The Law of Hyperbolic Growth of the Human Population

For most of recorded history, the number of people on Earth did not grow at a steady percentage rate. It grew faster and faster, and the growth rate itself kept rising as the population got bigger. A curve with that property is called hyperbolic. If you draw a hyperbola forward in time it does not just get steep. It shoots off to infinity at a definite date. In 1960 three engineers at the University of Illinois fitted such a curve to estimates of world population and found that the date of infinity fell on Friday, 13 November 2026. They called their paper "Doomsday."

That date is now only days away, and the world has roughly 8 billion people, not an infinite number. The forecast failed. But the pattern behind it, now known as the law of hyperbolic growth of the human population, is still one of the most discussed findings in long-run demography, economic history and the study of technological change. The interesting questions are why the old data fit a hyperbola so well, what process could produce that shape, and why the pattern broke down in the 1960s, right around the time it was discovered.

This article covers what the law says, where it came from, the mathematics in plain terms with worked numbers, the main explanations people have offered, the evidence for and against them, and the common ways the idea gets misread.

What the law says

The law is an empirical generalisation, a description of a pattern in data rather than a law of nature like gravity. In its simplest form it says that world population N at year t is approximately

N(t) = C / (t₀ − t)

where t₀ is a fixed year, the singularity, and C is a constant. As t gets closer to t₀, the gap (t₀ − t) shrinks toward zero, so N grows without limit.

The same law can be written as a rule about growth. If you differentiate the formula above, you get

dN/dt = N² / C

In words: the number of people added each year is proportional to the square of the population. Double the population and you quadruple the absolute annual increase. Equivalently, the percentage growth rate is proportional to the population itself. A world of 1 billion people grows, in percentage terms, twice as fast as a world of 500 million.

That is the key difference from the more familiar exponential growth. In exponential growth, the percentage growth rate is constant. A population growing at 1 percent a year adds 1 percent whether it holds a million people or a billion. In hyperbolic growth, the percentage rate climbs as the population climbs.

The English Wikipedia article on the subject attributes the discovery to Heinz von Foerster and records that the hyperbolic pattern stopped operating in the 1960s and 1970s. Since then, absolute annual growth has also fallen, and most long-range projections now expect population to level off or decline in the late twenty-first century.

Three ways a population can grow

It helps to set the hyperbola beside the two growth curves most people learn in school.

The logistic curve was the standard model for animal populations in the mid-twentieth century. Von Foerster's 1960 paper discusses it explicitly: it notes that logistic curves had been fitted with reasonable success to fruit flies in milk bottles and bacterial colonies in petri dishes, where more individuals mean more competition and lower survival. The point of the paper was that the human record looked like the opposite case. More people seemed to make each person's prospects better, not worse.

Doubling times tell the story

A simple way to tell these curves apart is to look at how long a population takes to double.

• Exponential growth: the doubling time is fixed.
• Logistic growth: the doubling time gets longer as the population nears its ceiling.
• Hyperbolic growth: the doubling time shrinks, and in the pure model each doubling takes about half as long as the one before, because the time left until the singularity halves each time the population doubles.

Our World in Data, using the HYDE historical database, Gapminder and the United Nations World Population Prospects, gives these doubling periods for world population:

From the fifteenth century to the mid-twentieth, each doubling took roughly a third to a half as long as the one before. That is the hyperbolic signature. The 36-year doubling ending in 1986 was the fastest in history. After that the doubling time started to lengthen again, which is what you would expect from a curve bending toward a ceiling, not toward a singularity.

Where the idea came from

Von Foerster, Mora and Amiot, 1960

The paper that made the pattern famous is "Doomsday: Friday, 13 November, A.D. 2026," by Heinz von Foerster, Patricia M. Mora and Lawrence W. Amiot, published in Science volume 132, pages 1291 to 1295, on 4 November 1960. The authors were on the staff of the Department of Electrical Engineering at the University of Illinois, Urbana, and the work was supported in part by a National Institutes of Health grant. Von Foerster was a physicist best known as a founder of second-order cybernetics, and the paper reads partly as a serious modelling exercise and partly as a provocation.

The argument runs in steps.

1. In an idealised "paradise" with unlimited food and no hazards, a population's growth depends only on the fertility and mortality of its members, and it grows exponentially.
2. In the real world, the usual assumption is that crowding lowers productivity, the net growth per individual. That produces the logistic curve.
3. But crowding only lowers productivity for creatures that compete without cooperating. The authors argued that a species able to communicate could form coalitions, until the whole population acted, in game-theory terms, "as a single person playing a two-person game with nature as its opponent." In that case more members could make the coalition more effective, so productivity would rise with population rather than fall.
4. They therefore tried a productivity that rises slowly with population, a = a₀N^(1/k), and derived the growth curve that follows from it.

The resulting curve has the form N = K / (t₀ − t)^k. The authors named t₀ "doomsday," the date at which N goes to infinity and, in their phrase, "the clever population annihilates itself."

To test it, they gathered 24 published estimates of world population covering roughly 100 generations, from the time of Christ to 1958. They checked the estimates for independence, threw out those that seemed merely to repeat each other, and used least squares to fit the three parameters. Their results were:

The root-mean-square deviation of the data from the curve was about 7 percent. Because k came out almost exactly 1, the fitted curve is very nearly a pure hyperbola. The decimal 2026.87 falls in mid-November, which is where the famous Friday the 13th comes from.

From the fit they derived two handy rules. The productivity, or annual percentage growth rate, is about 0.99 divided by the number of years left until doomsday. The doubling time is about 0.445 times the number of years left until doomsday.

The paper also contains a few memorable flourishes. The authors calculated that Charlemagne, with the evidence available to him, could have predicted doomsday within 300 years, Elizabeth I within 110 years, and Napoleon within 30 years, while in 1960 the date could be pinned within about 10 years because the extrapolation reached only 4 percent beyond the last data point. Running the curve backwards, they found about 200,000 people a million years ago, and noted with some humour that the date at which N equals 1, a hypothetical "Adam," came out at about 200 billion years ago, far older than the universe. They took that as a sign that the formula should not be pushed into the deep past, where the assumption of a communicating, cooperating population loses its meaning.

Their conclusion was blunt. Humanity, they wrote, had always managed to develop the technology to feed itself, so the "pessimists" who expected starvation were wrong: "our great-great-grandchildren will not starve to death. They will be squeezed to death." Their proposed remedy was a "peoplo-stat," a deliberate social control system to hold population steady. Since nobody was going to try to shorten lifespans, the only lever was fertility, and they estimated that holding population constant would require cutting the birth rate to about half its 1960 value, or an average family of "just a little above two children."

Von Hoerner and the simple hyperbola

A later and widely quoted version of the law comes from the astronomer Sebastian von Hoerner, writing in 1975. As summarised on Wikipedia, he used the pure hyperbola N = C/(t₀ − t) with t₀ = 2025 and C = 200 billion person-years. With k fixed at 1 the formula becomes very easy to use, which is one reason this version appears in so many later discussions. Von Hoerner also thought the law could apply across the whole of human existence, whereas von Foerster had fitted only the period since the start of the Common Era.

Later development

Other researchers extended, explained and criticised the pattern over the following decades. The main names, and what they emphasised, are summarised below.

The mathematics without the jargon

Why a hyperbola blows up

Take von Hoerner's version, N = 200 billion / (2025 − t). In 1925 the gap to 2025 is 100 years, so N = 2 billion. In 1975 the gap is 50 years, so N = 4 billion. In 2000 the gap is 25 years, so N = 8 billion. Every time you halve the remaining distance to 2025, the population doubles. Since you can halve a distance forever without reaching zero, the population can double forever, faster and faster, until at 2025 itself the formula divides by zero.

That is what mathematicians call a finite-time singularity. It is quite different from exponential growth, which can get very large but never reaches infinity at any finite date.

The growth rate rises with population

From the same formula, the percentage growth rate in any year is simply 1/(t₀ − t). In 1925 that is 1/100, or 1 percent a year. In 1975 it is 1/50, or 2 percent. Because N is also C/(t₀ − t), the growth rate equals N/C. For von Hoerner's constant of 200 billion, a world of 2 billion people grows at 1 percent a year, and a world of 4 billion grows at 2 percent. This is the formal sense in which "the growth rate is proportional to the population."

Worked example: the model against the record

The table below compares the two classic fitted formulas with population estimates from Our World in Data. The model values are calculated directly from the published parameters: von Foerster's N = 1.79 × 10¹¹ / (2026.87 − t)^0.99, and von Hoerner's N = 2 × 10¹¹ / (2025 − t). The observed figures are rounded milestones from the Our World in Data charts.

For the 1700 row: 1.79 × 10¹¹ divided by 326.87^0.99 is about 0.58 billion, and 2 × 10¹¹ divided by 325 is about 0.62 billion.

The fit up to about 1960 is impressive given how rough the historical estimates are. After 1960 the model and reality part company quickly. By 2000 von Hoerner's formula overshoots by about a third, and by 2020 both formulas are wildly wrong. That divergence is the whole story of the modern demographic transition, written in one column.

There is also a subtler mismatch. The pure hyperbola puts the 1960 growth rate at 1/(2025 − 1960), or about 1.5 percent a year. The actual peak growth rate, reached in the early 1960s, was higher: Our World in Data gives 2.3 percent in 1963 in its text, while its chart annotation gives 2.2 percent in 1964. The population level fitted well, but growth in the 1950s and early 1960s was briefly running ahead of even the hyperbola, before it collapsed. Fast mortality decline after the Second World War is the usual explanation for that burst.

Why would population grow hyperbolically?

A curve fitting the data is not an explanation. Since the 1960s, researchers have tried to find a mechanism that would make growth depend on the square of population. Most proposals share one idea: more people produce more of something that helps everyone, and that something feeds back into more people.

Cooperation as a single player

Von Foerster's own explanation was game-theoretic. A population that communicates well enough can act as one coalition against nature, and a bigger coalition is more versatile and effective. The Wikipedia summary of his work stresses this point: hyperbolic growth is possible only if humanity behaves "as a single player," which implies some form of cooperation among all people. Von Foerster pointed to the "steady social build-up during historical time," urbanisation, and mass communication as evidence that humans fit this description.

The idea is suggestive but vague. It does not say what exactly the coalition produces or how the effect scales.

Ideas are non-rival: Kremer's model

The most influential economic explanation came from Michael Kremer in "Population Growth and Technological Change: One Million B.C. to 1990," published in the Quarterly Journal of Economics in 1993 (volume 108, issue 3, pages 681 to 716). Kremer later shared the 2019 Nobel Memorial Prize in Economic Sciences for work on development experiments, but this paper is about the very long run.

The model rests on two assumptions.

1. Technology limits population. This is the Malthusian part. At any time, the population is about as large as the current level of technology can support. Better technology means more people.
2. More people produce more technology. This is the endogenous-growth part. An idea, once discovered, can be used by everyone at once without being used up, which economists call non-rivalry. If each person has a roughly constant chance of coming up with a useful invention, then a larger population produces more inventions per year, and every invention benefits the whole population.

Put the two together and you get a loop. More people mean faster technological progress, which raises the population ceiling, which means more people. Kremer showed that in the simplest version of this model, the growth rate of population is proportional to its level, which is exactly the hyperbolic law. In a footnote he observed that this would produce an infinite population in finite time "if it were not for the demographic transition."

Kremer did not just assert the model. He tested it in two ways.

Time-series test. Using world population estimates from McEvedy and Jones's Atlas of World Population History (1978) for 10,000 B.C. to 1900, Edward Deevey's estimates for earlier periods, and United Nations figures from 1920 on, he regressed population growth rates on population levels. The null hypothesis, that technological change is independent of population, predicts no relationship. The data strongly rejected it. He also showed the pattern was not confined to the industrial era. He reported long-run average growth of less than 0.00073 percent a year from 200,000 B.C. to 10,000 B.C., 0.037 percent from 10,000 B.C. to year 1, and 0.073 percent from year 1 to 1600. The rate kept rising as population rose, long before factories existed.

Natural experiment. Kremer's most famous evidence comes from the end of the last ice age. Around 10,000 B.C., rising seas flooded land bridges and almost cut off contact between five regions: the Old World (including sub-Saharan Africa, which had some contact across the Sahara), the Americas, mainland Australia, Tasmania and Flinders Island. If more people really generate more technology, then by 1500, when Columbus's voyages restored contact, the regions should be ranked in technological level by their size. Kremer's Table VII gives:

The technological ranking matched the size ranking. The Old World had the most advanced technology. The Americas followed, with agriculture, cities and the elaborate calendars of the Aztec and Maya. Mainland Australia had hunter-gatherer societies. Tasmania, slightly smaller than Ireland, lacked many technologies found on the Australian mainland, including the boomerang and the spear-thrower, and its people had lost the ability to make bone tools that archaeology shows they once had. Flinders Island, about 680 square kilometres, appears from radiocarbon evidence to have lost its last inhabitants about 4,000 years after being cut off. Kremer pointed out that if technological change were unrelated to population, the chance that the four inhabited regions would line up in the same order as their land area is only 1 in 24, and 1 in 120 including Flinders Island.

Kremer was careful about the limits. He called his quantitative calibration "heroic." The model underpredicted the population of the Americas relative to the Old World, and he listed several possible reasons, from underestimated early population to differences in land quality.

Accumulated knowledge and information: Kapitsa

The Russian physicist Sergei Kapitsa built a phenomenological model of human population growth over the entire human past. Instead of starting from economics, he treated humanity as a system whose growth depends on its collective, accumulated knowledge, passed on by culture and language. In his model the hyperbolic regime extends back an extremely long way; Wikipedia records that Kapitsa dated the start of the law to 1.6 million years ago. His model also builds in the end of hyperbolic growth, smoothing out the singularity so that the curve passes through an inflection point, which Kapitsa placed around 2005, and then levels off.

Kapitsa's work was influential in Russia and widely popularised, but the Wikipedia article notes it has been criticised for "excessive physicalism," treating human history as if it were a physical system governed by a single equation.

Life-saving technologies: Podlazov

A. V. Podlazov emphasised a narrower mechanism: technologies that save lives. On this view, what matters most is not technology in general but specifically the inventions and practices that reduce mortality, from food storage and sanitation to medicine, and these accumulate in proportion to the number of people who can discover and spread them.

Compact macromodels: Korotayev, Malkov and Khaltourina

The most extensive mathematical treatment in recent decades is by Andrey Korotayev and colleagues, especially in the three-volume Introduction to Social Macrodynamics by Korotayev, Artemy Malkov and Daria Khaltourina. Their first volume is subtitled Compact Macromodels of the World System Growth.

In a 2005 article in the Journal of World-Systems Research, "A Compact Macromodel of World System Evolution," Korotayev presented a two-equation model in the spirit of Kremer. One equation says population moves toward the level that technology can support: dN/dt = a(bK − N), where K is technology. The other says technology grows in proportion to both population and existing technology. With such a model, he wrote, one can reproduce "rather well the long-run hyperbolic growth of world population before 1962–3."

The fits he reported are very tight:

Two things stand out. First, the fit is best for the most recent and best-documented centuries and weaker for the deep past. Korotayev interpreted that as evidence that the world was less integrated in the remote past, so the "single player" assumption fitted less well. Second, the fit stops at 1962 to 1963, the years when global growth peaked.

The same research programme argues that world GDP grew not hyperbolically but quadratically-hyperbolically, faster than population, because output growth includes both more people and more output per person. A review of Introduction to Social Macrodynamics reports the authors' finding that Angus Maddison's estimates of world GDP for 1 to 1973 CE are almost perfectly approximated by a quadratic-hyperbolic equation.

Korotayev and Markov also noted a similar hyperbolic pattern in the diversity of species over the history of the biosphere, before human influence. This is one of the more speculative extensions of the idea.

When did the law apply?

Researchers disagree, sometimes sharply, about the time span over which the hyperbola holds.

Wikipedia also notes that the parameters were not constant over time. Growth before the Common Era appears to have followed a different hyperbola, with a higher rate, than growth afterwards. Korotayev's own fits, quoted above, are good but noticeably weaker for 10,000 to 500 BCE than for the last few centuries.

The deep-past estimates are also very uncertain. Kremer stressed that while the percentage error in early estimates is large, the absolute numbers were tiny, which limits how much that error affects the overall fit. Still, it is fair to say that the hyperbola is strongly supported for roughly the last two thousand years and increasingly speculative before that.

The end of hyperbolic growth

What happened in the 1960s

The hyperbola broke down at almost exactly the moment it was published. Global population growth peaked in the early 1960s at about 2.2 to 2.3 percent a year, depending on the dataset, and has fallen ever since. Our World in Data gives about 0.9 percent for 2023. Absolute annual additions peaked later, and Wikipedia notes that absolute growth rates have declined since 1989.

The cause is the demographic transition, a pattern observed in country after country. In the first stage, both birth and death rates are high and population grows slowly. In the second, death rates fall, especially for children, while birth rates stay high, and population grows quickly. In the third, birth rates fall too, and growth slows toward zero or below. Our World in Data describes the global transition as now "closing."

The hyperbolic model has no room for this. It assumes that more people always mean more growth. But in the twentieth century, the same forces that had raised survival, such as better health, education and income, began to lower fertility. Families chose to have fewer children once most children survived, once schooling became expensive and valuable, and once women had more education and work opportunities. Population growth decoupled from the growth of technology.

Kremer anticipated this in his model. He showed that if population growth falls when income rises above a certain level, the model produces a demographic transition and eventually steady growth of technology and income, without infinite population.

Korotayev's school offers a further twist. They argue that the end of hyperbolic growth reflects the emergence, from the 1960s on, of a single integrated world system in which the old feedback loop changed character. Whether or not one accepts that framing, the timing is clear: the hyperbola describes the world up to about 1962, and not after.

Where population is heading now

The United Nations' World Population Prospects 2024, the twenty-eighth edition of the official UN estimates covering 237 countries or areas, projects that world population, about 8.2 billion in 2024, will peak at about 10.3 billion in the mid-2080s, reported as around 2084, and decline slightly to about 10.2 billion by 2100. The UN described the probability that world population peaks within this century as about 80 percent. The earlier peak compared with previous editions was attributed mainly to lower fertility in some of the largest countries, especially China. More than half of all countries now have fertility below the replacement level of about 2.1 births per woman.

Older estimates of the eventual plateau fall in a similar range. Jean-Noël Biraben's model put the limit at 10 to 12 billion people, and the Wikipedia article notes that other models suggest similar stabilisation levels, with some scenarios showing decline after a peak.

So instead of a hyperbola, the full curve of human population now looks more like an S that has bent over. The steepest part of the climb lies behind us. Our World in Data puts it simply: humanity "just lived through the steepest increase of that curve."

What the law does and does not show

What it shows

• Faster-than-exponential growth is real. Whatever the exact formula, the percentage growth rate of world population rose over the long run, from tiny fractions of a percent in prehistory to more than 2 percent in the 1960s. A constant-rate exponential model cannot capture that.
• A simple rule can summarise a complicated history. One equation with two or three numbers fits about two thousand years of rough population estimates to within about 7 percent. That is striking, and it demands an explanation.
• Population and technology appear linked. Kremer's regional comparison and the long-run time series both fit the idea that larger, connected populations generate technology faster.

What it does not show

• It does not predict the future. A finite-time singularity in a model is a signal that the model's assumptions must fail before that time. Real populations are limited by fertility choices, mortality, resources and many other factors. The singularity was never a forecast that infinitely many people would exist.
• It does not prove a single mechanism. Several quite different mechanisms, from von Foerster's cooperation to Kremer's non-rival ideas to Podlazov's life-saving technologies, can produce the same curve. The fit alone cannot tell them apart.
• It does not mean "more people is always better." The model describes a historical feedback between population and technology. It says nothing about whether faster growth today would raise welfare, and it describes the past under conditions, such as high mortality and a Malthusian link between technology and population, that no longer hold.

Criticisms and open questions

Data quality

Most of the data points before the last few centuries are informed guesses. McEvedy and Jones, Kremer's main source, built their post-200 B.C. estimates from regional figures derived largely from Roman and Chinese censuses, and their earlier figures from archaeological and anthropological evidence. Different authors' series disagree. Kremer showed his main result held across the alternative series of Deevey, Clark and Durand, which suggests the pattern is robust to this uncertainty. But the exact parameters of any hyperbola fitted to such data are not precise.

The constant inventor

Sergey Tsirel criticised the assumption, central to the Kremer-style models, that each person contributes a roughly constant probability of invention. He noted that most important inventions arose in a few small regions during special eras, not evenly across humanity. If innovation depends strongly on institutions, cities, trade networks and culture rather than raw headcount, the simple models are at best an average over very uneven reality.

Kremer himself acknowledged a version of this point. He wrote that his paper abstracted from fluctuations in research productivity per person because it focused on very long periods over which they might average out, but that the study of how institutions affect research productivity remains critical for understanding shorter periods and differences between countries.

Is the world one system?

The hyperbolic models treat the whole human population as a single unit. For much of history this is plainly false: the Americas and Eurasia had no contact for thousands of years, as Kremer's own natural experiment depends on. Defenders respond that the world-level fit works best when the world was most connected, and that before then the curve is weaker. Critics see this as making the argument partly circular.

Physics envy

Kapitsa's critics and others have objected to treating history like a physical system with universal constants. Human population depends on decisions about marriage, contraception, migration and war that no one equation captures. The success of the hyperbolic fit can be read as evidence of a deep regularity, or as a reminder that smooth curves can be fitted to many noisy datasets.

What drives growth now?

Some authors, according to the Wikipedia summary, expect future development to centre not on population but on the growth of information, perhaps in a combined human-machine "supermind" built on the Internet. Ray Kurzweil's 2005 book The Singularity Is Near is among the works cited in this context. These are speculative extrapolations from the population-technology feedback, not tested predictions, and they are best read as ideas rather than findings.

Common misconceptions

"Scientists predicted the world would end on 13 November 2026." Not literally. Von Foerster and his co-authors used "doomsday" to name the date at which their formula goes to infinity. Their actual message was that population growth could not continue on its historical path and that society would need to control fertility. The paper's tone is partly satirical, with lines about space travel and the lack of a "re-entry permit to earth" for emigrants.

"The model was a failure." As a forecast of the future, yes. As a description of the past, it was remarkably good, and it correctly flagged that population growth was accelerating in a way that could not continue. The forecast failed partly because the trend it described ended, and that ending is exactly what demographers now call the demographic transition.

"Hyperbolic means the same as exponential." No. Exponential growth has a constant percentage rate and never reaches infinity at a finite time. Hyperbolic growth has a rising percentage rate and does reach infinity at a finite time. On a graph of the logarithm of population against time, exponential growth is a straight line, while hyperbolic growth curves upward.

"The law proves population will keep rising." The opposite is now expected. The UN projects a peak in the 2080s and a slow decline afterward.

"Hyperbolic growth was caused by the Industrial Revolution." Kremer's evidence suggests the rising growth rate long predates industrialisation. He found population significantly predicting growth in India by 1600 and China by 1750, and long-run growth rising from prehistory to the year 1 to 1600.

Frequently asked questions

Who discovered the law?

The pattern is credited to Heinz von Foerster, who published it with Patricia Mora and Lawrence Amiot in Science in 1960. Sebastian von Hoerner gave a simpler version in 1975, and Michael Kremer provided the best-known economic explanation in 1993.

What exactly happens on 13 November 2026?

Nothing, demographically. That is simply the date, 2026.87 in decimal years, where von Foerster's fitted formula divides by zero. The formula stopped describing reality in the 1960s.

Why does the doubling time get shorter?

In a hyperbola, the time to double is proportional to the time remaining until the singularity. Von Foerster's fit gave a doubling time of about 0.445 times the years left. In 1960, with about 67 years to go, that is about 30 years. In 1900, with about 127 years left, about 56 years. As the singularity approaches, the time left shrinks, and so does the doubling time.

What does the constant C mean?

In von Hoerner's version, C is 200 billion person-years. It sets the scale: the percentage growth rate equals N divided by C. With N = 2 billion, the rate is 1 percent a year; with N = 4 billion, 2 percent. It can also be read as the total number of person-years humanity would accumulate in the run-up to the singularity, though that interpretation is mainly a convenience.

Is there a modern version of the law that still works?

Not for population. Korotayev and colleagues argue that a quadratic-hyperbolic model fits world GDP up to 1973 very well, but population itself has been growing more slowly than any hyperbola for six decades. Most current models treat the hyperbolic era as a historical phase that ended with the demographic transition.

Does the law apply to individual countries?

It was formulated for the world as a whole, partly to average out migration and local disasters. Von Foerster explicitly chose world totals for that reason. Individual countries have very different histories and are strongly affected by migration, wars and famines, so a single national hyperbola is much less meaningful.

How does this relate to the "technological singularity"?

Both ideas use the mathematical notion of a singularity, a point where a quantity becomes infinite. Some writers have linked the hyperbolic growth of population and economic output to speculation about runaway technological change. The population law itself is a historical fit; claims about a coming technological singularity are separate and much more speculative.

Why is the law still studied if it no longer holds?

Because explaining why it held for so long, and why it stopped, gets at fundamental questions: how ideas spread, why some societies developed technology faster than others, how fertility responds to development, and whether long-run human history follows any regular laws at all. Kremer's paper, for example, is a standard reference in growth economics, and the hyperbolic fit remains a benchmark that any long-run model of human history has to account for.

Summary

The law of hyperbolic growth says that, over roughly the last two thousand years and possibly much longer, world population grew in such a way that its percentage growth rate was proportional to its size. A curve with that property heads toward infinity at a finite date, which von Foerster's 1960 fit placed in November 2026. The fit to the historical record was surprisingly good, within about 7 percent, and later analyses with more data found correlations above 0.99 for the period up to 1962.

The leading explanation is a feedback loop: technology sets the population ceiling, and a larger population produces technology faster because ideas can be shared by everyone. Kremer's comparison of regions isolated at the end of the ice age is the classic evidence. The loop broke in the 1960s when falling fertility severed the link between more technology and more people. Global growth peaked at about 2.2 to 2.3 percent a year in the early 1960s and has since fallen below 1 percent, and the UN now projects a peak around 10.3 billion in the 2080s. The doomsday date arrives with the hyperbola long abandoned. What remains is a clear record of an extraordinary acceleration, and a set of still-argued theories about what drove it.

Sources

• Law of hyperbolic growth of the human population (Wikipedia)
• Von Foerster, Mora and Amiot, "Doomsday: Friday, 13 November, A.D. 2026," Science 132 (1960), 1291–1295 (scanned copy)
• Kremer, "Population Growth and Technological Change: One Million B.C. to 1990," Quarterly Journal of Economics 108(3), 1993 (IDEAS/RePEc record)
• Kremer 1993, full text (gwern.net archive)
• Korotayev, "A Compact Macromodel of World System Evolution," Journal of World-Systems Research 11(1), 2005
• Review of Korotayev, Malkov and Khaltourina, Introduction to Social Macrodynamics (Social Evolution & History)
• Our World in Data: Population growth over time
• Our World in Data: World population growth, past and future
• UN DESA Population Division: World Population Prospects 2024
• Dawn: World population to peak at 10.3 billion in 2080s, United Nations says (2024)