Warning: naive philosophical logic-chopping ahead. – Harry
I first ran across the doomsday argument many years ago (under another name, if I remember correctly) in the novel Manifold: Time by Steven Baxter. It’s one of those interesting ideas – it’s obviously nonsense, but it’s very hard to pin down exactly where the reasoning fails. At the time I think I pretty much dismissed it, but it appeared again in an online article a year or two back and I’ve been musing about it on and off ever since. It turns out to be even more interesting than I thought, because once you do pin down the logical flaw, you realize that it depends on a metaphysical assumption. This means that, arguably, the Doomsday Argument is the first ever experimental test of a metaphysical idea, which I find fascinating.
For those unfamiliar with the Doomsday Argument, the short version is that, using Bayesian statistics, you can take the fact that you were born now (rather than, say, a thousand years into the future) and use it to predict that the human race is almost certain to become extinct in the near future. For more information, I’d recommend this site by Professor Nick Bostrom of Oxford University. There is also a Wikipedia entry but personally I found it disorganized and confusing.
So where do I think the flaw lies? Well, the DA requires you to reason like this: if the human race is not about to become extinct, what was the probability of my being born now rather than at some point in the future? The subjective nature of this question is essential to the argument – you can’t, for example, imagine being a visiting alien pointing to someone and saying “what was the probability of such-and-such a person [or people] being born now?”. It just doesn’t work that way.
Now, that question sounds reasonable. But consider this: is it really meaningful to ask whether you might have been born at a different time? What, exactly, does the question actually mean? If somebody were born in 3000AD with your exact genetic code, would that person be you? I argue that (s)he would not, because your experiences and memories are at least as critical to your identity as your genetic code. Even identical twins are not the same person.
It should also be kept in mind that the chances of a person being born with an identical genetic code to you at some point in the future are unaffected by whether or not you were in fact born now. So if you would consider a person born with your genetic code in 3000AD instead of now to be yourself, you would have to also consider a person born with your genetic code in 3000AD as well as now as being yourself. In which case there would have been two yous. So to speak. 
At first glance, this would seem to handily refute the Doomsday Argument, but there’s a catch. What about your soul? If we suppose that people have souls, and that these souls already exist in some sense before a person is born, then suddenly the DA works again – you can ask “what were the odds that I [i.e., my soul] would be placed in the world at this time, rather than a thousand years in the future?” perfectly meaningfully.
It is, at this point, very tempting to assert that if humanity does not become extinct in the near future that would disprove the existence of the soul. This would however be dishonest, for several reasons: for one, my reasoning only applies if souls are (so to speak) created ahead of time, which is not necessarily the case; for another, the DA might have one or more other flaws (I believe Professor Bostrom discusses some possibilities in detail, though I must admit I haven’t actually read the book); and finally and perhaps most fatally, the DA also requires you to suppose that your soul is placed in the world at random, which I think according to most religions is unlikely to be the case.
Despite these caveats, I personally find this situation fascinating … as I may have said before. If you’ve gotten this far without giving up in dismay, bless you, and I hope I haven’t bored you completely to tears! My next post should be back to normally scheduled programming.
 English really isn’t a very suitable language for discussing this sort of hypothesis, is it?