Mathematics, that breathtaking invention of ours that reveals the tiniest particles of matter and takes us to the outermost reaches of the cosmos, is found by many people to be intimidating. In The Universe and the Teacup, K. C. Cole demystifies mathematics and shows us-with humor and wonderfully accessible stories-why math need not be frightening. Using the O. J. Simpson trial, the bell curve, and Emmy Noether, the nineteenth-century woman scientist whose work was essential for Einstein's theory of relativity, Cole helps us see that more than just being a tool, math is a key to understanding the beauty of everything from rainbows to relativity.
|Publisher:||Houghton Mifflin Harcourt|
|Edition description:||First Edition|
|Product dimensions:||5.31(w) x 8.00(h) x 0.61(d)|
About the Author
A popular science columnist for the Los Angeles Times and teacher at UCLA, K.C. Cole is a recipient of the 1995 American Institute of Physics Award for Best Science Writing. She is also the author of the internationally bestselling The Universe and the Teacup, First You Build a Cloud, and The Hole in the Universe. Cole lives in Santa Monica, California.
Read an Excerpt
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.
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 fate 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 (Godel'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.
Table of ContentsACKNOWLEDGMENTS.....................................................IX
INTRODUCTION The Sentimental Fruits................................1
CHAPTER 1 What's Math Got to Do with It?............................5
PART I / WHERE MIND MEETS MATH
PART II / INTERPRETING THE PHYSICAL WORLD
PART III / INTERPRETING THE SOCIAL WORLD
PART IV / THE MATHEMATICS OF TRUTH
SELECTED BIBLIOGRAPHY..............................................203 INDEX..............................................................209
Exclusive Author Essay
Physicist Frank Oppenheimer used to say that artists and scientists were the official "noticers" of society. Their job was to notice things that other people either had never been taught to see or had learned to ignore -- then to go out and tell the world about what they'd found.
I realize now that this is precisely what I've become: an official noticer. I get paid to be the ultimate voyeur. I peer over scientists' shoulders as they build machines of almost unfathomable proportions that re-create -- albeit on a small scale -- the creation of the universe. I eavesdrop as they struggle to find the unifying principles of nature in ten-dimensional space. I hang out in laboratories and lecture halls where scientists try to decode the messages written on the walls of the universe or streaming from the deep throats of black holes.
I certainly didn't start out as someone who was interested in science. I wanted to understand the way the world works. And I thought the way to do that was to study the social sciences: psychology, sociology, anthropology, political science. Like most people with an interest in human affairs, I never gave much of a thought to math or physics -- fields that seemed to have little to do with the kinds of things that interested me.
One of the reasons, of course, was that most of the science I learned in school was crammed into rigid boxes labeled geometry, biology, physics -- as if they have nothing to do with each other, much less with human experience. It was science stripped of all the wonderful ambiguity that pervades the real practice of science: the wrong answers; the right answers to the wrong questions. Most of all, it didn't reflect the role -- the critical role -- that our understanding of the physical universe plays in shaping our emotional and philosophical one.
I first started making these connections in a series of "Hers" columns for The New York Times. These grew into my first science book: First You Build a Cloud: Reflections on Physics as a Way of Life. It explores what hard science has to say about quasi-philosophical questions such as the nature of right and wrong, cause and effect, aesthetics, disorder, and the use and abuse of metaphor.
The next book, similarly, grew mostly from articles I had written for The Los Angeles Times that linked mathematics with everything from the O. J. Simpson trial to fairness in divorce settlements. It is called The Universe and the Teacup: The Mathematics of Truth and Beauty, and it's a celebration of mathematics as a singular set of rules for seeing the truth.
The subsequent book is perhaps less obviously philosophical. Yet The Hole in the Universe: How Scientists Peered Over the Edge of Emptiness and Found Everything (you can see I have a fondness for long subtitles) also links physics, mathematics and perception in an exploration of the invisible forces that shape everything. We only call them "nothings" because we aren't aware of their existence. But they hold up the universe just the same. Like physics and philosophy, something and nothing are two sides of the same coin.
--K. C. Cole
Most Helpful Customer Reviews
Being a person that is not very good at mathmatics and always dreads the thought of not only solving math problems, but now reading about the abstract ablitilys of math and the universe i was not thrilled. At first glance i suspects this book would be boring and not the least bit interesting, but author K.C Cole proved me wrong. This book actually simplifies the abstract ideas of math which can be used in many examples in the universe. Cole seems to understand that most people see mathmatics as difficult and pointless in the beginning of the book then he beings to break the ideas into smaller concepts that make math seem more important than many would think. He goes on to explain that mathmatics deals with many polictical and social inventions and it even works hand in hand with science. Ideas such as history, religion, family and pyshcology all deal with math. Even justice systems and fairness are all solved with mainly mathmatics. This book proves that math is a way to better understand reality. It is worth the read.
"The Universe and the Teacup" deeply examines the complex field of mathematics and simplifies it by applying math to many entities in the world. In her book K.C. Cole writes about the importance of understanding mathematics not only as a way to add and subtract numbers but also as a way of thinking. "The Universe and the Teacup" is divided into four sections; "Where Mind Meets Math," "Interpreting the Physical World," "Interpreting the Social World," and "The Mathematics of Truth." In the first section Cole examines the great difficulties that people encounter with interpreting how large and how small a number really is in comparison to others. Cole also proves how we avoid certain "risks" that really are not very harmful to us at all (statistically, that is), and how the "small" risks we do take are really much more dangerous to us, regardless of contrary belief. In "Interpreting the Physical World" the author focuses on measurements and discusses how every measurement we use boils down to quantity. The third section of this book deals with the issue of fair voting, and here Cole explains to the reader that our voting system in the United States really is not very fair by alluding to the studies done by law professor Lani Guinier. Finally, the fourth section of this book focuses on chance and why things happen. "The Universe and the Teacup" is a great book which really makes the reader question our world. It provides the reader with a whole new perspective of mathematics and how it really does apply to everything we do.
In her non-fiction book titled "The Universe and the Teacup: The Mathematics of Truth and Beauty," K.C. Cole, a science writer for the Los Angeles Times, takes her readers on a beautiful journey through the ins and outs of science, astrology and math. Science and math become attainable to any reader as she relates these often intimidating topics with other more beloved topics such as psychology, biology, anthropology, sociology, law, theology and poetry. I highly recommend this book for math or science educators. Math becomes so much more relevant as Cole points out its place in every day aspects of life, in the beauty of nature, and how we use math to quantify and qualify our lives. She is inspiring and can easily be referenced to inspire students who may be struggling to see the relevance of math and science in their lives.
This is a book that anyone, who is willing to learn, should read. This opens up a person's mind to what math can do for people in everyday situations and a better understanding of math. It's very interesting how Cole connects such topics with math, and they match perfectly! It's also very well written; not too complicated, and not too easy.
As student who has never really been into Astronomy, I really liked this book. It made me look at the universe (and everything inside it) in a different way than I usually see it. The numbers that make up our universe surprised me. For example, the fact that many people cannot determine the difference between one million and one billion, or one million to one trillion. The fact that when placed on a number line, one to one trillion, one million and one trillion would not be any where near each other. Another thing that grabbed my attention was the idea that everything in the universe is symmetrical but the way that things work is because of broken symmetry; "stars, water droplets, neurons," are similar, "but add up to complex [things like] galaxies, clouds, minds." I was also intrigued by the fact that the only way for astronomers to find new stars or get closer to stars is by clearing away the "noise." Like you would clear the noise of a room to listen to one specific conversation, astronomers clear away particles from the sky to see more clearly. I really enjoyed reading this book and changing the way that I see our universe.