A statistical claim that your birth rank among all humans ever born lets you estimate how many come after you, and the answer is uncomfortably soon.
Documented
Carter's lever, Leslie's book, Gott's wall
Documented: Carter circulated the argument in the 1980s; Leslie published The End of the World (1996) as its full case; John Gott independently ran the same logic on the Berlin Wall, predicting its lifespan from when he visited, and on humanity, and published in Nature (1993). The mechanics are sampling theory: absent other information, your birth rank is drawn uniformly from the total, so the total is probably within a modest multiple of the births so far.
Assessment: Documented as a formally valid application of Bayesian reasoning that professionals have spent forty years failing to cleanly kill. The argument is sound given its prior; the war is over whether the prior is legal.
Stories
The generational vertigo
The folk version is the unease anyone feels realizing they might be ordinary: every civilization before us assumed it was mid-story and most were right, because middles are where the population is. It circulates now as secular eschatology, doom without a prophet, just a percentile.
Assessment: Documented as the argument's emotional engine. It flatters no one and consoles no one, which paradoxically lends it credibility with the temperamentally grim. Popper noted our own paradox file: a tolerant person must still check the math.
Interesting claims
The counterfire
The interesting claims are the escapes: the self-sampling assumption may be incoherent (SIA versus SSA, the sleeping-beauty wars); observers may not be randomly positioned; reference classes are chosen, not found, humans, persons, minds, each giving different dooms; and the argument retroactively predicts the past too, which should embarrass it. The anthropic shadow literature extends the fight to extinction-risk estimation generally.
Assessment: Documented unresolved philosophy of probability. The argument survives every refutation by becoming more technical, which is either the mark of depth or the mark of a trap built from definitions. Forty years, no verdict, filed accordingly.
Sources and further reading
Carter, circulated papers on the anthropic principle (1980s), the origin
Leslie, The End of the World: The Science and Ethics of Human Extinction (1996)
Gott, Implications of the Copernican Principle for Our Future Prospects, Nature (1993)
Bostrom, Anthropic Bias (2002), the self-sampling wars
The Sleeping Beauty problem literature, the adjacent battlefield
Worth asking
If your position in line predicts the length of the line, is that statistics or a horoscope in a lab coat?