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CHAPTER 1

What's Math Got to Do with It?

Understanding is a lot like sex. It's got a practical purpose, but that's not why people do it normally.

— Frank Oppenheimer

Finding out what's true is a central passion of human activity. It's a question that dominates the stage and the dinner table, the classroom and the courtroom, the scientific laboratory and the spiritual retreat. And yet, with the explosion of information reverberating in our brains, it becomes harder and harder to hear the clear ring of truth through the competing facts and philosophies.

As it turns out, mathematics offers a singular set of tools for seeing truth. Indeed, it brings surprising clarity to an astonishing range of issues, from cosmic questions (the fete of the universe) to social controversy (O. J.'s guilt) to specific matters of public policy (race and IQ scores).

People outside the sciences rarely pick up these tools — in part because math seems intimidating. Even if people are aware that such tools exist, they don't know how to apply the tools to things they care about.

But mathematics already underlies many of society's most-cherished political and social inventions: Ideas about cause and effect, fairness and justice, selfishness and cooperation, balancing risks, spending on welfare or national defense, even the nature of scientific discovery itself.

True, our ideas about the physical and social world do spring from sources other than numbers: religion, history, family, psychology. We accept the "truths" revealed by these sources as intuitively commonsensical, or obviously right; our Declaration of Independence describes them as "self-evident."

But math — that most logical of sciences — shows us that the truth can be highly counterintuitive and that sense is hardly common.

Mathematics is a way of thinking that can help make muddy relationships clear. It is a language that allows us to translate the complexity of the world into manageable patterns. In a sense, it works like turning off the houselights in a theater the better to see a movie. Certainly, something is lost when the lights go down; you can no longer see the faces of those around you or the inlaid patterns on the ceiling. But you gain a far better view of the subject at hand.

William Thurston, the director of the Mathematical Sciences Research Institute (and by some accounts the world's greatest living geometer) calls math a kind of "mindware." It allows us to see and articulate concepts we can't handle in any other way. Ingrid Daubechies — the MacArthur Award-winning Princeton mathematician who resurrected wavelet analysis (a tool for doing everything from storing fingerprints to seeing stars) — says it's akin to poetry: a way of taking a big idea and condensing and honing it until it communicates exactly the right information.

Mathematics can function as a telescope, a microscope, a sieve for sorting out the signal from the noise, a template for pattern perception, a way of seeking and validating truth. It is a lens that can clarify the obscure, or obscure and distort what was seemingly clear. It can take you into the core of a star or to the edge of the universe, give you the outcome of an election or the result of pumping carbon dioxide into the atmosphere for a hundred years. You can extrapolate to the end of time, or back to its beginning. You can get there from here.

Mathematicians do not see their art as a way of simply calculating or ordering reality. They understand that math articulates, manipulates, and discovers reality. In that sense, it's both a language and a literature; a box of tools and the edifices constructed from them.

Once I was flying in a plane back from the Boston area, where I had been talking with a cosmologist at MIT about the universe and all that. I looked down from my window and saw islands that were clearly connected under the shallow water by strips of land. On the ground, those links would have been invisible, the islands completely unconnected. From the air, the paths between them were laid out as clearly as road maps. There's a reason, I thought, that a lot of fundamental physics requires looking in higher dimensions. You can see more from an elevated point of view.

In the same way, the tools of mathematics allow one to see otherwise invisible patterns and connections. Mathematics has revealed hidden trends (HIV infection), new kinds of matter (quarks, dark matter, antimatter), and crucial correlations (between smoking and lung cancer). It does this by exposing the bare bones of a situation, overcoming the commonsense notions that so often lead us astray. Math allows you to strip off the coverings and get right down to the skeleton. What is going on underneath that accounts for what you see on the surface? What's holding it up? If you dig deep enough, what do you find?

In some sense, the unfolding story of the universe is a history of finding hidden connections. The nature of light was discovered when a certain number (the speed of light) kept popping out of equations linking electricity to magnetism. Light was exposed as an electromagnetic fluctuation — an understanding that allowed experimenters to go looking for others of its same species. Radio signals, for example, ride on light that vibrates more slowly than the eye can see; X rays vibrate faster.

Equations speak volumes, teasing out economic trends, patterns of disease, growth of populations, and the effects of prejudice and discrimination. Math produces a quite literal expansion of consciousness. It allows us to see more. With these tools, we can extrapolate into the future (but there are hazards) and see invisible things (curved space).

"What do we really observe?" asked Sir Arthur Eddington in 1959, summing up the lessons of the century's recent revolution in physics: "Relativity theory has returned the answer — we only observe relations. Quantum theory returns another answer — we only observe probabilities."

What we observe, in other words, are mathematical relationships.

Since mathematics is so good at exposing the truth, it's curious how often it's used to perpetuate misunderstandings and lies. Math has power because we give more weight to numbers than we do to words. "Figures often mislead people," says mathematician Keith Devlin. "There is no shame in that: words can mislead as well. The problem with numbers is our tendency to treat them with some degree of awe, as if they are somehow more reliable than words ... This belief is wholly misplaced."

People often look to mathematics as an objective line of argument that will rescue them from the uneasiness of ambiguity. If only we put things in terms of numbers, we hope, perhaps truth will out. But math only articulates these ambiguities; it is no lifeboat out of the sea of confusion — only the buoy that marks the shoals. After all, it was a mathematical theorem (Gödel's theorem) that proved some truths can't be reached by the road of pure logic at all.

A prime case of intimidation by the numbers is the book The Bell Curve, a treatise so controversial that a half dozen books were published in response. Written by Charles Murray of the American Enterprise Institute and the late Richard Herrnstein of Harvard, the book wheels out an arsenal of mathematical artillery to bolster the proposition that intelligence is mostly inherited, that blacks have less of it, and that little can be done about it. Reviewers — not to mention readers — admitted to shell shock in the face of such a barrage of statistics, graphs, and multiple-regression analyses.

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.

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 / 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.

(Continues…)

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Excerpted from "The Universe and the Teacup"
by .
Copyright © 1997 K. C. Cole.
Excerpted by permission of Houghton Mifflin Harcourt Publishing Company.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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