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Thedoomsday argument (DA), orCarter catastrophe, is aprobabilistic 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 theastrophysicistBrandon Carter in 1983,^[1] leading to the initial name of the Carter catastrophe. The argument was subsequently championed by thephilosopherJohn A. Leslie and has since been independently conceived byJ. Richard Gott^[2] andHolger Bech Nielsen.^[3]

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 theCopernican principle suggests that any one human is equally likely to find themselves in any position$n$ of the total population$N$.

$f$ 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.

If Leslie's figure^[4] 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 of80 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.

Assume, for simplicity, that the total number of humans who will ever be born is 60 billion (N₁), or 6,000 billion (N₂).^[5] 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 beforeX, and arrive at say 59,854,795,447, which would necessarily placeX among the first 60 billion humans who have ever lived.

It is possible to sum theprobabilities for each value ofN and, therefore, to compute a statistical 'confidence limit' onN. For example, taking the numbers above, it is 99% certain thatN is smaller than 6 trillion.

Note that as remarked above, this argument assumes that the prior probability forN is flat, or 50% forN₁ and 50% forN₂ in the absence of any information aboutX. On the other hand, it is possible to conclude, givenX, thatN₂ is more likely thanN₁ if a different prior is used forN. 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 isN. If we conclude thatN₂ is much more likely thanN₁ (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 ofN. A further, more detailed discussion, as well as relevant distributions P(N), are given below in theRebuttals section.

The doomsday argument doesnot 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 becomeextinct. An abbreviated form of the argumentdoes 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.)

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 specifically proposes the functional form for theprior distribution of the number of people who will ever be born (N). Gott's DA used thevague prior distribution:

${\displaystyle P(N)=\frac{k}{N}}$.

where

• P(N) is the probability prior to discoveringn, the total number of humans who haveyet been born.
• The constant,k, is chosen tonormalize the sum of P(N). The value chosen is not important here, just the functional form (this is animproper prior, so no value ofk gives a valid distribution, butBayesian inference is still possible using it.)

Since Gott specifies theprior distribution of total humans,P(N),Bayes' theorem and theprinciple of indifference alone give usP(N|n), the probability ofN humans being born ifn is a random draw fromN:

${\displaystyle P(N\mid n)=\frac{P(n\mid N)P(N)}{P(n)}.}$

This isBayes' theorem for theposterior probability of the total population ever born ofN,conditioned on population born thus far ofn. Now, using the indifference principle:

${\displaystyle P(n\mid N)=\frac{1}{N}}$.

The unconditionedn distribution of the current population is identical to the vague priorN probability density function,^{[note 1]} so:

${\displaystyle P(n)=\frac{k}{n}}$,

giving P (N |n) for each specificN (through a substitution into the posterior probability equation):

${\displaystyle P(N\mid n)=\frac{n}{N^2}}$.

The easiest way to produce the doomsday estimate with a givenconfidence (say 95%) is to pretend thatN is acontinuous variable (since it is very large) andintegrate over the probability density fromN =n toN =Z. (This will give a function for the probability thatN ≤Z):

${\displaystyle P(N\le Z)={\int }_{N=n}^{N=Z}P(N|n)dN}$${\displaystyle =\frac{Z-n}{Z}}$

DefiningZ = 20n gives:

${\displaystyle P(N\le 20n)=\frac{19}{20}}$.

This is the simplestBayesian 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 avague prior distribution seems well-motivated as it assumes as little knowledge as possible aboutN, 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 putat 200,000. Also, Gott tried to give a 95% confidence interval between aminimum 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 withZ = 40n, andn = 200,000 years:

${\displaystyle P(N\le 40[200000])=\frac{39}{40}}$

This is how Gott produces a 97.5% confidence of extinction withinN ≤ 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 isBertrand'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

${\displaystyle N\ge 200000\times \frac{40}{39}\approx 205100\text{years}}$;

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 theBerlin Wall andBroadway and off-Broadway plays.^[6]

Leslie's argument differs from Gott's version in that he does not assume a vague prior probability distribution forN. 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 forN. He calls this theprobability 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 ofJesus to 1958, with only 7% of thevariance left unexplained. Several follow-up letters (1961, 1962, ...) were published inScience 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.^{[note 2]}

Thereference class from whichn is drawn, and of whichN is the ultimate size, is a crucial point of contention in the doomsday argument argument. The "standard" doomsday argumenthypothesis 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 onpractical andphilosophical reasons. According toNick Bostrom,consciousness is (part of) the discriminator between what is in and what is out of the reference class, and thereforeextraterrestrial intelligence might have a significant impact on the calculation.^{[citation needed]}

The following sub-sections relate to different suggested reference classes, each of which has had the standard doomsday argument applied to it.

Nick Bostrom,considering observation selection effects, has produced aSelf-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 givesN < 20n with 95% confidence (the standard doomsday argument). However, he hasrefined this idea to apply toobserver-moments rather than just observers. He has formalized this as:^[7]

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 thatN < 10n (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.

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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 anearly 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 thedistribution of human characteristics that places modern and historic humans outside the mainstream implies that it is already known, before examiningn, that it is likely to be very early inN. This is an argument for changing the reference class.

For example, if one is certain that 99% of humans who will ever live will becyborgs, but that only a negligible fraction of humans who have been born to date are cyborgs, one could be equally certain that at least one hundred times as many people remain to be born as have been.

Robin Hanson's paper sums up these criticisms of the doomsday argument:^[8]

All else is not equal; we have good reasons for thinking we are not randomly selected humans from all who will ever live.

Thea posteriori observation thatextinction level events are rare could be offered as evidence that the doomsday argument's predictions are implausible; typically,extinctions of dominantspecies happen less often than once in a million years. Therefore, it is argued thathuman extinction is unlikely within the next ten millennia. (Anotherprobabilistic argument, drawing a different conclusion than the doomsday argument.)

In Bayesian terms, this response to the doomsday argument says that our knowledge of history (or ability to prevent disaster) produces a prior marginal forN with a minimum value in the trillions. IfN is distributed uniformly from 10¹² to 10¹³, for example, then the probability ofN < 1,200 billion inferred fromn = 60 billion will be extremely small. This is an equally impeccable Bayesian calculation, rejecting theCopernican principle because we must be 'special observers' since there is no likely mechanism for humanity to go extinct within the next hundred thousand years.

This response is accused of overlooking thetechnological threats to humanity's survival, to which earlier life was not subject, and is specifically rejected by most^{[by whom? – Discuss]} academic critics of the doomsday argument (arguably exceptingRobin Hanson).

Robin Hanson argues thatN's prior may beexponentially distributed:^[8]

${\displaystyle N=\frac{e^{U(0,q]}}{c}}$

Here,c and q are constants. Ifq is large, then our 95% confidence upper bound is on the uniform draw, not the exponential value ofN.

The simplest way to compare this with Gott's Bayesian argument is to flatten the distribution from the vague prior by having the probability fall off more slowly withN (than inverse proportionally). This corresponds to the idea that humanity's growth may be exponential in time with doomsday having a vague priorprobability density function intime. This would mean thatN, the last birth, would have a distribution looking like the following:

This priorN distribution is all that is required (with the principle of indifference) to produce the inference ofN fromn, and this is done in an identical way to the standard case, as described by Gott (equivalent to${\displaystyle \alpha }$ = 1 in this distribution):

${\displaystyle Pr(n)={\int }_{N=n}^{N=\mathrm{\infty }}Pr(n\mid N)Pr(N)dN={\int }_n^{\mathrm{\infty }}\frac{k}{N^{(\alpha +1)}}dN=\frac{k}{\alpha n^{\alpha }}}$

Substituting into the posterior probability equation):

${\displaystyle Pr(N\mid n)=\frac{\alpha n^{\alpha }}{N^{(1+\alpha )}}.}$

Integrating the probability of anyN abovexn:

${\displaystyle Pr(N>xn)={\int }_{N=xn}^{N=\mathrm{\infty }}Pr(N\mid n)dN=\frac{1}{x^{\alpha }}.}$

For example, ifx = 20, and${\displaystyle \alpha }$ = 0.5, this becomes:

${\displaystyle Pr(N>20n)=\frac{1}{\sqrt{20}}\simeq 22.3\mathrm{\%}.}$

Therefore, with this prior, the chance of a trillion births is well over 20%, rather than the 5% chance given by the standard DA. If${\displaystyle \alpha }$ is reduced further by assuming a flatter priorN distribution, then the limits on N given byn become weaker. An${\displaystyle \alpha }$ of one reproduces Gott's calculation with a birth reference class, and${\displaystyle \alpha }$ around 0.5 could approximate his temporal confidence interval calculation (if the population were expanding exponentially). As${\displaystyle \alpha \to 0}$ (gets smaller)n becomes less and lessinformative aboutN. In the limit this distribution approaches an (unbounded)uniform distribution, where all values ofN are equally likely. This is Page et al.'s "Assumption 3", which they find few reasons to reject,a priori. (Although all distributions with${\displaystyle \alpha \le 1}$ are improper priors, this applies to Gott's vague-prior distribution also, and they can all be converted to produceproper integrals by postulating a finite upper population limit.) Since the probability of reaching a population of size 2N is usually thought of as the chance of reachingN multiplied by the survival probability fromN to 2N it follows that Pr(N) must be amonotonically decreasing function ofN, but this doesn't necessarily require an inverse proportionality.^[8]

Another objection to the doomsday argument is that theexpected total human population is actuallyinfinite.^[9] The calculation is as follows:

The total human populationN =n/f, wheren is the human population to date andf is our fractional position in the total.

We assume thatf is uniformly distributed on (0,1].

The expectation ofN is$${\displaystyle E(N)={\int }_0^1\frac{n}{f}df=n[\ln (f){]}_0^1=n\ln (1)-n\ln (0)=+\mathrm{\infty }.}$$

For a similar example of counterintuitive infinite expectations, see theSt. Petersburg paradox.

One objection is that the possibility of a human existing at all depends on how many humans will ever exist (N). If this is a high number, then the possibility of their existing is higher than if only a few humans will ever exist. Since they do indeed exist, this is evidence that the number of humans that will ever exist is high.^[10]

This objection, originally byDennis Dieks (1992),^[11] is now known byNick Bostrom's name for it: the "Self-Indication Assumption objection". It can be shown that someSIAs prevent any inference ofN fromn (the current population).^[12]

The SIA has been defended by Matthew Adelstein, arguing that all alternatives to the SIA imply the soundness of the doomsday argument, and other even stranger conclusions.^[13]

TheBayesian argument byCarlton M. Caves states that the uniform distribution assumption is incompatible with theCopernican principle, not a consequence of it.^[14]

Caves 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:

Your friendly enquiry about the age of the celebrant elicits the reply that she is 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'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.(See Caves' online paperbelow.)

Cave's example example exposes a weakness inJ. Richard Gott's "Copernicus method" DA: it does not specify when the "Copernicus method" can be applied. But this criticism is less effective against more refined versions of the argument.Epistemological refinements of Gott's argument byphilosophers such asNick Bostrom specify that:

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 (viaactuarial tables) Caves' Birthday party age-estimate could not 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 would need to completely exclude all prior knowledge of likely human life spans; in principle this could be done (e.g.: 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 behidden prior to the survival estimate being made. (Although this is no longer exactly the DA, it is much more comparable to it.)

Without knowing the lady's age, the DA reasoning produces arule to convert the birthday (n) into a maximum lifespan with 50% confidence (N). Gott'sCopernicus method rule is simply: Prob (N < 2n) = 50%. How accurate would this estimate turn out to be? Westerndemographics are now fairlyuniform across ages, so a random birthday (n) could be (very roughly) approximated by a U(0,M] draw whereM is the maximum lifespan in the census. In this 'flat' model, everyone shares the same lifespan soN =M. Ifn happens to be less than (M)/2 then Gott's 2n estimate ofN will be underM, its true figure. The other half of the time 2n underestimatesM, 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.

Some philosophers have suggested that only people who have contemplated the doomsday argument (DA) belong in the reference class "human". If that is the appropriate reference class,Carter defied his own prediction when he first described the argument (to theRoyal Society). An attendant could have argued thus:

Presently, only one person in the world understands the Doomsday argument, so by its own logic there is a 95% chance that it is 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 aparadox for the doomsday argument:^[9]

If a member of the Royal Society 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 be a 5% chance that 40 or more people would actually be interested. Also, of course, ignoring something because you only expect a small number of people to be interested in it is extremely short sighted—if this approach were to be taken, nothing new would ever be explored, if we assume noa priori knowledge of the nature of interest and attentional mechanisms.

Various authors have argued that the doomsday argument rests on an incorrect conflation of future duration with total duration. This occurs in the specification of the two time periods as "doom soon" and "doom deferred" which means that both periods are selected to occurafter the observed value of the birth order. A rebuttal in Pisaturo (2009)^[15] argues that the doomsday argument relies on the equivalent of this equation:

${\displaystyle P(H_{TS}|D_{p}X)/P(H_{TL}|D_{p}X)=[P(H_{FS}|X)/P(H_{FL}|X)]\cdot [P(D_p|H_{TS}X)/P(D_p|H_{TL}X)]}$,

where:

X = the prior information;

D_p = the data that past duration ist_p;

H_FS = the hypothesis that the future duration of the phenomenon will be short;

H_FL = the hypothesis that the future duration of the phenomenon will be long;

H_TS = the hypothesis that thetotal duration of the phenomenon will be short—i.e., thatt_t, the phenomenon'stotal longevity, =t_TS;

H_TL = the hypothesis that thetotal duration of the phenomenon will be long—i.e., thatt_t, the phenomenon'stotal longevity, =t_TL, witht_TL >t_TS.

Pisaturo then observes:

Clearly, this is an invalid application of Bayes' theorem, as it conflates future duration and total duration.

Pisaturo takes numerical examples based on two possible corrections to this equation: considering only future durations and considering only total durations. In both cases, he concludes that the doomsday argument's claim, that there is a "Bayesian shift" in favor of the shorter future duration, is fallacious.

This argument is also echoed in O'Neill (2014).^[16] In this work O'Neill argues that a unidirectional "Bayesian Shift" is an impossibility within the standard formulation of probability theory and is contradictory to the rules of probability. As with Pisaturo, he argues that the doomsday argument conflates future duration with total duration by specification of doom times that occur after the observed birth order. According to O'Neill:

The reason for the hostility to the doomsday argument and its assertion of a "Bayesian shift" is that many people who are familiar with probability theory are implicitly aware of the absurdity of the claim that one can have an automatic unidirectional shift in beliefs regardless of the actual outcome that is observed. This is an example of the "reasoning to a foregone conclusion" that arises in certain kinds of failures of an underlying inferential mechanism. An examination of the inference problem used in the argument shows that this suspicion is indeed correct, and the doomsday argument is invalid. (pp. 216-217)

Gelman and Robert^[17] assert that the doomsday argument confuses frequentistconfidence intervals with Bayesiancredible intervals. Suppose that every individual knows their numbern and uses it to estimate an upper bound onN. Every individual has a different estimate, and these estimates are constructed so that 95% of them contain the true value ofN and the other 5% do not. This, say Gelman and Robert, is the defining property of a frequentist lower-tailed 95% confidence interval. But, they say, "this does not mean that there is a 95% chance that any particular interval will contain the true value." That is, while 95% of the confidence intervals will contain the true value ofN, this is not the same asN being contained in the confidence interval with 95% probability. The latter is a different property and is the defining characteristic of a Bayesian credible interval. Gelman and Robert conclude:

the Doomsday argument is the ultimate triumph of the idea, beloved among Bayesian educators, that our students and clients do not really understand Neyman–Pearson confidence intervals and inevitably give them the intuitive Bayesian interpretation.

• Anthropic principle
• Human overpopulation
• German tank problem
• Global catastrophic risk
• Doomsday event
• Fermi paradox
• Measure problem (cosmology)
• Mediocrity principle
• Quantum suicide and immortality
• Simulated reality
• Survival analysis
• Survivalism
• Technological singularity

• John A. Leslie,The End of the World: The Science and Ethics of Human Extinction, Routledge, 1998,ISBN 0-41518447-9.
• J. R. Gott III,Future Prospects Discussed, Nature, vol. 368, p. 108, 1994.
• This argument plays a central role inStephen Baxter's science fiction book,Manifold: Time, Del Rey Books, 2000,ISBN 0-345-43076-X.
• The same principle plays a major role in theDan Brown novel,Inferno, Corgy Books,ISBN 978-0-552-16959-2
• Poundstone, William,The Doomsday Calculation: How an Equation that Predicts the Future Is Transforming Everything We Know About Life and the Universe. 2019 Little, Brown Spark.Description &arrow/scrollable preview. Also summarised in Poundstone's essay,"Math Says Humanity May Have Just 760 Years Left".The Wall Street Journal, updated June 27, 2019.ISBN 9783164440707

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• The Doomsday argument category on PhilPapers
• A non-mathematical, unpartisan introduction to the DA
• Nick Bostrom's response to Korb and Oliver
• Nick Bostrom's annotated collection of references
• Kopf, Krtouš & Page's early (1994) refutation based on theSIA, which they called "Assumption 2".
• The Doomsday argument and the number of possible observers by Ken Olum In 1993J. Richard Gott used his "Copernicus method" to predict the lifetime of Broadway shows. One part of this paper uses the same reference class as an empirical counter-example to Gott's method.
• A Critique of the Doomsday Argument by Robin Hanson
• A Third Route to the Doomsday Argument by Paul Franceschi,Journal of Philosophical Research, 2009, vol. 34, pp. 263–278
• Chambers' Ussherian Corollary Objection
• Caves' Bayesian critique of Gott's argument. C. M. Caves, "Predicting future duration from present age: A critical assessment", Contemporary Physics 41, 143-153 (2000).
• C.M. Caves, "Predicting future duration from present age: Revisiting a critical assessment of Gott's rule.
• "Infinitely Long Afterlives and the Doomsday Argument" by John Leslie shows that Leslie has recently modified his analysis and conclusion (Philosophy 83 (4) 2008 pp. 519–524): Abstract—A recent book of mine defends three distinct varieties of immortality. One of them is an infinitely lengthy afterlife; however, any hopes of it might seem destroyed by something like Brandon Carter's 'doomsday argument' against viewing ourselves as extremely early humans. The apparent difficulty might be overcome in two ways. First, if the world is non-deterministic then anything on the lines of the doomsday argument may prove unable to deliver a strongly pessimistic conclusion. Secondly, anything on those lines may break down when an infinite sequence of experiences is in question.
• Mark Greenberg, "Apocalypse Not Just Now" in London Review of Books
• Laster: A simple webpage applet giving the min & max survival times of anything with 50% and 95% confidence requiring only that you input how old it is. It is designed to use the same mathematics asJ. Richard Gott's form of the DA, and was programmed bysustainable development researcher Jerrad Pierce.
• PBS Space Time The Doomsday Argument

• Probabilistic arguments
• 1983 introductions
• Doomsday scenarios

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