One ingredient is the Copernican principle, which states that you should not expect to be special in any regard: our galaxy is not the center of the universe, the earth is not the center of the Milky Way, and Waterloo is not in the middle of nowhere. A typical example is the number of planets in our solar system. The Copernican principle tells you that we are an average solar system in an average galaxy, and the number of planets is not exceptional but rather common. You then expect other solar systems to have a similar number of planets.
It is quite natural for human beings to adopt the Copernican principle. Typically, we assume everybody else experiences the world similar as we do. Needless to say, this is the cause of countless problems, and the reason why the men in my life state frequently they don't understand me... ah, sorry, getting distracted...
To briefly summarize the doomsday argument: consider the lives of humans as distributed over time, until doomsday, after which there are no humans any more. Then the Copernican principle comes into play. Neither you nor me is special in any regard, and the conclusion is that with equal probability I could be any of these human beings. The most common versions of the doomsday argument that I know are
A) If one assumes that population is growing, the largest number of humans will live directly before doomsday. Thus, if I am a random choice among all the humans that will ever been born, the probability is the highest that doomsday is within my lifetime.
B) I consider I am a random choice and want to know how much time is left before the end of the world. I assume a confidence of typically 95% with which I am among the last human beings that will ever been born. Since I roughly know how many people have lived before me this gives an estimate about the total number of people that will ever been born, and the time left to doomsday - with a confidence of 95%.
Okay, nice mathematical trick. Now here is what I don't understand about it.
One of the lessons from stochastic I can recall is that coincidence has no memory. Consider I flip a coin, the probability for each outcome is 1/2, and repeated flips are completely uncorrelated. I flip and it's heads. I flip again and it's heads again. I do that, say, 120 times, and the result is always the same. Then you have to make a bet on the next flip. What would you bet?
Well, okay, by now you have either fallen asleep or decided I am cheating somehow and wouldn't want to bet with me (thereby employing the Copernican principle which states that I am rather average... okay, okay, being somewhat cynical here). But let's assume this is a fair game, and I am totally unable to cheat. You'd be tempted to bet next flip does not continue this awfully unlikely series of heads, no? After all, the saying goes 'lightning never strikes twice'.
However, the probability for the next flip to show heads is (drums please) 1/2. As it was all the time. Yes, the probability for the 120 heads is a tiny (1/2)120, but the probability for the next flip is still 1/2. Coincidence has no memory. Think about Mike who enters the room after flip # 119. He sees # 120 showing heads, and the whole room goes 'ooooooh'. Mike would conclude these people are totally nuts.
Things are different if you take a bag and place in it 120 red and 120 blue marbles and pick them blindly without putting back. If you picked 120 red ones, you know the next one has to be blue. In this case, the probability depends on how many marbles you have already picked.
Now let's come back to mankind. To be clear we will state the following assumption:
1) There is a probability p that doomsday is tomorrow.
And we specify: we don't know when it is, but it's some universal property. That is to say, I am having one of my better days and I am willing to assume the presence of mankind on this planet does not increase the probability of the world ending tomorrow.
Okay. Then quantify the number of all humans that will ever been born as N and distribute them over the time prior to doomsday with function N(t), consider you are number n out of N. Take the derivative of N(t), multiply it with the speed of light, integrate it, subtract two, invert it and turn the paper upside down. Think hard about it and then tell me what is the probability that the world will end tomorrow?
Well. By assumption 1), the probability is p.
And it is in no relation with the number of people on that planet whatsoever. How come the doomsday argument suggests it is related with the number of people alive? To make the argument, the number of people living is treated as a random variable distributed over time with a probability (density), out of which you 'pick' your live span. You might picture it as a bag that contains all the CVs of all the people that will ever live, and you have to draw one. However, the way that the Copernican principle is used, there is no mentioning of how many CVs already have been distributed, so your pick has to be understood as independent of this.
This one can do for stuff like number of planets but not for events with a causal connection - like lives, or everything that has a time evolution. Even though the time evolution itself does not have to be deterministic, the total number of people at a given time is not a random variable over time. For example, if one just assumes some probability and distributes all the N people according to it, there is the small but non-vanishing probability that all the N people are born in the years 0 +/- 10 - which is impossible because it is in conflict with the evolution law for the population growth.
What one has for the population is is not a probability distribution over time, but a stochastic differential equation, that yields the probability for so-and-so many people living at a timestep t from that at timestep t-1. If one knows stochastics better than I, one can calculate a time-dependent average value
To come back to the two doomsday arguments.
A) Relies on the number of humans being a random variable instead of having a stochastic evolution. For an evolution law however, the Copernican principle does not make sense. Obviously, the number of people living at time t does depend on the number of people living before that. That is, you can't pick your CV out of the bag without making sure that your parents were born before you. There are causal correlations between the elements in that distribution.
B) Is a psychologically very interesting reformulation. One replaces one unknown parameter, the time when doomsday is with another unknown parameter, that is the confidence of being among the last humans that will ever live. Both parameters are related to each other. But both are still unknown. The conclusion doesn't yield any insights, it just sounds surprising.
Hmm. Running late. Have to get on that highway again. Sigh. A nice weekend to all of you :-)