty humans take with decimals dozens of times a day. But Lorenz was not doing ordinary work. He was running a simple computer model of the atmosphere on the Royal McBee, an ungainly vacuum-tube machine that occupied most of a small room on the third floor of Building 24 at the Massachusetts Institute of Technology. The numbers he had typed were the starting conditions for a simulation of weather. And when he returned to look at the output, expecting a sequence that closely resembled his previous run—the same atmospheric drama, perhaps slightly abbreviated—he found instead something stranger and, in the end, more consequential than almost any single result in the history of twentieth-century science.

The two runs diverged. Not a little. Catastrophically. Weather that had begun looking like a recognizable wintry pattern in the first simulation, the one with six significant figures of precision, became something unrecognizable in the second simulation, the one with only three. Sun yielded to storm and back again. Pressures inverted. Temperatures lurched. Within a few simulated "days," the two forecasts had no qualitative relationship to each other. They were, the meteorologist realized as he stared at the printout, two different weathers. They had begun, as far as his program was concerned, with exactly the same laws of motion and exactly the same equations—only the initial numbers had been infinitesimally altered. And yet here was divergence so complete that the smaller difference had grown, by every step of the integration, into a chasm.

Lorenz was not a man given to grand pronouncements. He did not march down the corridor announcing that he had overturned Newtonian mechanics. He sat, instead, and thought. He had already been unsettled for some time by a question that the success of numerical weather prediction seemed to raise and then elude: if we know the equations of motion for the atmosphere, and we know how to integrate them, and we know how to measure the present state of the atmosphere with reasonable accuracy, why is it that our two-day forecasts are useful but our five-day forecasts often wrong, and our ten-day forecasts are nearly worthless? Something was wrong with the picture. The picture itself—that the atmosphere was a deterministic system governed by known laws, and that its future flowed inexorably from its present—was so deeply embedded in the scientific worldview that to question it required either great courage or great naïveté. Lorenz had neither, but he had something that was, in the long run, more useful: he had a machine that printed lies.

The lesson he eventually drew from that winter morning, and that he would spend the rest of his career articulating, was simple and devastating. The atmosphere is deterministic. Its equations are known. But it is not, and cannot be, predictable—not in the strong sense that Laplace had once imagined, the sense in which a sufficiently complete knowledge of the present would yield the future as a matter of calculation. The limit is not technological. It is mathematical. It lives in the equations themselves. Two states of the atmosphere that are arbitrarily close in our representation can evolve into states that are arbitrarily far apart in reality. Tiny causes can carry enormous consequences. This property has come to be called *sensitive dependence on initial conditions*. Its most evocative name—the one that everyone knows, the one that even newspaper editors can pronounce—is the butterfly effect: a butterfly beating its wings in Brazil can, in principle, set the atmosphere on a course that culminates, weeks later, in a tornado in Texas.

The phrase sounds whimsical. It is in fact a precise statement about a feature of certain dynamical systems that distinguishes them sharply from the clocks and planets of classical mechanics. And it is the seed from which an entire science of complexity has grown. To understand why, we have to do something that Lorenz himself eventually did, which is to step back from the immediate phenomenon and ask what kind of world would have made it possible, what kind of world would have made it impossible, and what kind of world we actually inhabit.

The World Before Chaos: Laplace's Demon and the Dream of Complete Predictability

Stand in a great cathedral of science for a moment, the way Newton might have stood in the chapel of Trinity College in the winter of 1666, and consider what was achieved. A new way of seeing had been born. The planets moved in ellipses, the apple fell at a rate proportional to a constant, the moon's orbit could be computed from the same universal gravitation that governed the falling stone. The world was not capricious. It was lawful. And those laws, written in the language of differential equations, allowed one to do something unprecedented: starting from a description of how things are now, calculate how things will be at any future moment. The clockwork universe was a new kind of cosmos. It was a place where, in the words of Alexander Pope's famous epitaph,

> Nature and Nature's laws lay hid in night: > God said, "Let Newton be!" and all was light.

A century and a half later, Pierre-Simon Laplace, the great French mathematician and astronomer, made the metaphysical claim that came to define the ambition of modern science. Given a sufficiently complete description of the state of every particle in the universe at a single instant, and given the laws of motion, nothing would be hidden. The future would unfold as a necessary consequence. The past would be recoverable. Nothing would be uncertain except our ignorance.

Laplace did not write this in a private notebook. He wrote it in the introduction to his *Essai philosophique sur les probabilités* (1814), an enormously influential work that effectively founded mathematical probability as a serious science. The relevant passage, which has since been called Laplace's demon, is worth quoting in spirit, because the exact wording, in his French original, has shaped how generations of scientists and philosophers have thought about determinism:

> We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all the forces of nature, and the respective situations of all things at a given instant, could—if its comprehension were vast enough—comprehend in a single formula the motions of the largest bodies in the universe and those of the lightest atom. For such an intelligence, nothing would be uncertain; the future, like the past, would be present to its eyes.

Notice what Laplace is claiming. He is not claiming that we can predict the universe. He is claiming that we can, in principle, contain it in a thought. The limitation is our comprehension, not the laws. Given unlimited knowledge and unlimited computational power, every fact about the future is determined, and prediction becomes a kind of reading.

This picture was extraordinarily productive. It underwrote the success of celestial mechanics through the nineteenth century: the discovery of Neptune from irregularities in the orbit of Uranus, the precise return of comets, the prediction of eclipses. It underwrote thermodynamics and statistical mechanics, even when those fields shifted attention to averaged quantities and probability: a gas was understood as a vast collection of particles, each deterministic in its trajectory, whose macroscopic behav