Doomsday: Friday, 13 November, A.D. 2026. A Status Report

Translated, with modifications, from the Russian original published at N + 1 on July 9, 2015.

One of the main jobs of science is to build theories with predictive power. Once upon a time, for instance, accurate predictions of solar eclipses were in high demand. The story goes that the Chinese emperor Zhong Kang of the Xia dynasty, some two thousand years BC, executed two court astrologers because they got drunk and failed to predict an eclipse on time, so the whole country plunged into horror and panic (the all-caps kind). Eclipses have been sorted out since then and don’t cause panic anymore (and scientists get executed less often, too: those in power have found other ways to humiliate them), but the problems of celestial mechanics are still quite relevant. Say, for the Internet to work well, one needs to predict precisely where a communication satellite launched from a given point with a given velocity will go. Otherwise there is a real risk of finding it, some time later, at the bottom of the ocean.

Mechanics, though, was more or less figured out several hundred years ago. Mechanical systems are relatively easy to study, because you can run all kinds of experiments with pendulums, balls, springs, and other characters of high-school problems. And these experiments are perfectly reproducible: a pendulum in Russia, Australia, and the US, yesterday, today, and a hundred years ago, behaves exactly the same way. Things are much harder in the sciences where experiments are difficult and expensive (and often downright impossible), and where the actors, unlike balls and springs, have a will of their own. Yet it is precisely from these young sciences that study people and society, such as economics, sociology, or political science, that we now await predictions with bated breath: economic, social, and political crises affect our lives far more than solar eclipses do.

Crises, though, are small fry. There are bigger questions. How long will humanity last, anyway? What will happen to the population of the planet, and how soon will we run out of resources? What can be predicted about all this, and how, if we use mathematical models instead of coffee grounds? That’s what today’s post is about.

Amoebas, rabbits, and people

This story usually begins with Thomas Malthus. The model that now bears his name says: the growth rate of a population is proportional to the size of that population. Suppose, for example, that a jar is full of amoebas, and every minute each amoeba splits in two. Then the number of amoebas will grow in geometric progression. If initially there was one amoeba in the jar, after one minute there will be two, after two minutes four ($2 \times 2 = 2^2$), after three minutes eight ($2 \times 2 \times 2 = 2^3$), and so on. After $t$ minutes there will be $2^t$ of them.

In practice, though, amoebas (and people) don’t reproduce only at fixed moments of time, and a more convenient model is one with continuous time, described by a differential equation. Let $x(t)$ be the size of the population at time $t$. Then the growth rate is the time derivative $dx(t)/dt$. Translating the previous paragraph into the language of differential equations, we get this:

$$\frac{dx(t)}{dt} = k\,x(t).$$

Here $k$ is some constant (the coefficient of proportionality) that stands for the relative growth rate, the “fertility”, which differs from species to species.

To begin with, let’s assume $k = 1$. The resulting equation $dx/dt = x$ can be rendered in plain language as “the derivative of a function equals the function itself”. As the great science of calculus teaches us, the function with this property is $x(t) = e^t$, also known as the exponential. A similar function, $2^t$, was discussed above; the only difference is that the base is not 2 but the irrational number $e \approx 2.718281828\ldots$ It is worth noting here that a differential equation always has many solutions: depending on the initial size of the population (let’s denote it $x_0$) we get different functions $x(t)$. It turns out that all the solutions of our equation (the original one, with $k$) are given by the formula

$$x(t) = x_0 e^{kt}.$$

That this function is a solution can be checked by differentiating it directly; that there are no other solutions follows from the main result of the general theory of ordinary differential equations, the existence and uniqueness theorem for the initial value problem.

On the interactive plot below you can play with the solution: drag the point at $t = 0$ up and down to change $x_0$, and use the slider to change $k$.

Note that $k$ can also be negative: this corresponds to the population dying out. For positive $k$, however, the population grows without bound, and does so fairly quickly: at least faster than any linear function $x = at + b$. Malthus believed that the resources available to humanity grow exactly linearly (in arithmetic progression), and therefore humanity, which naturally grows in geometric progression, is doomed to suffering (wars, for instance) that will keep its growth in check (unless people come to their senses and limit it themselves).

Was Malthus right?

As for suffering, there is certainly plenty of it. But how accurately does the exponential law describe the growth of the Earth’s population? Let’s check. The plot below shows the number of people (in millions) at various moments from 1000 BC to roughly our days.¹ Again, you can change the parameters of the model: the slider sets the fertility $k$, and dragging the point at $t = 0$ (that is, at the very beginning of the Common Era) sets the initial population. Try to pick them so that the blue theoretical curve passes close to all the orange points.

Did it work? For me, not really. You can make the approximation work well over a long stretch of the past (from 1000 BC to about 1500 AD), but then the real population curve shoots upward and our theoretical curve can’t keep up.

What’s going on?

People multiply too fast!

Faster than amoebas (and even rabbits)!

This is the conclusion that Professor Heinz von Foerster reached in the middle of the last century, after carefully analyzing (together with his colleagues) the data available at the time. He proposed the following explanation: people, using their ability to communicate, organize their lives in more efficient and comfortable ways, and because of this the “fertility” $k$ is not a constant (as in the Malthusian model) but increases as the population grows.

In his 1960 paper in Science², von Foerster points to the human population as a typical example of such growth, citing the steady social development throughout history, the rapid urbanization of recent centuries, and the explosive development of mass communication in recent decades. Indeed: socio-economic achievements (the division of labor, say) and scientific ones let people obtain food and withstand external threats far more efficiently. Child mortality falls and life expectancy grows thanks to medicine. The more people live on Earth, the more resources they can spend on this development, which in turn further loosens the constraints on life, and hence on reproduction. Thus the size of the population becomes an additional resource for its own growth.

But that’s all poetry. What can be tested is not abstract reasoning but concrete mathematical laws. Let’s return to the equation $dx/dt = kx$, but now let the fertility $k$ be proportional to the size of the population: $k = sx$, where $s$ is a new parameter (a constant). The differential equation of our new model looks like this:

$$\frac{dx}{dt} = s\,x^2.$$

Its solution is not an exponential but a function of the form

$$x(t) = \frac{x_0}{1 - x_0 s t}.$$

Let’s try our luck with the new function.

Much better! It seems that our new law (it is called hyperbolic), with suitably chosen parameters, really does describe the growth of the Earth’s population in the past quite well. But what happens if we apply it to predict the future?

The Big Boom

The exponential grows fast. The function we obtained in the previous section grows even faster. So fast that as $t$ approaches a certain fixed moment, it becomes arbitrarily large! Indeed, when $t$ is close to $t_0 = 1/(x_0 s)$, the denominator of the fraction $x_0/(1 - x_0 s t)$ becomes very small. The numerator doesn’t change, and dividing it by a very small denominator gives a very, very, VERY large number. The population grows at an ever-accelerating pace and tends to infinity in finite time! In the picture, this corresponds to a vertical line (an asymptote) that the graph of our function approaches. Here is what it looks like:

So what happens at the moment $t_0$? According to the model, an infinite number of people will be living on Earth. “Our great-great-grandchildren will not starve to death. They will be squeezed to death,” writes Professor von Foerster in Science. Truly, that is not the end we were expecting!

When will this moment come? Careful calculations by von Foerster and his coauthors show: it will happen precisely on Friday, November 13, 2026 (give or take five years or so).³ That is, today.

Are we dead yet?

Not yet. It is Friday, November 13, 2026, as I write this, and the population of the Earth is about 8.3 billion. Still finite. Worse for the model, the number is growing at less than one percent a year, and every year a little slower.

What happened? Von Foerster’s paper came out in 1960, and every population statistic collected before then agreed with it. But by the end of the 1960s the growth rate of the world population had passed its all-time high, and from the early 1970s it was falling in the developing countries too, a decade after the developed ones. Not because the Earth had run out of food, as Malthus would have expected: because people, first in the rich countries and then everywhere else, decided to have fewer children. Social and technological changes, not famine. A hyperbola knows nothing about contraception, and our simple model could not have predicted such a turn. The UN now expects the population to peak at about 10.3 billion in the 2080s and then to shrink.

Phew.

Does this mean the paper was wrong? No. Any mathematical model has a limited domain of applicability, and using it mechanically for predictions can lead to wrong results. But this doesn’t diminish the value of von Foerster’s paper: the mechanisms it discussed apparently really did operate for thousands of years, and the paper itself started a line of research into hyperbolic growth that is still alive today (hello, Singularity).