Faster

The Law of Small Numbers

James Gleick

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.

The Philosopher: There is no such thing as time, only life

A. C. Grayling

AN AVERAGE human life lasts less than 1,000 months. One third of those months are spent asleep, so a conscious human existence averages between merely 600 and 700 months. A lifetime is thus a truly fleeting thing, lodged between a sleep and a forgetting; and there scarcely seems time to draw breath in it, before its last breath is drawn.
The first mystery of time, then, is how little of it any individual has. The second is how unimaginably vast is the universe's history by comparison. If that history were compressed into an hour, the time that humankind has existed would barely fit into its last split second. If humanity were to extinguish itself through ecological disaster or war, its presence in this corner of the cosmos would be unnoticeable between the massive weights of time that stretched before and after it. Can time be explained?
St Augustine pinpointed the problem when he said, "If you tell me to meet you at such a time, I have no problem; but if you ask me what time is, I cannot answer." As a bishop St Augustine should not, of course, have worried, for time does not matter to theology; the deity is eternal, "eternal" means "outside time", and because eternity is the true reality, it follows that time is unreal. Why worry about what does not exist?
Such reflections are as unpersuasive as the theology that prompts them. Time is all too real a feature of the human condition; memory and hope, regret and anxious expectation, just as the vividness of the present moment in the dental chair or the lover's embrace are crystalline facts of experience. Time is always with us: whether dragging too slowly or slipping through our fingers, it can't be avoided. Time is remarkably elastic. It flows differently for different people, even when they are doing the same thing. A couple at a play can have sharply contrasting senses of its passage. Suppose one likes and the other dislikes the play; time will flash by for the former, limp by for the latter. This protean quality of time is a significant matter for the question of how best to live. Days spent in the enjoyment of beauty, pleasure or learning will seem like miniature lifetimes, even though, when they are over, they will appear to have raced away. Travel to fascinating places feels like this too - while there the hours feel endless, afterwards they seem a moment.
Both are true; for time is not an objective or absolute measure which ticks away in equal quanta independently of experience. Instead, time is made of experience, and it is the quality of experience that determines the quantity of the time one lives. To prove this, think of idling fruitlessly at home over a long weekend, and compare it with flying to an exciting city and filling the same days with art gallery visits, dinners, the opera, and sightseeing. The quantity of life packed into such a weekend far exceeds the quantity of life lived in idle days. This is how time is truly to be measured: by the amount of life that fills it. Accordingly, some people live more than one lifetime in one lifetime, while others, through timidity, or lack of energy or imagination, spend an entire lifetime living less than one lifetime. It is as if these latter eat their soup with a fork; they miss the opportunity to grasp what experience has to offer, and to take from it its best to fill each moment. In a famous essay Walter Pater defined the best life as one in which time, far from being wasted, is created by vigour of living: "A counted number of pulses only is given to us of a variegated, dramatic life," he wrote. "How may we see in them all that is to be seen by the finest senses? How shall we pass most swiftly from point to point, and be present always where the greatest number of vital forces unite in their purest energy? To burn always with this hard, gemlike flame, to maintain this ecstasy, is success in life."
Isaac Newton thought of time as an absolute measure, ticking away independently of what is happening in the universe - even if nothing is. Einstein saw that the rate of time's flow depends upon observers' points of view. In his General Theory of Relativity he combined space and time into a single four-dimensional continuum, showing that the universe and spacetime are the same thing. It is not so remote an analogy to see his theory as echoing the fact that how much time you have depends upon how much living you do - this is the same as saying: there is no such thing as time, only life.

Copyright 2004 <http://www.timesonline.co.uk/section/0,,549,00.html> Times Newspapers Ltd.  November 23, 2004

The study of time measurement is called horology. People have always sought accurate measurements of time. Ancient people found that the Sun, moon, and stars move in predictable cycles at regular intervals; they used this observation to produce accurate calendars for measuring days, months, seasons, and years. See also equation of time.More complex societies have discovered ways to measure time even more precisely. Sundials enabled ancient people to divide the daytime up into smaller pieces. Civilizations in Egypt, China, and Greece invented water clocks that could keep fairly accurate time in the dark too. Traditionally, aboard ships, a system of hourglasses and ship's bells were used to mark watches and for navigation. Mechanical clocks were developed in Europe in the 14th Century. Today, time can be measured on very accurate clocks, often called chronometers. The best available clocks are atomic clocks.At first, people set their clocks based on the noon sun in their locality. The invention of time zones, north-south strips of the Earth in which everyone's clocks are coordinated, made time measurement standardized worldwide. With only a few exceptions, every place on Earth is part of a standard time zone connected with Greenwich Mean Time (because the benchmark for the world's time zones is the time in Greenwich, England).The development of human understanding of the nature and measurement of time, through the work of making and improving its measurements, (calendars, clocks) and its intuitive concepts (spacetime, general relativity), has been a major engine of scientific discovery since the beginnings of civilization.

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.
[Metro Jul16,2005]


Anybody out there? Regarding the new theory on time travel (Metro,Thu) why not invite inventors to come back to  Jun22,2005 to prove it's possible? Think of the publicity the Metro will get if someone did return.
Paul Walton,Northumberland


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