The doomsday argument (DA), or Carter catastrophe, is a probabilistic argument that aims to predict the total number of humans who will ever live. It argues that if a human's birth rank is randomly sampled from the set of all humans who will ever live, it is improbable that one would be at the extreme beginning. This implies that the total number of humans is unlikely to be much larger than the number of humans born so far.
The doomsday argument was originally proposed by the astrophysicist Brandon Carter in 1983, leading to the initial name of the Carter catastrophe. The argument was subsequently championed by the philosopher John A. Leslie and has since been independently conceived by J. Richard Gott and Holger Bech Nielsen. Summary The premise of the argument is as follows: suppose that the total number of human beings who will ever exist is fixed. If so, the likelihood of a randomly selected person existing at a particular time in history would be proportional to the total population at that time. Given this, the argument posits that a person alive today should adjust their expectations about the future of the human race because their existence provides information about the total number of humans that will ever live.
If the total number of humans who were born or will ever be born is denoted by , then the Copernican principle suggests that any one human is equally likely to find themselves in any position of the total population .
is uniformly distributed on (0,1) even after learning the absolute position . For example, there is a 95% chance that is in the interval (0.05,1), that is . In other words, one can assume with 95% certainty that any individual human would be within the last 95% of all the humans ever to be born. If the absolute position is known, this argument implies a 95% confidence upper bound for obtained by rearranging to give .
If Leslie's figure is used, then approximately 60 billion humans have been born so far, so it can be estimated that there is a 95% chance that the total number of humans will be less than 2060 billion = 1.2 trillion. Assuming that the world population stabilises at 10 billion and a life expectancy of 80 years, it can be estimated that the remaining 1140 billion humans will be born in 9120 years. Although estimates vary depending on projections of world population over the coming centuries, the argument maintains that it is unlikely that more than 1.2 trillion humans will ever live.
Aspects Assume, for simplicity, that the total number of humans who will ever be born is 60 billion (N1), or 6,000 billion (N2). If there is no prior knowledge of the position that a currently living individual, X, has in the history of humanity, one may instead compute how many humans were born before X, and arrive at say 59,854,795,447, which would necessarily place X among the first 60 billion humans who have ever lived.
It is possible to sum the probabilities for each value of N and, therefore, to compute a statistical 'confidence limit' on N. For example, taking the numbers above, it is 99% certain that N is smaller than 6 trillion.
Note that, as remarked above, this argument assumes that the prior probability for N is flat, or 50% for N1 and 50% for N2 in the absence of any information about X. On the other hand, it is possible to conclude, given X, that N2 is more likely than N1 if a different prior is used for N. More precisely, Bayes' theorem tells us that P(N|X) = P(X|N)P(N)/P(X), and the conservative application of the Copernican principle tells us only how to calculate P(X|N). Taking P(X) to be flat, we still have to assume the prior probability P(N) that the total number of humans is N. If we conclude that N2 is much more likely than N1 (for example, because producing a larger population takes more time, increasing the chance that a low probability but cataclysmic natural event will take place in that time), then P(X|N) can become more heavily weighted towards the bigger value of N. A further, more detailed discussion, as well as relevant distributions P(N), are given below in the Counterarguments section.
The doomsday argument does not say that humanity cannot or will not exist indefinitely. It does not put any upper limit on the number of humans that will ever exist nor provide a date for when humanity will become extinct. An abbreviated form of the argument does make these claims, by confusing probability with certainty. However, the actual conclusion for the version used above is that there is a 95% chance of extinction within 9,120 years and a 5% chance that some humans will still be alive at the end of that period. (The precise numbers vary among specific doomsday arguments.)
Variations This argument has generated a philosophical debate, and no consensus has yet emerged on its solution. The variants described below produce the DA by separate derivations.
Gott's formulation: "vague prior" total population Gott specifically proposes the functional form for the prior distribution of the number of people who will ever be born (N). Gott's DA used the vague prior distribution: . where P(N) is the probability prior to discovering n, the total number of humans who have yet been born. The constant, k, is chosen to normalize the sum of P(N). The value chosen is not important here, just the functional form (this is an improper prior, so no value of k gives a valid distribution, but Bayesian inference is still possible using it.)
Since Gott specifies the prior distribution of total humans, P(N), Bayes' theorem and the principle of indifference alone give us P(N|n), the probability of N humans being born if n is a random draw from N:
This is Bayes' theorem for the posterior probability of the total population ever born of N, conditioned on population born thus far of n. Now, using the indifference principle:
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The unconditioned n distribution of the current population is identical to the vague prior N probability density function,The only probability density functions that must be specified a priori are: Pr(N) - the ultimate number of people that will be born, assumed by J. Richard Gott to have a vague prior distribution, Pr(N) = k/N Pr(n|N) - the chance of being born in any position based on a total population N - all DA forms assume the Copernican principle, making Pr(n|N) = 1/N
From these two distributions, the doomsday argument proceeds to create a Bayesian inference on the distribution of N from n, through Bayes' rule, which requires P(n); to produce this, integrate over all the possible values of N which might contain an individual born nth (that is, wherever N > n):
This is why the marginal distribution of n and N are identical in the case of P(N) = k/N so:
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giving P (N | n) for each specific N (through a substitution into the posterior probability equation):
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The easiest way to produce the doomsday estimate with a given confidence (say 95%) is to pretend that N is a continuous variable (since it is very large) and integrate over the probability density from N = n to N = Z. (This will give a function for the probability that N ≤ Z):
Defining Z = 20n gives:
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This is the simplest Bayesian derivation of the doomsday argument: The chance that the total number of humans that will ever be born (N) is greater than twenty times the total that have been born is below 5%
The use of a vague prior distribution seems well-motivated as it assumes as little knowledge as possible about N, given that some particular function must be chosen. It is equivalent to the assumption that the probability density of one's fractional position remains uniformly distributed even after learning of one's absolute position (n).
Gott's "reference class" in his original 1993 paper was not the number of births, but the number of years "humans" had existed as a species, which he put at 200,000. Also, Gott tried to give a 95% confidence interval between a minimum survival time and a maximum. Because of the 2.5% chance that he gives to underestimating the minimum, he has only a 2.5% chance of overestimating the maximum. This equates to 97.5% confidence that extinction occurs before the upper boundary of his confidence interval, which can be used in the integral above with Z = 40n, and n = 200,000 years:
This is how Gott produces a 97.5% confidence of extinction within N ≤ 8,000,000 years. The number he quoted was the likely time remaining, N − n = 7.8 million years. This was much higher than the temporal confidence bound produced by counting births, because it applied the principle of indifference to time. (Producing different estimates by sampling different parameters in the same hypothesis is Bertrand's paradox.) Similarly, there is a 97.5% chance that the present lies in the first 97.5% of human history, so there is a 97.5% chance that the total lifespan of humanity will be at least
In other words, Gott's argument gives a 95% confidence that humans will go extinct between 5,100 and 7.8 million years in the future.
Gott has also tested this formulation against the Berlin Wall and Broadway and off-Broadway plays.
Leslie's argument differs from Gott's version in that he does not assume a vague prior probability distribution for N. Instead, he argues that the force of the doomsday argument resides purely in the increased probability of an early doomsday once you take into account your birth position, regardless of your prior probability distribution for N. He calls this the probability shift.
Heinz von Foerster argued that humanity's abilities to construct societies, civilizations and technologies do not result in self-inhibition. Rather, societies' success varies directly with population size. Von Foerster found that this model fits some 25 data points from the birth of Jesus to 1958, with only 7% of the variance left unexplained. Several follow-up letters (1961, 1962, ...) were published in Science showing that von Foerster's equation was still on track. The data continued to fit up until 1973. The most remarkable thing about von Foerster's model was it predicted that the human population would reach infinity or a mathematical singularity, on Friday, November 13, 2026. In fact, von Foerster did not imply that the world population on that day could actually become infinite. The real implication was that the world population growth pattern followed for many centuries prior to 1960 was about to come to an end and be transformed into a radically different pattern. Note that this prediction began to be fulfilled just in a few years after the "doomsday" argument was published.See, for example, Introduction to Social Macrodynamics by Andrey Korotayev et al.
Reference classes The reference class from which n is drawn, and of which N is the ultimate size, is a crucial point of contention in the doomsday argument. The "standard" doomsday argument hypothesis skips over this point entirely, merely stating that the reference class is the number of "people". Given that you are human, the Copernican principle might be used to determine if you were born exceptionally early, however the term "human" has been heavily contested on practical and philosophical reasons. According to Nick Bostrom, consciousness is (part of) the discriminator between what is in and what is out of the reference class, and therefore extraterrestrial intelligence might have a significant impact on the calculation.
The following sub-sections relate to different suggested reference classes, each of which has had the standard doomsday argument applied to it.
SSSA: Sampling from observer-moments