Perfect Figures: The Lore of Numbers and How We Learned to Count
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Perfect Figures: The Lore of Numbers and How We Learned to Count
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Overview
Since the beginning of civilization, numbers have been more than just a way to keep count. Perfect Figures tells the stories of how each number came to be and what incredible associations and superstitions have been connected to them ever since. Along the way are some of the great oddities of numbers' past as:
-a time when finger-counting was a sign of intelligence (the Venerable Bede could count to a million on his hands)
-the medieval Algorists, who were burnt at the stake for their use of Arabic rather than Roman numerals
-the Bank of England, which stubbornly kept accounts on notched wooden sticks until 1826
Filled with Crumpacker's eloquent wit and broad intelligence, Perfect Figures brings the history of numbers to life just as Bill Bryson did for the English language in The Mother Tongue.
Product Details
| ISBN-13: | 9781429935470 |
|---|---|
| Publisher: | St. Martin's Publishing Group |
| Publication date: | 08/07/2007 |
| Sold by: | Macmillan |
| Format: | eBook |
| Pages: | 288 |
| File size: | 337 KB |
About the Author
BUNNY CRUMPACKER is the author of The Sex Life of Food, as well as two books based on historic food and recipe pamphlets. Her book reviews appear in The Washington Post. She lived in Valley Cottage, New York.
Bunny Crumpacker (1933-2010), a New York native, has been a professional caterer, editor, newspaper columnist, and school community relations officer. Her book reviews appear in The Washington Post. She is the author of How to Slice an Onion, The Sex Life of Food, Perfect Figures, and two cookbooks based on food and recipe pamphlets issued from 1875 to 1950--a chronicle of American cooking in those years. She and her husband, a record producer, lived in the Hudson River Valley region, just north of New York City.
Read an Excerpt
Perfect Figures
The Lore of Numbers and How We Learned to Count
By Bunny Crumpacker
St. Martin's Press
Copyright © 2007 Bunny CrumpackerAll rights reserved.
ISBN: 978-1-4299-3547-0
CHAPTER 1
ONE IS ALL
One is the beginning, the single starting place. It's the universe at the big bang: There was that enormous event, that unthinkable noise, and suddenly, in a fraction of a second, whatever there was — had it been One? — had shattered. It became a billion million stars, galaxies after galaxies of stars, stars with planets and moons and meteors and asteroids, each one containing everything again, atoms and molecules, charm and quark, and each thing — each atom, each galaxy — was still one, one again. One after one to infinity.
At our own beginning, there were no numbers, not even one. We had no need for numbers — no need to count, no need to know how many. Sufficient unto itself, and for our survival, was each person, each thing, each moment.
We knew the day, and the darkness that came after the day, night after day and then day after night. But eventually the time must have come when someone wanted to keep track of yesterday, today, and tomorrow, and count what fills those days: moons and meals and springtimes. And eventually, someone wanted to count what was his — animals, perhaps, or arrows, or seeds, oil, and grain. Perhaps someone wanted to know what was coming — how many days it would be necessary to wait until the floods would come again, or the moon would disappear and then come back, how long before the baby would be born, or how long before the sun returned from its trip to the edge of the world and the cold passed and the days slowly began to grow longer again.
There is an unexpected story in the history of how we learned to count — from our first recognition of numero uno, the one we mean when we point to ourselves, to the vast numbers we think of when we look at the stars on a moonless night.
Counting is a natural process, almost inevitable, and numbers are organic. They begin with the single line of our bodies, the psychic feel of ourselves. They grow, one by one by one, but they always remain intimately connected to our physical being, from the twoness of our eyes and arms and legs to our ten fingers and — should we need them — our ten toes.
Counting is as natural as numbers. We count each other, and then our children, the things we own, the days we've passed through. Numbers count the things of the world, which must have become less untamable as we numbered its parts and learned to give them names. When we drew an animal on the wall of a cave, we made this one, and then that one — one and two — and we gave the animal its name. When we made designs of the stars over our heads and told their stories to each other, we remembered how many stars there are in Orion's belt or in Cassiopeia's chair. Everywhere, we learned numbers this naturally, and taught ourselves to count, because we always needed to know how many.
Once, we wanted to know how many cattle left in the morning and came home at night, how many seeds it was necessary to save for next year's planting, or how many days there were from full moon to full moon. Now we need zip codes and Social Security numbers, phone numbers and license plates. Our passports and our houses are numbered, as are our charge accounts and checks and telephones. Numbers have grown away from the simplicity of you and me and the baby. Now, they define us with increasing complexity, and we lose track of how they began, one by one by one — this one, that one, those, and me, you, and the rest of the world.
In another way, perhaps God taught Adam to count when he crafted Eve from Adam's rib and suddenly, where there had been one, lo! there were two. Did Adam and Eve keep track of their children by counting? Did they subtract one from two when Abel disappeared and only Cain was left? Apparently not. But clearly, God could count.
Once we left the garden, learning to count for ourselves was a slow process — it took millennia to learn to answer, in all the varied ways, the basic questions of mathematics: how many? how much? Like magic, numbers became visible as we needed to know them. Along the way, people counted in different ways in different places — by twos, fours, fives, twelves, twenties, sixties, and finally, by tens. Everywhere, numbers became part of civilization, and then came to mean more than just the amount they stood for — they meant good luck and bad, wishes and fairy tales, religion and a way of forecasting the future.
Although we learned numbers at different times and places, we always began with one. Here, we used piles of pebbles as equivalents for numbers, and there, we made notches on sticks or bones, but everywhere, even today, men and women have used their fingers to count. We forget how organic numbers are — how they grew out of our bodies as we learned to hold one finger up to mean one, or point to our eyes for two, and to find higher numbers on our fingers and our toes.
It isn't necessary to count the way we do now in order to answer the basic question of how many. Little children can say numbers from one to eleventy and beyond, but they have different ways of telling you how many toys they have — this one and that one and the other one, maybe, or just twenty-teen-two.
COUNTING WITHOUT NUMBERS
Do animals count the same way we do? No matter how well we think of ourselves, we aren't alone in grasping the concept of number. Even the lowly rat can learn to press a lever a specific number of times, or take the third right turn (not the second or fourth) in a maze that leads to a reward. Pigeons can learn to peck at a target a specific number of times — thirty-five, not forty, or twenty-three, not nineteen. Chimpanzees will choose a tray that holds seven chocolates over one that holds six. Clever Hans, the famous counting horse, used almost subliminal cues from his trainer and figured out from that — not from the number — how many times to stamp his hoof, but all these animals are, in some way, counting.
Recent research leads us to believe that counting may be simply hard-wired into our brains, whether we're pigeons or people. Monkeys in a study at the Massachusetts Institute of Technology linked computer frames containing a number of dots to frames that were different but contained the same number of dots. Neurons in the monkeys' prefrontal cortex (the part of the brain that makes rapid decisions) rewired themselves when the first frames were shown, and when the same numbers reappeared, the neurons were quickly reactivated — the monkeys recognized the number, even though the frames were different.
But how do cicadas keep track of the years they need underground? They rise through the green grass and into the trees once every seventeen years to make a new generation, and then return through pale roots to the dark underground, to wait and count through all those slow winters before the precise springtime comes and they are ready to rise again, thousands of them all at once, to find their mates.
How do ants find their way home? Or, for that matter, how do they know the way to return to a food source, once they've found it? Researchers trained desert ants to travel from their nest to a food source, and then, to test whether the number of steps they had to take affected their ability to locate the food, glued stiltlike extensions to the legs of some ants and shortened (by cutting) the legs of others. The newly tall ants took the same number of steps they had already learned, and went right past their food. The shortened ants traveled only part of the way to their food goal. According to a 2006 issue of Science, as the ants became used to their new leg length, they adjusted their internal pedometers, in there counting away, and learned how many steps they now needed to take from the nest to the food and back again.
A group of wasps places dead bugs in the cells of developing wasps to be used as food. Different kinds of wasps use different numbers of bugs, but each is consistent, always using five, or ten, or twenty-four. In one group, ten bugs are always placed with female eggs but only five with males, because the female wasps are larger than the males. Somebody is counting.
Perhaps the bugs were arranged in patterns, to be recognized in the same way that we recognize the number of dots on a pair of dice. We don't actually count one-two-three-four, up to six; we look at the pattern and we know. Maybe that's what the wasps do.
Dolphins can recognize strings of as many as eight abstract figures. People, smart as we are, can go up to only six, or at the most, seven. We have to put hyphens in our telephone numbers — one after the area code, and another after what used to be called the exchange — because it's easier to remember a group of three numbers, another group of three, and a group of four, than it is to remember an uninterrupted group of ten. Zip codes stop at five; the added specific code of four more numbers is separated from the zip numbers by a hyphen. Is it counting to remember a phone number? Or is it the memorization of a group of words that happen to mean a number?
We don't usually stop and count anything we're looking at until the group goes higher than four or five. We look at the pattern — a square with something at every corner for four, and with something in the middle for five, or the shape of a triangle with something at each of the angles — and we know the number. We're recognizing the pattern, and we know the pattern means a number. A two-car garage has two doors and holds two cars, and that's as much as most of us need to know about it. But six and seven and eight and, of course, beyond — that's too many to count at a glance. We're not dolphins.
There was a group of Indians in South America without any number words beyond three. But they could recognize groups of things, and knew whether they were complete or not. When they traveled, they were accompanied by their dogs — and while they didn't know how many dogs they had, they knew at a glance if one was missing, and they'd call until the dog returned. Teachers on a class trip work in somewhat the same way. They have a feel for the way the group should look, and if it doesn't look right, then they stop and count. That kind of glance — the pile is smaller than it should be — gives you important information (something is missing), but it doesn't give you a number, it doesn't say how many you have. This kind of counting, the pattern at a glance (there's actually a word for it: subitizing), works only with relatively small numbers. If the teacher has thirty or more kids with her (and no parents, alas), she'll be hard put to know without counting when one is missing.
Up to a point, birds can do the same thing. Take one egg from a nest of several and the bird pays no attention. Take two, and the nest is abandoned. A nineteenth-century astronomer and mathematician, Sir John Lubbock, wrote about a landowner who was bothered by a crow that had chosen to nest in his watchtower. He'd go into the tower to shoo away the bird, and the bird would fly outside and wait until the man had left — and then it would fly back to the nest. After a while, the landowner thought of a way to fool the bird. Two men were sent to the tower with instructions for one to wait inside while the other left; the expectation was that when the bird saw a man leaving, it would feel safe and fly back inside, where the remaining man could deal with it. But the bird was smart enough to wait until they had both left before it returned to the nest. The next day, three men went into the tower and two men left. The clever bird wasn't fooled; it waited patiently again until all three men had left, and then flew happily back inside the tower. Four men entered the tower; three left; the bird waited for the last man before flying back home. At last, five men entered the tower; four left; and this time the bird flew back in again while the fifth man was still there. Conclusion: the bird could count to four, but not to five.
The bird was aware of quantity, even if not of number, even if up to only four. This is the beginning of counting. To be aware of the self is the first step toward being aware of the other; and knowing that there are two — me and you — is the first step toward counting the rest of the world. Eventually, we'll know that counting has no end — that it goes on forever, and even then, there can still be one more. In order to do that, to count to forever, we'll have to have the numbers that will make it possible to do so.
We can reach all our numbers, to count to forever, using very few numerals and words. We only have ten digits to work with, no matter what we're counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Each of those numerals has its own name — one, two, three, and so on, past ten to twelve. After that, for a long time number names are just a combination of the words that have gone before: thirteen is three and ten; twenty is two tens. Ninety-nine is nine tens and nine. When we add one more, we reach a hundred, the first new number name since twelve. The next is thousand, and the next after that — a long way away — is million — so few words and digits for so many numbers!
In exactly the same way, there are only twenty-six letters in the English alphabet, enough to encompass both Shakespeare and the Marx Brothers and everything before and after. A dictionary, said Anatole France, is the universe in alphabetical order. There was only one Shakespeare; there were three principal Marx Brothers; there are millions of words in the dictionary and numbers in the universe. But there are still only twenty-six letters in the English alphabet and a mere ten digits. Those ten numerals are enough to count all the words ever uttered by anybody, and all the letters in all the words. Further, the ten digits are all we need to count every grain of sand, every star, every thing — even every no thing, as the distance of empty space — in the universe.
ONE WAS FIRST
The first number to be thought of, inevitably — because it was the first to be needed — was one, that inward number. One might be enough for a while: one is today, one is now, one is self, one is me, one is first.
As long as one is the only number you need, life is simple. If you have only one of anything, you needn't count it. It's either there or it isn't. You know right away: you see it or you don't. It's not until you add another and another — and another — that keeping track becomes complicated and you need more numbers. Numbers are a sign of plenty.
Even with plenty, though, you can get by with using just one for a while. The notches on ancient bones, the first written countings we know about, are simply lines of one, like notches on a gun stock or scratches made on a jailhouse wall when the sun goes down.
Those lines carefully etched into an animal bone must have come after the very first countings, the ones that we made on our fingers. The index finger points when it's held horizontally; it counts when it's upright. One finger; one number. Most counting systems are based either on five (one hand), ten (two hands), or twenty (fingers and toes). The Latin word for fingers is digiti; our fingers are digits, and numbers are digits as well. In the Middle Ages, digiti was used to mean the single numbers of the decimal system, while articuli ( joints) meant the tens; then digiti came to mean the numerals themselves. When we talk about digital computing, we mean computing by numbers — rather quickly.
There are great advantages to counting with your fingers. With any luck, they're always there. They're clearly visible — other people can see them if you want them to. You can feel what you count. They're portable; wherever you go, they go with you. The problem is, though, that they aren't permanent. You hold up two fingers, and everybody understands that you mean two. When you put your fingers down, two disappears. It leaves no record. You can't walk around for days with two fingers in the air — you'll need them for other things. Unless you have more hands than most of us do, you can't count very high and you can count only one kind of thing at a time — birds or bananas, but not both. (More on finger counting, which can be surprisingly complex, in later chapters.)
There had to be better ways. And there were. Everywhere, always, there was someone who found a better way.
(Continues...)
Excerpted from Perfect Figures by Bunny Crumpacker. Copyright © 2007 Bunny Crumpacker. Excerpted by permission of St. Martin's Press.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
Title Page,ACKNOWLEDGMENTS,
1 - ONE IS ALL,
2 - TWO IS YOU AND YOUR IMAGE IN THE MIRROR,
3 - THREE IS A MAGIC NUMBER,
4 - FOUR IS A SOLID NUMBER,
5 - FIVE IS FOR FINGERS,
6 - SIX IS PERFECT,
7 - SEVEN IS A HAPPY NUMBER,
8 - EIGHT IS FOREVER,
9 - NINE IS A NEW NUMBER,
0 - ZERO IS NOT NOTHING,
10 - TEN MEANS TWO HANDS,
11 - ELEVEN IS A LEFTOVER NUMBER,
12 - TWELVE IS THE LAST OF THE BASIC NUMBER NAMES,
100 - HUNDRED IS A STRANGE WORD,
1,000 - THOUSAND IS A SWOLLEN HUNDRED,
1,000,000 - MILLION IS NOT YET FOREVER,
And One More - THE MANIFEST DESTINY OF NUMBERS,
OTHER BOOKS BY BUNNY CRUMPACKER,
BIBLIOGRAPHY,
INDEX,
Notes,
Copyright Page,