Los Angeles Times

Sunday, February 1, 1998

Amazing Grace
THE UNIVERSE AND THE TEACUP: The Mathematics of Truth and Beauty. By K.C. Cole . Harcourt Brace: 210 pp., $21


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On an acknowledgments page, Times science writer K.C. Cole says she was surprised when her editor told her she had written a book about mathematics. She had intended to write a book that surveyed trends in contemporary science, and she has, delightfully, but math is the common thread that binds it all together. What do the largest galaxy and the smallest teacup have in common? Answer: mathematical structure. Cole's book is a loving paean to the awesome power and beauty of mathematics.
     Cole's early chapter on giant numbers stresses how hard it is to grasp their applications. Our galaxy contains some 200 billion stars, and there are more than 200 billion galaxies. "Number numbness" sets in when you contemplate the vastness of space and time. A geologist, after chalking on the blackboard a line that runs from zero to a trillion, asks where to put the point that indicates a billion. Amazingly, it is extremely close to zero! "Compared to a trillion," Cole comments, "a billion is peanuts."
     Few realize how rapidly numbers grow by a simple process of doubling. Put two grains of wheat on the first square of a chessboard, four on the next, then eight and so on. You can't get very far. After only 10 steps, a square requires 1,024 grains. The last square needs more grains than were ever produced on Earth. Big numbers, Cole tells us, "creep up on us unawares." Disaster looms when population grows exponentially on a planet with a surface as finite as a chessboard.
     Widespread failure to understand simple probabilities receives careful treatment. We worry more about alar in apples, Cole writes, than the greater threat of cigarette smoke. A woman may avoid fatty meats, yet not mind sleeping with strangers. Everywhere there is what Cole calls "skewed perception of risks."
     In a chapter on measurements, Cole introduces the central mystery of quantum mechanics. How can an electron have no definite properties, such as position or spin, but acquire precise attributes as soon as they are measured? She likens this to a spinning coin, which is neither heads nor tails until it falls flat. Difficulties in measuring intelligence are also considered.
     There are also severe limits on how large certain things can be. No teacup can be as big as Jupiter because gravity would pull it into a sphere. Gravity would fracture the thighs of a person 60 feet tall. A flea can jump high in relation to its size, but the jump's height, approximately one meter, is about the same as what an average person can achieve. Cole calls it an "interesting invariant" that most animals can't leap higher than that.
     A thousand other fascinating facts and shrewd observations crowd into Cole's lyrical praise of science, from the wonders of galaxies to the invisible creatures that live on our eyelashes. She is good at emphasizing how unpredictable properties continually emerge as the universe evolves from primeval simplicity to the enormous complexity of a human brain. What could be more unlike a gas than water? Yet when two gases, hydrogen and oxygen, combine, the more complex properties of water emerge. Stars and rocks result from even more complex combinations of molecules. At some time in the distant past, life sprang into existence from the complexity of a self-replicating molecule. As it evolved from simple one-cell forms, the magic of increasing complexity produced trees, butterflies, dinosaurs, frogs and elephants.
     You and I somehow emerged, after billions of years, from the quantum fields that predated the big bang. Biology is not applied physics, Cole reminds us, and psychology is not applied biology. Knowledge of sounds reveals nothing about a Mozart symphony. "You can learn everything there is to know about the atoms that make up a cat," Cole writes, "and that still will not tell you whether it will scratch the furniture or sleep on your head."
     Slow increments of change lead to "tipping points," where a qualitative change occurs suddenly. Water cools gradually, then presto: It turns into ice. Tipping points are all over the human scene: an unanticipated drop in crime, a plunge of the Dow, an angry man's murder of his wife. Chaos theory studies systems that start out orderly, then by altering in a completely deterministic way, suddenly cross a threshold to become unpredictably random.
     "The Signal in a Haystack" is Cole's title for a chapter on how science does its best to filter significant facts from vast amounts of irrelevant information that often contaminate research results. Astronomical data are still not free enough from cosmic noise to determine how fast the universe is expanding or exactly how old it is. Are there really black holes at the centers of galaxies? Does star wobbling indicate planets? Are wiggly forms on a meteorite fossils from Mars? "The sky is like a cocktail party," Cole writes, "with too many conversations going on at once." Scientists listen to the babble in the hope of hearing what the universe is trying to tell them. As Cole writes: "That's exactly how the search for truth is supposed to work. You see something, and then you try everything you can think of to make it go away; you turn it upside down and inside out, and push on it from every possible angle. If it's still there, maybe you've got something."
     "The Universe and the Teacup" contains informative chapters on the inevitable flaws in every voting system and on how things can be divided fairly. Fair division is a piece of cake when only two people want half of a piece of cake. One person cuts; the other chooses. The trick has endless applications to social and political conflicts, such as who gets what after a divorce, assuming each party wants to be "envy free." If more than three players are involved, the task of fair division becomes more difficult. Ingenious solutions by various experts are skillfully outlined.
     The conflict between altruism and egoism also gets Cole's attention. Does self-sacrifice have survival value for an evolving animal species? Can bitter conflicts between business firms or nations be settled amicably by applying game theory? The famous tit-for-tat strategy is explained. This involves competition between two "players," such as the effort of one nation to establish free trade with another nation or to obtain mutual disarmament. The tit-for-tat strategy is to make the first move in the desired direction. After that, do whatever your adversary does. Eventually it dawns on both players that in the long run, cooperation maximizes self-interest.
     Near her book's close, Cole tackles the question of whether scientific truth is always fallible in contrast to the certainty of math and logic. Kurt Godel's famous undecidability theorem (it states that every formal system of mathematics contains true statements that cannot be proved true within the system) is considered and so is the current fad of fuzzy logic. The book's final chapter is a tribute to Emmy Noether, an eminent German mathematician. She proved that behind every invariant in physics lurks a symmetry that can be defined by an algebraic structure called a group: an insight essential to relativity theory and particle physics. Immediately after the big bang, the universe was inconceivably hot and perfectly symmetrical. As it cooled, various symmetries were shattered to fashion the cool and broken universe we know and love.
     I have touched on only a tiny fraction of the topics covered in this dense and passionate book. You will put it down sharing Cole's awe and wonder at the vastness and intricacy of what G. K. Chesterton once said an atheist must view as the most exquisite mechanism ever constructed by nobody.
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Martin Gardner Is the Author of Many Books About Science, Mathematics, Philosophy and Literature. "The Night Is Large," a Collection of His Essays, Was Published Last Year by St. Martin's Press

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