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The Law of Small Numbers
In 1876, when people thought the world was beginning to grow pretty large, Colorado was admitted to American statehood and called itself the Centennial State. Back east, the centennial was also celebrated with the first international trade fair, the Philadelphia Centennial Exposition. Thirteen years later, the French organized a Centennial Exposition and built Gustave Eiffel's tower to commemorate their own revolution. Four years after that, in 1893, a Chicago fair celebrated the fourth centennial of Christopher Columbus's discovery of America (no one had thought to mark any of the first three centennials). In 1939 a crowd in Cooperstown, New York, dedicated a new Hall of Fame and declared a somewhat fictional centennial of the invention of baseball. In 1976, United States football celebrated the centennial of its playing rules. The centennial impulse has ancient roots. It draws on our love of round numbers, our enjoyment of celebrating, and perhaps our slightly wishful view of the human life span. In 1617 Protestants across Europe marked the hundredth anniversary of Martin Luther's posting of the Ninety-five Theses. The centennial is an odd creature nonetheless. It is self-referential, hermeneutic, an excuse for parties, a 'pseudo-event" in the sense of Daniel Boorstin - an occasion that exists only for the sake of publicity. A pseudo-event does not actually happen, and in the course of not happening it can consume considerable money and public attention. Oddly enough, in our accelerating and crowded age, the centennial may be headed for a collapse under its own weight. The year 2000, a very round number, will engender more than a few centennials. No doubt the proprietors of the Guide Michelin will observe the hundredth anniversary of its birth. So will the Métro in Paris. So will the National Automobile Show, the Davis Cup in tennis, the American League in baseball, and the International Ladies Garment Workers Union. The multiplication of centennials outpaces the multiplication of actual events, because people and institutions typically provide not one but many choices of suitable dates, beginning with birth and death. Concert halls and theaters in 2000 will surely observe the centennials of Aaron Copland (born 1900), Oscar Wilde (died), Kurt Weill (born), Arthur Sullivan (died), and Louis Armstrong (born). Kodak will have occasion to remind its customers about the birth of the Brownie hand-held camera. Psychiatric associations may note the centennial of The Interpretation of Dreams. Scientists could plausibly choose 2000 as the year to mark the hundredth anniversary of many different achievements, from blood typing to quantum physics. The world really is growing larger, and the effects are peculiar. Richard K. Guy, a mathematician in Alberta, Canada, asks you to think about sequences of numbers. Even if we're nonmath we make room for some of these in our heads. Any New Yorker, for example, will recognize:
· 14, 18, 23, 28, 34, 42, 50, 59, 66, 72,...
There are people who spend their lives analyzing and cataloguing these sequences - tens of thousands of them. Some are ordinary. Some are sublime. Some are ridiculous.
· 1, 4, 9, 16, 25,... These are the square numbers (lx 1,2x2, and so on). · 1, 2, 4, 8, 16,... The powers of two, of course. · 1, 20, 400, 8902, 197281,... The number of possible chess games n moves long (0 moves, 1 move, 2 moves, and so on). · 1,4, 11, 16,24,... Aronson's Sequence, defined by: T is the first, fourth, eleventh, sixteenth, twenty-fourth letter in this sentence.
The second sequence is also the number of layers in a piece of paper folded in half n times. That is not a coincidence. The sequence also happens to represent the number of regions into which a circle is divided by lines connecting n points spread around its rim.
That, however, is a coincidence, and the coincidence does not last. The powers of two continue in their merry way ... .32, 64, 128...), while the circle-slicing sequence goes off on its own ( ... .31, 57, .9....). If you looked only at the small numbers in the sequence, you might be fooled about the pertinent mathematical rule. The mathematicians who track these things have noticed how often the small-number sequences appear to be doing double duty; or worse. Here is another one: 1, 1, 2, 5, 14,... This could be the number of different ways of folding a strip of n postage stamps into a little stack.
Or it could be the number of distributions of n distinguishable objects in indistinguishable boxes, with at most three objects in a box.
Or it could be the number of different groups, up to isomorphism, of order 2n. (Sorry, no picture.) You would have to be a very sharp mathematician to know that these three sequences are fundamentally different despite their identical start. The postage stamps continue:
· 1,1,2,5,14,39,120,358, 1176,3527,11622,...
The distributions:
· 1,1,2,5,14,42,132,429, 1430,4862,16796,... As for the groups of order 2n:
1, 1, 2, 5, 14, 51, 267, and after that no one is frankly quite certain.
So that innocent-looking sequence 1, 1, 2, 5, 14 bears a heavy burden. A growing collection of observations of this kind led Guy to formulate and "prove by intimidation" what he calls the Strong Law of Small Numbers. It is not a purely mathematical observation. To put it simply: "There aren't enough small numbers to meet the many demands made of them." This law is the enemy of mathematical discovery, Guy says. A mathematician sees a pattern. Sometimes the pattern persists forever. Sometimes the pattern is a figment, and disappears when we reach the realm of large numbers. In focusing on sequences of numbers, mathematicians are studying, analyzing, and classifying the ways in which the purest tendrils of things unfold as they go from small to large. These are patterns made of logic. Anyway, as a rule, the realm of small numbers is misleading. One-fourth of the first one hundred numbers are prime numbers; one-tenth are perfect squares. These nice things quickly get rarer among the large numbers. For that matter, if you looked only at the small numbers, you would think numbers were very likely to be Fibonacci numbers, Bell numbers, Catalan numbers, Motzkin numbers, and even perfect numbers. Of course, you would be wrong. No less a master than Pierre de Fermat looked at the first numbers in the sequence of powers of powers of two plus one (22n + 1: 3, 5, 17, 257, 65537,...) and determined that they were, and all their successors would be, prime. He was wrong. Many errors in the same family have followed along behind. Is there a message for us here? Back in the real world, in a simpler time, the Columbia Broadcasting System, the German Nazi party, and the Daimler-Benz and Chrysler car companies chose iconic forms for their company logos. Now marketing specialists have more trouble finding simple, memorable, geometric shapes suitable for logos; logo creation has become a multimillion-dollar business. You could probably identify the Chrysler icon, the pentagon with five equiangular spokes, even out of context. The Mercedes icon, the circle with three spokes, has also been spoken for. For a car maker hoping to lay claim to a memorable little icon to be milled in metal and stuck on hoods, how many possibilities are there? A small number. Also at mid-century, the number of varieties of prepared mustard available within ten miles of the average resident of the industrial world-a number that stood at zero, of course, through most of human history-was rising toward one. Now, in the most ordinary condiment aisle, after Gulden's and Dijon come English, Bavarian, ball-park, Habafiero, dipping, island, Creole, horseradish, brown, tarragon, honey, Indian, classic, three-alarm, purple-no, we will not go there. Let's just say the number was formerly small and now is large. Thus choosing the proper mustard takes time, not to mention savoring it. To name a new medicine once required a few moments thought. Now a pharmaceutical company knows that it will conduct a proprietary name evaluation as part of its labeling review at the Food and Drug Administration. It will be prudent to request an early consultation with the FDAs Labeling and Nomenclature Committee, never forgetting the Patent and Trademark Office and the United States Adopted Names Council. The domain of drug names is densely overcrowded, and the density carries particular dangers because confusion can be deadly. Is Rezulin the new insulin enhancer or the old anti-acne medication? Is Dynacin the antibiotic and Dynacirc the antihypertensive, or is it the other way around? The package designers may wish to use the letters NS for their nasal spray; will they know that doctors also use them as shorthand for "normal saline"? The process of inventing a drug name routinely takes many months' work by expensive consultants-Brand Institute, Name Lab, Lexicon-even before considering cross-language difficulties. Another sign of this overcrowded name space is the tendency to capitalize brand names in the middle: market a high-speed network-access method and call it PeRKinet, as though we now had a fifty-two-letter alphabet. Anyone who has tried to find a fresh or unique name for a brand, an Internet domain, a children's book, a rock band, a space vehicle, or a perfume has stepped into a packed, finite space and bumped into the Strong Law of Small Numbers. This intensity, this swarming, comes with our greater reach. Our choice of shoes is global; cobbler, farewell. Just as computers make it possible to see larger Mersenne primes than Fermat imagined, they make it possible to link the world's number theorists and would-be number theorists. You could join the Great Internet Mersenne Prime Search and subscribe to the Mersenne Prime Mailing List, so that hours would not pass before you would learn of the discovery in 1997 of the thirty-sixth, an 895,932-digit number, by a man in England using a computer program written in Florida, or the thirty-seventh, a number that would fill an even thicker book, the next year. Mann Mersenne himself made quite a few errors, it turns out. Then again, he lived in a cloister. You do not; or, if you do, your cloister has its own Web site. When you stretch out an arm to buy mustard, no delicatessen on earth is too far. When one hundred years roll around, you have a plenitude to celebrate. The Strong Law of Small Numbers tells us something about the increasing complexity that so often triggers that sense of hurriedness. Like the small numbers, the words of two syllables and the basic condiments and the central television networks bear a heavy burden. They are placed under strain by access to the varied words and tastes and video programming sources that lie beyond. All our information sources evolve toward complexity. No software program gets simpler in release 2.01. No television-news anchor or daily newspaper holds its former central position as announcer to a whole nation. Instead citizens awaken each day with a multitude of experiences to divide one from the other-last night's five hundred channels and million Web sites. Yet these complex strands sometimes return to a simple point of origin. The focal points of national obsession become, if anything, more furious and intense: the trial of O. J. Simpson, the perils of Monica Lewinsky, the coming of-dare we say it-the millennium. Andy Warhol is less famous for any art he may have left behind than for his observation, "In the future, everyone will be famous for fifteen minutes." The World Brain's attention span may seem short. Its ability to focus on any one celebrity may seem to have waned, but that is because the pageant flitting before its eyes is so crowded and multifarious, not because fame is so easily had. The ranks of the unfamous and invisible have also swelled. Woody Allen's gloss is correct: "Almost nobody will be famous for even one minute." The connections between complexity and speed-between variety and time pressure-are not always obvious, but they are real. In 1996 the American trademark authorities noted 234 applications for different products to be named with the word millennium. The next year the number rose to 404. By early 1998 the pace of applications had more than doubled again. Companies attempted to reserve for themselves the label of "official" airline, candy, hole-in-one prize company, light bulb, water, vending machine, baby, souvenir, retirement planner, Champagne, public-relations firm, vacation, Web site, and sponsor of rapid tooling and prototyping stereolithography, of, in, and for the millennium. They asked to trademark Millennium sewing machines, mutual funds, air fresheners, popcorn, metal, magic, fluids, collectibles, great moments, bathing systems, bottling services, golf tournaments, moon monsters, heroes, dynamics, coins, bombs, minutes, law firms, injury law firms, and personal injury law firms. They grabbed and invented spellings for Billenium, Hillenium, Malenium, Pharmillenium, Mealleaniyumm, Millenion, Millenifix, Milleniatron, and Mil-Looney-Um. Slogans already spoken for include Have a Nice Millennium, New Millennium Madness, Navigating the New Millennium, Rock the Millennium, Working Straight Through the Millennium, We Survived the Second Millennium, This Is the Millennium, and Only One Company Offers So Much on This Side of the Millennium. Then there is the land rush around the terms Y2K and twenty-first century and the number 2000. Did you hurry? Or did you wait till the last moment? Confronted with complexity; our instincts seek order, pattern, simplicity. We humans are geniuses at distillation-we automatically take the buzzing, teeming richness of experience and find a manageable set of objects or laws. Or is that set manageable after all? The Strong Law of Small Numbers says it tends to get overloaded. What could be more reassuring than a round number like 2K? Yet the millennial panic starts welling up... We are small-number people in a large-number world.
You cannot change the past after all
TIME travel is possible but you cannot go back and change your future, according
to a new theory. While the laws of physics seem to permit journeys across
time, the new research suggests the basic features of quantum theory will
put a block on paradoxical situations such as time travellers going back
to prevent their own births. Their ideas make a nonsense of the adventures
of Michael J Fox in the Back To The Future films. Now, Daniel Greenberger
of New York City University and Karl Syozil, of Vienna University, have
re-examined Einstein's theory of relativity which suggests the space-time
dimension curves back on itself. That would make it impossible to go back
and meet versions of ourselves. But their study of quantum waves show the
paradoxes implied by Einstein's equations would never arise,
New Scientist reported.
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