Hello everyone; I'm new to the forum, and have been advised to post this in the "discussion" section. Hope this is OK.
I've found some references to discussions here on Brandon Carter / John Leslie's "Doomsday Argument" and they seemed well-informed. One thing I've noticed about the argument though (but haven't seen discussed before) is that it can be made much sharper by assuming that we are making random *observations*, rather than just that we are a random *observer*.
For those who know the literature, this is a form of Nick Bostrom's Strong Self-Sampling Assumption as opposed to the (basic) Self-Sampling Assumption. Oddly enough, Bostrom discusses SSSA quite a lot in connection with the Doomsday Argument, but I can't see that he's done quite the analysis below.
So here goes:
In the "random observer" model (the Self-Sampling Assumption with the widest reference class of "all observers"), we discover that we are in a human civilization and there have been ~100 billion observers before us in that civilization. We should then predict (crudely) that there will be about ~100 billion observers coming after us in that civilization; also we should predict that a typical civilization of observers won't have much more than ~100-200 billion observers in total (otherwise we'd be in one of the much bigger ones, rather than in a smaller one). So typical civilizations don't expand beyond their planets of origin, and don't even last very long on their planets of origin.
Further, since there are currently ~150 million human births per year that would imply the end of the human race in ~700 years at current population size and birth-rates. Doom soon-ish but not very soon.
But what about the "random observation" model? One difference here is that a large portion of the ~100 billion humans living before us died very young (high infant mortality rate) so made very few observations. For instance, Carl Haub, who calculated the 100 billion number (see http://www.prb.org/Articles/2002/HowManyPeopleHaveEverLivedonEarth.aspx) reckons that for most of human history, life expectancy at birth has been little more than 10 years. By contrast, recent observers have had a life expectancy of 60+ years, so are making many more observations through their lives than average. This means that *observations* are much more concentrated in the present era than *observers*.
Working with Haub's population numbers, there have been about 1-2 trillion "person-years" of observations before our current observations (in January 2012). Also, that estimate is very stable even when we make quite different estimates about birth-rate. (The reason is that the overall population at different stages in history is easier to estimate than the overall birth-rate, so integrating population through time to give person-years is easier than integrating birth-rate through time to give births).
Under the "random observation" model, we would predict a similar number of person-years of observations to come in the future of our civilization. At a human population size of ~7 billion, there are only around 1-2000 / 7 or ~200 years until human extinction: doom rather sooner. And if population climbs to 10 or 14 billion before flattening out (as some demographers predict) then doom even sooner still.
What's also quite striking is that over 20% of all observations *so far* have happened since 1900, and under a "doom soon" model the *majority* of all observations would happen in the period of multi-billion population sizes. So our current observations look very typical in this model.
Now I'm aware that Bostrom thinks the SSSA is a way out of the Doomsday Argument, since by relativizing the "reference class" (to something other than all observations, or all human observatioons) then we get a less "doomish" prediction. All we can conclude is that the reference class we are part of (whatever that is) will terminate soon, whereas observers in general can carry on. I'm also aware of a number of criticisms of the whole SSA/SSSA approach.
On the other hand, it is quite striking that a very simple reference class (all observations), coupled to a very simple population model for observers (exponential growth -> short peak -> collapse) predicts more or less exactly what we are seeing now.