A Sensitive Dependence on Initial Conditions
by Kim Stanley Robinson (1991)
The covering law model of historical explanation states that an event is explained if it can be logically deduced from a set of initial conditions, and a set of general historical laws. These sets are the explanans, and the event is the explanandum. The general laws are applied to the initial conditions, and the explanandum is shown to be the inevitable result. An explanation, in this model, has the same structure as a prediction.
On the morning of August 6th, 1945, Colonel Paul Tibbets and his crew flew the Enola Gay from Tinian Island to Hiroshima, and dropped an atomic bomb on the city. Approximately a hundred thousand people died. Three days later, another crew dropped a bomb on the outskirts of Nagasaki. Approximately seventy thousand people died. The Japanese surrendered.
President Harry Truman, in consultation with his advisors, decided to drop the bombs. Why did he make these decisions? Because the Japanese had fiercely defended many islands in the South Pacific, and the cost of conquering them had been high. Kamikaze attacks had sunk many American ships, and it was said that the Japanese would stage a gigantic kamikaze defense of the home islands. Estimated American casualties resulting from an invasion of the home islands ranged as high as a million men.
These were the conditions. General laws? Leaders want to end wars as quickly as possible, with a minimum of bloodshed. They also like to frighten potential postwar enemies. With the war in Europe ended, the Soviet Army stood ready to go wherever Stalin ordered it. No one could be sure where Stalin might want to go. An end to the Japanese war that frightened him would not be a bad thing.
But there were more conditions. The Japanese were defenseless in the air and at sea. American planes could bomb the home islands at will, and a total naval blockade of Japan was entirely possible. The Japanese civilian population was already starving; a blockade, combined with bombing of military sites, could very well have forced the Japanese leaders to surrender without an invasion.
But Truman and his advisors decided to drop the bombs. A complete explanation of the decision, omitted here due to considerations of length, would have to include an examination of the biographies of Truman, his advisors, the builders of the bomb, and the leaders of Japan and the Soviet Union; as well as a detailed analysis of the situation in Japan in 1945, and of American intelligence concerning that situation.
President Truman was re-elected in 1948, in an upset victory over Thomas Dewey. Two years later the United States went to war in Korea, to keep that country from being overrun by Communists supported by the Soviet Union and China. It was only one of many major wars in the second half of the twentieth century; there were over sixty, and although none of them were nuclear, approximately fifty million people were killed.
Heisenberg’s uncertainty principle says that we cannot simultaneously determine both the velocity and the position of a particle. This is not a function of human perception, but a basic property of the universe. Thus it will never be possible to achieve a deterministic prediction of the movement of all particles throughout spacetime. Quantum mechanics, which replaced classical mechanics as the best description of these events, can only predict the probabilities among a number of possible outcomes.
The covering law model of historical explanation asserts that there is no logical difference between historical explanation and scientific explanation. But the model’s understanding of scientific explanation is based on classical mechanics. In quantum reality, the covering law model breaks down.
The sufficient conditions model of historical explanation is a modification of the covering law model; it states that if one can describe a set of initial conditions that are sufficient (but not necessary) for the event to occur, then the event can be said to be explained. Deduction from general law is not part of this model, which is descriptive rather than prescriptive, and “seeks only to achieve an acceptable degree of coherent narrative.”
In July of 1945, Colonel Tibbets was ordered to demonstrate his crew’s ability to deliver an atomic weapon, by flying a test mission in the western Pacific. During the takeoff Tibbets shut down both propellers on the right wing, to show that if this occurred during an armed takeoff, he would still be able to control the plane. The strain of this maneuver, however, caused the inboard left engine to fail, and in the emergency return to Tinian the Enola Gay crashed, killing everyone aboard.
A replacement crew was chosen from Tibbets’ squadron, and was sent to bomb Hiroshima on August 9th, 1945. During the run over Hiroshima the bombardier, Captain Frank January, deliberately delayed the release of the bomb, so that it missed Hiroshima by some ten miles. Another mission later that week encountered cloud cover, and missed Kokura by accident. January was court-martialed and executed for disobeying orders in battle. The Japanese, having seen the explosions and evaluated the explosion sites, surrendered.
January decided to miss the target because: he had a visionary dream in which he saw the results of the bombing; he had not been in combat for over a year; he was convinced the war was over; he had been in London during the Blitz; he disliked his plane’s pilot; he hated Paul Tibbets; he was a loner, older than his fellow squadron members; he had read the Hornblower stories in the Saturday Evening Post; he once saw a truck crash into a car, and watched the truck driver in the aftermath; he was burned on the arm by stove oil when a child; he had an imagination.
The inboard left engine on the Enola Gay failed because a worker at the Wright manufacturing plant had failed to keep his welding torch flame on a weld for the required twenty seconds. He stopped three seconds too soon. He stopped three seconds too soon because he was tired. He was tired because the previous night he had stayed up late, drinking with friends.
In 1948, President Truman lost to Thomas Dewey in a close election that was slightly influenced by a political group called the January Society. The Korean conflict was settled by negotiation, and in February of 1956 a treaty was signed in Geneva, banning the use and manufacture of nuclear weapons.
Light behaves like either wave or particle, depending on how it is observed. The famous two-slit experiment, in which interference in wave patterns causes light shining through two slits in a partition to hit a screen in a pattern of light and dark bars, is a good example of this. Even when photons are sent at the slits one at a time, the pattern of light and dark bars still appears, implying that the single quantum of light is passing through both slits at the same time, creating an interference pattern with itself.
History is an interference pattern, says the covering law model. The conditions are particles; the laws are waves.
The necessary conditions model states that historical explanation requires merely identifying the kind of historical event being explained, and then locating among its initial conditions some that seem necessary for the event to take place. No general laws of history can help; one can only locate more necessary conditions. As William Dray writes in Laws and Explanation in History, an explanandum is explained when we “can trace the course of events by which it came about.”
Tibbets and his crew died in a training flight crash, and the Lucky Strike was sent in Enola Gay’s place. The bombardier, Captain Frank January, after much frantic thought on the flight there, performed just as Tibbets’ bombardier would have, and dropped the bomb over the T-shaped Aioi Bridge in Hiroshima. Approximately a hundred thousand people died. Three days later Nagasaki was bombed. The Japanese surrendered. Truman was re-elected. The Korean War led to the Cold War, the assassination of Kennedy on November 22nd, 1963, the Vietnam War, the collapse of the Soviet bloc in the fall of 1989. Replacing one crew with another made no larger difference.
Richard Feynman’s notion of a “sum over histories” proposes that a particle does not move from point A to point B by a single path, as in classical mechanics, but rather by every possible path within the wave. Two numbers describe these possible paths, one describing the size of the wave, the other the path’s position in the crest-to-trough cycle. When Pauli’s exclusion principle, which states that two particles cannot occupy the same position at the same velocity within the mathematical limits of the uncertainty principle, is applied to the sum over histories, it indicates that some possible paths cause interference patterns, and cancel each other out; other paths are phased in a reinforcing way, which makes their occurrence more probable.
Perhaps history has its own sum over histories, so that all possible histories resemble ours. Perhaps every possible bombardier chooses Hiroshima.
The weak covering law model attempts to rescue the notion of general historical laws by relaxing their rigor, to the point where one can no longer deduce the explanandum from the explanans alone; the laws become not laws but tendencies, which help historians by providing “guiding threads” between events and their initial conditions. Thus the uncertainty principle is acknowledged, and the covering law model brought into the twentieth century.
But can any historical model explain the twentieth century? Tibbets crashed, the Lucky Strike flew to Hiroshima, and Captain January chose to spare the city. He was executed, the war ended, Dewey won the 1948 election; the Korean conflict was resolved by negotiation; and nuclear weapons were banned by treaty in February of 1956.
But go on. In November of 1956, conflict broke out in the Middle East between Egypt and Israel, and Britain and France quickly entered the conflict to protect their interests in the Suez Canal. President Dewey, soon to be replaced by President-elect Dwight Eisenhower, asked Britain and France to quit the conflict; his request was ignored. The war spread through the Middle East. In December the Soviet Army invaded West Germany. The United States declared war on the Soviet Union. China launched assaults in Indochina, and the Third World War was under way. Both the United States and the Soviet Union quickly assembled a number of atomic bombs, and in the first week of 1957, Jerusalem, Berlin, Bonn, Paris, London, Warsaw, Leningrad, Prague, Budapest, Beirut, Amman, Cairo, Moscow, Vladivostok, Tokyo, Peking, Los Angeles, Washington, D.C., and Princeton, New Jersey (hit by a bomb targeted for New York) were destroyed. Loss of life in that week and the year following was estimated at a hundred million people.
At normal energies, the strong nuclear force has a property called confinement, which binds quarks tightly together. At the high energies achieved in particle accelerators, however, the strong nuclear force becomes much weaker, allowing quarks and gluons to jet away almost like free particles. This property of dispersion at high energies is called “asymptotic freedom.”
History is a particle accelerator. Energies are not always normal. We live in a condition of asymptotic freedom, and every history is possible. Each bombardier has to choose.
In The Open Society and Its Enemies Karl Popper writes: “If two armies are equally well-led and well-armed, and one has an enormous numerical superiority, the other will never win.” Popper made this proposition to demonstrate that any historical law with broad explanatory power would become so general as to be trivial. For the school of thought that agrees with him, there can be no covering laws.
In June of 1945, seven of the scientists who had worked on the Manhattan Project submitted a document called the Franck Report to the Scientific Panel of the Interim Committee, which was overseeing the progress of the bomb. The Franck Report called for a demonstration of the bomb before observers from many countries, including Japan. The Scientific Panel decided this was a possible option and passed the Report on to the Committee, which passed it on to the White House. “The Buck Stops Here.” Truman read the report and decided to invite James Franck, Leo Szilard, Niels Bohr, and Albert Einstein to the White House to discuss the issue. Final consultations included Oppenheimer, Secretary of War Stimson, and the military head of the Manhattan Project, General Leslie Groves. After a week’s intense debate Truman instructed Stimson to contact the Japanese leadership and arrange a demonstration drop, to be made on one of the uninhabited islands in the Izu Shichito archipelago, south of Tokyo Bay. An atomic bomb was exploded on Udone Shima on August 24th, 1945; the mushroom cloud was visible from Tokyo. Films of the explosion were shown to Emperor Hirohito. The Emperor instructed his government to surrender, which it did on August 31st, one day before Truman had declared he was going to begin bombing Japanese cities.
Truman won the election of 1948. In 1950 North Korean troops invaded the south, until a series of six so-called Shima blasts, each closer to the north’s advance forces, stopped them at the 38th parallel. In 1952 Adlai Stevenson became president, and appointed Leo Szilard the first presidential science advisor. In 1953 Stalin died, and in 1956 Szilard was sent to Moscow for a consultation with Khrushchev. This meeting led to the founding of the International Peace Brigade, which sent internationally integrated teams of young people to work in underdeveloped countries and in countries still recovering from World War Two. In 1960 John Kennedy was elected president, and he was succeeded in 1968 by his brother Robert. In 1976, in the wake of scandals in the administration, Richard Nixon was elected. At this point in time the postwar period is usually considered to have ended. The century itself came to a close without any further large wars. Though there had been a number of local conflicts, the existence of nuclear weapons had ended war as practiced in the first half of the century. In the second half, only about five million people died in war.
The great man theory considers particles; historical materialism considers waves. The wave/particle duality, confirmed many times by experiment, assures us that neither theory can be the complete truth. Neither theory will serve as the covering law.
The defenders of the covering law model reply to its various critiques by stating that it is irrelevant whether historians actually use the model or not; the fact remains that they should. If they do not, then an event like “the bottle fell off the table” could be explained by either “the cat’s tail brushed it,” or “the cat looked at it cross-eyed,” and there would be no basis for choosing between the two explanations. Historical explanation is not just a matter of the practice of historians, but of the nature of reality. And in reality, physical events are constrained by general laws—or if they are not laws, they are at least extraordinarily detailed descriptions of the links between an event and those that follow it, allowing predictions that, if not deterministically exact, are still accurate enough to give us enormous power over physical reality. That, for anyone but followers of David Hume, serves as law enough. And humans, as part of the stuff of the universe, are subject to the same physical laws that control all the rest of it. So it makes sense to seek a science of history, and to try to formulate some general historical laws.
What would these general laws look like? Some examples:
- If two armies are equally well-led and well-armed, and one has an enormous numerical superiority, the other will never win.
- A privileged group will never relinquish privilege voluntarily.
- Empires rise, flourish, fall and are replaced, in a cyclical pattern.
- A nation’s fortunes depend on its success in war.
- A society’s culture is determined by its economic system.
- Belief systems exist to disguise inequality.
- Lastly, unparalleled in both elegance and power, subsuming many of the examples listed above: power corrupts.
So there do seem to be some quite powerful laws of historical explanation. But consider another:
- For want of a nail, the battle was lost.
For instance: on July 29th, 1945, a nomad in Kirgiz walked out of his yurt and stepped on a butterfly. For lack of the butterfly flapping its wings, the wind in the area blew slightly less. A low-pressure front therefore moved over east China more slowly than it would have. And so on August 6th, when the Enola Gay flew over Hiroshima, it was covered by ninety percent cloud cover, instead of fifty percent. Colonel Tibbets flew to the secondary target, Nagasaki; it was also covered. The Enola Gay had little fuel left, but its crew was able to fly over Kokura on the way back to Tinian, and taking advantage of a break in the clouds, they dropped the bomb there. Ninety thousand people died in Kokura. The Enola Gay landed at Tinian with so little fuel left in its tanks that what remained “wouldn’t have filled a cigarette lighter.” On August 9th a second mission tried Hiroshima again, but the clouds were still there, and the mission eventually dropped the bomb on the less heavily clouded secondary target, Nagasaki, missing the city center and killing only twenty thousand people. The Japanese surrendered a week later.
On August 11th, 1945, a child named Ai Matsui was born in Hiroshima. In 1960 she began to speak in local meetings on many topics, including Hiroshima’s special position in the world. Its citizens had escaped annihilation, she said, as if protected by some covering angel (or law); they had a responsibility to the dead of Kokura and Nagasaki, to represent them in the world of the living, to change the world for the good. The Hiroshima Peace Party quickly grew to become the dominant political movement in Hiroshima, and then, in revulsion at the violence of the 1960s in Vietnam and elsewhere, all over Japan. In the 1970s the party became a worldwide movement, gaining the enthusiastic support of ex-President Kennedy, and President Babbitt. Young people from every country joined it as if experiencing a religious conversion. In 1983 Japan began its Asian Assistance League. One of its health care programs saved the life of a young woman in India, sick with malaria. The next year she had a child, a woman destined to become India’s greatest leader. In 1987, the nation of Palestine raised its flag over the West Bank and parts of Jordan and Lebanon; a generation of camp children moved into homes. A child was born in Galilee. In 1990 Japan started its African Assistance League. The Hiroshima Peace Party had a billion members.
And so on; so that by July 29th, 2045, no human on Earth was the same as those who would have lived if the nomad in Kirgiz had not stepped on the butterfly a century before.
This phenomenon is known as the butterfly effect, and it is a serious problem for all other models of historical explanation; meaning trouble for you and for me. The scientific term for it is “sensitive dependence on initial conditions.” It is an aspect of chaos theory first studied by the meteorologist Edward Lorenz, who, while running computer simulations of weather patterns, discovered that the slightest change in the initial conditions of the simulation would quickly lead to completely different weather.
So the strong covering law model said that historical explanation should equal the rigor of scientific explanation. Then its defenders, bringing the model into the quantum world, conceded that predictions can never be anything but probabilistic at best. The explanandum was no longer deducible from the explanans; one could only suggest probabilities.
Now chaos theory has added new problems. And yet consider: Captain Frank January chose to miss Hiroshima. Ten years later, nuclear weapons were universally banned. Eleven years later, local conflicts in the Middle East erupted into general war, and nuclear weapons were quickly reassembled and used. For it is not easy to forget knowledge, once it is learned; symmetry T, which says that physical laws are the same no matter which way the time arrow is pointed, does not actually exist in nature. There is no going back.
And so by 1990, in this particular world, the bombed cities were rebuilt. The Western industrial nations were rich, the Southern developing nations were poor. Multinational corporations ruled the world’s economy. The Soviet bloc was falling apart. Gigantic sums of money were spent on armaments. By the year 2056, there was very little to distinguish this world from the one in which January had dropped the bomb, in which Tibbets had bombed Hiroshima, in which Tibbets had made a demonstration, in which Tibbets bombed Kokura.
Perhaps a sum over histories had bunched the probabilities. Is this likely? We don’t know. We are particles, moving in a wave. The wave breaks. No math can predict which bubbles will appear where. But there is a sum over histories. Chaotic systems fall into patterns, following the pull of strange attractors. Linear chaotic figures look completely non-repetitive, but slice them into Poincaré sections and they reveal the simplest kinds of patterns. There is a tide, and we float in it; perhaps it is the flux of the cosmos itself; swim this way or that, the tide still carries us to the same destination. Perhaps.
So the covering law model is amended yet again. Explanations still require laws, but there are not laws for every event. The task of historical explanation becomes the act of making distinctions, between those parts of an event that can be explained by laws, and those that cannot. The component events that combine to create an explanandum are analyzed each in turn, and the historian then concentrates on the explicable components.
Paul Tibbets flies toward Hiroshima. The nomad steps out of his yurt.
Lyapunov exponents are numbers that measure the conflicting effects of stretching, contracting, and folding in the phase space of an attractor. They set the topological parameters of unpredictability. An exponent greater than zero means stretching, so that each alternative history moves farther and farther apart as time passes. An exponent smaller than zero means contraction, so that alternatives tend to come back together. When the exponent equals zero, a periodic orbit results.
What is history’s Lyapunov exponent? This is the law that no one can know.
Frank January flies toward Hiroshima. The nomad stops in his yurt.
It is said that the historian’s task requires an imaginative reconstruction of the thinking of people who acted in the past, and of the circumstances in which they acted. “An explanation is said to be successful when the historian gets the sense of reliving the past which he is trying to explain.”
You are flying toward Hiroshima. You are the bombardier. You have been given the assignment two days before. You know what the bomb will do. You do not know what you will do. You have to decide.
There are a hundred billion neurons in the brain. Some of the neurons have as many as eighty thousand synaptic endings. During thought, neurotransmitter chemicals flow across the synaptic clefts between one neuron’s synaptic knobs and another’s dendritic spines, reversing a slight electric charge, which passes on a signal. The passage of a signal often leaves changes in the synapses and dendrites along the way, forever altering the structure of the brain. This plasticity makes memory and learning possible. Brains are always growing; intensely in the first five years, then steadily thereafter.
At the moment of choice, then, signals fly through a neural network that has been shaped over a lifetime into a particular and unique structure. Some signals are conscious, other are not. According to Roger Penrose, during the process of decision quantum effects in the brain take over, allowing a great number of parallel and simultaneous computations to take place; the number could be extraordinarily large, 1021 or more. Only at the intrusion of the “observation,” that is to say a decision, do the parallel computations resolve back into a single conscious thought.
And in the act of deciding, the mind attempts the work of the historian: breaking the potential events down into their component parts, enumerating conditions, seeking covering laws that will allow a prediction of what will follow from the variety of possible choices. Alternative futures branch like dendrites away from the present moment, shifting chaotically, pulled this way and that by attractors dimly perceived. Probable outcomes emerge from those less likely.
And then, in the myriad clefts of the quantum mind, a mystery: the choice is made. We have to choose, that is life in time. Some powerful selection process, perhaps aesthetic, perhaps moral, perhaps practical (survival of the thinker), shoves to consciousness those plans that seem safest, or most right, or most beautiful, we do not know; and the choice is made. And at the moment of this observation the great majority of alternatives disappear without trace, leaving us in our asymptotic freedom to act, uncertainly, in time’s asymmetrical flow.
There are few covering laws. Initial conditions are never fully known. The butterfly may be on the wing, it may be crushed underfoot. You are flying toward Hiroshima.