by David Orrell
LIMITED VS UNLIMITED
The object of the Beagle’s expedition was “to complete the survey of Patagonia and Tierra del Fuego . . . to survey the shores of Chile, Peru, and of some islands in the Pacific—and to carry a chain of chronometrical measurements round the world.”5 During the five-year cruise, Darwin made copious observations of geology, plants, insects, animals, fossils, and the local human beings. He also produced some great travel writing: “In the morning we got under way, and stood out of the splendid harbour of Rio de Janeiro. . . . We saw nothing particular, excepting on one day a great shoal of porpoises, many hundreds in number. The whole sea was in places furrowed by them; and a most extraordinary spectacle was presented, as hundreds, proceeding together by jumps, in which their whole bodies were exposed, thus cut the water. . . . One dark night we were surrounded by numerous seals and penguins, which made such strange noises that the officer on watch reported he could hear the cattle bellowing on shore. On a second night we witnessed a splendid scene of natural fireworks; the mast-head and yard-armends shone with St. Elmo’s light; and the form of the vane could almost be traced, as if it had been rubbed with phosphorous. The sea was so highly luminous, that the tracks of the penguins were marked by a fiery wake, and the darkness of the sky was momentarily illuminated by most vivid lightning.”6
Back in the less distracting surroundings of his home in England, Darwin evolved the theory of evolution by natural selection, which was based on the idea that variety plus selection equals evolution:
Variety + Selection = Evolution
Offspring are similar, but not identical to, their parents. Each individual therefore has different qualities. Every species produces far more offspring than can survive. Those that are better suited to their environment are more likely to live to have children of their own. As a result, the species gradually evolves and becomes better suited becomes better suited to its environment.
Darwin’s theory implied that we—like seals, penguins, porpoises, and all other creatures on the planet—are merely the end product of a long line of gradual, mechanistic changes and improvements.
That life evolves in some way had been proposed by other scientists, including Jean Baptiste de Lamarck, who first coined the word “biology” and believed that species developed directly in response to their surroundings. A proto-giraffe straining to reach the upper leaves might not make it himself, but he would pass on to his offspring a longer neck. Darwin’s mechanism differed in that the variability was assumed to be purely random. He believed the environment acted as a selection mechanism, and therefore gave a direction to evolution, so that only favourable changes were retained over generations. (The theory didn’t say exactly what caused the variation, but it was later traced to the genes.)
Darwin’s theory was strongly influenced by the work of two economists. The first was Rev. Thomas Malthus, who in his 1798 Essay on the Principle of Population made a notorious prediction: that the human race would eventually outstrip its food supply. His essay didn’t prove its points with equations, but it nevertheless presented a kind of mathematical model: “Population, when unchecked, increases in a geometric [i.e., exponential] ratio. Subsistence increases only in an arithmetical [i.e., linear] ratio. A slight acquaintance with numbers will show the immensity of the first power in comparison with the second.” Suppose, for example, that every twenty-five years, on average, a couple produces four children who survive long enough to have children themselves. Since the replacement rate needed to maintain a stable population is only two people per couple, this means that every twenty-five years, the population will increase by a factor of two (see figure3.1). We would say that the population grows in an exponential fashion (known to Malthus as geometric), with a doubling time of twenty-five years.
Sustenance, on the other hand, relies on increasing the harvest, which Malthus argued would increase only at a linear rate. Since exponential growth always overtakes linear growth, no matter how much land or how many people one starts with, the population will eventually become too large to support itself. Dire consequences follow: “Sickly seasons, epidemics, pestilence, and plague, advance in terrific array, and sweep off their thousands and ten thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world.”
Malthus’s somewhat pessimistic conclusion was out of step with the expansionist mood of Victorian Britain, where large families were positively encouraged. (Darwin, for example, had ten children, though three died in childhood.) It helped lead to economics being labelled the dismal science, and turned him, in the words of one biographer, into “the best abused man of his age. . . . For thirty years it rained refutations.”7 Karl Marx, himself a determined determinist, called Malthus’s essay a “libel on the human race.”8 Darwin, however, realized that this imbalance between the production of offspring and the resources of the environment would provide a kind of mathematical dynamic for his theory of evolution. With such intense competition for resources, only those best suited to the environment would survive and pass their attributes on to the next generation, while the weak would perish.
FIGURE 3.1. if population grows exponentially but resources linearly, then population will eventually outstrip resources, leading to famine.
The exponential growth in population identified by Malthus is a result, to use a modern term, of positive feedback; the rate of increase itself increases with time. It’s like a bank account where the interest earned each year is reinvested; the amount of money available to earn interest the next year therefore increases, leading to the exponential growth that bankers call the “miracle of compound interest” (unless the account earns only a fraction of a percent a year, in which case it is the “miracle of high bank profits”). With a growing population, there are always more people to have babies, so the total rate of births increases, and so on. Growth begets growth.
In fact, most species have a much faster turnaround than humans, and some can grow at truly explosive rates. While the bacteria E. coli were unknown to Darwin—he was not unknown to them, since they resided in his gut—the species can replicate itself at a rate of about one generation every twenty minutes. In twelve hours a single cell, given ideal growth conditions, could grow to 236, or 68 billion cells. In less than one day, the colony could grow large enough to fill a room.
As Malthus pointed out, there is a regulatory mechanism that will eventually limit and control this growth. E. coli doesn’t take over the world because as the colony increases in size, it runs out of food. The individual members come into competition, both with each other and with other species, for the available resources. The higher the population, the fiercer the competition, so continued growth makes future growth less likely—negative feedback. The population does not therefore continue exponentially, but instead tapers off at some limit. If it exceeds that level, the force of competition will restore it to equilibrium.
An example of such regulation in a different context was James Watt’s invention of the flyball governor (see page 93). Darwin, an enthusiastic trader of company shares, was probably more influenced by the work of a second economist, Adam Smith. In The Wealth of Nations, Smith identified a regulating mechanism in free markets, which he referred to as the “invisible hand” of capitalism. If a shortage of some product occurred, for example, the price would spike up; however, as profits rose, more suppliers would enter the market, thus driving the price down once again. Conversely, if prices were too low, suppliers would leave or go broke. Prices would always therefore be restored to the “natural price,” which would reflect the cost of production, as an automatic result of people’s selfinterest: “It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love.” While positive feedback accentuates changes, be they increases or decreases, negative feedback acts
as a stabilizer that damps out perturbations.
THE FLYBALL GOVERNOR
A mechanical example of negative feedback is James Watt’s 1788 invention, the flyball governor, which was used to control the speed of a steam engine. The device consists of two heavy metal balls suspended by movable arms from a central revolving spindle, which is driven by the engine. As the engine speeds up, the two metal balls spread apart and lift under the centrifugal force. They are linked to the throttle in such a way that if the engine runs too quickly, the power cuts back. Conversely, if the speed is too slow, the balls sink down, making the throttle accelerate. It’s like a hand on the tiller of a boat, keeping a steady course.
The apparently simple mechanism results in quite complicated non-linear dynamics—perhaps ironically, systems designed for stable behaviour can be very hard to model. The physicist James Clerk Maxwell had to invent a new branch of mathematics, known as control theory, to analyze Watt’s invention. Without the governor, steam engines would have blown up, so control systems were as important to the Industrial Revolution as coal. Positive and negative feedbacks are ubiquitous not just in engineering, but also in atmospheric, biological, and economic systems, which is one of the reasons predicting them is difficult.
Darwin viewed the natural world as a kind of self-regulating dynamical system of opposing forces. Positive and negative feedback were two manifestations of an underlying creative principle. Variation among individuals and survival of the fittest created a trialand- error selection mechanism, which meant that species’ traits did not remain static but were in a constant state of dynamic flux and self-improvement. Malthus saw a system spiralling off into disaster, but the Darwinian world was limited, kept in check, and constantly improved by competition. Just as a healthy economy was maintained by the invisible hand of price competition, nature was a well-tended garden, with the law of selection doing the pruning and weeding.
In one sense, then, Darwinism was Malthus with a happy ending. In another sense, though, the theory of evolution was as radical as Copernicus’s claim that the earth went around the sun. The heliocentric model had dealt the human ego a huge blow, since it meant that rather than occupying a favoured position at the centre of the universe, our planet was just rotating through space with the others. Evolution implied that there was also nothing particularly special about the human race. We were evolving as well as revolving. Like Copernicus, Darwin was aware of the upsetting consequences of his work, and perhaps as a result, he didn’t publish On the Origin of Species until 1859, twenty-eight years after his voyage, and then only because the naturalist Alfred Russel Wallace had come up with the same idea. One compensation of the theory was that it was widely agreed that English gentlemen, such as Darwin and his peers, were themselves the pinnacle of evolution.
THE ARROW OF TIME
By the end of the nineteenth century, science had itself evolved into a grand world-view, a kind of secular faith that rivalled, or for increasing numbers of people replaced, religion. Nature—for so long a source of awe and mystery, but also of fear—had been placed in the dock, interrogated, broken down. The laws had been applied. In Darwin’s theory of evolution, science even had its own creation myth: nature had been shown to be the result of her upbringing. It followed that she could be predicted and controlled.
Newton had believed that the universe was rather like a game of pool. As he wrote in the Principia, “It seems probable to me that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles.” Even light, he believed, was a stream of such particles (the colour balls). God was relegated to the role of prime mover, the one who breaks the pack with a brilliant cue shot and lets it go; after that, everything worked on automatic. But if all natural phenomena could be explained by physical laws, even the evolution of our own species, then there seemed to be little need for a divine force at all. When Napoleon asked Laplace why God did not play a role in his calculations for the solar system, he is said to have replied, “I have no need of that hypothesis.”
This clockwork, materialistic view of the world applied also to the insides of living beings. Diseases that had once been viewed as random events, or the result of witchcraft, were shown by biologists such as Louis Pasteur to be caused by micro-organisms. Our own bodies were structured like a complex machine: the heart was a pump, veins and arteries the plumbing, the nervous system a network of electrical cables. Newton’s doctor, Richard Mead, believed that one day medicine would reduce to a set of equations.9
Science also extended its cold, mechanical arm into the humanities. The fields of economics, political science, and sociology developed into quasi-deterministic sciences that mimicked the equations of physics in their modelling of human behaviour. Ensconced in the reading room at the British Library, Karl Marx developed his theory of class struggle, which would culminate, he believed, in the ideal state of communism—a kind of heaven for true believers. At Marx’s burial service, his collaborator, Friedrich Engels, claimed, “Just as Darwin discovered the law of evolution in organic nature, so Marx discovered the law of evolution in human history.”10 Institutes such as the London School of Economics modelled society and the economy as Darwin-like competitions for limited resources. August Comte introduced the term “sociology” to describe the scientific study of social behaviour, but he preferred the name “social physics.”
Even the workings of the mind could be explained in machinelike terms. Sigmund Freud shocked the Victorians in 1900 with The Interpretation of Dreams. The human mind, he suggested, was a complicated system of drives and forces, the most powerful of which was the sexual urge. If repressed, it sought release in dreams or, sometimes, in deviant behaviour. Like a blocked drainpipe in the plumbing of the soul, every psychological problem had a precise cause that could be resolved by calling in the analyst.
The revolution in science marched apace with the Industrial Revolution, which spread both scientific technology and Newton’s concept of “absolute time” into factories and sweatshops around the world. Once, time had been a somewhat elastic, non-linear concept. Days were broken down into regular intervals, but these would be shorter in winter and longer in summer, to reflect the changing length of the day.11 While Newton was building his mathematical theories, however, the Royal Greenwich Observatory was going up outside London. Its construction, funded by the sale of 690 barrels of gunpowder, was to ensure British naval supremacy, which relied on accurate navigation, which in turn required accurate clocks.12 In 1884, the Greenwich Meridian became the world’s prime meridian, dividing the globe, by a straight line, into East and West, so time and space were both on the same Cartesian grid. Uniform, linear
time spread around the world, along with language and empire. Soon everyone was marching to the same Greenwich Mean beat. Time was parsed to smaller and smaller intervals, and people’s lives became dominated by the absolute law of the clock and the worship of the twin gods of time and money (though Benjamin Franklin argued that time was money).
While the power attributed to nature or divine forces was on the decline, the power of the scientist was on the rise. Books such as History of the Conflict between Religion and Science (1874), by the New York University scientist John Draper, and A History of the Warfare between Science and Theology in Christendom (1896), by Andrew Dickson White from Cornell University, directly challenged the role of religion. To many people, science now seemed to hold the answers to the big questions: Who are we? Where do we come from? And especially, where are we going? The future of the world was not determined by random forces or destiny or the will of God, but was the natural consequence of physical laws. In the mechanistic church, the bible was the laws of physics as encoded in the scientific literature; the priests who interpreted these laws were the scientists; and the cathedrals were the laboratories, universities, and new research institutes like the Massachusetts Institute of Technology. The linear march of science seemed unstoppable—at least until, at the dawn of the twentieth century
, it hit the first in a sequence of bumps in the road.
CHAOS
The success of reductive science lay in its ability to “divide every difficult problem into small parts, and to solve the problem by attacking these parts.” For example, the methods of calculus, when used to predict the path of a falling stone, involve breaking down time and space into smaller and smaller increments. This works because time is, for the purposes here, uniform. One instant is like another—it’s all the same stuff. Greenwich Mean Time is unaffected by the fact that someone is dropping a stone from a tower. In the case of time, it seemed true that one could divide every problem into small parts.
The divide-and-conquer approach also works for combinations of simple forces. If the falling object in figure 2.4 (see page 81) also moves horizontally—say, it has been shot from a cannon—then the velocity can be parsed into horizontal and vertical components, and each treated separately. In general, the systems that yield most easily to this approach are those that are described as linear. While the term “linear” is often used to describe the straightforward, cause-and-effect logic championed by Aristotle, it also has a more precise mathematical meaning. A function of some variable is linear if the plot of the function versus the variable is a straight line.13 An example is a perfect spring, which is defined as one for which the restorative force varies linearly with displacement. Place a small weight such as a piece of straw on the spring, and it will compress a certain amount. Double the weight, and it compresses the same amount more. The initial condition of the spring doesn’t matter— it changes the same extra amount whether there is a weight already there or not. To know the deflection caused by two or more straws, we can measure the deflection of a single straw and multiply by the total number. In a linear system like this, the whole is the same as the sum of the parts.
