The Doomsday argument is a probabilistic argument that claims to predict the number of future members of the human species 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 are about halfway through it.
The probabilistic math on the Doomsday Argument gives us a 95 percent chance that we are among the last 95 percent of humans to born. So if an estimated 70 billion humans have already lived, than there is a high percentage chance than no more than 1.4 trillion humans (70 billion times 20) will ever live--a point we'd reach at current population growth rates in 10,000 years.
This io9 story summarizes some scientists' refutations against the Doomsday Argument, including this strong case against it:
"One of the more potent criticisms came in 1998 by K.B. Korb and J.J. Oliver who essentially argued that the DA is a gross oversimplification and that it violates reasonableness. They also argued that a sample size of one (i.e. oneself) is too small to make a serious difference to one’s rational beliefs, and that the DA could also be applied to one’s own life span."
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