doomsday argument

The Doomsday argument is a probabilistic argument that claims to predict the future lifetime of the human race given only an estimate of the total number of humans born so far. It was first proposed by the astrophysicist Brandon Carter in the 1980s and was subsequently championed by the philosopher John Leslie. It has since been independently discovered by J. Richard Gott and H. B. Nielsen. Similar theories predicting an end to the world from population statistics were proposed earlier by Heinz von Foerster, among others. This article primarily follows Gott's development of the argument.

The Doomsday argument

Let us imagine our fractional position f = n/N along the chronological list of all the humans who will ever be born, where n is our absolute position from the beginning of the list and N is the total number of humans. Assuming that we are equally likely (along with the other N humans) to find ourselves at any position n, we can assert that our fractional position f is uniformly distributed on the interval (0,1] prior to learning our absolute position. This is an example of the Copernican principle. Let us further assume that our fractional position f is uniformly distributed on (0,1] even after we learn of our absolute position n. This is equivalent to the assumption that we have no prior information about the total number of humans, N. Now, we can say with 95% confidence that f = n/N is within the interval (0.05,1]. In other words we are 95% certain that we are within the last 95% of all the humans ever to be born. Given our absolute position n, this implies an upper bound for N obtained by rearranging

n / N > 0.05

to give

N < 20n.

If we assume that 60 billion humans have been born so far (Leslie's figure) then we can say with 95% confidence that the total number of humans, N, will be less than 20·60 = 1200 billion. Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, one can calculate how long it will take for the remaining 1140 billion humans to be born. Thus we find the argument predicts, with 95% confidence, that mankind will disappear within 9120 years. Depending on your projection of world population in the forthcoming centuries, your estimates might vary, but the main point of the argument is that mankind is likely to disappear rather soon.

Remarks

A precise formulation of the argument requires the Bayesian interpretation of probability, which is widely, if not universally, accepted.

The argument assumes no 'prior' knowledge on the distribution of N. While this may not be an unreasonable assumption for 'in principle' reasoning, it would be rejected by many Bayesians.

The argument makes the implicit assumption that N is finite: otherwise all humans will have a position close to 0 in the range (0,1]. In principle there seems to be no reason why we must assume the prior existence of some finite upper bound to our position, which suggests that there is a fundamental problem with the argument.

The total number of humans born so far may depend on one's definition of "human".

Simplification

Here is a simplified version of the argument, based on A refutation of the Doomsday Argument by Korb and Oliver. Assume for simplicity that there are two possible numbers for N, the total number of humans who will ever be born: either N = 60 billions, or N = 6 000 billions. Now, you have no a priori knowledge of your position in the history of humanity, so you decide to compute how many humans have been born before you. It turns out that you are human #59 854 795 447, i.e. one of the first 60 billions. Now, if in fact N = 60 billions, the probability that you were in the first 60 billions is 100%, of course. However, if N = 6 000 billions, then the probability that you were in the first 60 billions is only 1%. Therefore, it is more likely that N = 60 billions (although it is not certain). In fact, N is monotonically less probable as it grows larger. 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 95% certain that N is smaller that 1200 billion.

Other versions

This argument has generated a lively philosophical debate, and no consensus has yet emerged on its solution. 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 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. In fact the use of a vague prior distribution of the form P(N) = 1/N seems well-motivated as it assumes as little knowledge as possible about N. As mentioned above, it is in fact equivalent to the assumption that the probability density of one's fractional position remains uniformly distributed even after learning of one's absolute position.

Singularity

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 fit 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.

Rebuttals

The possibility of not existing at all

One objection originally by Dennis Dieks, and expanded by Ken Olum, is that the possibility of you existing at all depends on how many humans will ever exist. If this is a high number, then the possibility of you existing is higher than if only a few humans will ever exist. Since you do indeed exist, this is evidence that the number of humans that will ever exist is high. As it happens, this precisely cancels out the effect of the Doomsday Argument, and therefore, one's birth position gives no information about the total number of humans that will exist. On the other hand, this argument seems to suggest that, all other things being equal, any theory which postulates a high number of conscious beings in the universe is more likely true than a theory which does not. This is, to say the least, controversial.

Many worlds

The problem with the argument might lie in its implicit assumption of a pre-determined linear timeline. The many-worlds interpretation of quantum mechanics suggests that time has a network-like structure with many actually-occurring pasts merging into each present moment and many actually-occurring futures branching from each present moment. The apparent linearity of time is due to the fact that our memories are consistent with only one past. It has been suggested that a generalized form of the argument, in which all finite values of total population size are realized in different futures, avoids both the prior assumption of a finite upper bound to our birth position and also any correlation between our present position and a particular future total population size that we experience should we live long enough to see Doomsday.

Caves' rebuttal

Caves (see his on-line paper at External Links below) uses Bayesian arguments to show that the uniform distribution assumption is, in fact, incompatible with the Copernican principle, not a consequence of it. He gives a number of examples to show that Gott's rule is implausible. One example is reproduced here Suppose you are going to a meeting of your book club, to be held at a member’s house that you’ve never been to before. You find the right street, but having forgotten the street address, you choose between two houses where there is evident activity. Knocking at one, you are told that the activity within is a birthday party, not a book-club meeting. Your friendly enquiry about the age of the celebrant elicits the reply that she is celebrating her 50th birthday. According to Gott, you can predict with 95% confidence that the woman will survive between 50/39 = 1.28 years and 39*50 = 1, 950 years into the future. Since the wide range encompasses reasonable expectations regarding the woman's survival, it might not seem so bad, till one realizes that Eq(4) {Gott's rule} predicts that with probability 1/2 the woman will survive beyond 100 years old and with probability 1/3 beyond 150. Few of us would want to bet on the woman’s survival using Gott's rule.

External links

References

John Leslie, The End of the World: The Science and Ethics of Human Extinction, Routledge, 1998, ISBN 0-415-18447-9.
J. R. Gott III, Implications of the Copernican Principle for our Future Prospects, Nature, vol. 363, pp. 315-319, 1993.
J. R. Gott III, Future Prospects Discussed, Nature, vol. 368, p. 108, 1994.
This argument plays a central role in Stephen Baxter's science fiction book, , Del Rey Books, 2000, ISBN 0-345-43076-X.