Through the mirror: The world’s population is expected to reach about 8.3 billion people by the middle of next month, far less than the mathematically infinite population predicted in a 1960 forecast. The failed prediction has relevance for debates about artificial intelligence. It shows how a model can track an accelerating trend for years without finding that the trend will continue indefinitely. Predictions of rapid technological progress face the same question: How long will the conditions for today’s growth last?
Heinz von Foerster and his colleagues achieved their prediction using a hyperbolic growth model. Unlike exponential growth, in which a population grows at a constant percentage rate, hyperbolic growth assumes that the rate itself increases with population growth.
This feedback leads to ever faster expansion. Finally, the model reaches a singularity where the calculated population diverges to infinity in finite time. The researchers placed this point in November 2026.
Your arithmetic wasn’t the problem. The population changes captured by the model did not continue.
The forecast was based on estimates of world population from about AD 1 to 1958. The researchers published it in Science under the title “End of the World: Friday, November 13, 2026 AD.” They also chose the date because it was Foerster’s birthday.
Population figures gave the model some credibility. Humanity reached around a billion people around 1800. This number doubled by 1925 and then doubled again to 4 billion by 1975. It took less than another 50 years for the number to reach 8 billion.
The shrinking intervals suggested faster growth than an exponential curve. Conventional demographic forecasts consistently underestimated population increases in the 1960s and 1970s. The hyperbolic model performed better and accurately tracked population changes until the late 1970s.
Critics nevertheless questioned its physical plausibility and criticized the lack of explicit consideration of economic and technological adjustments. An accurate adjustment to population numbers could not prove that people would continue to reproduce under the same conditions.
The model was relatively simple and transparent. But that simplicity also left out changes that would ultimately undermine the prediction.
The relationship between population size and growth rates began to weaken in the mid-1960s. In the 1980s, developed countries found themselves in a period when fertility was below replacement levels, while demographic change accelerated in developing countries.
The total population continued to increase, but larger populations were no longer consistently growing faster in percentage terms. This distinction explains why the model could stay close to observed population numbers for a while while losing the basis for its long-term forecast.
Economist Michael Kremer looked at hyperbolic growth again in 1993. His work showed that faster-than-exponential growth could help explain broad patterns in population history going back about a million years.
More recent data complicates this determination. Adding United Nations population figures through 2023 weakens the positive relationship between population size and growth rates. Although these more recent observations cover only a small portion of the historical timeline, they include the largest populations and therefore have a significant impact on the results.
Another problem is the older data. Early population figures are estimates based on limited archaeological evidence and assumptions about technology, land use and population density.
For much of human history, population growth was extremely slow. Long periods of stability or gradual growth were punctuated by periods of rapid growth, including the Neolithic Revolution and the modern population explosion. A single curve can obscure the differences between these periods.
Hyperbolic forecasts also rely heavily on the data used to create them. An observer in 200 B.C. AD, extrapolating Neolithic population trends into the future, would have predicted that humanity would reach its current size around the year 1000.
The choice of historical period can therefore dramatically change the forecast. A model can describe a growth phase well without explaining what happens when that phase ends.
In the late 1950s, John von Neumann discussed the acceleration of technological change and suggested that it might reach a point at which prediction becomes impossible. However, he did not mention an equation or a date.
The population forecast makes it clear how difficult this jump is. Observed acceleration can support useful predictions without proving that growth will continue to accelerate.
The same limitation applies to forecasts based on AI scaling laws. Extending existing relationships indefinitely can make explosive technological growth seem inevitable. However, these relationships may depend on conditions specific to the time period in which they are measured.
This does not rule out major advances in AI. This means that evidence of rapid progress and evidence of an approaching singularity are different things.
