Randall Munroe has given a very good summary of the argumentation:
Humans will go extinct someday. Suppose that, after this happens, aliens somehow revive all humans who have ever lived. They line us up in order of birth and number us from 1 to N. Then they divide us divide them into three groups--the first 5%, the middle 90%, and the last 5%:
Now imagine the aliens ask each human (who doesn't know how many people lived after their time), "Which group do you think you're in?"
Most of them probably wouldn't speak English, and those who did would probably have an awful lot of questions of their own. But if for some reason every human answered "I'm in the middle group", 90% of them will (obviously) be right. This is true no matter how big N is.
Therefore, the argument goes, we should assume we're in the middle 90% of humans. Given that there have been a little over 100 billion humans so far, we should be able to assume with 95% probability that N is less than 2.2 trillion humans. If it's not, it means we're assuming we're in 5% of humans--and if all humans made that assumption, most of them would be wrong.
To put it more simply: Out of all people who will ever live, we should probably assume we're somewhere in the middle; after all, most people are.
If our population levels out around 9 billion, this suggests humans will probably go extinct in about 800 years, and not more than 16,000.He goes on to state that most people immediately conclude that the idea is obviously wrong, but "the problem is, everyone thinks it's wrong for a different reason. And the more they study it, the more they tend to change their minds about what that reason is."
Well, there are two reasons why that could be so. One is that the argument is really quite clever but most people don't realise it. The other is that there is so much wrong with it that people discover new layers of wrongness every time they look at it.
I guess I would have to be counted among those who think that the Doomsday Argument is, indeed, idiotic. Admittedly I cannot come up with a super-deep Bayesian counter-argument such as are referenced in the linked Wikipedia article. But I don't think that is necessary because this does not look like a job for probabilistic reasoning anyway.
A lot of the intuitive force that the Doomsday Argument has for its proponents appears to derive from its similarity of the to the German Tank Problem, which is nearly always mentioned in the same breath (as in the Wikipedia article and Munroe's post, of course). Given a small sample of serial numbers, one can make an educated guess how large the series approximately must be. If you know five serial numbers of 3, 6, 11, 21 and 25, you can guess that it is somewhat unlikely that there will be more than forty items in total, and more than one thousand are vanishingly unlikely. But if you have already seen numbers above one thousand in your sample, well...
But the problem of considering the German Tank Problem analogous to the Doomsday Argument should be blindingly obvious: Humans do not come with serial numbers. The former gets its intuitive and probabilistic force from information that is quite simply not available to the latter.
And that makes it rather easy to construct a reductio ad absurdum for the Doomsday Argument. If it had been advanced 50,000 years ago, its force would have been precisely the same, and one would have had to conclude that we would surely be extinct by now. If it had been advanced 3.2 billion years ago, it could have been used to conclude that the likelihood of any life forms still being around 3.1 billion years ago was so close to zero as not to matter. But wouldn't you know it, life forms reproduce, and we are still here.
Although at this point not strictly necessary, this last observation leads seamlessly to a closer examination of some premisses of the argument as described by Munroe. "Humans will go extinct someday." Now I am not saying that this is not a sentence that we could assign a high probability. Most species that have ever existed were unsuccessful side branches of the tree of life, and comparatively few of the species that existed, say, 100 million years ago have extant descendants. What is more, we are large, complex organisms; those in particular are easily wiped out by some catastrophe, allowing smaller, more generalist vertebrates to fill the same niches again.
On the other hand, there is no precedent for the level of intelligence we have evolved. We really are much more flexible and adaptable than most other vertebrates. There are currently seven billion of us, on every continent, living very different lifestyles. If necessary, we will change what we do instead of insisting on the same one food plant or type of nesting space as so many other animals do. If necessary, we can eat insects and dig up wild roots. Even if climate oscillations, diseases, wars, resource depletion and suchlike do their worst, even if a K/T style meteorite crashes into us, what really is the chance that all of us get wiped out? Remember, it is enough if a couple thousand humans survive and start over.
That means there is also the possibility that we (that is, some of us) will just go on and on, evolving over time, perhaps even splitting into distinct species along the way, and our descendants will still be around in twenty million years - even if they appear no more human to us than an orang-utan does now.
The thing is, we just don't know which of these it will be. "Humans will go extinct someday" is a fairly bold pronouncement about the future, at least seen across the time frame that the discussion is about, and it should at least be phrased somewhat more carefully to avoid being question-begging.
Another matter-of-factly stated premiss is "given that there have been a little over 100 billion humans so far." This being the blog of a phylogenetic systematist, it is perhaps unsurprising what my objection will be: Where do you start counting humans, and why there? The claim appears to derive from this article which assumes an age of the human species of 50,000 years and then lists estimated population sizes throughout the millenia, the oddest of which is "2" as the starting point. (Seriously, Adam and Eve?)
But well, why 50,000 years ago? That seems awfully arbitrary. Surely we were the same species lineage hundreds of thousands of years earlier. But even that is arbitrary. The point is that there is no moment at which we suddenly turned human; instead there is an unbroken chain of subtly changing ancestors going back to the progenitor of all life more than three billion years ago.
That is our lineage, all of it, not only one arbitrarily chosen part at the end. So why is that not our "reference class", to use a term that is thrown around by Bayesians in this context? Surely there is no objective cut-off anywhere on the way. And if we add up all those ancestors, apes, tree shrews, fish, worms, microbes with generation times of less than a day and all, how many members has our lineage had so far? Punching that number into the silly probabilistic exercise, how many more members of our lineage would we expect in the future?
But of course my intention is not to claim that there will be seven fantastillion more humans, and that we will not go extinct until the sun explodes. No idea, maybe we will go extinct in a few thousand years. Rather my intention is to show that the Doomsday Argument doesn't make sense, regardless of what conclusions one would like to draw from it.