The Universe And The Teacup Page 2

  Yet the fearless few who plunged into the statistics headlong found that the numbers which seemed to speak so clearly swept crucial qualifications under the rug, making much of the mathematics meaningless.†

  The question I get asked most frequently is: How can you ever find out what’s true short of becoming a mathematician yourself? The answer is: You don’t have to. You merely need the confidence to ask the questions that were probably on your mind anyway. Such as: How do you know? Based on what evidence? Compared to what else? Like the woman who spent a day exploring exhibits at the Exploratorium in San Francisco—then went home and wired a lamp. There was nothing in the world-renowned science museum that taught her how to wire a lamp. What she found there was simply the belief in her own abilities to figure things out.

  Used correctly, math can expose the glitches in our perceptual apparatus that lead to common illusions—such as our inability to perceive the true difference between millions and billions—and give us relatively simple ways of protecting ourselves from our own ignorance. As the physicist Richard Feynman once said: “Science is a long history of learning how not to fool ourselves.” A knowledge of the mathematics behind our ideas can help us to fool ourselves a little less often, with less drastic consequences.

  In short, math matters—a lot more than most people think. We have to make life-and-death decisions based on what numbers tell us. We cannot afford to remain dumb about mathematical ideas simply because we hated them in high school—any more than we can remain dumb about computers, or AIDS. Mathematics is essential, not peripheral, knowledge.

  As someone who started out interested in social questions, I am particularly impressed at the power of math to help sift through evidence and decide what is true in a wide variety of situations. Some of the tools may be obvious (like probability) while others are more subtle and even obscure (like the relationship between symmetry, truth, and things that never change, no matter what).

  Many different kinds of truths lie in numbers, and exploring them is the purpose of this book. What does it mean when one number can be correlated with another? Say: IQ and intelligence, or math scores with big feet? If one thing makes another thing more probable, is it fair to call it a cause? What is the most effective strategy for winning at games? Is endless economic growth really a good thing (or even possible)? Was there life on ancient Mars? What’s the fairest way to divide the national budget, or the best way to survive a game of “chicken”? What is the probability of getting killed by a terrorist? Getting married after forty? Running into your brother-in-law in Manhattan? In Nome? What, if anything, do these numbers we attach to things mean?

  No doubt about it, mathematics embodies great power. It’s no wonder that the physicist Sir James Jeans concluded: “The Great Architect of the Universe now begins to appear as a pure mathematician.”

  At the same time, it is far from foolproof. Like all science, it grows and thrives in cultures and is heavily influenced by their peculiarities. This book focuses on various mathematical guides to the truth that can be applied to a wide range of questions, from issues in the news to matters of purely philosophical or aesthetic interest.

  What I personally like best is the way that truth and beauty come together in the work of Emmy Noether and Albert Einstein: How deep truths can be defined as invariants—things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that.

  The journey begins here.

  PART I

  Where Mind Meets Math

  Mathematics did not appear out of nothing and nowhere—as some cosmologists believe the universe did. It was created (or discovered, if you prefer) by human beings. As such, math reflects many aspects of humanity, including physical characteristics, psychology, and culture. The ways our brains and bodies work have molded not only the study of mathematics, but also our everyday perceptions of quantitative things. It is, after all, human nature—in its broadest psychological and physiological sense—that creates the sophisticated mathematics that revealed curved space-time and quarks and that led to the creation of everything from computers to gene therapy. This same human nature, however, limits our ability to understand phenomena that may be critical to our survival as a species—including risks, population growth, and national budgets.

  In what has become a recurring theme in human evolution, the same strategies that serve as stepping-stones to some truths become the obstacles we trip over in pursuit of others in different contexts.

  To take an obvious example of how mathematics is shaped by human physiology, most number systems used in societies around the world are built on multiples of ten because virtually all human beings are born with ten fingers and ten toes. (Some cultures expand upon this idea by making use of wrists, elbows, shoulders, and chest.)

  Less obvious, our brains appear to be calibrated rather like the scales used to measure the strength of earthquakes, where a small increase on the scale (say, from a 7 to an 8 magnitude) designates an enormous increase in destructive power (on the order of ten times stronger). This may well account for the inability of people—no matter how well schooled in mathematics—to comprehend the true difference between a million and a billion.

  In addition, the world beyond our physical bodies is sculpted by forces that produce omnipresent mathematical objects. Geometry really does grow on trees. Because of the way gravitational, electrical, and nuclear forces operate, everything Moon size or larger is round or roundish. Water always falls from a fountain in parabolas. Soap bubbles meet at 120-degree angles, and the two hydrogens and oxygen in water molecules meet at 105 degrees—giving shape to bubbles and snowflakes. Trees and blood vessels and rivers all branch in strikingly similar ways.

  By the same token, our measures and notions of time are based on the revolutions of our planet and the time it takes to wind its way around the Sun. We orient ourselves in space along Earth’s spin axis (north-south) and magnetic poles. We think of down as a fixed direction, although down for me is up for someone living on the other side of the globe. In truth, down is the direction of the greatest pull of gravity. If you are alone in space, there is no such thing as down, up, east, or west.

  Many mathematical operations—like adding and subtracting— directly derive from our physical experience: adding two apples or cutting a pie into thirds or figuring out the circumference of a circle from its diameter.

  But mathematical concepts also go beyond experience. The perfect circles and right angles favored by geometers do not exist in the natural world. Numbers can do things that things cannot. There is an old tale that illustrates how some things just don’t add up: A man stands on a street corner begging for change by holding up a sign that reads: 2 wars; 1 leg; 2 wives; 3 children; 2 wounds. Total: 10. And things can do things numbers cannot. If you add hydrogen to oxygen in a 2 to 1 proportion under the right conditions, you do not get three units of gas; you get water.

  Still, almost every attempt to take numbers beyond experience has met with extreme cultural backlash. When negative numbers were first introduced, people thought they were absurd. Since it made no sense to have minus two apples, what could minus two possibly mean? The introduction of zero was as fiercely contested as the Copernican Sun-centered view of the solar system. And if the stories about the Pythagoreans are true, people actually lost their lives as a result of the discovery of irrational numbers—like pi—that cannot be expressed as fractions. Indeed, our use of the word irrational—meaning completely nonsensical and off-the-wall—reflects quite accurately how people felt about these numbers when they were first discovered.

  Mathematicians today rely on all kinds of strange objects that were once thought completely out of the bounds of common sense: various kinds of infinities; imaginary and transcendental numbers; higher dimensional geometries; and so forth.

  But however far away mathematics gets from human experience, our physical world—including our own physical makeup—continues to play a pivotal role in how we perceive mathematical ideas.

  Chapter 2

  EXPONENTIAL AMPLIFICATION

  The greatest shortcoming of the human race is

  our inability to understand the exponential function.

  —physicist Albert A Bartlett

  Consider the extreme difficulty we have with very large or very small numbers. Anyone who has ever mixed up a billion and a trillion knows that after a while, all big numbers begin to look alike. Daily, we are bombarded with incomprehensible sums:

  The national debt has grown to trillions of dollars. The Milky Way galaxy contains 200 billion stars, and there are 200 billion other galaxies in the universe. The chemical reactions that power everything from fire to human thought take place in femtoseconds (quadrillionths of a second). Life has evolved over a period of roughly 4 billion years.

  What are we to make of such numbers? The unsettling answer is, not much. Our brains, it appears, may not be engineered to cope with extremely large or small numbers. Douglas Hofstadter coined the term “number numbness” to describe this syndrome, and almost everyone suffers from it After all, it’s so easy to confuse a million and a billion; there’s only one lousy letter difference. Except that a million is an almost imperceptible one-thousandth of a billion—a teeny tiny slice.

  No one, apparently, is immune. As Donald Goldsmith pointed out in the Wall Street Journal, President Bill Clinton managed to lose track of 90,000 physician visits in a speech on health care. He multiplied 500 children by 200 doctors and came up with 10,000 visits, 90,000 short
.

  All of us have trouble grasping how inflation at 5 percent can cut income in half in a decade or so, or how a population that’s growing at even 2 percent can rapidly overtake every inch of space on Earth. From the incredible shrinking dollar to the explosive power of nuclear bombs, things add up in ways that humans find hard to get a handle on. And yet, the consequences of this built-in number blindness are enormous.

  If we can’t readily grasp the real difference between a thousand, a million, a billion, a trillion, how can we rationally discuss budget priorities? We can’t understand how tiny changes in survival rates can lead to extinction of species, how AIDS spread so quickly, or how small changes in interest rates can make prices soar. We can’t understand the smallness of subatomic particles or the vastness of interstellar space. We haven’t a clue how to judge increases in population, firepower of weapons, energy consumption.

  Fortunately, scientists and mathematicians have come up with all manner of metaphors and tricks designed to give us a glimpse at those huge and tiny universes whose magnitudes seem quite beyond our comprehension. University of California, Berkeley, geologist Raymond Jeanloz, for example, likes to impress his students with the power of large numbers by drawing a line designating zero on one end of the blackboard and another marking a trillion on the for side. Then he asks a volunteer to draw a line where a billion would fall. Most people put it about a third of the way between zero and a trillion, he says. Actually, it falls very near the chalk line that marks the zero.

  Compared to a trillion, a billion is peanuts. The same goes for the difference between a millionth and a billionth. If the width of this page represents a millionth of something, then a billionth of it would be much less than a pencil line.

  S. George Djogvski, writing in Caltech’s Engineering and Science, offers this analogy to help us imagine the vast distances of space. If the Sun were an inch across and five feet away from our vantage point on Earth, “the solar system would be about a fifth of a mile across. The nearest star would be 260 miles away, almost all the way to San Francisco [from Los Angeles], and our galaxy would be 6 million miles across. The next nearest galaxy would be 40 million miles away. At this point, you begin to lose scale, even with this model—the nearest cluster would be 4 billion miles away, and the size of the observable universe would be a trillion miles. If you were to ride across it at five dollars per mile, you could pay off the national debt.”

  Not that we can comprehend national debt any better than these numbers. The late physicist Sir James Jeans—a great popularizer of Einsteins theories—wrote about how seemingly impossible it was for people to imagine a range of sizes that goes “from electrons of a fraction of a millionth of a millionth of an inch in diameter, to nebulae whose diameters are measured in hundreds of thousands of millions of miles.” He tries to help out with the following: “If the Sun were a speck of dust 1/300 of an inch in diameter, it [that is, the specksized Sun] would have to extend 4 million miles in every direction to encompass even a few neighboring galaxies.”

  And also: “Empty Waterloo Station of everything except six specks of dust, and it is still far more crowded with dust than space is with stars.”

  And also: “The number of molecules in a pint of water placed end to end ... would form a chain capable of encircling the Earth over 200 million times.”

  And, finally, he offers a way to imagine the stupendous heat involved in nuclear fusion. A pinhead heated to the temperature of the center of the Sun, writes Jeans, “would emit enough heat to kill anyone who ventured within a thousand miles of it.”

  These images carry emotional lessons that numbers alone cannot. They give us a sense—as well as knowledge—of what truly large numbers are about.

  One of the main reasons that large numbers grow so explosively is that multiplication is a powerful engine for growth—even when the only number you happen to be multiplying is insignificantly puny, like the number two.

  There’s an old legend about the mathematician who invented chess that illustrates this very well. The king liked the game so well, the story goes, that he offered the mathematician any prize she wanted. She asked only for two grains of wheat to be placed on the first square of the chessboard, four on the second, eight on the third ... and so forth, doubling the number of grains of wheat for each of the sixty-four squares on the chessboard.

  How much grain did she win? More than human beings had produced in the entire history of the world. That’s the power of doubling.

  An even more vivid story comes from physicist Albert A. Bartlett, who has launched a one-person crusade in support of exponential literacy. The recipient of a Distinguished Service citation from the American Association of Physics teachers, Bartlett is currently professor emeritus at the University of Colorado, Boulder.

  Here’s the story he uses to demonstrate the precarious state of our natural resources, even in times of apparent wealth:

  Imagine an average colony of bacteria, living in Bacterialand. They go off to found a new colony—in a Coke bottle found buried in the earth. They excavate it and make themselves at home. Let’s say we start with two intrepid explorers who settle this new colony. Let’s say they double their population once each minute. Let’s say they start at eleven A.M., and by twelve noon their bottle is full, and they’re out of space and resources.

  What time would it be, Bartlett asks, when even the most farsighted bacteria saw an overpopulation problem on the horizon? Certainly not before 11:58, he answers, because at that point the bottle would only be one-quarter full. (Two doublings away from full.) Even at 11:59, it would be only half full, and you can just hear the bacteria politicians singing platitudes like: No need to worry, folks! WE HAVE MORE SPACE LEFT IN OUR HOMELAND THAN WE’VE USED IN ALL THE HISTORY OF THIS COLONY!

  Nevertheless, they decide to explore offshore for more Coke bottles. They find three! Wow! How long does this give the bacteria colony before it runs out of room again? Answer: Two minutes.

  In fact, everything that grows exponentially doubles sooner or later. Compound interest at 7 percent doubles your money in ten years. Population growth at 7 percent doubles the number of people in ten years. Let’s say we call the first two bacteria in the bottle Adam and Eve. The human population is expected to double in fifty to sixty years (by the most optimistic calculations). But it’s unlikely that people will get too perturbed about it anytime soon, because the bottle appears to be still far from full—especially if you live in Montana.

  Exponential growth is good news, of course, for people with money in the bank, or better, the stock market. One lousy dollar invested at a return of 5 percent a year compounded continually for five hundred years will flow in at the rate of $114 per second.

  It’s bad news for people with fixed incomes, like people on pensions or salaries that don’t increase. It explains why a typical market basket of goods that cost $100 in 1982 cost 50 percent more—$142—in 1992. The same basket back in 1946 cost $19.50.

  It’s important to note that these numbers add up so fast only if the gains are compounded. That is, every time the population of bacteria doubled, the number that was multiplied by two was the sum total of all the previous multiplications. So what’s doubling is rapidly growing, and the growth gets bigger the bigger it gets.

  Compare, for example, the interest you’d earn if you put $1,000 in the bank at 10 percent interest for one hundred years. If you simply added $100 in interest each year, after a hundred years you’d have $11,000. But if you take 10 percent of the total principal plus interest each year, you wind up with more than $22 million.

  These numbers creep up on us unawares in part because of the way our brains are engineered. Our minds don’t perceive the explosive magnitudes of exponential amplification because our brains appear to be calibrated like the Richter scale,* which measures the power of earthquakes.