by Carl Sagan
2S = 2 + 22 + 23 + 24 + … + 263 + 264
Subtracting the first equation from the second gives us
2S – S = S = 264 – l,
which is the exact answer.
How much is it roughly in ordinary base-10 notation? 210 is close to 1,000, or 103 (within 2.4 percent). So 220 = 2(10X2) = (210)2 = roughly (103)2 = 106, which is 10 multiplied by itself 6 times, or a million. Likewise, 260 = (210)6 = roughly (103)6 = 1018. So 264 = 24 × 260 = roughly 16 × 1018, or 16 followed by 18 zeros, which is 16 quintillion grains. A more accurate calculation gives the answer 18.6 quintillion grains.
—
Another common appearance of exponentials is the idea of half-life. A radioactive “parent” element—plutonium, say, or radium—decays into another, perhaps safer, “daughter” element, but not all at once. It decays statistically. There is a certain time by which half of it has decayed, and this is called its half-life. Half of what is left decays in another half-life, and half of the remainder in still another half-life, and so on. For example, if the half-life were one year, half would decay in a year, half of a half or all but a quarter would be gone in two years, all but an eighth in three years, all but about a thousandth in ten years, etc. Different elements have different half-lives. Half-life is an important idea when trying to decide what to do with radioactive waste from nuclear power plants or in contemplating radioactive fallout in nuclear war. It represents an exponential decay, in the same way that the Persian Chessboard represents an exponential increase.
Radioactive decay is a principal method for dating the past. If we can measure the amount of radioactive parent material and the amount of daughter decay product in a sample, we can determine how long the sample has been around. In this way we find that the so-called Shroud of Turin is not the burial shroud of Jesus, but a pious hoax from the fourteenth century (when it was denounced by Church authorities); that humans made campfires millions of years ago; that the most ancient fossils of life on Earth are at least 3.5 billion years old; and that the Earth itself is 4.6 billion years old. The Cosmos, of course, is billions of years older still. If you understand exponentials, the key to many of the secrets of the Universe is in your hand.
If you know a thing only qualitatively, you know it no more than vaguely. If you know it quantitatively—grasping some numerical measure that distinguishes it from an infinite number of other possibilities—you are beginning to know it deeply. You comprehend some of its beauty and you gain access to its power and the understanding it provides. Being afraid of quantification is tantamount to disenfranchising yourself, giving up on one of the most potent prospects for understanding and changing the world.
CHAPTER 3
MONDAY-NIGHT HUNTERS
The hunting instinct has [a] … remote origin in the evolution of the race. The hunting and the fighting instinct combine in many manifestations.… It is just because human bloodthirstiness is such a primitive part of us that it is so hard to eradicate, especially where a fight or a hunt is promised as part of the fun.
WILLIAM JAMES,
Psychology, XXIV (1890)
We can’t help ourselves. On Sunday afternoons and Monday nights in the fall of each year, we abandon everything to watch small moving images of 22 men—running into one another, falling down, picking themselves up, and kicking an elongated object made from the skin of an animal. Every now and then, both the players and the sedentary spectators are moved to rapture or despair by the progress of the play. All over America, people (almost exclusively men), transfixed before glass screens, cheer or mutter in unison. Put this way, it sounds stupid. But once you get the hang of it, it’s hard to resist, and I speak from experience.
Athletes run, jump, hit, slide, throw, kick, tackle—and there’s a thrill in seeing humans do it so well. They wrestle each other to the ground. They’re keen on grabbing or clubbing or kicking a fast-moving brown or white thing. In some games, they try to herd the thing toward what’s called a “goal”; in other games, the players run away and then return “home.” Teamwork is almost everything, and we admire how the parts fit together to make a jubilant whole.
But these are not the skills by which most of us earn our daily bread. Why should we feel compelled to watch people run or hit? Why is this need transcultural? (Ancient Egyptians, Persians, Greeks, Romans, Mayans, and Aztecs also played ball. Polo is Tibetan.)
There are sports stars who make 50 times the annual salary of the President; some who are themselves, after retirement, elected to high office. They are national heroes. Why, exactly? There is something here transcending the diversity of political, social, and economic systems. Something ancient is calling.
Most major sports are associated with a nation or a city, and they carry with them elements of patriotism and civic pride. Our team represents us—where we live, our people—against those other guys from some different place, populated by unfamiliar, maybe hostile people. (True, most of “our” players are not really from here. They’re mercenaries and with clear conscience regularly defect from opposing cities for suitable emolument: A Pittsburgh Pirate is reformed into a California Angel; a San Diego Padre is raised to a St. Louis Cardinal; a Golden State Warrior is crowned a Sacramento King. Occasionally, a whole team picks up and migrates to another city.)
Competitive sports are symbolic conflicts, thinly disguised. This is hardly a new insight. The Cherokees called their ancient form of lacrosse “the little brother of war.” Or here is Max Rafferty, former California Superintendent of Public Instruction, who, after denouncing critics of college football as “kooks, crumbums, commies, hairy loudmouthed beatniks,” goes on to state, “Football players … possess a clear, bright, fighting spirit which is America itself.” (That’s worth mulling over.) An often-quoted sentiment of the late professional football coach Vince Lombardi is that the only thing that counts is winning. Former Washington Redskins’ coach George Allen put it this way: “Losing is like death.”
Indeed, we talk of winning and losing a war as naturally as we do of winning and losing a game. In a televised U.S. Army recruitment ad, we see the aftermath of an armored warfare exercise in which one tank destroys another; in the tag line, the victorious tank commander says, “When we win, the whole team wins—not one person.” The connection between sports and combat is made quite clear. Sports fans (the word is short for “fanatics”) have been known to commit assault and battery, and sometimes murder, when taunted about a losing team; or when prevented from cheering on a winning team; or when they feel an injustice has been committed by the referees.
The British Prime Minister was obliged in 1985 to denounce the rowdy, drunken behavior of British soccer fans who attacked an Italian contingent for having the effrontery to root for their own team. Dozens were killed when the stands collapsed. In 1969, after three hard-fought soccer games, Salvadoran tanks crossed the Honduran border, and Salvadoran bombers attacked Honduran ports and military bases. In this “Soccer War,” the casualties numbered in the thousands.
Afghan tribesmen played polo with the severed heads of former adversaries. And 600 years ago, in what is now Mexico City, there was a ball court where gorgeously attired nobles watched uniformed teams compete. The captain of the losing team was beheaded, and the skulls of earlier losing captains were displayed on racks—an inducement possibly even more compelling than winning one for the Gipper.
Suppose you’re idly flipping the dial on your television set, and you come upon some competition in which you have no particular emotional investment—say, off-season volleyball between Myanmar and Thailand. How do you decide which team to root for? But wait a minute: Why root for either? Why not just enjoy the game? Most of us have trouble with this detached posture. We want to take part in the contest, to feel ourselves a member of a team. The feeling simply sweeps us away, and there we are rooting, “Go, Myanmar!” Initially, our loyalties may oscillate, first urging on one team and then the other. Sometimes we root for the underdog. Other times, shamefully, we ev
en switch our allegiance from loser to winner as the outcome becomes clear. (When there is a succession of losing seasons, fan loyalties tend to drift elsewhere.) What we are looking for is victory without effort. We want to be swept up into something like a small, safe, successful war.
In 1996, Mahmoud Abdul-Rauf, then a guard for the Denver Nuggets, was suspended by the National Basketball Association. Why? Because Abdul-Rauf refused to stand for the compulsory playing of the National Anthem. The American flag represented to him a “symbol of oppression” offensive to his Muslim beliefs. Most other players, while not sharing Abdul-Rauf’s beliefs, supported his right to express them. Harvey Araton, a distinguished sports writer for the New York Times, was puzzled. Playing the anthem at a sporting event “is, let’s face it, a tradition that is absolutely idiotic in today’s world,” he explains, “as opposed to when it began, before baseball games during World War II. Nobody goes to a sporting event to make an expression of patriotism.” On the contrary, I would argue that a kind of patriotism and nationalism is very much what sporting events are about.*
The earliest known organized athletic events date back 3,500 years to preclassical Greece. During the original Olympic Games, an armistice put all wars among Greek city-states on hold. The games were more important than the wars. The men performed nude: No women spectators were allowed. By the eighth century B.C., the Olympic Games consisted of running (lots of running), jumping, throwing things (including javelins), and wrestling (sometimes to the death). While none of these events was a team sport, they are clearly central to modern team sports.
They were also central to low-technology hunting. Hunting is traditionally considered a sport, as long as you don’t eat what you catch—a proviso much easier for the rich to comply with than the poor. From the earliest pharaohs, hunting has been associated with military aristocracies. Oscar Wilde’s aphorism about English fox hunting, “the unspeakable in full pursuit of the uneatable,” makes a similar dual point. The forerunners of football, soccer, hockey, and kindred sports were disdainfully called “rabble games,” recognized as substitutes for hunting—because young men who worked for a living were barred from the hunt.
The weapons of the earliest wars must have been hunting implements. Team sports are not just stylized echoes of ancient wars. They also satisfy an almost-forgotten craving for the hunt. Since our passions for sports run so deep and are so broadly distributed, they are likely to be hardwired into us—not in our brains but in our genes. The 10,000 years since the invention of agriculture is not nearly enough time for such predispositions to have evolved away and disappeared. If we want to understand them, we must go much further back.
The human species is hundreds of thousands of years old (the human family several millions of years old). We have led a sedentary existence—based on farming and domestication of animals—for only the last 3 percent of that period, during which is all our recorded history. In the first 97 percent of our tenure on Earth, almost everything that is characteristically human came into being. So a little arithmetic about our history suggests we can learn something about those times from the few surviving hunter-gatherer communities uncorrupted by civilization.
—
We wander. With our little ones and all our belongings on our backs, we wander—following the game, seeking the water holes. We set up camp for a time, then move on. In providing food for the group, the men mainly hunt, the women mainly gather. Meat and potatoes. A typical itinerant band, mainly an extended family of relatives and in-laws, numbers a few dozen; although annually many hundreds of us, with the same language and culture, gather—for religious ceremonies, to trade, to arrange marriages, to tell stories. There are many stories about the hunt.
I’m focusing here on the hunters, who are men. But the women have significant social, economic, and cultural power. They gather the essential staples—nuts, fruits, tubers, roots—as well as medicinal herbs, hunt small animals, and provide strategic intelligence on large animal movements. Men do some gathering as well, and considerable “housework” (even though there are no houses). But hunting—only for food, never for sport—is the lifelong occupation of every able-bodied male.
Preadolescent boys stalk birds and small mammals with bows and arrows. By adulthood they have become experts in weapons procurement; in stalking, killing, and butchering the prey; and in carrying the cuts of meat back to camp. The first successful kill of a large mammal marks a young man’s coming of age. In his initiation, ceremonial incisions are made on his chest or arms and an herb is rubbed into the cuts so that, when healed, a patterned tattoo results. It’s like campaign ribbons—one look at his chest, and you know something of his combat experience.
From a jumble of hoofprints, we can accurately tell how many animals passed; the species, sexes, and ages; whether any are lame; how long ago they passed; how far away they are. Some young animals can be caught by open-field tackles; others with slingshots or boomerangs, or just by throwing rocks accurately and hard. Animals that have not yet learned to fear men can be approached boldly and clubbed to death. At greater distances, for warier prey, we hurl spears or shoot poisoned arrows. Sometimes we’re lucky and, by a skillful rush, drive a herd of animals into an ambush or off a cliff.
Teamwork among the hunters is essential. If we are not to frighten the quarry, we must communicate by sign language. For the same reason, we need to have our emotions under control; both fear and exultation are dangerous. We are ambivalent about the prey. We respect the animals, recognize our kinship, identify with them. But if we reflect too closely on their intelligence or devotion to their young, if we feel pity for them, if we too deeply recognize them as relatives, our dedication to the hunt will slacken; we will bring home less food, and again our band may be endangered. We are obliged to put an emotional distance between us and them.
—
So contemplate this: For millions of years, our male ancestors are scampering about, throwing rocks at pigeons, running after baby antelopes and wrestling them to the ground, forming a single line of shouting, running hunters and trying to terrify a herd of startled warthogs upwind. Imagine that their lives depend on hunting skills and teamwork. Much of their culture is woven on the loom of the hunt. Good hunters are also good warriors. Then, after a long while—a few thousand centuries, say—a natural predisposition for both hunting and teamwork will inhabit many newborn boys. Why? Because incompetent or unenthusiastic hunters leave fewer offspring. I don’t think how to chip a spearpoint out of stone or how to feather an arrow is in our genes. That’s taught or figured out. But a zest for the chase—I bet that is hardwired. Natural selection helped mold our ancestors into superb hunters.
The clearest evidence of the success of the hunter-gatherer lifestyle is the simple fact that it extended to six continents and lasted millions of years (to say nothing of the hunting proclivities of nonhuman primates). Those big numbers speak profoundly. After 10,000 generations in which the killing of animals was our hedge against starvation, those inclinations must still be in us. We hunger to put them to use, even vicariously. Team sports provide one way.
Some part of our beings longs to join a small band of brothers on a daring and intrepid quest. We can even see this in role-playing and computer games popular with prepubescent and adolescent boys. The traditional manly virtues—taciturnity, resourcefulness, modesty, accuracy, consistency, deep knowledge of animals, teamwork, love of the outdoors—were all adaptive behavior in hunter-gatherer times. We still admire these traits, although we’ve almost forgotten why.
Besides sports, there are few outlets available. In our adolescent males, we can still recognize the young hunter, the aspirant warrior—leaping across apartment rooftops; riding, helmetless, on a motorcycle; making trouble for the winning team at a postgame celebration. In the absence of a steadying hand, those old instincts may go a little askew (although our murder rate is about the same as among the surviving hunter-gatherers). We try to ensure that any residual zest for killing d
oes not spill over onto humans. We don’t always succeed.
I think of how powerful those hunting instincts are, and I worry. I worry that Monday-night football is insufficient outlet for the modern hunter, decked out in his overalls or jeans or three-piece suit. I think of that ancient legacy about not expressing our feelings, about keeping an emotional distance from those we kill, and it takes some of the fun out of the game.
Hunter-gatherers generally posed no danger to themselves: because their economies tended to be healthy (many had more free time than we do); because, as nomads, they had few possessions, almost no theft, and little envy; because greed and arrogance were considered not only social evils but also pretty close to mental illnesses; because women had real political power and tended to be a stabilizing and mitigating influence before the boys started going for their poisoned arrows; and because, when serious crimes were committed—murder, say—the band collectively rendered judgment and punishment. Many hunter-gatherers organized egalitarian democracies. They had no chiefs. There was no political or corporate hierarchy to dream of climbing. There was no one to revolt against.
So, if we’re stranded a few hundred centuries from when we long to be—if (through no fault of our own) we find ourselves, in an age of environmental pollution, social hierarchy, economic inequality, nuclear weapons, and declining prospects, with Pleistocene emotions but without Pleistocene social safeguards—perhaps we can be excused for a little Monday-night football.
TEAMS AND TOTEMS
Teams associated with cities have names: the Seibu Lions, the Detroit Tigers, the Chicago Bears. Lions and tigers and bears … eagles and seahawks … flames and suns. Allowing for the difference in environment and culture, hunter-gatherer groups worldwide have similar names—sometimes called totems.
A typical list of totems, mainly from the era before European contact, was recorded by the anthropologist Richard Lee in his many years among the !Kung “Bushmen” of the Kalahari Desert in Botswana (see below at far right). The Short Feet, I think, are cousins to the Red Sox and White Sox, the Fighters to the Raiders, the Wildcats to the Bengals, the Cutters to the Clippers. Of course there are differences—due to technological differences and, perhaps, to varying endowments of candor, self-knowledge, and sense of humor. It’s hard to imagine an American sports team named the Diarrheas (“Gimme a ‘D’ …”). Or—my personal favorite, a group of men with no self-esteem problems—the Big Talkers. And one in which the players are called the Owners would probably cause some consternation in the front office.