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An illustration of an ominous clock before the Earth. Javier Zarracina/Vox

A math equation that predicts the end of humanity

How much longer till we all die off? 760 years, give or take.

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The most mind-boggling controversy in the contemporary philosophy of science is the “doomsday argument,” a claim that a mathematical formula can predict how long the human race will survive. It gives us even odds that our species will meet its end within the next 760 years.

The doomsday argument doesn’t tell what’s going to kill us — it just gives the date (very, very approximately).

When I first came across this idea, I thought it was absurd. A prediction must be founded on data, not math! That is by no means an uncommon reaction. One critic, physicist Eric J. Lerner, branded doomsday “pseudo-science, a mere manipulation of numbers.”

Yet I now believe the doomsday prediction merits serious attention — I’ve written my latest book about it. Start with J. Richard Gott III. He’s a Princeton astrophysicist, one of several scholars who independently formulated the doomsday argument in the last decades of the 20th century. (Others are physicists Holger Bech Nielsen and Brandon Carter and philosopher John Leslie.) In 1969, Gott was a physics undergraduate fresh out of Harvard, spending the summer in Europe. At a visit to the Berlin Wall, he did a quick calculation and announced to a friend: The Berlin Wall will stand at least 2 and 2/3 more years but no more than 24 more years.

Demolition on the wall began 21 years later. This motivated Gott to write his method up. He published it in the journal Nature in 1993. There, Gott wrote of the future of humanity itself. He forecast a 95 percent chance that the human race would cease to exist within 12 to 18,000 years.

Not all Nature readers were convinced. “‘There are lies, damn lies and statistics’ is one of those colourful phrases that bedevil poor workaday statisticians,” biostatistician Steven N. Goodman complained in a letter to Nature. “In my view, the statistical methodology of Gott … breathes unfortunate new life into the saying.”

Yet Gott and his predictions also received favorable attention in the New York Times and the New Yorker (where a profile of Gott was titled “How to Predict Everything”). Gott is an engaging storyteller with a Kentucky accent that’s survived decades in the Ivy League. He has become a sort of scientific soothsayer, successfully predicting the runs of Broadway plays and when the Chicago White Sox would again win the World Series (they did in 2005).

Can it really be that easy to predict “everything”? It quickly became clear that 1) most scholars believe the doomsday argument is wrong, and 2) there is no consensus on why it’s wrong. To this day, Gott’s method, and a related one developed by Carter and Leslie, inspire a lively stream of journal articles.

The logic of the doomsday equation, explained

Gott calls his prediction technique the Copernican method. Copernicus, the great Renaissance astronomer, asserted that the Earth is not the center of the universe. Over the past centuries, astronomers have generalized this claim to the credo that humanity’s position in the universe is unlikely to be central or special. Our sun is an ordinary star in an ordinary galaxy. The cosmic “you are here” dot says we’re smack in the middle of Randomville.

The Copernican principle is normally uncontroversial when applied to an observer’s location in space. Gott’s idea was, why not apply it to a location in time?

You can sketch Gott’s logic on a napkin. Represent the Berlin Wall’s existence through time as a bar, like the time bar of a video. It’s got a beginning, a middle, and an end.

William Poundstone

Gott’s insight was that learning the past duration of the wall gives a clue to its future duration. For a random tourist, that past duration is likely to be a substantial fraction of the wall’s past-and-future existence. This allows an order-of-magnitude estimate of the future duration.

Suppose you had visited the wall at the 25 percent point in the timeline. At that moment, the wall’s future would have been three times as long as its past (75 percent is three times 25 percent).

Or pretend you visited at the 75 percent point. Then the future would have been only one-third the past duration.

Now take a deep breath. Imagine that a tourist wanted to make this prediction:

“The future duration of the Berlin Wall will be between one-third and three times as long as its past duration.”

This statement would have turned out to be true for anyone who visited the wall in the shaded part of the diagram. Because the shaded region is half the bar, we can say that, for half the days of the Berlin Wall’s existence, this prediction would have been correct.

Gott made that prediction, except that he also made use of the knowledge that the wall was then eight years old. He computed the most likely future duration to be between 8/3 and 8×3 (2.67 and 24) more years.

He reasoned that this prediction had a 50 percent chance of being right. You may feel that 50 percent is too wishy-washy and Gott just got lucky. No problem: The method can supply predictions with any degree of confidence you choose. To achieve 95 percent confidence, you’d make a diagram with the shaded region covering the middle 95 percent of the bar. The prediction range would be wider (from 1/39 to 39 times the past duration). Had Gott used this formulation, his prediction for the wall’s ceasing to exist would have been 0.21 to 312 years after his visit. This is less impressive, given the extremely wide range — but it would have been correct, too.

In short, the Copernican method is a mathematical parlor trick that does what it claims to do. You must encounter something of unknown duration at a random point in time (both important conditions!). But if you meet those conditions, it works.

So, how much longer do we have?

Homo sapiens has been around for about 200,000 years. There has been a huge population explosion in the past few millennia. Thus, as a random observer of my own species, I am far more likely to be living at a time when more humans are living (such as right now). This needs to be taken into account. The easiest way to do that is to use human lives, rather than years, as the marker of time.

Imagine a complete, chronological list of the human race: every person who ever lived or will have lived, sorted by time of birth. I will again represent it as a horizontal bar.

William Poundstone

Half the people who will ever live are in the first half of the list. Half are in the second half. These statements are necessarily right, no matter how long or short the list may end up being.

I’m curious as to where my name falls in that list. I could be relatively early, if humanity has a long, populous future ahead. I could also be late in the list — if something catastrophic is about to happen, and there won’t be many generations after mine.

Suppose that some people want to make this prediction:

“The number of future births will be less than the number of past births.”

This claim will be true for the people in the second half of the list (the shaded area). Is it true for me? I can’t say, other than that, according to Gott’s assumptions, there’s a 50 percent chance it is.

Demographers have estimated the total number of people who ever lived at about 100 billion. That means that about 100 billion people were born before me. Currently, about 130 million people are born each year. At that rate, it would take only about 760 years for another 100 billion more people to be born. That’s the basis of the claim that there’s a 50 percent chance that humans will become extinct within about 760 years. The flip side of the claim is there’s also a 50 percent chance we’ll survive past 760 years, possibly long past that.

As Holger Bech Nielsen pointed out, the latter part of this estimation isn’t airtight. A sharp decrease in the birthrate could postpone doomsday. Yet it’s hard to put an upbeat spin on that. It might mean a global catastrophe leaving a handful of post-apocalyptic survivors.

There has been speculation about how future technology might change the human condition. Genetically or digitally enhanced humans could live for centuries and have few children. The doomsday calculation’s notion of “human lives” may need reworking to allow for that.

Yet even this does not seem to offer an easy out. What the doomsday argument says, fundamentally, is that the human future is not so long and populous as we generally think. Will we resolve our differences, save the planet, and go on to explore the galaxy? Gott’s Magic 8 Ball says, “VERY DOUBTFUL.”

Why critics object to the doomsday argument

Criticisms of the doomsday argument are legion. Steven Goodman felt that Gott was misusing the “principle of indifference.” This says that, when you know nothing about which possible outcome will arise, you should assign them equal probabilities.

An illustration: Alice has no idea whether a coin toss will land heads or tails. She assigns both the same 50 percent chance — using the principle of indifference. Ironically, it’s Alice’s ignorance about the toss that justifies this.

Ben happens to know a trick coin is being used, and he says the chance of heads is 100 percent. Both Alice and Ben are being reasonable; they just know different things.

Gott’s version of doomsday derives from one big, bold assumption: that we can know nothing about where we stand in the ultimate timeline of human existence. Goodman objected that one could, in principle, have any sort of knowledge about our position in the human timeline. If so, Gott’s math need not apply.

Another objection centers on the “self-indication assumption.” As proposed by physicist Dennis Dieks and others, this says that we should favor hypotheses that have more intelligent beings over those that have fewer. Let’s say I’m trying to decide between hypothesis (1), which says 200 billion humans will be born before doomsday, and (2), which says that 200 trillion humans are destined to exist. You can then make the metaphysical case: “I’m a unique human being. The chance of me existing is 1,000 times greater with (2) than (1). That gives me reason to think (2) is more likely to be right.”

Should you accept the self-indication assumption, it cancels out the doomsday argument. That is an appealing prospect, but philosopher Nick Bostrom has raised compelling objections to the assumption. For instance: It is currently theorized that our universe is part of a multiverse containing an infinite number of other universes, and presumably an infinite number of intelligent observers. Invoke the self-indication assumption, and there is an overwhelming statistical case for the multiverse theory! But no one accepts that, nor should they. Dieks himself accepted Bostrom’s argument that the assumption is flawed.

How I came to accept the doomsday math

Writing a book about other people’s conflicting beliefs is an invitation to reexamine your own. There is a simple way of evading the doomsday prediction, one I mentioned above. Should I (reasonably) believe I’m not at a random point in human existence, then the doomsday math doesn’t apply to me.

Technological optimists say we have a long, populous future ahead (and therefore we’re still early in that glorious destiny). It’s an attractive thought. I kept asking myself: Why should I believe we have a long future? The best I could come up with is that we’ve gotten out of all the scrapes we’ve been in in the past. Humanity has survived mammoths, malaria, and atom bombs. Nothing’s killed us off yet.

But that’s like the joke about the optimist who’s fallen out of a 100-story building. He’s passed 90 stories and says to himself, “So far, so good!” We would, of course, have to find that we’ve surmounted all extinction threats — right up to the moment of doomsday.

The doomsday argument pulls back the curtain on technocratic optimism. It forces us to contemplate the possibility that we, and the universe, are more random than we like to think. The fact that our species is capable of a long future does not mean this is probable. It may be something that has to be earned by being smarter, wiser, kinder, more careful — and luckier — than we’ve ever had to be before.

William Poundstone is the author of 16 books, including The Doomsday Calculation: How a Formula That Predicts the Future Is Transforming Everything We Know About Life and the Universe. He has written for the New York Times, Harper’s, and Harvard Business Review, among other publications. He lives in Los Angeles. You can read more about the doomsday debate on Quora.

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