Doomsday argument

Almost everyone who hears this argument immediately sees something wrong with it. The problem is, everyone thinks it's wrong for a different reason. And the more they study it, the more they tend to change their minds about what that reason is.

The Doomsday argument is an argument, purely from probability theory, that purports to show that humanity will probably go extinct sometime within the roughly foreseeable future, possibly less than one thousand years. It is extremely controversial, both in professional and lay circles.

It was first formulated by the applied mathematician Brandon Carter in 1983, popularised by Richard Gott in Nature in 1993, and analysed in depth by John Leslie in the 1990s.

There has been a fair degree of interest from philosophers and mathematicians, but it often involves quite obscure mathematics heavily dependent on intuitions about probabilities and statistics, so good luck understanding it. But the best counterarguments attempt to formalise the intuition that it doesn't make sense to argue from something that is 100% true (regardless of future events) to something uncertain in the future.

Of tanks and humans
The argument has its roots in the German tank problem, which was the attempt by the Allies during World War II to determine how many tanks Germany was producing purely by examining the serial numbers of the tanks that they were able to observe. While it's of course impossible to determine such a number with complete accuracy, if you get four tanks and they all have numbers under twenty, there probably aren't more than twenty of them; and similarly if the numbers have five digits apiece, there are probably tens of thousands of them. Thus we can make a rough guess as to how many there are. (This ignores the fact that you can fake serial numbers, which was a problem for the German tank study but isn't one for the Doomsday argument.)

If we have just one such number, we can make a very rough guess that the number we have is probably about half of the total. If we're looking at tank number twelve, there are probably not 600,000 tanks. Statistically, if there were that many, we would get something with six digits.

The Doomsday argument applies this to the human race. In the entire history of our species, there have been about one hundred billion humans so far. If the human race were to continue to survive for thousands of years hence, and maintained the current increase in reproduction, the total number of humans would be far greater; on the order of the high trillions or even the quadrillions. This would mean that we are observing humanity in its infancy, which is statistically unlikely. The argument therefore concludes that we're in the "middle" of the human timeline and, extrapolating from current population growth, that timeline will end roughly within the next thousand years.

Typical person
The argument assumes that the person thinking it is a typical human being. Leslie acknowledged this weakness but suggested that absent grounds for knowing otherwise, we should assume uniformity. Some critics think this is not terribly convincing.

William Eckhardt ponders whether we should consider ourselves in the sequence of human beings, or mammals or vertebrates; Leslie suggests that because the argument only concerns human survival it is only relevant to consider us qua humans. The depressing nature of the argument depends in part on the exponential growth in human population; if applied to all vertebrates it would predict a less rapid end.

Determinism
The Carter-Leslie argument assumes that the calculation of the probability will not affect the actual lifespan of the human race, but it is at least theoretically possible that people could look at the result of the Doomsday argument and then work to change history, thus invalidating the argument. This isn't a temporal paradox, but rather a problem with the assumptions in the Doomsday argument.

Margin of error
It's one thing to have a statistical prediction, but all predictions come with a margin of error. Korb and Oliver suggest that in classical probability theory it is impossible to assign a margin of error to a single sample, and therefore it is impossible to know the margin of error for the prediction, which weakens its predictive force.

Random sampling
There is some uncertainty about whether the selection of us, here, now can be considered genuinely random. Sowers notes that Leslie and Nick Bostrom have attempted to defend randomness. Eckhardt also criticises it based on classical statistics on the basis that we are not really selecting a sample at random (equiprobable sampling) because we cannot select a null sample. This depends rather a lot on your intuitions about statistics, and the more you look into the philosophical underpinning of statistics and probability theory, the worse it gets.

Bayesian argument
The original argument is Bayesian in nature, based on our beliefs about probability dependent on certain facts. The basis of Bayesian arguments against Doomsday lie in the criticism (as George F Sowers expresses it): "There seems something inherently wrong with accepting the idea that a datum which is entailed by all possible futures (the number of persons who have existed in the past) can be used to distinguish between those futures." Attempts to formalise this using Bayesian probabilities seem to show that the evidence used in Doomsday argument has no value in predicting the future.

Eckhardt uses Bayesian probability to show that for the argument to increase our estimate of the probability of being near the end of the world, the sample rank (position in history) must have some kind of dependence on the doomsday rank (total number of people), or else the Bayesian probability will not change at all; he suggests there is no such dependence. George F. Sowers Jr uses a similar argument to show that unless the sample is non-random, the sampling process provides no information.

We can consider the Bayesian question of how the argument should affect our estimate of the likelihood end of the world: if we don't already have a strong view of where we are in the human timeline, does the Doomsday argument significantly increase our belief in the probability of the end of the world? Korb and Oliver use a Bayesian argument which considers the probability that we occur in different positions in human history (which you might divide up into cohorts of a billion: are we in the 60th billion, the 120th billion, the 2000th billion...). Absent a strong initial reason to believe we are in one cohort, the Doomsday argument provides only weak grounds to prefer one cohort over another. They note that typical formulations of the Doomsday argument attempt to adjudicate between two possible positions in the timeline (we are near the beginning or near the end), in which case it appears persuasive, but if you open it up to many more possibilities and do a big calculation over many possible cohorts, Bayesian calculations suggest there is only a low posterior probability of the imminent end of the world. And increasing the number of cohorts you consider only decreases the force of the Doomsday argument.

Extensions
As well as being in the middle of human civilisation, could we be in the middle of a long sequence of universes, each of which will create the human race? This might suggest one of several cyclical cosmological theories, such as that of Roger Penrose.

Self-fulfilling prophecy?
If everybody hears the argument and gets really depressed or violent, it could become true.

This depends on people understanding the argument, which is unlikely. Or on them listening to philosophers, which is even more unlikely.