Explain the Doomsday Argument to me

Factual Questions
1

So I was reading some loonytunes 2012 websites after watching “End of Days” with our governor and was, as you can tell, getting into the whole apocalypse thing. (In retrospect, I wish I was listening to “Armaggedon it” by Def Leppard; nobody’s perfect).

Then I came across the “Doomsday Argument,” and, though I’m admittedly not much of a mathmologist, I am having trouble with it.

As I understand it: The Doomsday Argument claims that there’s a 95% chance that we’re within the last 95% of humans ever born, which I already don’t quite understand, so I’d love that explained. Beyond this, at least according to the wikipedia article:

A big question here for me is doesn’t this argument change due to population? As in, if I posited the Doomsday Article in 1804, when we at 1 billion people, wouldn’t this then argue that the extinction of humans is much liklier, much earlier? Or, say, if we make this argument twenty years from now? I just feel like I’m missing something here. I’m sure when I hear it, I will be mighty ashamed.

So, please, bring on the shame.

2

Well, I wouldn’t be too ashamed about not understanding it, because I don’t think it makes much sense.

As near as I can understand it, with the fancy math taken out it boils down to :
First, assume that the human race will go extinct in a finite time, having had a finite number of human beings alive (call that total number of humans N). Second assume that the human population keeps growing at the same kind of exponential rate it has been, so that the population of humans is largest right before the catastrophe.

Now if some near-omiscient immortal being numbered each human born, from the first true Homo Sapiens to the last surviving individual, and randomly picked a number between 1 and N, the most likely period for that individual to be from is the period with the most humans alive. If the most humans are alive right before the catastrophe, then the most likely period for that randomly-chosen human to be alive is right before the catastrophe. (This bit is straightforward, if 1 and 2 are true).

Thirdly, we assume that YOU are a randomly-chosen individual from the entire history of the human species. That implies you are most likely from the period right before catastrophe.
Now, I think you make a pretty good point against this argument by pointing out that a person in 1805 could have exactly the same argument to prove that the human race would likely go extinct before the year 2008. And also for a person in 5 AD to argue nobody will be alive for the millenium year, and so on.

The thing is, assuming that You are a random sample from the entire history of the species doesn’t work in this case.
(Hint: if YOU are a random sample, and I am a random sample, then the fact that we’re both alive right now makes it twice as likely that the catastrophe is about to happen. And if we think that the person next to me on the bus is also a random sample, that makes it three times as likely. Etc.)

3

Or, written even simpler (assuming I understand it correctly), we only know that humankind has lasted this long, so we can only guess that an equal period of time is continuable. The ability for our race to last longer than that is unproven.

The total population of humankind in all that is a bit off-the-point in my mind.

4

So you’re saying that the argument that we’re probably near the end times must obviously be wrong, because the same argument implies that we’re probably near the end times?

And yes, one could have made the same argument a thousand years ago, but the argument wouldn’t have been as strong a thousand years ago as it is now, because many fewer people were around then than there are now, so a randomly selected person is much more likely to be around now than around in 1009.

5

I am not a mathematician. We need one, STAT. (Or a statistician–stat.)

Meantime, here’s my take:

Going by the wiki article, it appears that the argument has almost nothing to do with humans whatsoever–it’s an exercise in pure probability.

It starts with the Copernican Principle, an assumption that we humans hold no special place of observation in space or time. Therefore, just as we assume that the Earth is not the center of the universe, we assume that we are not living in any particular period of human history (beginning, middle, end). Therefore, if you assign each human being a sequential serial number, starting with 1 for the first ever Homo sapiens in Africa 200,000 years ago right up through today, and continue on until the last human who is ever born (assigned serial number N), you would have to assume that we humans who are discussing the problem today hold no special place in this order of serial numbers.

The distribution is assumed to be uniform (why, exactly? need help here), so our fractional position (f) in this sequence is stated as f = n/N.

Now, with this uniform distribution and all, there is a 95% chance that our serial numbers fall within the last 95% of all serial numbers. That means f > 0.05. Flipping the equation around gives you this:

f = n/N
f > 0.05
n/N > 0.05
N < 20n

Now we have a probable upper bound for N, the serial number of the last human being. It’s 20 times n, which is our serial number.

Now we bring in some historical numbers, and apply this theory to humans specifically. John A. Leslie posited the total number of all humans who had ever lived up until the present time to be 60 billion. In other words, all of our serial numbers are in the neighborhood of 60 billion. (Hey, 61,000,090,812! Lookin’ good!) Given this, we can plug “60 billion” in for “n” and get the following:

N < 20n
N < 20 * 60 billion
N < 1.2 trillion

So now we can say that it’s probable that the last human being to ever live will have a serial number that is less than 1.2 trillion.

Then, given our current serial number count (circa 60 billion) and extrapolating on likely population trends (birth rates, lifespan, maximum carrying capacity of the Earth, etc.), you get an upper bound for reaching that 1.2 trillion digit within the next 9,120 years.

In other words, there is a 95% chance of human extinction by 11,125 AD (give or take a few years). Not a certainty, but a probability.

Where I start to get confused is when the argument starts talking about uniform distribution. I’d love a mathematician to explain that for me a bit better, and show how it fits into everything here.

After that, it seems you get into the head-spinning nature of probability in general. So, for the 3 billionth human, the human race was 95% likely to have ended by now. And yet, for us, it’s STILL 95% likely to end within 20 times our current count (1.2 trillion).

I feel like I’m trapped in the Monty Hall problem.

6

I think that the argument assumes that we know nothing else about human beings. But one thing that we do know about the species is that it is rapidly increasing in numbers, and has been increasing in numbers for a while now. Is a species that is so successful on that test likely to start declining very soon?

7

I think it doesn’t even assume anything about species. It seems a purely mathematical construct. As far as I can tell, it might as well be applied to candy bars coming off a factory line.

I think it’s saying: Given a randomly-chosen representative from a sequence, you have a 95% chance that said representative’s position lies within the final 95% of the sequence. (Makes sense, right?) Given therefore that n/N > 0.05, and therefore N < 20n, there is a 95% chance that the sequence will end before it reaches 20 times the position of the selected representative.

So, per this reasoning, if you watch the 60 billionth (<–totally made-up number; I don’t know how many Hershey Bars have ever been produced) Hershey Bar roll off the line, there is a 95% chance that fewer than 1.2 trillion Hershey bars will EVER be made. Extrapolating from current production trends, you can calculate how many more years it would take to produce that many bars, and state, therefore, that there is a 95% chance that no Hershey bars will be made after July 15, 2057 (<-- totally made-up date; I know nothing about Hershey’s production methods or sales trends).

8

The way I see it, human population has been on a constant rise. Sure we may be at the high point of human population now - but we’ve always been at the high point of human population throughtout history. That’s a basic fact of being on a constant slope.

The other possibility is that there is a maximum capacity of human beings (which seems a reasonable assumption). Even if we’ve reached it, there’s no mathematical reason to assume that that maximum number will decline. A more plausible alternative is that we’ve just increased in population until the our possible environmental niches are full and now that it’s full it’ll remain that way. Human population has reached a plateau and will now coast along at its current level indefinitely.

9

I think that one flaw in the argument is that we AREN’T at a random point in history. We are in the present day, because the future does not yet exist. So if the present population stays the same or increases for, say, a million more years the chance of us living then is zero, despite the far larger number of people who’ll live in the future.

10

Wait a second-- what flippin’ catastrophe? That just popped up out of nowhere.