The Doomsday Calculation

Diana and Charles

Diana Spencer met Charles, Prince of Wales, at a garden party in 1977. The couple fell in love and, after due diligence by their families, wed at St. Paul’s Cathedral in July 1981.

American artist Mark Tansey incorporated Diana in his 1986 painting Achilles and the Tortoise. She is shown planting a hemlock, a sapling version of the mature tree behind her. Diana was often photographed planting trees, among them an apple tree she planted in honor of Isaac Newton.

In 1993 Diana came to the attention of American astrophysicist J. Richard Gott III. Gott had devised a mathematical formula for predicting the future. He wanted to test it on a celebrity marriage, and he chose Charles and Diana’s because a magazine reported they were the most famous couple of the time. Gott’s formula predicted a 90 percent chance that the royal marriage would end in as little as 1.3 more years. At the time, a royal divorce was considered almost unthinkable.

In December 1995 Queen Elizabeth II, incensed by tabloid reports of the couple’s extramarital affairs, wrote a letter advising Charles and Diana to divorce. The split was formalized on August 28, 1996. The following year, on August 31, 1997, Diana had a champagne supper in Paris with her new romantic interest, film producer Dodi Fayed. After leaving the restaurant, Diana and Dodi were killed when their alcohol-impaired chauffeur challenged paparazzi to a street race.

Tansey’s picture contains at least four other portraits. To the right of Diana is mathematician Mitchell Feigenbaum holding a bottle of champagne, whose bubbles epitomize chaos theory. Feigenbaum, a pioneer of that theory, demonstrated that many phenomena are fundamentally unpredictable. In 1996 he founded Numerix, a firm using Bayesian probability to price financial derivatives for the so-called rocket scientists of Wall Street.

To the right of Mitchell, though easily missed, is the familiar face of Albert Einstein, shown in profile. The speeding rocket and slow-growing hemlock allude to Einstein’s thought experiments of racing trains and light beams, used to develop his theory of relativity. Standing in front of Einstein is Benoit Mandelbrot, the IBM mathematician who described the concept of fractals. The hemlock tree and rocket blast are fractals, complex shapes in which each part resembles the whole.

Zeno of Elea, a Greek philosopher whose features are known from ancient busts, dangles a cigarette. Zeno propounded the paradox of Achilles and the Tortoise. Swift Achilles challenges the Tortoise to a footrace. The Tortoise demands a head start. Whenever Achilles catches up to where the Tortoise was, he still has a little farther to go. Thus, Zeno argued, Achilles can never overtake the Tortoise. For Zeno’s followers, the paradox was proof that something is deeply wrong about our understanding of space, time, and reality.

This book tells the story of another mind-boggling idea, the doomsday argument. As advanced by Gott and other scholars, it is a mathematical scheme to predict how long the human race will survive. The idea seems incredible to almost everyone at first encounter, but as we will see, it is not easily dismissed. In the following chapters I will present the cases for and against this provocative idea and attempt to evaluate them. I will show how the type of reasoning used in the doomsday argument has many potential applications. The argument has caused bright people to reflect on our fragile existence, our hopes, and our obligation to future generations—and to reexamine the nature of evidence and the place of humans in the universe.

How to Predict Everything

Six-year-old Helen Gregg, her nine-year-old sister, Frances, and their nine-year-old cousin, Ella Davies, never saw the atomic bomb that hit their playhouse. They were about six hundred feet away, in the South Carolina woods, on that bright spring day of March 11, 1958. The bomb was egg-shaped with stabilizing fins, a near-twin of the “Fat Man” bomb that struck Nagasaki. It annihilated the playhouse that Helen and Frances’s father had built for the girls, leaving a crater seventy-five feet across and thirty feet deep.

All the tons of earth thrown up in the air came back down in a hellish rain. It was that that injured the three girls, parents Walter and Effie Gregg, and their son Walter Jr. There were no deaths aside from a few chickens. The Greggs lived in a town called Mars Bluff. Today, sixty summers later, the crater is still visible.

Albert Madansky was a young statistics PhD from the University of Chicago, recruited by the RAND Corporation, a Santa Monica think tank contracting to the Pentagon. RAND wanted Madansky to tackle a problem that was easy to state but difficult to answer: What is the probability of a nuclear weapon detonating by accident?

The Mars Bluff incident, occurring the year after Madansky began work at RAND, was a prime topic of discussion. Madansky learned what the public had not. A B-47 Stratojet had left Hunter Air Force Base, Georgia, as part of a drill in handling atomic weapons. Early in the flight a red warning light came on in the cockpit, indicating that the bomb wasn’t properly secured.

Copilot Bruce Kulka banged the warning light with the butt of his service revolver. The light went off. Later it came back on. Kulka went to the bomb bay to fix the problem. He reached around the bomb to engage a lock, hitting the wrong button. The weapon came loose, crashing through the bomb bay doors and plummeting fifteen thousand feet.

A fission bomb contains chemical explosives, TNT in this case, surrounding a core of uranium or plutonium. Unspeakable tragedy was avoided only because the bomb was unarmed, without any fissile material. The ground impact detonated the TNT, however, creating a massive conventional explosion.

Accidents like Mars Bluff had been happening for some time. Madansky was allowed to see a top secret list of sixteen “dramatic incidents” that had occurred between 1950 and 1958.

RAND’s people worried about other scenarios. What if a bomb was lost and a civilian found it? What if an angry or unstable officer launched an atomic bomb without authorization? There were no statistics on such events because they had never happened.

In conventional statistical thinking, you can’t assign a probability to something that has never happened. Whereof one has no data, one must remain silent.… But Madansky had studied statistics at Chicago with Leonard “Jimmie” Savage. Savage had been born with the name Ogashevitz, though it was generally agreed that Savage fit him better. He was brutally critical of anyone he judged less brilliant than himself, a group that seemed to cover just about everyone in the fields of mathematics and economics. Savage was a contrarian by nature. One of his most contrary pet ideas was Bayes’s theorem—an obscure formula, named for an obscure minister of eighteenth-century England. Madansky was able to see that Bayes’s theorem offered exactly what RAND needed: a way to assign a probability to doomsday.

RAND’s 1958 report (authored by Madansky and colleagues Fred Charles Iklé and Gerald J. Aronson, and declassified in 2000) noted that the US atomic arsenal was growing rapidly, multiplying the opportunities for an accident. At the height of the Cold War, the Strategic Air Command intended to keep about 270 B-52 bombers in the air at all times, ready to launch a nuclear attack on word from the president.

“A probability that is very small for a single operation, say one in a million, can become significant if this operation will occur 10,000 times in the next five years,” the RAND report warned. With more bombs being transported more miles, the authors computed that a major catastrophe was near-inevitable in just a few years.

The report sketched countermeasures, ranging from the mundane to the bizarre. It proposed electrifying the bomb’s arming switches, so that anyone touching them would get a mild shock, lessening the chance of accidentally hitting the wrong button. As to the Dr. Strangelove scenario of a deranged individual starting World War III, the report argued for psychological screening of all who worked with the bombs. The most practical ideas were to put combination locks on bombs and to arrange that two individuals must act simultaneously to arm a bomb.

The RAND group was reporting to General Curtis LeMay, a no-nonsense war hero who fretted about American leadership being too politically correct to use its nuclear weapons. To Madansky’s relief, LeMay immediately grasped the seriousness of the problem. The general ordered the combination locks and the two-person system.

In folk wisdom, lightning never strikes the same place twice. Yet on January 24, 1961, the Carolina low country had another nuclear close call. One of LeMay’s B-52s developed a fuel leak and began to break up in midair near Goldsboro, North Carolina. As the tail sheared off, two bombs slid out of the bomb bay and plunged to earth. Three crew members died, and five parachuted to safety.

There wouldn’t have been any safety had the bombs gone off. This B-52 was carrying hydrogen bombs. Had either of them detonated, the fallout plume would have reached Philadelphia.

One of the bombs was discovered suspended from a tree by its parachute. It had barely kissed the earth. The “arm/safe” switch was still on “safe.”

The other bomb’s parachute failed to deploy. This bomb broke apart, and the fragments fell into a swampy area with enough water to soften the impact and spare the conventional explosives.

Bomb disposal expert Lieutenant Jack ReVelle was called in to find the pieces. “Until my death,” ReVelle said, “I will never forget hearing my sergeant say, ‘Lieutenant, we found the arm/safe switch.’ And I said, ‘Great.’ He said, ‘Not great. It’s on “arm.”’”

“You’re the Product”

Thomas Bayes, the nonconformist minister of Tunbridge Wells, England, drew his last breath on April 17, 1761. For reasons not clear he left his life’s greatest achievement filed away, unpublished and unread. It was another mathematically inclined minister, Richard Price, who found Bayes’s manuscript after his death and recognized its importance. Price counted among his acquaintances a notorious group: the American revolutionaries Thomas Paine, Thomas Jefferson, and Benjamin Franklin, as well as Mary Wollstonecraft, the feminist who married an anarchist and gave birth to the author of Frankenstein.

Price sent the Royal Society of London “an essay which I have found among the papers of our deceased friend Mr. Bayes, and which, in my opinion, has great merit.”

This essay described what we now call Bayes’s theorem (or rule or law). It addresses a fundamental question of the Enlightenment worldview: How do we adjust our beliefs to account for new evidence?

To put it in modern terms, you start with a prior probability (“prior,” for short). This is an estimate of the likelihood of something happening, based on everything already known. This estimate is then adjusted up or down for new data, according to a simple formula.

Price praised Bayes’s ingenuity but offered this warning: “Some of the calculations… no one can make without a good deal of labour.”

Partly for that reason Bayes’s theorem was neglected. Repeated calculations were tedious to do by hand—but that changed in the twentieth century with the invention of the computer. Bayes’s theorem was adopted by insurance companies, the military, and the technology industry. It is no exaggeration to say that the Reverend Bayes’s long-forgotten rule is behind much of Silicon Valley’s wealth.

“If you’re not paying for it, you’re the product being sold.” This is a maxim of our digital economy. Google, Facebook, Instagram, Twitter, YouTube—all our entrancing and addictive apps—are free products that come with a Faustian bargain. To use these services we allow their providers to collect so-called personal information—information that is valuable because of Bayes’s theorem. In the aggregate, as “big data,” personal information allows marketers to predict what you will buy, how much you will pay, and whom you will vote for. These Bayesian predictions, updated with every click, swipe, post, or GPS coordinate, are the secret sauce of many a tech company.

This success story is, however, only the prologue to the stranger one that concerns us. In recent years it has been recognized that Bayesian methods can shed light on deep mysteries of existence, including the future of the human race itself.

Ozymandias

This is the sonnet “Ozymandias” (1818) by Romantic poet Percy Bysshe Shelley, husband of Frankenstein author Mary Shelley, daughter of feminist Mary Wollstonecraft, friend of minister Richard Price, promoter of the intellectual property of Thomas Bayes. The theme of “Ozymandias” is that glory is fleeting. Nothing lasts.

In the summer of 1969, J. Richard Gott III celebrated his Harvard graduation with a tour of Europe. He visited the supreme monument of Cold War anxiety, the Berlin Wall. Standing in the shadow of the landmark, he contemplated its history and future. Would this symbol of totalitarian power one day lie in ruins?

This was a matter discussed by diplomats, historians, op-ed writers, TV pundits, and spy novelists. Opinions varied. Gott, who was planning postgraduate work in astrophysics, brought a different perspective. He devised a simple trick for estimating how long the Berlin Wall would stand. He did the math in his head and announced his prediction to a friend, Chuck Allen. The wall would stand at least two and two-thirds more years but no more than twenty-four more years, he said.

Gott went back to America. In 1987 President Ronald Reagan demanded, “Mr. Gorbachev, tear down this wall!” From 1990 to 1992 the wall was demolished. That was twenty-one to twenty-three years after Gott’s prediction and within the range he announced.

Gott called his secret the “delta t argument.” “Delta t” means change in time. It’s also known as the Copernican method, after Nicolaus Copernicus, the great Polish astronomer of the Renaissance. Copernicus’s leap of imagination was that the Earth is not the center of the universe. It is only one of a number of planets circling the sun. This thinking led to a simpler model of the solar system, one that agreed better with observation.

To astronomers, Copernicus’s insight has been a gift that keeps on giving. Over the past five centuries it has been established again and again that humanity does not occupy a central or special place in the scheme of things. Our sun is an ordinary star in an ordinary galaxy. It is not at the center of the galaxy but well off to the margins. Our galaxy does not occupy a special place in the cluster of galaxies to which it belongs, and this cluster has no special place in the universe as we know it. Even the whole of the observable universe is now widely believed to be an insignificant speck in a yet-greater multiverse. The cosmic “you are here” dot says we’re smack in the middle of nowhere.

The Copernican principle is generally applied to an observer’s location in space, but the delta t argument applies it to an observer’s location in time. Gott began with the assumption that his visit to the Berlin Wall had not taken place at any special moment in the wall’s history. That premise allowed Gott to predict the wall’s future without any expertise on Cold War geopolitics. His 1969 prediction was that there was a 50 percent chance that the wall would stand at least another 2.67 years after his visit but no more than 24 years.

Gott published his method in the prestigious journal Nature in 1993, and it ignited a controversy that still burns white hot. Many insisted that Gott’s method could not possibly be valid. They cited erudite (and remarkably different) reasons. Some discerned in Gott’s article a symptom of a jaded intellectual culture. “In the age of Quantum Mechanics, we often embrace a fantastic conclusion simply because it is fantastic and shocking,” complained George F. Sowers Jr. “Our sensibilities have been numbed. But the world is not so topsy-turvy that we can reason à la doomsday.”

Still others reported that they had tried Gott’s method, and it worked. A group of British mathematicians used Gott’s idea to compute how much longer the Conservative Party would remain in power. In line with their prediction, the party was ousted three and a half years later.

How Long Will Love Last?

Gott is a colorful character, literally. When I met him he was wearing a turquoise jacket of almost fluorescent hue and a tan fedora. He is a natural storyteller, with a Kentucky twang that has survived decades in the Ivy League, and a droll sense of humor. In the years after the appearance of his Nature article he became a minor celebrity as a sort of scientific soothsayer. In 1997 Gott invited readers of New Scientist to use the arrival time of the magazine to estimate how long they would be with their present boyfriend, girlfriend, or spouse. The principle can apply just as well to readers of this book.

You are now reading these words at a random moment in the course of your romantic relationship. It can hardly be otherwise. This isn’t a book about how to tell if he or she is really into you. It’s not a book about how to find a good divorce attorney. This book might have come into your life at almost any time. That’s the unromantic Copernican assumption. There is nothing at all special about this moment.

Chances are, then, that you are not at the very beginning of the relationship, nor at the very end. You’re somewhere in the middle. If you accept this premise, the past duration of your relationship gives a very, very rough idea of its future duration.

You may recognize this as common sense. If you met someone five days ago, it wouldn’t be surprising for the affair to be over five days from now. It’s too early for a tattoo or a deposit on a beach house for next summer. You may find this kind of estimation amusing or depressing or both. But the real question is, how accurate should we expect such estimates to be?

Gott realized that you don’t need fancy math to calculate that. All it takes is a diagram you can sketch on a napkin.

Draw a horizontal bar representing your love affair’s duration in time. Think of it as the scroll bar of a movie. The relationship’s beginning is at the left, and its end is at the right. Since no one knows how long love will last, we can’t mark the bar in hours, days, or years. Instead we’ll mark it in percentage points. The relationship’s beginning is at 0 percent, and its end is at 100 percent (however long that is in real time). The present moment must fall somewhere between 0 and 100 percent, but we don’t know where.

Still with me?

I have shaded half the bar. It’s the middle half, running from 25 to 75 percent. The present moment can be represented by a map pin (“You are here”). We’ll assume it’s equally likely to fall anywhere along the bar’s length. That could be in the shaded part or the unshaded part. But because the shaded region is exactly 50 percent of the bar, we can say that the odds are 50:50 that the current moment falls within the shaded part.

I’ve put two sample pins on the diagram. They mark the ends of the shaded region. The left pin is at 25 percent. There is no reason to believe that this pin corresponds to where you are in your relationship’s timeline. But suppose for the sake of argument that it does. Then your love has lasted 25 percent of its total duration, and it still has another 75 percent to go. The future is three times longer than the past.

The pin on the right is at 75 percent. Should that be the correct position, the future (the 25 percent remaining) is only one-third as long as the past (75 percent).

Because these two pins bound the middle half of the bar, it’s even odds that the present moment falls inside this range. That means there’s a 50 percent chance that your relationship’s future will be somewhere between one-third and three times as long as its past. Gott used this calculation with his Berlin Wall prediction.

This prediction is one of many similar ones you might make. In his Nature article, Gott adopted the 95 percent confidence level that is widely used in science and statistics. To publish a result in a scientific journal, it is generally necessary to show a 95 percent or greater probability that the result is not due to sampling error. You don’t have to be a scientist to appreciate that 95 percent is pretty confident. Is this Mr. Right or is it just Mr. Right Now? You’re never 95 percent sure of that. Nor are you often 95 percent confident of tomorrow’s weather or the winner of the next election.

I’ve made another diagram with the middle 95 percent of the bar shaded. This time the shaded region runs from 2.5 to 97.5 percent. Should you find yourself at the left pin, you have 2.5 percent of the duration behind you and 97.5 percent ahead. The future is 97.5/2.5 or 39 times as long as the past.

At the right pin, the future is only 1/39 as long as the past. Thus the range for 95 percent confidence, in this or any other Copernican estimate of future duration, is 1/39 to 39 times the past duration.

past time/39 < future time < past time*39

For example, let’s say you met someone a month ago. You can be 95 percent confident that this relationship will end in no less than 1/39 month and no more than 39 months. That spans about eighteen hours to a little over three years. You can be reasonably sure you won’t miss a break-up text when you switch off your ringer for a movie. You should also expect that you won’t be involved with this person five years from now—so say Gott’s statistics of love.

Lindy’s Law

Over the years, Gott and others have claimed diverse applications of the Copernican method. Take Wall Street’s famous weasel words: past performance is no guarantee of future results. Nonetheless an incredible amount of effort goes into divining future stock performance from (what else?) the past.

Statistics on corporate survival—and on tenure on ranked lists or indexes like the Fortune 500 or the S&P 500—show a Copernican effect. How long a company has existed (or been on the ranked list) is a rough predictor of how long it will survive (remain on the list).

The Copernican principle has some relation to the survivor bias that plagues stock investors. At any given time, an index fund or portfolio tends to be weighted with stocks that have done well in the immediate past, but that are unlikely to perform comparably well in the long run. Investors are always grabbing gold that crumbles to ashes in their hands.

A Broadway show is a special type of business. Like corporations, plays run for as long as their investors can hope to make a profit. But compared to corporations, Broadway shows are mayflies, with lifespans measured in weeks. Gott realized that that offered him a chance to make a testable prediction. On the day his 1993 Nature article was published, he identified forty-four plays and musicals that were then running in New York, including hits like Cats as well as productions that were quickly forgotten. Four years later thirty-six of the forty-four plays had closed, all within Gott’s prescribed 95 percent confidence intervals.

It was recently reported that 79 percent of Broadway musicals are flops, closing before they recoup their costs. Tax write-offs notwithstanding, it appears that many backers of plays overestimate runs. Gott’s prediction method does not factor in playwright, stars, casts, or reviews; nor does it consider advance ticket sales, celebrity buzz, advertising campaigns, or what people are willing to pay or do to score a ticket. He nonetheless found that how long a show had already run was a better predictor of its future run than much informed opinion is. The New Yorker’s editors were impressed enough with Gott and his methods that they commissioned Timothy Ferris to write a profile of Gott. The 1999 article ran with the title “How to Predict Everything.”

A grad student gazes, Zen-like, at a wall and gains enlightenment. Can it really be that easy to predict “everything”?

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