An essay "Doomsday Argument" was written about the doomsday arguments on a German's blog. If I understand her well, I think that the two of us agree. But it may be useful to mention which of the assumptions are rational or justified and which of them are not. Let me start with the
The principle says that the Earth is not a privileged celestial body, the Sun is not a privileged star, and the Milky Way is not a privileged galaxy. A generalization of the principle says that the humans are not a privileged species. A more ambitious version of the principle, the mediocrity principle, is the actual driving force of many people who believe the anthropic principle. All these people must believe in some kind of integalactic democracy that gives the same voice to every single intelligent observer within the Galaxy or even the multiverse.
Is the principle universally true and profound?
Well, I wouldn't say so. What would you expect from a person who doesn't even believe in trans-national democracy and who thinks that even the equality within one country is just a convenient law that can never quite work de facto?
In my opinion, the original version of the Copernican principle was mainly an ideological tool to oppose a wrong theory of astronomy, namely a theory that has always been driven by religious and political forces rather than scientific arguments. The Catholic black-buttockers used to insist that the Earth had to play as special a role in the Universe as it played in their favorite book, the Old Testament. While Christianity has made many great things for the Western civilization, its stubborn belief in certain ancient astrophysical theories - and later also biological theories - turned out to be counterproductive at a certain point. The Copernican principle is a symbol of the revolt against the religious dogmas.
In the case of the Earth, Copernicus and his soulmates were right. The physical parameters of our planet don't differ from the parameters of other planets in some spectacular qualitative way.
However, we should ask: Was this conclusion inevitable? Is there a permanent physical principle that tells you that things always turn out to be less special than they are believed at the beginning? My answer is a resounding "No". If you declare that things are not special but they are rather generic, you only choose a probability distribution that describes your ignorance more faithfully.
But it is not true that as we continue to learn how the real world works, objects are constantly becoming less special. Sometimes they are becoming more special and less equal. All values of wavelengths of light may a priori look equally natural and likely but the 21.1 centimeter line turns out to be rather special in astrophysics, after all.
Billions of other insights of all sciences - essentially all insights we have - may be viewed as additional examples of objects and numbers becoming more special than they used to be before you have a learned a new fact or before you have written down a new theory. When we look more carefully, the Earth has certain features that also make it somewhat special, after all. You can also find features of homo sapiens that make our species more viable and more intelligent than many others.
As you can see, I view all kinds of mediocrity arguments to be a tool to construct quasi-realistic "priors" - expectations that you insert as a starting point for your logical inference. These arguments represent a tool to avoid unjustified dogmas. These arguments seem to be a more fair description of your primordial ignorance. However, they are definitely not unbreakable constraints that the final answers must confirm.
Naturalness is another example. If you deal with an effective field theory, are all dimensionless couplings inevitably comparable to one? Once again, the answer is clearly "No". Why do we expect that is should "normally" be so? Well, it's because we describe our ignorance by a uniform prior. With a uniform probability distribution for a certain dimensionless parameter, it is unlikely that the parameter will be extremely tiny.
But the uniform probabilistic distribution is not a God-given law of the Cosmos. It is just a convenient trick to make balanced expectations - expectations that often turn out to be wrong anyway as soon as we figure out how the system works in more detail, as soon as we discover new reasons that make some special expectations more meaningful. Naturalness is thus not an unbreakable law of physics either. Even if you are a huge optimist, it is just a useful tool to quantify how unexpected the values of numerical parameters within a certain framework are.
The doomsday argument is an example of the mediocrity reasoning that is even worse than just an unjustified prejudice: I think that in this case, the conclusions of a mediocrity argument are manifestly flawed.
The argument assumes that you should be a generic observer. Because the number of people on the Earth grows exponentially, most people during the history live right before the collapse of the civilization. That's why the doomsday argument leads many people to predict that the humankind will collapse pretty soon. Jehovah's Witnesses as well as Anthropogenic Global Warming bigots, among dozens of similar groups, surely consider the doomsday arguments to be a general weapon that strengthens their predictions about the judgment day.
Are they right?
We can't be quite sure whether their particular predictions are going to be right - except for the predictions that have already been falsified - but we can be absolutely sure that the method with which they have reached their conclusions are completely irrational. Why? It's because we actually know the laws that will decide about the collapse of the civilization. More precisely, we know them partially but well enough to falsify certain oversimplified doomsday calculations.
Whether or not men will be around in 2100 and whether or not the future civilization is going to be stronger than ours will depend on the success of our fight against Islamic terrorists, North Korean communists, environmental terrorists, diseases, political correctness, shrinking fossil fuel resources, mutated viruses, left-wing nutcases who want to cripple the world's economy just for the sake of it, dangerous asteroids, or dozens of other potential threats you could think about. When you understand how these things work, you may offer qualified estimates of the probability that all of us will be screwed by 2100.
Even though we are clearly not able to make perfect predictions about the world in 2100 - not even good predictions - it seems clear to me that the method of looking into particular threats and their internal mechanisms is much more rational and reliable a way to deduce the future of mankind than some general doomsday arguments. I believe that if we understand the microscopic mechanisms of the dangerous processes in depth, we can make essentially accurate predictions of certain phenomena.
There is absolutely no reason to think that such predictions calculated from an increasingly detailed and accurate microscopic description of these phenomena will agree with some simple stochastic predictions based on the doomsday arguments or the mediocrity arguments. And if the two approaches to make predictions disagree, be sure that your humble correspondent prefers the microscopic analysis of the terrorists, asteroids, or viruses.
Such a contradiction means that one of the frameworks to predict has to be wrong. It is the mediocrity framework that is wrong. The only acceptable reason why a mediocrity argument should be right is a causal mechanism that is included among the laws of Nature. Such a mechanism would have to be somewhat analogous to thermalization: thermalization is a process that naturally makes all microstates with the same values of macroscopic parameters equally likely.
But as soon as we find out that the answers to various questions are actually decided by mechanisms different from such generalized thermalization or as soon as we find out that such hypothetical thermalization mechanisms would contradict causality and other well-established principles, these thermalization mechanisms that were invented to produce uniform distributions are simply falsified.
In the case of the doomsday arguments, we simply know that the answer to the question "When will the last humans die?" manifestly depends on other questions than those that enter the doomsday argument calculations.
Another recent example have been the Jovian citizens - those who live on Jupiter. Should the number of Jovian beings predicted by a theory influence our confidence that a particular theory is right before we actually count how many people live there? I have argued, together with Hartle and Srednicki, that the answer is "No". Theories become more acceptable or less acceptable once we compare their predictions to phenomena that we can already check and once we see an agreement or a disagreement. This is the right method to refine the probability estimates that a theory is correct. Predictions that we are not yet able to test - confirm or rule out - can't influence our confidence in a theory as long as we remain rational.
Analogously, if someone believes that the world should collapse because of some mediocrity argument, he or she is just a victim of another irrational prejudice. This prejudice might have the opposite flavor than the typical prejudices imposed upon our ancestors by Christianity. But this opposite flavor doesn't make these prejudices correct. They are equally irrational as the Christian dogmas.
Moreover, there exist methods to use the mediocrity arguments that lead to very different conclusions. For example, a mediocrity argument may be used to argue that the present can't be a special moment in the history of the Universe (or mankind) because all moments are created equal. Such a conclusion sharply contradicts the gloomy predictions of the usual doomsday arguments.
If you allow me to summarize, I view all kinds of mediocrity arguments to be nothing else than a very rough method to decide about your expectations long before you know anything about the system you study. Once you start to understand how various systems work, you may instantly realize that the expectations based on the mediocrity principle were just wrong. There exists no God-given rule that an egalitarian viewpoint based on uniform distributions is closer to the truth than the viewpoint of someone who avoids the mediocrity arguments.
Also, if you can prove that there can exist no mechanism that would be creating certain uniform distributions but that would still be consistent with other well-known facts about the Universe, the mediocrity argument is ruled out, too. Various types of thermal equilibrium of intelligent observers who live in different parts of a huge multiverse and at different moments are probably incompatible with the rest of physics we know.
We should carefully avoid unjustified dogmas about the special nature of our environment or our species or ourselves but we should also avoid unjustified dogmas claiming that these things can't be special.
And that's the memo.