Doom For The Doomsday Argument

Heard of the Doomsday Argument, sometimes also called the Carter Catastrophe? It claims to be able to name the date, with a certain confidence, at which mankind dies off. It has been the subject of a great many discussions and scientific papers. Its disturbing results have many very concerned.

Mathematical author William Poundstone used it to say there’s a “50 percent chance that humans will become extinct within about 760 years.” Not a lot of time left.

All of us will be dead before we can check his prediction. But we still have time to see whether this calculation makes any sense.

It turns out it does not. And the reason it does not is extremely important, because the mistake made in the argument is found all too frequently in science.

The first assumption in the Doomsday Argument (DA) is that there will be N total people, adding up all the already born and those still to come, whatever this number is. That much is true.

The second assumption is that your birth was nothing special. This seems to be true, too. You were not the first person born, and, if you can read these words, it means you were not the last person to be born, either. So you are somewhere in the great middle.

That second assumption then morphs and becomes the claim you could have been born at any point among everybody ever born. Which is to say, the assumption claims it was equally likely you could be the first person born, or the second, or third, or any, up to the last.

This is clearly false on the observation you were not the first born, or even the second! But the assumption might be approximately true if N is large. For instance, if you were the 100 billionth person to be born (and several estimates put the total so far at that number), and N is 1,000 trillion, then 99.99% of all people are yet to be born. So you are close enough to be “the first” as to make no difference.

If we continue with the second approximate assumption, and say you could equally well have been born at any point in man’s history, then there is a 50% chance half of all people will come after you. If there have been n people so far, and there will be an eventual total of N, then some simple math shows that there is a 50% chance that N ≤ 2 * n.

Since there have been about 100 billion people so far, there is a 50% chance that the total ever born will be less than or equal to 2 * 100 billion = 200 billion.

This means there are 100 billion of us yet to come. If, similar to now, 150 million are born each year, then 100 billion / 150 million per year = 666 years left, which is close to Poundstone’s estimate.

Yet this can’t be right. Have you spotted the error? Many don’t.

It’s not the 50%. There is nothing special about the 50%; we could (and will) have used any percentage between 0% to 100%.

Consider the first man born. Call him Adam. For him, n = 1. He looks at the math of the DA and calculates there is a 50% chance N ≤ 2 * 1. So he will not be surprised when he comes across Eve. Adam would, though, be shocked to learn that some 100 billion more were to arrive.

Maybe Adam could have used a tighter probability than 50%. Suppose he wanted to be 99% certain instead of just 50-50. Then our equation becomes N ≤ 100 * n.

That means Adam was 99% certain that the total number of people who ever live would be less than or equal to 100—not a hundred billion, one hundred period. But there were, so far, 100 billion. Off by (as mathematician’s say) nine orders of magnitude.

Then consider man number 200 billion, who, we calculated gave a 50% chance will be born 666 years from now. He, not knowing he is the last, would calculate the DA and conclude there is a 50% chance N ≤ 400 billion! And the 400 billionth man would answer 800 billion, and so on.

It turns out when you are born, how many have come before you, makes an enormous difference. That second assumption in the DA does not hold. After reflection, it seems absurd to discard the information that 100 billion souls preceded you, and assume that you could have been born at any time from Adam to the end. You simply could not have been born before you were.

The real argument is this: we know 100 billion have lived so far; how many more are to come?

There is no way to know. Not just using your place among births. How many people there will be is a good question, but like all questions, and all probabilities, depends on what information is assumed. If you change the assumptions, the probabilities can change radically.

This small example shows how easy even sophisticated math can do wrong. There is much more to it, and for those who have a deeper interest, I have an article which lays out all the mathematics involved.

Subscribe or donate to support this site and its wholly independent host using credit card click here. Or use the paid subscription at Substack. Cash App: $WilliamMBriggs. For Zelle, use my email: matt@wmbriggs.com, and please include yours so I know who to thank.