Table Of Content-W
I
ARCHIMEDES'
REVENGE
The Joys and Perils
of Mathematics
Paul Hoffman
FAWCETT CREST * NEW YORK
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A Fawcett Crest Book
Published by Ballantine Books
Copyright 0 1988 by Paul Hoffman
All rights reserved under International and Pan-American
Copyright Conventions. Published in the United States by
Ballantine Books, a division of Random House, Inc., New
York, and simultaneously in Canada by Random House of
Canada Limited, Toronto.
Portions of this book appeared in another form in the New
York Times Magazine and in Science Digest.
Cover photo by Ernie's Studio, Vegreville, Alberta, Canada
ISBN 0-449-21750-7
This edition published by arrangement with W. W. Norton &
Company, Inc.
Manufactured in the United States of America
First Ballantine Books Edition: September 1989
Seventh Printing-January 1995
For Martin Gardner
CONTENTS
Introduction I
PART I: NUMBERS 7
1. 666 and Friends 9
2. Archimedes' Revenge 26
3. Prime Prostitution 34
4. The Cryptic Case of a Swarthy Stranger 48
PART II: SHAPES 79
5. Adventures of an Egg Man 81
6. The M6bius Molecule 108
7. The Case of the Missing Three-Holed Hollow
Sphere with One Handle 122
PART III: MACHINES 135
8. Turing's Universal Machine 139
9. Did Willy Loman Die in Vain? 151
10. The Machine Who Would Be King 167
11. A Boy and His Brain Machine 190
PART IV: "ONE MAN, ONE VOTE" 213
12. Is Democracy Mathematically Unsound? 215
13. The Quantum Congress 249
Suggestions for Further Reading 261
Index 263
ARCHIMEDES'
REVENGE
INTRODUCTION
There was more imagination in the head of Ar-
chimedes than in that of Homer.
-VOLTAIRE
When Isaac Newton made his famous understatement "If
I have seen further than [others], it is by standing upon
the shoulders of giants," he surely had in mind Archi-
medes of Syracuse, the greatest mathematician of antiq-
uity. Archimedes, however, was also a mechanical genius,
inventing, among other gadgets, the water snail, or Ar-
chimedean screw, a helical pump for raising water for
irrigation. Although little is known about Archimedes'
life or about his assessment of his own work, most com-
mentators suspect that he valued his theoretical mathe-
matical discoveries more than his practical inventions.
Plutarch, for one, writes, "And yet Archimedes pos-
sessed such a lofty spirit, so profound a soul, and such
a wealth of scientific theory, that, although his inventions
had won for him a name and fame for superhuman sa-
gacity, he would not consent to leave behind him any
treatise on this subject, but regarding the work of an en-
gineer and every art that ministers to the needs of life as
ignoble and vulgar, he devoted his earnest efforts only to
those studies the subtlety and charm of which are not
affected by the claims of necessity. " Other commentators
add that even when he dealt with levers, pullies, or other
machines, he was seeking general principles of mechan-
ics, not practical applications.
How much Archimedes truly preferred the theoretical
1
2 ARCHIMEDES' REVENGE
to the practical may never be known. It is clear, however,
that in his work there is tension between theory and ap-
plication, a tension that still pervades mathematics
twenty-two centuries later.
My aim in this book is to sketch the range and scope
of mathematics. I do not pretend that this book is com-
prehensive. Indeed, it is quirky in its choice of subjects.
But it couldn't be otherwise. Mathematics is a discipline
practiced in every university in the world, and it is at
least as broad a field as biology, in which one researcher
tries to understand the AIDS virus while another studies
the socialization of wombats.
I approach mathematics as I do a Chinese menu, trying
dishes here and there, recognizing both common ingre-
dients and distinctive flavors. After only one Chinese
meal, you'll hardly be an expert on Chinese cuisine, but
you'll know much more about Chinese food than some-
one who has never eaten it at all. So it is with mathe-
matics. By dipping into a handful of mathematical topics,
you'll not learn everything that's important in mathe-
matics, but you'll have a much better feel for the subject
than someone who hasn't taken the plunge.
Many books have been written about the philosophical
underpinnings of mathematics, about the extent to which
it is the science of certainty, in that its conclusions are
logically unassailable. Many other works have rhapso-
dized at length about the nature of infinity and the beauty
of higher dimensions. Such philosophical and poetic ex-
cursions have their place, but they are far from the con-
cerns of most working mathematicians. In this book I
give a glimpse of some of the things that mathematicians,
pure and applied, actually do.
I also want to counter a misconception: that any result
in mathematics can be achieved just by laboriously doing
enough computation-that, in other words, if you want
to solve a mathematical problem, it's merely a matter of
doing enough arithmetic. Granted, you and I lack the
Introduction 3
computational skills to attack complex mathematical
problems, but we suspect that those in the know-those
who understand mathematical symbols-can grind out an
answer to almost any problem if they choose to. After
all, we are taught to believe that mathematics is syllogis-
tic, that deducing a mathematical result is as straightfor-
ward as drawing the conclusion "Socrates is mortal"
from the premises "All men are mortal" and "Socrates
is a man." If only mathematics were that simple!
; One of my aims is to convey a sense of the limits of
mathematical knowledge. In each area of mathematics
that we examine, I'll point out what is known and not
known. Sometimes our knowledge- is limited because a
field is young and not many mathematicians have devoted
themselves to it. In other cases, little is known because
the problems are extraordinarily difficult. In still other
cases, there are more fundamental reasons for the math-
ematician's limited knowledge; it can be shown that the
problems are simply immune to quick mathematical so-
lutions.
Mathematics is full of surprises. Number and shape are
among humanity's oldest concerns, and yet much about
them is still not understood. What could be simpler than
the concept of a prime number-an integer greater than
I like 3, 5, 17, or 31 that cannot be evenly divided by an
integer other than I and itself? The ancient Greeks knew
that the supply of primes is inexhaustible, but no one
knows whether the supply of twin primes-pairs of
primes, such as 3 and 5, that differ by 2-is infinite, too.
No one knows whether there's an infinite number of per-
fect numbers, integers like 6 that are equal to the sum of
all of their divisors except, of course, the integer itself
(in this case, 3, 2, and 1). And no one knows if a perfect
number can be odd. Paul Erdos, the great Hungarian
number theorist who is a master of proving basic theo-
rems about primes-at the age of eighteen, he came up
with a celebrated proof that there is always a prime be-
4 ARCHIMEDES' REVENGE
tween every integer greater than I and its double-
believes that mathematicians are nowhere near under-
standing the integers, let alone other kinds of numbers.
"It will be another million years, at least," says Erdos,
"before we understand the primes."
The mathematical understanding of shape is no more
advanced. In two dimensions, many questions remain un-
answered about what shapes can be used to tile a surface,
given certain basic constraints. The three-dimensional
analogue of the tiling problem, the packing of shapes as
densely as possible into a given space, is not solved for
many basic shapes. Lack of theoretical knowledge, how-
ever, need not always stand in the way of pragmatists, as
the designer Ronald Resch demonstrated when he built a
three-and-a-half-story Easter egg.
With fundamental questions about number and shape
still unsettled, it is no wonder that there is much dis-
agreement and confusion about what the computer-a
very complex mathematical tool-can and cannot do. I
have tried to stay clear of mushy metaphysical issues
about the nature of man and machine in favor of pre-
senting what little is known about the theoretical limits
on computing. I discuss the surprising power of Turing's
universal computing machine-a strip of paper divided
into cells. And I look at a probable limitation: computer
scientists think they'll be able to prove that a certain class
of simple-sounding computational problems-including
that of the traveling salesman who wants to choose the
shortest route between a bunch of cities-can never be
efficiently solved by machine (or mathematician). Mov-
ing from theory to practice, I look at the efforts of Hans
Berliner and Danny Hillis to design, respectively, a chess-
playing machine and a general-purpose computer that
takes the idea that "two heads are better than one" to an
amazing extreme. It is much too early to see completely
how these efforts will pan out, but both machines are
Description:An introduction to the delights and challenges of modern mathematics.
