by David Orrell
Now, speaking metaphorically, replace the spring with a camel. As you add pieces of straw onto the camel, at first little happens— the camel gradually sags lower and curses you in camel language. But suddenly you come to the “straw that breaks the camel’s back,” and you get what mathematicians term a “non-linear response.” The impact of the last straw is much greater than the impact of the first. With non-linear systems, initial conditions matter. This fact was discovered in a rather unsettling fashion by Henri Poincaré, a professor at the University of Paris.
In 1889, to commemorate the sixtieth birthday of King Oscar II of Sweden and Norway, a competition was held to reward the best research in celestial mechanics related to the stability of the solar system. While we all know that the sun rises in the east and sets in the west, it isn’t entirely obvious that this is a permanent state of affairs. Who is to say that the earth couldn’t take a detour one morning and head off into outer space, with the sun just becoming a gradually shrinking dot? This was the kind of nagging question to which aging monarchs turned their attention.
The competition was won by Poincaré, who showed that there was no straightforward solution to a system with even three bodies. A small difference in the initial positions could lead to radically different outcomes. The reason for this mess was non-linearities in the equations. As if involved in a complicated, triangular love affair, each planet was affected in a non-linear way by interaction with the other bodies. Depending on the initial configuration, a planet could either stay in the system or be ejected from it. Furthermore, the trajectories were so complicated that Poincaré couldn’t even draw them. He had discovered chaos.
When I was a kid, my parents gave me a pendulum, a rod with a magnet on its end, which swung just above a metal plate. There were three additional small magnets that could be positioned anywhere on the plate. When the pendulum was released, it would swing in a wild and unpredictable fashion as it was attracted to or repelled by the magnets, until it eventually came to rest because of friction in the rod. Poincaré’s system was similar, except that instead of competing magnetic fields, there were gravitational fields and no friction.
Since the earth’s orbit is constantly perturbed by the other planets, Poincaré’s result implied that the solar system might be less stable than it appears. Astronomers later confirmed that the solar system is probably weakly chaotic, but the perturbations from regular orbits are small, so we won’t crash into Venus any time soon.
Non-linear systems are not unusual. Just as the irrational numbers outnumber, so to speak, the rationals, so most systems are nonlinear, rather than linear. There are more ways to draw a crooked line than a straight one. Chaos is a somewhat rarer phenomenon, but it’s still easily produced. As a simple example, imagine that you take a lump of bread dough, stretch it to twice its length, then cut it in half and put one piece on top of the other. Then you repeat, as shown in figure3.2.The procedure is similar to the usual kneading, although there you would fold the dough back onto itself without cutting it in half.
FIGURE 3.2. Two yeast cells in the dough are initially separated by a distance d. The dough is cut in half, the two halves are stacked together, and then the dough is stretched so the distance between cells doubles to 2d. After another iteration, the distance is 4d, and so on. This continues until one or both cells cross to the right-hand side of the dough.
Consider two yeast cells in the dough, perhaps a mother and a daughter (yeast reproduces by budding). Each time the dough is stretched, the distance between the mother and the daughter increases by a factor of two. The system therefore shows sensitivity to initial condition: the mother and the daughter started at almost the same position, but after only a few iterations, they are tragically and irreversibly separated.
The system is known as the shift map, for reasons discussed in Appendix I. Like an ODE, a map takes you from a starting point into the future, but it does so in a discrete, step-wise fashion. In fact, when ODEs are solved numerically, as a sequence of steps of length ID="68"> t, they are equivalent to maps. Chaotic ODEs show both the sensitivity to initial condition and even the same kind of mixing or kneading behaviour as the shift map.
To see the effect that chaos has on predictability, suppose we try to predict the future location of the daughter cell based on the position of the mother cell. This small error in the initial condition will be magnified exponentially until the prediction becomes useless. Even with only a small uncertainty in the initial condition, any prediction performed by applying the shift map is soon no better than random. In fact, as shown in Appendix I, a better guess is to use as our “model” the long-term average of all possible states (i.e., we look for the yeast cell in the middle of the dough). This is sometimes referred to as the climatological forecast: it is like predicting the temperature for a day next week by dispensing with a dynamical model altogether and just determining the average from historical records for that particular day of the year. This illustrates how, even for a simple system, prediction strategies can be complicated by the effects of chaos.
The word “chaos” was actually first used to describe such systems by the mathematician James Yorke in 1975.14 For the ancient Greeks, chaos was the original formless void from which the cosmos was born. Pythagoras associated it with the unlimited, indeterminate aspect of the universe, while the Roman writer Ovid described it as an unordered primordial mass. It therefore evokes a somewhat frightening image of complete randomness and emptiness of meaning, and indeed its existence threw a spanner into the nineteenth century’s clock-like universe. Chaos implied that having an accurate model of a system wasn’t enough to predict its future evolution. Even the slightest error in our knowledge of the initial state would eventually grow. And in a non-linear system, the strategy of breaking a whole down into its components would no longer apply because, like a haiku poem, the whole would be more than the sum of the parts.
Once chaos was discovered, though, it was mostly ignored. Powerful techniques existed to solve linear problems, but nonlinear equations were hard or impossible to tackle, so scientists and engineers simply avoided them, at least until the invention of fast computers in the 1960s. Many phenomena could still be reasonably approximated by the linear approach, particularly engineered systems (which were often designed specifically to work in a linear regime). A metal beam with only small displacements was almost linear, as was the flow of water around a surface at slow speed. While chaos made the job of Laplace’s demon harder, it did not make it impossible, nor did it call into question the mechanistic view of the universe. Even if there was uncertainty in the initial condition, it would still be possible to make probabilistic predictions by taking this variability into account.
HARD AND SOFT
Just as the mechanistic philosophy had managed to bypass chaos, though, and seemed to be taking over, robot-like, the entire world, it stumbled into another small obstacle, which caused it to pause. This was the awkward development, from two extraordinary 1905 papers by Albert Einstein, of the theories of relativity and quantum mechanics. These changed physics—the first at very high speeds, the second at very small scales. The world machine was not as deterministic as it seemed.
Galileo once argued that so long as you travel smoothly at a constant rate, it is impossible to tell whether you are moving. Suppose you are on a boat that sets off so smoothly that you don’t notice its motion. If you look out the window and see a docked boat passing by, you might mistakenly think that it is moving and you are not. And so long as you are moving at a constant speed, you’ll find that if you drop your book to the floor, it will fall straight down, exactly as it would if you were outside on the dock. The laws of physics apply whether you are moving or not. But according to the equations of James Clerk Maxwell, the speed of light should also be constant. Therefore, if a person on the dock flashes a light at you, the beam will appear to both of you to be travelling at the same speed, even though you are moving relative to each other. But this doesn’t make
sense—unless, as Einstein argued in 1905 with his special theory of relativity, time and motion are somehow linked. He predicted that time should flow at a slightly different rate depending on whether you sit still or whiz around very fast. (Scientists tested this in 1971 by comparing two highly accurate clocks, one on the ground and the other on an aircraft. The time difference recorded after a long flight, measured in millionths of seconds, was consistent with predictions made using Einstein’s equations.) His theory was later extended to the general theory of relativity, which showed that gravity was a kind of distortion in the space-time fabric. Newton’s reductionist law of gravity was thus reduced even further, to a by-product of relativity.
As an encore, Einstein suggested that a number of paradoxes associated with the classical theory of electromagnetic radiation could be resolved by assuming that it existed in discrete packets, or quanta. The full implications of quantum mechanics, as it became known, occupied the best physics minds of a generation, but the eventual conclusion was that matter and energy are interrelated (by Einstein’s famous equation, E = mc2) and have a dual particle/wave nature. In Newtonian dynamics, the state of a particle could be exactly described in state space by a single position and momentum. But in quantum dynamics, this was replaced by a wave function, which only specified the probability that the particle could exist in a particular region of state space. Furthermore, Heisenberg’s uncertainty principle stated that the more accurately the position was measured, the greater the uncertainty in momentum, and vice versa. Matter was both determinate and indeterminate, limited and unlimited, hard and soft. At a fundamental level, nature did not observe such distinctions. The wave model and the particle model were incomplete: like breathing in and breathing out, they were two aspects of a unified, dynamic process.
Together, relativity theory and quantum mechanics unravelled much of the certainty that underlay classical physics. Time was not absolute and independent, but was itself a dynamic quantity that depended on the observer. The uncertainty principle showed that even Laplace’s demon could never know the exact position and momentum of a particle. It was as if the universe had a built-in random component. If it wasn’t possible to know the initial conditions of a system, and if time itself depended on the observer, then how could someone predict the future? Properties such as non-locality, where particles can influence each other instantaneously across space, seemed to undermine even the idea of cause and effect.
Such implications were fiercely resisted by some scientists. Einstein, who was religious, insisted that “God does not play dice with the universe,” and pursued until his death the Pythagorean dream of a “theory of everything” that would reduce the universe to a small set of equations. He also spearheaded a rapprochement between science and religion, like a high priest capable of interpreting the mind of God. As he once remarked, “In this materialistic age of ours the serious scientific workers are the only profoundly religious people.”15 Others embraced quantum physics as the way to a new science that equated matter with a kind of spirit (see, for example, Fritjof Capra’s Tao of Physics).16 The predictive sciences, however, mostly stayed out of the debate. While relativity and quantum dynamics were serious issues for systems on an atomic scale, they could safely be ignored for systems that were moving at speeds much slower than the speed of light, and that involved a large number of molecules, such as the weather. The mechanistic approach also gained further currency in biology with the discovery of DNA, as will be discussed in Chapter 5.
Somewhat paradoxically, the biological and social sciences, which modelled themselves after the physical sciences, have clung to the deterministic, mechanistic approach long after it was abandoned in atomic physics. The zoologist Richard Dawkins, for example, describes human beings as “lumbering machines” controlled by our genes.17 The computer scientist Rodney Brooks writes, “We are machines, as are our spouses, our children, and our dogs.”18 But as the physicist David Bohm points out, “At the end of the nineteenth century, physicists widely believed that classical physics gave the general outlines of a complete mechanical explanation of the universe. Since then, relativity and quantum mechanics have overturned such notions altogether. . . . Is it not likely that modern molecular biology will sooner or later undergo a similar fate?”19
GOOD VS EVIL
Determinism raises the issue of free will. The ancient Greeks seem to have believed that human destiny was ruled by fate, which could sometimes be divined by consulting with the oracles. With Christianity, it was believed that much of the future was preordained, but that humans had at least some control over their destiny. (Hardliners like the Lutherans, who believed in predestination, were the exception. They felt that since God knows everything, the future had already been decided. Johannes Kepler, a committed Lutheran, was repelled by this notion, and he got in trouble with his church by arguing against it.)
The notion of free will has continued to bounce back and forth like a philosophical tennis ball. Thomas Hobbes, in the mid-sixteenth century, was against it, since he believed that the human mind was ruled by deterministic, cause-and-effect mechanisms. Immanuel Kant, a century later, whacked it back, asserting that at least part of our minds had the capability for free reasoning. Laplace was against it, because of the success of deterministic science, but quantum mechanics reopened the debate. In a system with sensitivity to initial conditions, as the philosopher Karl Popper argued, quantum effects could be magnified to produce a fundamental indeterminacy in our actions. The path of human history could therefore not be predicted or controlled. Popper dedicated The Poverty of Historicism to “the countless men and women of all creeds or nations or races who fell victims to the fascist and communist belief in Inexorable Laws of Historical Destiny.”20
While one interpretation of quantum mechanics implies that matter is fundamentally indeterministic and can be described only in terms of probabilities, another possibility is that it just appears this way to us because we don’t understand the underlying dynamics. For example, if an electron is in a particular quantum state, there is a certain probability that it will jump to another state with a different energy level. To us, it may seem that this is a purely free, spontaneous event, but since we cannot claim to understand the full workings of the atom, we don’t know if there is a subtle, undetectable force that controls the electron. Does the atom jump, or is it pushed?
If we believe in free will, we might think that when the alarm goes off in the morning, we decide to go to the bathroom, choose to make toast and a cup of tea, and make up our minds what to wear. One could argue, though, that our actions are no less automatic than that alarm going off. Our life is like a movie that is predetermined. It just happens to be very convincingly made, so we go along with it, get into character.
The question of free will has obvious legal implications: if criminals are the product of their environment, then their crimes are not really their fault and the aim of prison should be rehabilitation. People are not intrinsically good or evil. On the other hand, if criminals freely choose their vocation, then justice means retribution rather than therapy.
Science is seen by most of its practitioners as ethically and politically neutral. The scientific process aims to be rational and objective, and therefore eschews subjective value judgments. However, many people choose science over more lucrative careers because they want to contribute to the greater good—by probing the nature of matter, curing disease, or helping the environment. Like priests, they feel they are answering a higher calling. The development of quantum physics, and particularly the discovery that mass can be converted into energy in devices such as atomic bombs, muddied not only the waters of determinism but also the hands of scientists. It’s hard to be objective when you have the power to destroy the world. After the bomb was dropped, Einstein said, “If I knew they were going to do this, I would have become a shoemaker.”21 While some physicists did decide to change fields, the invention of the atomic bomb actually led to vast increases in recruitmen
t for high-energy physicists. The most reliable way to get funding for a scientific project is still to show that, like Galileo’s telescope, it has a military application.
FEATHERED SERPENT II: RETURN FROM THE DEAD
Nothing does as much to shake one’s faith in rational prediction as the arrival of sudden, unforeseen calamities. In Europe, the Great War of 1914–18 destroyed towns, cities, and entire empires. Out of the chaos emerged art movements such as Dadaism and Surrealism, which celebrated irrationality and spontaneity.
The war was followed by an even more deadly event— the flu pandemic of 1918. Known as the Spanish flu, it was actually first recorded in a military camp in Kansas, where in March 1918 it infected over 500 people in two days. The disease spread through the world’s population in months, eventually infecting about a fifth and killing tens of millions (estimates vary). Nowhere was spared, with the exception of a few isolated towns and islands, like Marajo in Brazil. Eighteen months later, it disappeared as mysteriously as it had arrived. It was later found to have been caused by an avian virus, normally resident in wild birds, that had mutated to a human-transmissible form. Not as devastating as smallpox was to the Aztecs, but a feathered serpent nonetheless.
Is it possible to predict the course of such pandemics? One method is to assume that the future will resemble the past. In this spirit, scientists in 2005 reconstituted the virus from segments of genetic material recovered from preserved tissue samples of long-dead victims. They hoped to identify predictable mutations in avian flu that lead to virulence and transmissibility in humans. The problem is that the future never does quite copy the past; like financial crises, each epidemic is a little different from the one that came before.
