What Is Random?: Chance and Order in Mathematics and Life

217 Pages·1999·4.861 MB·English

by  Edward Beltrami (auth.)

What Is Random?
What Is Random?
chance and order
.
In
mathematics
and life
••••
Edward Beltrami
c
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
©1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1999
Softcover reprint of the hardcover 1s t edition 1999
On the cover: "Reflection of the Big Dipper" by Jackson Pollack.
Credit: Stedelijk Museum, Amsterdam, Holland/Superstock.
All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or ttansmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher.
Library ofCongress Cataloging-in-Publication Data
Beltrami, Edward J.
What is random? : chance and order in mathematics and life I
Edward BeitramÎ.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4612-7156-7 ISBN 978-1-4612-1472-4 (eBook)
DOI 10.1007/978-1-4612-1472-4
1. Probabilities. 2. Chance. 1. Tirle.
QA273.B395 1999
519.2-DC21 99-18389
CIP
Printed on acid-free paper.
987654321
SPIN 10708901
Contents
[1] The Taming of Chance 1
From Unpredictable to Lauful 2
Probability 9
Order in the Large 14
The Normal Law 18
Is It Random? 25
More About the Law ojLarge Numbers 30
Where We Stand Now 33
[2] Uncertainty and Information 35
Messages and Information 36
Entropy 40
Messages, Codes, and Entropy 44
Approximate Entropy 53
v
Contents
Again, Is It Random? 58
The Perception of Randomness 62
[3] Janus-Faced Randomness 65
Is Determinism an Illusion? 66
Generating Randomness 77
Janus and the Demons 82
[4] Algorithms, Information, and Chance 91
Algorithmic Randomness 92
Algorithmic Complexity and Undecidability 102
Algorithmic Probability 109
[5] The Edge of Randomness 117
Between Order and Disorder 118
Self-Similarity and Complexity 129
What Good is Randomness? 143
Sources and Further Readings 145
Technical Notes 153
Appendix A: Geometric Sums 179
Appendix B: Binary Numbers 183
Appendix C: Logarithims 189
References 191
Index 197
vi
A Word About Notation
It is convenient to use a shorthand notation for certain mathe
matical expressions that appear often throughout the book.
For any two numbers designated as a and b the product "a
times b" is written as ab and sometimes as a·b, while "a divided
by b" is a/b. The product of "a multiplied by itselfb times" is
ab, so that, for example, 210 means 1024. The expression 2-n is
synonymous with 1/2n.
In a few places I use the standard notation Va to mean
"the square root of a," as in V25 = 5.
"All numbers greater than a and less than b" is expressed as
(a, b). If "greater than" is replaced by "greater than or equal to"
then the notation is [a, b).
vii
A Word About Notation
A sequence of numbers such as 53371 ... is generally in
dicated by at a2 a3 ... , in which the subscripts 1, 2, 3, ... in
dicate the "first, second, third, and so on" terms of the se
quence, which can be finite in length or even infinite (such as
the unending array of all even integers).
VIII
Preface
We all have memories of peering at a TV screen as myriad lit
tle balls chum about in an urn until a single candidate, a num
ber inscribed on it, is ejected from the container. The hostess
hesitantly picks it up and, pausing for effect, reads the lucky
number. The winner thanks Lady Luck, modem descendant
of the Roman goddess Fortuna, blind arbiter of good fortune.
We, as spectators, recognize it as simply a random event and
know that in other situations Fortuna's caprice could be
equally malicious, as Shirley Jackson's dark tale "The Lottery"
chillingly reminds us.
The Oxford Dictionary has it that a "random" outcome is
one without perceivable cause or design, inherently unpre
dictable. But, you might protest, isn't the world around us
ix
Preface
governed by rules, by the laws of physics? If that is so, it should
be possible to detennine the positions and velocities of each
ball in the urn at any future time, and the uncertainty ofw hich
one is chosen would simply be a failing of our mind to keep
track of how the balls are jostled about. The sheer enonnity of
possible configurations assumed by the balls overwhelms our
computational abilities. But this would be only a temporary
limitation: A sufficiently powerful computer could conceiv
ably do that brute task for us, and randomness would thus be
simply an illusion that can be dispelled. After thinking about
this for a while you may begin to harbor a doubt. Although na
ture may have its rules, the future remains inherently un
knowable because the positions and velocities of each ball can
never really be ascertained with complete accuracy.
The first view of randomness is of clutter bred by compli
cated entanglements. Even though we know there are rules,
the outcome is uncertain. Lotteries and card games are gener
ally perceived to belong to this category. More troublesome is
that nature's design itself is known imperfectly, and worse, the
rules may be hidden from us, and therefore we cannot specity
a cause or discern any pattern of order. When, for instance, an
outcome takes place as the confluence of totally unrelated
events, it may appear to be so surprising and bizarre that we say
that it is due to blind chance. Jacques Monod, in his book
Chance and Necessity, illustrates this by the case of a man hurry
ing down a street in response to a sudden phone call at the
same time that a roofw orker accidentally drops a hammer that
hits the unfortunate pedestrian's head. Here we have chance
x
Preface
due to contingency, and it doesn't matter whether you regard
this as an act of divine intervention operating according to a
predestined plan or as an unintentional accident. In either case
the cause, if there is one, remains indecipherable.
Randomness is the very stuff oflife, looming large in our
everyday experience. Why else do people talk so much about
the weather, traffic, and the financial markets? Although un
certainty may contribute to a sense ofa nxiety about the future,
it is also our only shield against boring repetitiveness. As I will
argue in the final chapter, chance provides the fortuitous acci
dents and capricious wit that give life its pungency. It is im
portant, therefore, to make sense of randomness beyond its
anecdotal meanings. To do this we employ a modest amount
of mathematics in this book to remove much of the vagueness
that encumbers the concept of random, permitting us to quan
tifY what would otherwise remain elusive. The mathematics
also provides a framework for unifying our understanding of
how chance is interpreted from the diverse perspectives ofp sy
chologists, physicists, statisticians, computer scientists, and
communication theorists.
In the first chapter I tell the story of how, beginning a few
centuries ago, the idea of uncertainty was formalized into a
theory of chance events, known today as probability theory.
Mathematicians adopt the convention that selections are made
from a set of possible outcomes in which each event is equally
likely, though unpredictable. Chance is then asked to obey
certain rules that epitomize the behavior of a perfect coin or an
ideal die, and from these rules one can calculate the odds.
xi