" T HE NO T H I N G T H A T
IS
ROBERT KAPLAN
THE
NOTHING
THAT IS
A Natural History of Zero Illustrations by Ellen Kaplan
OXFORD
UNIVERSITY PRESS
20OO
OXFORD
UNIVERSITY PRESS Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paulo Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan
Copyright © 1999 by Robert Kaplan Illustrations copyright © 1999 by Ellen Kaplan
Originally published in the United Kingdom by Allen Lane/The Penguin Press, 1999 Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Kaplan, Robert, 1933The nothing that is: a natural history of zero /Robert Kaplan p. cm. Included index ISBN 0-19-512842-7 1. Zero (The number) I. Title QA141.K36 1999 511.2'11dc21 99-29000
9876543 Printed in the United States of America on acid-free paper
To Frank Brimsek 3 hours 51 minutes 54 seconds
How close to zero is zero?
BRITISH DEPUTY PRIME MINISTER JOHN PRESCOTT
CONTENTS
ACKNOWLEDGEMENTS
x
A NOTE TO THE READER xii ZERO
THE L E N S I
ONE
MIND PUTS ITS S T A M P ON M A T T E R 4
TWO
THE G R E E K S HAD NO W O R D FOR IT 14
THREE TRAVELERS' TALES
FOUR EASTWARD FIVE
DUST
28
36
50
SIX
I N T O THE U N K N O W N 57
vii
SEVEN A PARADIGM SHIFTS EIGHT
68
A MAYAN INTERLUDE: THE DARK SIDE OF C O U N T I N G
NINE MUCHADO
1 Envoys of Emptiness 2 A Sypher in Augrim 3 This Year, Next Year, Sometime, Never 4 Still It Moves
80
90
90 93 103 106
TEN
ENTERTAINING ANGELS I The Power of Nothing 2 Knowing Squat 3 The Fabric of This Vision 4 Leaving No Wrack Behind 116 116 120 129 137
viii
ELEVEN
ALMOST NOTHING I Slouching Toward Bethlehem 2 Two Victories, a Defeat and Distant Thunder 144 144 160
TWELVE
IS I T O U T THERE? 175
THIRTEEN
B A T H - H O U S E WITH S P I D E R S 190
FOURTEEN
A L A N D W H E R E IT WAS ALWAYS AFTERNOON 195
FIFTEEN
WAS LEAR R I G H T ? 203
SIXTEEN
THE U N T H I N K A B L E 216
INDEX 220
ix
ACKNOWLEDGEMENTS
First and foremost, the two lighthouses from which I take my bearings: Ellen, whose drawings adorn and whose spirit informs this book; and Barry Mazur, whose verve and insights are endless. This book would have been nothing rather than about nothing had it not been for Christopher Doyle, Eric Simonoff and Dick Teresi. It has benefited immensely from Peter Ginna's humorous touch and inspired editing. My thanks as well to Stefan McGrath. There are many in the community of scholars to thank for the generosity of their time and the quality of their knowledge. Jon Tannenhauser has been lavish in his expertise and suggestions, as has Mira Bernstein. Peter Renz, whose store of information is larger even than his private library, has been invaluable. I'm very grateful for their help to Gary Adelman, Johannes Bronkhorst, Thomas Burke, Henry Cohn, Paul Dundas, Matthew Emerton, Harry Falk, Martin Gardner, Nina Goldmakher, Susan Goldstine, James Gunn, Raqeeb Haque, Takao Hayashi, Michele Jaffe, James Rex Knowlson, Takeshi Kukobo, Richard Landes, Boris Lietsky, Rhea MacDonald, Georg Moser, Charles Napier, Lena Nekludova, David Nelson, Katsumi Nomizu, Yori Oda, Larry Pfaff, Donald Ranee, Andrew Ranicki, Aamir Rehman, Abdulhamid Sabra, Brian A. Sullivan, Daniel Tenney III, Alf van der Poorten, Jared Wunsch, Michio Yano and Don Zagier. Finally, I can't thank enough, for their unfailing support when it most mattered, the Kaplans of Scotland; Tomas Guillermo, the Gilligans and the Klubocks of Cambridge; the HarrisonX
ACKNOWLEDGEMENTS
Mahdavis of Paris; the Franklins of iltshire; the Nuzzos of Chestnut Hill; the Zelevinskys of Sharon — and my students everywhere.
xi
A NOTE TO
THE READER
If you have had high-school algebra and geometry nothing in what lies ahead should trouble you, even if it looks a bit unfamiliar at first. You will find the bibliography and notes to the text on the web, at www.oup-usa.org/sc/0195128427/.
XII
ZERO
THE LENS
If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else and all of their parts swing on the smallest of pivots, zero. With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves. Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight. Zero's path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West. In this book you will see it appear in Sumerian days almost as an afterthought, then in the coming centuries casually alter and almost as casually disappear, to rise again transformed. Its power will seem divine to some, diabolic to others. It will just tease and flit away from the Greeks, live at careless ease in India, suffer our Western crises of identity and emerge this side of Newton with all the subtlety and complexity of our times. My approach to these adventures will in part be that of a naturalist, collecting the wonderful variety of forms zero takes on - not only as a number but as a metaphor of despair or delight; as a nothing that is an actual something; as the progenitor of
1
THE NOTHING THAT IS
us all and as the riddle of riddles. But we, who are more than magpies, feather our nests with bits of time. I will therefore join the naturalist to the historian at the outset, to relish the stories of those who juggled with gigantic numbers as if they were tennis balls; of people who saw their lives hanging on the thread of a calculation; of events sweeping inexorably from East to West and bearing zero along with them - and the way those events were deflected by powerful personalities, such as a brilliant Italian called Blockhead or eccentrics like the Scotsman who fancied himself a warlock. As we follow the meanderings of zero's symbols and meanings we'll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us. Its muse speaks only to those who ardently pursue her. And what is that pursuit? A mixture of tinkering and inspiration; an idea that someone strikes on, which then might lie dormant for centuries, only to sprout all at once, here and there, in changed climates of thought; an on-going conversation between guessing and justifying, between imagination and logic. Why should zero, that O without a figure, as Shakespeare called it, play so crucial a role in shaping the gigantic fabric of expressions which is mathematics? Why do most mathematicians give it pride of place in any list of the most important numbers? How could anyone have claimed that since 0x0 = 0, therefore numbers are real? We will see the answers develop as zero evolves. And as we watch this maturing of zero and mathematics together, deeper motions in our minds will come into focus. Our curious need, for example, to give names to what we create - and then to wonder whether creatures exist apart from their names. Our equally compelling, opposite need to distance ourselves ever further from individuals and instances, lunging always toward generalities and abbreviating the singularity of things to an Escher array, an orchard seen from the air rather than this gnarled tree and that. Below these currents of thought we will glimpse in successive
2
THE LENS
chapters the yet deeper, slower swells that bear us now toward looking at the world, now toward looking beyond it. The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than - or only a little less than - the angels in our power to appraise? Mathematics is an activity about activity. It hasn't ended has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway. One of the most visionary mathematicians of our time, Alexander Grothendieck, whose results have changed our very way of looking at mathematics, worked for years on his magnum opus, revising, extending - and with it the preface and overview, his Chapter Zero. But neither now will ever be finished. Always beckoning, approached but never achieved: perhaps this comes closest to the nature of zero.
3 3
ONE
MIND PUTS ITS S T A M P ON M A T T E R Zero began its career as two wedges pressed into a wet lump of clay, in the days when a superb piece of mental engineering gave us the art of counting. For we count, after all, by giving different number-names and symbols to different sized heaps of things: one, two, three ... If you insist on a wholly new name and symbol for every new size, you'll eventually wear out your ingenuity and your memory as well. Just try making up distinct symbols for the first twenty numbers - something like this:
and ask: how much is 7 plus 8? Let's see, it is And minus / ? Well, counting back / places from , it is 6. Or plus ? Unfortunately we haven't dreamed up a symbol for that yet - and were we to do so, we would first have to devise seven others. The solution to this problem must have come up very early in every culture, as it does in a child's life: group the objects you want to count in heaps all of the same manageable, named, size, and then count those heaps. For example,
and the unattractive
becomes
4
MIND PUTS ITS STAMP ON
SO
MATTER
of the
and
more.
The basic heaps tend to have 5 or 10 strokes in them, because of our fingers, but any number your eye can take in at a glance will do (we count eggs and inches by the dozen). No sooner do we have this short-cut (which brings with it the leap in sophistication from addition to multiplication), than the need for another follows: if + is altogether of the " and more, exactly what number is that? Won't we have to invent a new symbol after all? Different cultures came up with different answers. Perhaps from scoring across a stroke like these on a tally-stick, perhaps from handsignals wagged across the market-place, the Romans let X stand for a heap of ', V for . . , . ('V, that is, as half - the upper half — of 'X' - a one-hand sign) and so XV for three 5s, on the analogy of writing words from left to right. Instead of the cumbersome VVVV or XVV for four 5s, they wrote XX: two 10s. So our problem turned into: X + XVIII = XXVIII. This looks like a promising idea, but runs into difficulties when you grow tired of writing long strings of Xs for large numbers. At the very least, you're back to having to make up one new symbol after another. The Romans used L for 50, so LX was 10 past 50, or 60; and XL was 10 before 50, so 40. C was 100, D 500, M 1,000 and eventually - as debts and dowries mounted - a three-quarter frame around an old symbol increased its value by a factor of 100,000. So Livia left 50,000,000 sesterces to Galba, but her son, the Emperor Tiberius — no friend of anyone, certainly not of Galba (and anyway his mother's residual heir) - insisted that IDI be read as D - 500,000 sesterces, quia notata non perscripta erat summa, 'because the sum was in notation, not written in full'. The kind of talk we expect to hear from emperors. But this way of counting raised problems every day, and not just in the offices of lawyers.
5
THE N O T H I N G T H A T IS
What is 43 + 24? For the Romans, the question was: what is XLIII + XXIV, and no attempt to line the two up will ever automatically produce the answer LXVII. Representing large numbers was awkward (even with late Roman abbreviations, 1999 is MCMXCIX:
M
CM
XC
IX
1,000 100 before 1,000, so 900
10 before 100, so 90
1 before 10, so 9)
but working with any of them is daunting (picture trying to subtract, multiply or, gods forbid, divide). It needed one of those strokes of genius which we now take for granted to come up with a way of representing numbers that would let you calculate gracefully with them; and the puzzling zero - which stood for no number at all - was the brilliant finishing touch to this invention. The story begins some 5,000 years ago with the Sumerians, those lively people who settled in Mesopotamia (part of what is now Iraq). When you read, on one of their clay tablets, this exchange between father and son: 'Where did you go?' 'Nowhere.' 'Then why are you late?', you realize that 5,000 years are like an evening gone. The Sumerians counted by 1s and 10s but also by 60s. This may seem bizarre until you recall that we do too, using 60 for minutes in an hour (and 6 x 60 = 360 for degrees in a circle). Worse, we also count by 12 when it comes to months in a year, 7 for days in a week, 24 for hours in a day and 16 for ounces in a pound or a pint. Up until 1971 the British counted their pennies in heaps of 12 to a shilling but heaps of 20 shillings to a pound. Tug on each of these different systems and you'll unravel a history of customs and compromises, showing what you thought was quirky to be the most natural thing in the world. In the
6
MIND PUTS ITS S T A M P ON
MATTER
case of the Sumerians, a 60-base (sexagesimal) system most likely sprang from their dealings with another culture whose system of weights - and hence of monetary value - differed from their own. Suppose the Sumerians had a unit of weight call it 1 - and so larger weights of 2, 3 and so on, up to and then by 10s; but also fractional weights of1/4,1/3,1/2and 2/3. Now if they began to trade with a neighboring people who had the same ratios, but a basic unit 60 times as large as their own, you can imagine the difficulties a merchant would have had in figuring out how much of his coinage was equal, say, to 7| units of his trading-partner's (even when the trade was by barter, strict government records were kept of equivalent values). But this problem all at once disappears if you decide to rethink your unit as 60. Since 7f x 60 = 460, we're talking about 460 old Sumerian units. And besides,1/4,1/3,1/2and2/3of 60 are all whole numbers — easy to deal with. We will probably never know the little ins and outs of this momentous decision (the cups of beer and back-room bargaining it took to round the proportion of the basic units up or down to 60), but we do know that in the Sumerian system 60 shekels made a mina, and 60 minae a talent. So far it doesn't sound as if we have made much progress toward calculating with numbers. If anything, the Sumerians seem to have institutionalized a confusion between a decimal and a sexagesimal system. But let's watch how this confusion plays out. As we do, we can't but sense minds like our own speaking across the millennia. The Sumerians wrote by pressing circles and semi-circles with the tip of a hollow reed into wet clay tablets, which were then preserved by baking. (Masses of these still survive from those awesomely remote days — documents written on computer punchcards in the 1960s largely do not.) In time the reed gave way to a three-sided stylus, with which you could incise wedgeshaped (cuneiform) marks like this ; or turning and angling it differently, a 'hook' Although the Sumerians yielded to the Akkadians around 2500 BC, their combination of decimal
7
THE NOTHING T H A T IS
and sexagesimal counting remained intact, and by 2000 BC (Old Babylonian times now) the numbers were written like this:
1 2 3 4 5 6 7 8
9
or
For 10 they used one 'hook': so that 11 was '
and 12:
and so on - just like the later Roman invention of X, XI, XII (but there was no new symbol for 5, so 15 wasn't like the Roman
XV, but
20 was
30 was made with three then 40 with four,
hooks, variously arranged:
50 with five; and all the numbers in between were written just as you would expect:
34 was
and 59 was
Now the sexagesimal system intruded. 60 was one wedge again, but a bigger one: So writing numbers, smaller to
8
MIND PUTS ITS STAMP ON
MATTER
larger, from right to left (just as we do — thanks to them — although we write our words from left to right), 63 would You can construct the rest: 120 would be
137:
etc. If you want to travel briefly back in time (wet clay, a wooden stylus and a pervasive smell of sheep might help), try writing 217. Did you get:
Notice how important the size of the wedges is: the only difference between 62:
and 3:
is the larger first wedge. But
handwriting — even in cuneiform — changes; people are rushed or careless (try writing up the month's accounts with your stylus), and with thousands of records being kept by harried temple scribes of the names of donors and the number of sheep or fish or chickens each has brought as an offering, the large wedges may grow smaller and the small larger (perhaps from time to time there was a Tiberius factor too), and then where are we? Utter confusion - until someone comes up with the brilliant idea (or was it a makeshift or compromise that just worked its way into practice, as these things do?) of making the place where the wedges are written stand for their value. So large or small, and then two more. And means 202: three sixties, two tens means 182: 3 x 60 + 2.
9
THE NOTHING THAT IS
Once this positional system for writing numbers became common practice, it was inevitable that spacing would be called in for clarity, along with stylized groupings of wedges and hooks. So just as our '754' stands for (7 x 102) + (5 x 10) + (4 x 1),
and
This is wonderful. It not only allows us to write large numbers swiftly (1999 becomes
(33 x 60) = 1,980 +
19
but - much more important - lets us calculate with relative ease. We, for example, add
43 +14 ~57
by adding the 3 and 4 first, then the 4 and 1 tens. For the Babylonians,
And what about 'carrying', one of the sorrows of early childhood? For us,
10
MIND PUTS ITS S T A M P ON
82
MATTER
+ 41/123
123(2 and 1 units make 3 units, 8 and 4 tens make 12 tens, that is, 2 tens, then 1 hundred). For them,
six 10s three units make one 60 so with the 60 already there, we will now have two 60s For us, then, when you move a digit from a column to the one on its left, its value becomes ten times as large - for the Babylonians, 60 times larger. And when one column is full, you empty out its 10 - or 60 - units and put one more unit in the column to the left of it. No great thing, said Sophocles, comes without a curse. For all the brilliance of positional notation, how were the Babylonians to distinguish between 180 and 3: ? That is, how were they to know whether this '3' was in the units or the 60s column? How are the priests of the temple at E-Mach to know from the records whether two or 120 sheep were given last year as an offering to the goddess Ninmah? Clearly, by context; just as you know where to put the decimal point when you remember that a half gallon of milk costs one fifty-five, and your travel agent calls with a bargain flight to Toronto for one fifty-five. But life grows more complicated, the number of things inevitably larger, and context alone becomes a mumbling judge. After putting up with these ambiguities for a thousand years (is it the different rates of change ..."
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