Doomsday argument Totally Explained

Everything about The Doomsday Argument totally explained

The Doomsday argument (DA) 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. Simply put, it says that supposing the humans alive today are in a random place in the whole human history timeline, chances are we're about halfway through it. It was first proposed in an explicit way by the astrophysicist Brandon Carter in 1983, from which it's sometimes called the Carter catastrophe; the argument was subsequently championed by the philosopher John A. Leslie and has since been independently discovered by J. Richard Gott and Holger Bech Nielsen. Similar principles of eschatology were proposed earlier by Heinz von Foerster, among others.
   The Copernican principle suggests that we're equally likely (along with the other N-1 humans) to find ourselves at any position n, so assume our fractional position f is uniformly distributed on the interval (0,1] prior to learning our absolute position.
   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've no prior information about the total number of humans, N.
   Now, we can take an arbitrary number, say 95% confidence, that f = n/N is within the interval (0.05,1]. In other words we could assume that we could be 95% certain that we'd be 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 ť n / N > 0.05

to give. ť N < 20n.

If we take 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 billion = 1.2 trillion.
   Assuming that the world population stabilizes at 10 billion and a life expectancy of 80 years, one can calculate how long it'll take for the remaining 1140 billion humans to be born. The argument predicts, with 95% "confidence", that humanity will disappear within 9120 years. Depending on the projection of world population in the forthcoming centuries, estimates may vary, but the main point of the argument is that the human species may become extinct.

Remarks

The step that converts N into an extinction time depends upon a finite human lifespan. If immortality becomes common, and the birth rate drops to zero, N will never be reached.
The total number of humans born so far may depend on one's definition of "human".
A precise formulation of the DA requires the Bayesian interpretation of probability, which is widely, if not universally, accepted.
Even among Bayesians some of the assumptions of the argument's logic wouldn't be acceptable; for instance, the fact that it's applied to a temporal phenomenon (how long something lasts) means that N's distribution simultaneously represents an "aleatory probability" (as a future event), and an "epistemic probability" (as a decided value about which we're uncertain).
The U(0,1] f distribution is derived from two choices, which whilst being the default are also arbitrary:
- The principle of indifference, so that it's as likely for any other randomly selected person to be born after you as before you.
- The assumption of no 'prior' knowledge on the distribution of N.

Simplification: two possible total number of humans

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's no prior knowledge of the position that a currently living individual, X, has in the history of humanity, we may instead compute how many humans were born before X, and arrive at (say) 59,854,795,447, which would roughly place X amongst the first 60 billion humans who have ever lived.
Now, if we assume that the number of humans who will ever be born equals N₁, the probability that X is amongst the first 60 billion humans who have ever lived is of course 100%. However, if the number of humans who will ever be born equals N₂, then the probability that X is amongst the first 60 billion humans who have ever lived is only 1%. Since X is in fact amongst the first 60 billion humans who have ever lived, this means that the total number of humans who will ever be born is more likely to be much closer to 60 billion than to 6,000 billion. In essence the DA therefore suggests that human extinction is more likely to occur sooner rather than later.
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's 99% certain that N is smaller than 6,000 billion.

What the argument is not

The Doomsday argument (DA) does not say that humanity can't or won't exist indefinitely. It doesn't 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 DA's conclusion is: ť There is a 95% chance of extinction within 9120 years.

The DA gives a 5% chance that humans will still be thriving circa 11125 AD. (These dates are based on the assumptions above; the precise numbers vary among specific Doomsday arguments.)

Variations

This argument has generated a lively 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: ť

P(N) = frac , df = n ln (1) - n ln (0) = + infty .

This infinite expectation shows that, under the framework of the DA, humanity still has some chance of surviving an arbitrarily long time.
For a similar example of counterintuitive infinite expectations, see the St. Petersburg paradox.

SIA: The possibility of not existing at all

One objection is that the possibility of you existing at all depends on how many humans will ever exist (N). 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.
This objection, originally by Dennis Dieks (1992), is now known by Nick Bostrom's name for it: the "Self-Indication Assumption objection". It can be shown that some SIAs prevent any inference of N from n (the current population); for details of this argument from the Bayesian inference perspective see: Self-Indication Assumption Doomsday argument rebuttal.

Many worlds

John Eastmond's "Many-Worlds Resolution of the Doomsday Argument" claims that when the Doomsday Argument is extended from a form that deals with a single historic timeline into one dealing with the many bifurcating simultaneous histories suggested by the many-worlds interpretation of quantum mechanics then one finds that the generalized argument no longer makes any prediction about the future total size of the human race. More specifically, if each finite value of total population size is realized in a different future, then learning of our present position from the beginning of the human race doesn't change our prior belief about which particular total population size we'll find ourselves experiencing in one of humanity's many futures (assuming that versions of us live long enough to see versions of Doomsday).

Caves' rebuttal

Caves' Bayesian argument says that the uniform distribution assumption is incompatible with the Copernican principle, not a consequence of it.
   He gives a number of examples to argue that Gott's rule is implausible. For instance, he says, imagine stumbling into a birthday party, about which you know nothing:
ť Your friendly enquiry about the age of the celebrant elicits the reply that she's celebrating her (t_p = ) 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 [Gott'srule] 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. (See Caves' online paper below.)

Although this example exposes a weakness in J. Richard Gott's "Copernicus method" DA (that he doesn't specify when the "Copernicus method" can be applied) it isn't precisely analogous with the modern DA; epistemological refinements of Gott's argument by philosophers such as Nick Bostrom specify that: ť Knowing the absolute birth rank (n) must give no information on the total population (N).

Careful DA variants specified with this rule aren't shown implausible by Caves' "Old Lady" example above, because, the woman's age is given prior to the estimate of her lifespan. Since human age gives an estimate of survival time (via actuarial tables) Caves' Birthday party age-estimate couldn't fall into the class of DA problems defined with this proviso.
   To produce a comparable "Birthday party example" of the carefully specified Bayesian DA we'd need to completely exclude all prior knowledge of likely human life spans; in principle this could be done (for example: hypothetical Amnesia chamber). However, this would remove the modified example from everyday experience. To keep it in the everyday realm the lady's age must be hidden prior to the survival estimate being made. (Although this is no longer exactly the DA, it's much more comparable to it.)
   Without knowing the lady’s age, the DA reasoning produces a rule to convert the birthday (n) into a maximum lifespan with 50% confidence (N). Gott's Copernicus method rule is simply: Prob (N < 2n) = 50%. How accurate would this estimate turn out to be? Western demographics are now fairly uniform across ages, so a random birthday (n) could be (very roughly) approximated by a U(0,M] draw where M is the maximum lifespan in the census. In this 'flat' model, everyone shares the same lifespan so N = M. If n happens to be less than (M)/2 then Gott's 2n estimate of N will be under M, its true figure. The other half of the time 2n underestimates M, and in this case (the one Caves highlights in his example) the subject will die before the 2n estimate is reached. In this 'flat demographics' model Gott's 50% confidence figure is proven right 50% of the time.

Self-referencing doomsday argument rebuttal

Some philosophers have been bold enough to suggest that only people who have contemplated the Doomsday argument (DA) belong in the reference class 'human'. If that's the appropriate reference class, Carter defied his own prediction when he first described the argument (to the Royal Society). A member present could have argued thus: ť "Presently, only one person in the world understands the Doomsday argument, so by its own logic there's a 95% chance that it's a minor problem which will only ever interest twenty people, and I should ignore it."

Jeff Dewynne and Professor Peter Landsberg suggested that this line of reasoning will create a paradox for the Doomsday argument:
If a member did pass such a comment, it would indicate that they understood the DA sufficiently well that in fact 2 people could be considered to understand it, and thus there would a 95% chance that 40 people would actually be interested. Also, of course, ignoring something because you only expect a small number of people to be interested in it's extremely short sighted - if this approach were to be taken, nothing new would ever be explored, if we assume no a priori knowledge of the nature of interest and attentional mechanisms.
Additionally, it should be considered that because Carter did present and describe his argument, in which case the people to whom he explained it did contemplate the DA, as it was inevitable, the conclusion could then be drawn that in the moment of explanation Carter created the basis for his own prediction.

Math-free explanation by analogy

Think of the human race like a car driver. We've had some bumps, but no catastrophes, and our car (Earth) is still road-worthy, but we want insurance. We ask the cosmic insurer how much a millennium’s cover will be, but they haven't dealt with humanity before. How should they work out the premium? The Doomsday Argument says that all they've to ask is how long we've been on the road (at least 40,000 years without an accident), they should calculate our insurance based on us having a 50% chance of having a fatal accident inside another 40,000 years.
Insurance companies try to attract drivers with long accident-free histories not because they necessarily drive more safely than newly qualified drivers, but for statistical reasons: They calculate that each driver looks for insurance quotes every year, so that the time since the last accident is a random sample between accidents. The chance of being more than halfway through a random sample is half, and if they're more than half way between accidents then they're heading for an accident in less time than the time since their last. A driver who hasn't had a scratch in 40 years will be quoted a very low premium for this reason, but you shouldn't expect cheap insurance if you've only passed your test two hours ago (equivalent to the accident-free record of the human race in relation to 40 years of geological time.)

Analogy to the estimated final score of a cricket batsman

A random in-progress cricket test match is sampled for a single piece of information: the current batsman's run tally so far. If the batsman is dismissed (rather than declaring), what is the chance that he'll end up with a score more than double his current total? ť A rough empirical result is that the chance is half (on average).

The Doomsday argument (DA) is that even if we were completely ignorant of the game we could make the same prediction, or profit by offering a bet paying odds of 2-to-3 on the batsmen doubling his current score.
   Importantly, we can only offer the bet before the current score is given (this is necessary because the absolute value of the current score would give a cricket expert a lot of information about the chance of that tally doubling). It is necessary to be ignorant of the absolute run tally before making the prediction because this is linked to the likely total, but if the likely total and absolute value are not linked the survival prediction can be made after discovering the batter's current score. Analogously, the DA says that if the absolute number of humans born gives no information on the number that will be, we can predict the species’ total number of births after discovering that 60 billion people have ever been born: with 50% confidence it's 120 billion people, so that there's better-chance-than-not that the last human birth will occur before the 23rd century.
   It is not true that the chance is half, whatever is the number of runs currently scored; batting records give an empirical correlation between reaching a given score (50 say) and reaching any other, higher score (say 100). On the average, the chance of doubling the current tally may be half, but the chance of reaching a century having scored fifty is much lower than reaching ten from five. Thus, the absolute value of the score gives information about the likely final total the batsman will reach, beyond the “scale invariant”.
   An analogous Bayesian critique of the DA is that we somehow possess prior knowledge of the all-time human population distribution (total runs scored), and that this is more significant than the finding of a low number of births until now (a low current run count).
   There are two alternative methods of making uniform draws from the current score (n):

Put the runs actually scored by dismissed player in order, say 200, and randomly choose between these scoring increments by U(0, 200].

Select a time randomly from the beginning of the match to the final dismissal. The second sampling-scheme will include those lengthy periods of a game where a dismissed player is replaced, during which the ‘current batsman’ is preparing to take the field and has no runs. If we sample based on time-of-day rather than running-score we'll often find that a new batsman has a score of zero when the total score that day was low, but we'll rarely sample a zero if one batsman stayed at the crease, piling on runs all day long. Therefore, the fact that we sample a non-zero score would tell us something about the likely final score that the current batsman will achieve.
Choosing sampling method 2 rather than method 1 would give a different statistical link between current and final score: any non-zero score would imply that the batsman reached a high final total, especially if the time to replace batsman is very long. This is analogous to the SIA-DA-refutation that N's distribution should include N = 0 states, which leads to the DA having reduced predictive power (in the extreme, no power to predict N from n at all).

An interpretation of the argument

The Doomsday argument has to be interpreted on the basis of its own definition. The central concept is about the human race and a probabilistic estimation of its end taking into consideration its beginning. If Homo sapiens is described as the evolution from Homo erectus (or what it been), then the argument can be interpreted as an estimation on the evolution into Homo futurus (or what it been).

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