*Ring a ring o’ roses, A pocketful of posies. A-tishoo! A-tishoo! We all fall down.*

For this happy English nursery rhyme, children hold hands to form a circle, and then dance around, singing. Nice for a birthday party. At the end, they all fall down, laughing. However, many people believe this happy, innocent little song easily remembered by young children refers to the dreaded plague that killed hundreds of thousands all over Europe; at times, two-thirds of a community would perish. The “A-tishoo! A-tishoo!” may refer to the sneezing during the pneumonic phase of the disease that can develop after the initial, bubonic phase, known for its feared red spots and boils. The first phase alone led to tens—even hundreds—of thousands suffering an awful death. The frightening, painful deaths of the plague victims in the Middle Ages and in subsequent epidemics (notably the one in London in 1665) soon disappeared from the collective memory.

The worldwide wave of concern caused by the book "The Limits to Growth," a mere forty years ago, wherein the Club of Rome warned that our earthly resources are limited, seems to have suffered the same fate. It was soon forgotten. But if that concern was justified, by pushing it out of our mind, haven’t we lost much valuable time that could have been used to tackle the problem? The Club of Rome’s warning in 1972 did not have the immediate consequences of the plague; its consequences are longer term and will be felt by future generations, but they will have a much larger impact: the suffering and death of hundreds of millions of people. Of billions perhaps. What have we been doing since 1972? How can we have forgotten? Instead of reducing our numbers and resource use, we have stimulated them—deliberately. Since the 1970s, our reproductive rates have reached unprecedented heights, and per capita consumption has multiplied, particularly in the West. During the last ten thousand years, our numbers, demands, and reproductive rates have never been so high. Who worries?

We are now entering the second wave of concern about resource limitations, one to which concerns about energy supply and climate warming have been added. Other concerns haven’t penetrated the media that deeply yet, and discussing the increase in our numbers seems to be a taboo subject that has met with resistance. Meanwhile, the tone of the debate is more positive and optimistic than it was during the 1970s. For example, worldwide, ways of reducing the amount of carbon dioxide expelled into the air to fight climate warming are widely being discussed: how this can be done by more economical energy use, by replanting forests, or by burying the carbon dioxide. And we seem satisfied by the forecasts that our numbers will stabilize around 2050, not realizing that our resource use and its consequent waste production are not connected into a perfect cycle, but are linear. Stabilizing the world’s population while maintaining resource use at such an incredibly high level cannot but lead to rapid exhaustion and overpollution. We are getting closer and closer to those limits, and during the last thirty years the network of interactions has tightened into a fyke. Our higher numbers and demands give us less time to maneuver away from disaster.

In "The Limits to Growth," Dennis Meadows and others concluded from one calculation that the number of humans could crash suddenly rather than stabilize gradually. But none of the other calculations showed this effect; their results suggested that the numbers of humans on Earth had to be reduced gradually, and with them, the overuse of natural resources. It seemed that this single result was anomalous and could be ignored, although its cause remained unclear.

Twenty years later, however, in their 1992 follow-up book "Beyond the Limits," on the basis of calculations using data from the intermediate years, the authors reported that such crashes were no longer exceptional but had become the rule. Results without a population crash had become exceptional; crashes appeared to be normal and seemed not easily avoidable. This was a very different story. Without knowing the underlying causes, population crashes were now being attributed to delays in the fine tuning of interactions within the system and to the exceeding of limits of irreversible degradation.

But why a crash instead of a slower slide? What had changed in those twenty years? It remains difficult to understand. There are some known processes that produce such results and that can operate in concert, enhancing their effects.

Here’s a riddle: a pondweed like duckweed is assumed to grow in a pond, doubling its surface cover in a single day. As the surface area of the pond is finite, the duckweed can cover it completely in thirty days. So, how much of the pond has the duckweed covered by, say, the twenty-seventh day? The answer is astonishingly low: 12.5 percent. Hardly anything. Yet it’s true. Most people find it difficult to do the calculation, although the difficulty lies not in the calculation itself, but rather, in three ways of thinking that are unfamiliar. First, it is often difficult to calculate backward. It is easy to calculate the plant cover on day one, then on day two, and so on. That’s how we learned to solve such problems at school. In this case, however, you start by calculating the plant cover on day twenty-nine, which is 50 percent of the total cover on day thirty, then use that figure to calculate the cover on day twenty-eight, which is half of 50 percent, or 25 percent, and then do this again for day twenty- seven, which gives the 12.5 percent. Not difficult at all—just unfamiliar.

Second, growth is usually assumed to continue indefinitely, whereas in this case there is a definite end: the 100 percent cover of the pond. The growth process, therefore, is limited, just like our resources (the amount of forest, metals, and energy) are limited, or like the amount of waste we can dump into the environment. By contrast, there is no maximum temperature for Earth’s atmosphere. With enough carbon dioxide in the atmosphere, here on Earth it could become as hot as on Venus, where surface temperatures are 450°C. But like us, plants can tolerate temperatures only up to some point, beyond which they wilt and die. Neither plants nor humans could live on Venus. The same holds for the amount of waste the environment can tolerate before it becomes irretrievably polluted for plants or animals; we all know that there is some such threshold, but we don’t know how far pollution can proceed—we don’t know its limits.

Finally, we are not used to thinking in terms of doubling or tripling times. That is, most of us are not used to thinking in terms of exponential growth or reduction. We can add and subtract, and we can multiply once but not several times by the same number, which is what you do when calculating exponential growth. Moreover, from personal experience, we know that a family grows by adding children: one, two, or three. But when calculated over a population, later these children have families of their own and so the process changes from an additive to a multiplicative process, resulting in exponential growth. An average of three children per family produces a total of nine children in the next generation, then twenty seven in the third. This change in thinking from a concrete additive process to the abstract one of exponential growth can be difficult. But thinking in terms of exponential growth is essential in order to be able to understand many of the processes of resource use, the growth of industries, the spread of diseases, and the complexification of society.

We don’t know how or when such a crash develops, in what corner the problem will arise, or how fast it will go. It may be triggered by some minor problem, the effects of which amplify, pulling other sectors of society down, thereby rapidly, exponentially worsening the growing disaster. The only thing we do know is that many trends in society point in the wrong direction, making it increasingly more prone to collapse, such as our growing numbers and demands, the declining biodiversity, climate change, and the rapid depletion of sources of energy, nutrients, and water. What makes this worse is that we increasingly depend on an economy based on growth, growth considered both the cause and the cure of our problems. And we know that all of these trends are interdependent. For example, sustaining our growing numbers and demands depends on a regular and unlimited supply of energy—a supply that, nevertheless, is limited. Approaching its limits means that we need to develop ever more efficient technology to squeeze out the last remains of the fossil fuels from Earth, which, in turn, depends on ever larger investments. However, investors want to see their money back, which proves increasingly more difficult the closer we get to the point of depletion. They have to invest more and more, but as less fuel can be mined, the returns become less and less. Approaching the point of depletion, investors will gradually draw their money back in order to reinvest it into something more profitable: at this point, a positive feedback loop starts up; more and more investors withdraw their money, this loop destabilizing the societal system as a whole. Energy shortage will destabilize this system because energy is the main constituent of our body, our numbers, requirements, and infrastructural organization.

In fact, the decreasing trend in the energy returns on investment was already apparent in the early 1990s, a trend which continues to the present day and which may develop into the feared financial and economic positive feedback loop. Food will be more expensive to produce, leaving the poor in jeopardy. And so on. Similar trends in other basic requirements either occur already or are imminent. Because energy shortages may tighten, they can therefore also develop in the supply of nutrients or water; and the extraction, recycling, or desalination of seawater will require increasing amounts of energy. And this, in turn, can push us faster in the direction of energy depletion, and into that of the positive feedback loop of lesser energy returns on investment. And so forth. We know the trends, we know where they will lead and how, but we don’t know which of them will trigger the others to join into the one positive loop or the other, and when.

Most of us know from school how markets work; we learned the simple, two-factor systems of supply and demand. According to Adam Smith’s “invisible hand” of the late eighteenth century, prices will balance each other. In such systems, supply and demand balance each other through the price asked for a product. A greater supply leads to lower prices, which increases the demand. Greater demand, in turn, increases the price, which, again, increases the supply, which brings us back to the first step, a lowering of the prices. This results in an endless, wave-like process, undulating into eternity. The same idea can be found in ecology where two competing species or a predator and its prey would similarly balance each other; or in selection theory, where two parties would compete, although here the fittest eventually wins.

And in James Lovelock’s Daisyworld, over geological time producers and users of carbon dioxide would cause climate to fluctuate regularly around one stable level. In fact, the occurrence of a stable level is basic to the concept of carrying capacity, where abundances are also assumed to fluctuate around a certain stable level. This would also happen to the human population after 2050, when it is presumed to remain stable at the level of nine or ten billion people. All these concepts consider short-term or local dynamics underlying a long-term or global stability, never a collapse of the system. Such systems would be in equilibrium, or could even have two equilibriums at different levels from which it is difficult to escape. Fish stocks, for example, could switch to a lower equilibrium level after having been overfished for some time. It would then be almost impossible to switch back to the previous, higher level.

Although these processes do occur, reality is usually more unruly and complex. Usually, there are more than two system components interacting, and this is a point where the real problems arise. We can derive mathematical equations for the dynamics of two-component systems, but this is theoretically impossible for systems consisting of three components or more, such as in economic ones in human society, in ecologic ones in nature, or among the few planets rotating around the sun.

Because they are much of a jumble as well, societies can crash or collapse. Such crashes not only develop rapidly, but their cause, course, and timing are unpredictable. Mathematicians call this field of study deterministic chaos: unpredictability reigns, even when nothing happens by chance; chance within the process only gives additional unpredictability. Imagine, therefore, what happens when such systems contain an element of chance as well.

So, how does chance work, and does chance depend on the number of people making up society and its complexity? If so, does the chance of societal collapse increase over time as our numbers and their resulting societal complexity grow? Have our living conditions changed (gradual soil salination, or a sudden rise in the price of food due to drought in Australia or Russia, for example)?

Think for a moment of a die: what is the chance of throwing, say, a five? A die has six sides, each with the same chance of turning up. The chance of throwing a five is one in six, or 17 percent. Conversely, the combined chance of throwing any number other than five is five in six, or 83 percent. But how great is the chance of getting a five within two consecutive throws? That chance is obviously twice as large, or 33 percent, and the chance of getting any other number is 67 percent. Therefore, the more throws, the greater the chance of getting your preferred five at least once. And the chance of missing it reduces accordingly. The same reasoning applies to, say, the chance of some explosion happening in an oil pipe, though in this case you are interested in the chance of the event *not *happening. Now the chance that some disaster will not happen is made as small as possible, say, one in 10,000, and the chance of an explosion occurring is only one in 9,999. Obviously, these chances also depend on the length of the pipe, on the number of pipes, on the number of welds, or the number of pumping and control stations, that is, on the complexity of the pipe system, and also on the length of the period the system is operating: the longer the pipes and the more there are, the greater the complexity of the system they form and the longer the period of operation, the greater the chance of something going wrong, resulting in an explosion.

Moreover, all these mistakes and disasters have different chances of happening, and all these chances are superimposed. You can try out for yourself what happens by throwing different kinds of dice, the normal one with six sides, then one with four, eight, ten, twelve, twenty, and one with thirty sides. The result is a very wiggly line when you add the outcomes of these sets of dice for a number of throws together for each point on this line. Each new point is different from any of the previous ones and therefore is impossible to predict; it was already impossible to predict the outcome of one single die. Still, this curve resembles the real world in many respects where also many chance events occur, the one adding to another and each with a different chance of happening.

In reality, the chances have different and varying weights relative to the total process as well, and they interact both linearly and nonlinearly, which we all kept constant and independent when we threw our seven sets of dice. How can we predict the future of society but in general terms of depletion and pollution rates? These are our certainties, but we really can’t predict in detail what will happen and when as a social or economic result. For these societal effects we can only say that the chance of collapse increases with an increasing complexity of society, as well as with increasing stress from resource depletion, pollution, and social inequality.

Think of the decline of ancient Rome, which took centuries; nobody knows why it declined; we have more explanations than authors. Because of the great influence of chance in all aspects of society, whose behavior is unknowable and, hence, unpredictable—manageable only up to some point, after which further developments grow out of hand. Why the reason for a crash such as the decline of Rome is also unknowable, and why its crash was unmanageable, is that people usually look at only one process in isolation, such as the invasion of the Gothic tribes or the general poisoning of people by lead in the water pipes. In many cases, however, a disaster is triggered by the coinciding of a number of different events or processes, not by a single event or process. Therefore, as our numbers continue to grow exponentially, the size and complexity of society increases exponentially relative to those numbers. Consequently, the predictability of a particular crash developing from the occurrence of a certain combination of chance events or processes decreases.

Moreover, because many factors can be interdependent, a crash in one sector pulls others in its wake, making it a general crash in no time and also making it more difficult to manipulate or manage. Crashes of our socioeconomic system will therefore become more frequent and less easy to control.

I think that the collapse of the present human population, its numbers and quality of life, is likely, and also that the most humane way to weather this period is to design a strategy and follow it ourselves rather than sit back and wait complacently. Unfortunately, the time for old customs and cultural traditions or of long-held beliefs and trusts is over. As the latest calculations from 1992 by Meadows and colleagues in "Beyond the Limits*" *showed, our world can collapse, and this can happen even before any resource has definitively been depleted; collapse may come at any time and out of nowhere. It’s an inevitable, unavoidable result of the behavior of an oversized, complex, nonlinear system in which interdependent chance processes dominate.

The wave of large-scale deregulations because of the globalization of the last thirty years have only made this worse by allowing more positive feedback loops into the system. Nobody knows exactly how likely it is that our societal system will collapse or when. We know that this is theoretically inescapable, because all the local and national infrastructures and the global superstructure are based on abstractions. Moreover, system collapse follows from almost any simulation experiment based on relatively recent data—data that are now already twenty years old and are therefore too optimistic. In those twenty years, it has become even more likely that the conditions theoretically leading to system collapse will occur.

Excerpted with permission from "Wasted World: How Our Consumption Challenges the Planet." Copyright: 2012 University of Chicago Press.