Doomsday Argument

The doomsday argument (DA), or Carter catastrophe, is a probabilistic argument that claims to predict the future population of the human species based on an estimation of the number of humans born to date. 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. Similar principles of eschatology were proposed earlier by Heinz von Foerster, among others. A more general form was given earlier in the Lindy effect, which proposes that for certain phenomena, the future life expectancy is proportional to (though not necessarily equal to) the current age and is based on a decreasing mortality rate over time.

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 $N$, then the Copernican principle suggests that any one human is equally likely (along with the other $N-1$ humans) to find themselves in any position $n$ of the total population $N$so humans assume that our fractional position $f=n/N$ is uniformly distributed on the interval ,1before learning our absolute position.

$f$ is uniformly distributed on (0,1) even after learning the absolute position $n$. For example, there is a 95% chance that $f$ is in the interval (0.05,1), that is $f > 0.05$. 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 $n$ is known, this argument implies a 95% confidence upper bound for $N$ obtained by rearranging $n/N > 0.05$ to give $N < 20n$.

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 $N$ will be less than 20$\times$60 billion = 1.2 trillion. Assuming that the world population stabilizes 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. Depending on the projection of the world population in the forthcoming centuries, estimates may vary, but the argument states 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 (''N''₁), or 6,000 billion (''N''₂). 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 ''N''₁ and 50% for ''N''₂ in the absence of any information about ''X''. On the other hand, it is possible to conclude, given ''X'', that ''N''₂ is more likely than ''N''₁ 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 ''N''₂ is much more likely than ''N''₁ (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 Rebuttals 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.

Heinz von Foerster's prediction of humanity's disappearance on 13 November 2026

A 1960 issue of ''Science'' magazine included an article by Heinz von Foerster and his colleagues, P. M. Mora and L. W. Amiot, proposing an equation representing the best fit to the historical data on the Earth's population available in 1958:

Fifty years ago, ''Science'' published a study with the provocative title �
Doomsday: Friday, 13 November, A.D. 2026
��. It fitted world population during the previous two millennia with ''P'' = 179 × 10⁹/(2026.9 − ''t'')^0.99. This “quasi-hyperbolic” equation (hyperbolic having exponent 1.00 in the denominator) projected to infinite population in 2026—and to an imaginary one thereafter.
:—Taagepera, Rein
A world population growth model: Interaction with Earth's carrying capacity and technology in limited space
''Technological Forecasting and Social Change'', vol. 82, February 2014, pp. 34–41

In 1975, von Hoerner suggested that von Foerster's doomsday equation can be written, without a significant loss of accuracy, in a simplified hyperbolic form (''i.e.'' with the exponent in the denominator assumed to be 1.00):
:$\text=\frac,$
where
* 2026.9 is 13 November 2026 AD—the date of the so-called "demographic singularity" and von Foerster's 115th anniversary;
* ''t'' is the number of a year of the Gregorian calendar.

Despite its simplicity, von Foerster's equation is very accurate in the range from 4,000,000 BP to 1997 AD. For example, the doomsday equation (developed in 1958, when the Earth's population was 2,911,249,671World Population by Year
Worldometer) predicts a population of 5,986,622,074 for the beginning of the year 1997:
:$\frac=5986622074.$

The actual figure was 5,924,787,816.

The doomsday equation is called so because it predicts that the number of people living on the planet Earth will become maximally ''positive'' by 13 November 2026, and on the next moment will become ''negative''. Said otherwise, the equation predicts that on 13 November 2026 all humans will instantaneously disappear.

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:
:$P(N) = \frac$.
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'':

:$P(N\mid n) = \frac.$

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:

:$P(n\mid N) = \frac$.

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 ''n''th (that is, wherever ''N'' > ''n''):

:$P(n) = \int_^ P(n\mid N) P(N) \,dN = \int_^\frac \,dN$ $= \frac.$

This is why the marginal distribution of n and N are identical in the case of P(''N'') = ''k''/''N''
so:

:$P(n) = \frac$,

giving P (''N'' , ''n'') for each specific ''N'' (through a substitution into the posterior probability equation):

:$P(N\mid n) = \frac$.

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''):

:$P(N \leq Z) = \int_^ P(N, n)\,dN$ $= \frac$

Defining ''Z'' = 20''n'' gives:

:$P(N \leq 20n) = \frac$.

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 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'' = 40''n'', and ''n'' = 200,000 years:

:P(N \leq 40 00000 = \frac

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

:$N \geq 200000 \times \frac \approx 205100~\text$;

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''.

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

Nick Bostrom, considering observation selection effects, has produced a Self-Sampling Assumption (SSA): "that you should think of yourself as if you were a random observer from a suitable reference class". If the "reference class" is the set of humans to ever be born, this gives ''N'' < 20''n'' with 95% confidence (the standard doomsday argument). However, he has refined this idea to apply to ''observer-moments'' rather than just observers. He has formalized this as:

:The strong self-sampling assumption (SSSA): Each observer-moment should reason as if it were randomly selected from the class of all observer-moments in its reference class.

An application of the principle underlying SSSA (though this application is nowhere expressly articulated by Bostrom), is: If the minute in which you read this article is randomly selected from every minute in every human's lifespan, then (with 95% confidence) this event has occurred after the first 5% of human observer-moments. If the mean lifespan in the future is twice the historic mean lifespan, this implies 95% confidence that ''N'' < 10''n'' (the average future human will account for twice the observer-moments of the average historic human). Therefore, the 95th percentile extinction-time estimate in this version is 4560 years.

Counterarguments

We are in the earliest 5%, ''a priori''

One counterargument to the doomsday argument agrees with its statistical methods but disagrees with its extinction-time estimate. This position requires justifying why the observer cannot be assumed to be randomly selected from the set of all humans ever to be born, which implies that this set is not an appropriate reference class. By disagreeing with the doomsday argument, it implies that the observer is within the first 5% of humans to be born.

By analogy, if one is a member of 50,000 people in a collaborative project, the reasoning of the doomsday argument implies that there will never be more than a million members of that project, within a 95% confidence interval. However, if one's characteristics are typical of an early adopter, rather than typical of an average member over the project's lifespan, then it may not be reasonable to assume one has joined the project at a random point in its life. For instance, the mainstream of potential users will prefer to be involved when the project is nearly complete. However, if one were to enjoy the project's incompleteness, it is already known that he or she is unusual, before the discovery of his or her early involvement.

If one has measurable attributes that set one apart from the typical long-run user, the project doomsday argument can be refuted based on the fact that one could expect to be within the first 5% of members, ''a priori''. The analogy to the total-human-population form of the argument is that confidence in a prediction of the distribution of human characteristics that places modern and historic humans outside the mainstream implies that it is already known, before examining ''n'', that it is likely to be very early in ''N''. This is an argument for changing the reference class.

For example, if one is certain that 99% of humans who will ever live will be cyborg

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