Table Of ContentNEW f p i t i o n
D O E S G O D
PLAY D I C E ?
The New Mathematics of Chaos
PENGUIN BOOKS
DOES GOD PLAY DICE?
Ian Stewart was born in Folkestone in 1945. He graduated in mathematics from
CambridgeandobtainedaPh.D.fromtheUniversityofWarwick,whereheisnowPro-
fessor of Mathematics. He has also held visiting positions in Germany, New Zealand,
Connecticut and Texas. He is an active research mathematician with over a hundred
published papers, and he takes a particular interest in problems that lie in the gaps be-
tween pure and applied mathematics.
Ian Stewart has written or co-authored over sixty books, including Nature's Num-
bers, shortlisted for the 1996 Rhone-Poulenc Prize for Science Books; The Collapse of
Chaos; Fearful Symmetry; From Here to Infinity; Game, Set and Math; Another Fine
Math You've Got Me Into; The Problems of Mathematics; the bestselling Does God
Play Dice?; Nature's Numbers; From Here to Infinity; Figments of Reality; and Life's
OtherSecret(severalofwhicharepublishedinPenguin);andthreemathematicalcom-
ic books published in French. He has written for a wide range of newspapers and mag-
azines in the UK, Europe and the USA, including Nature, Focus, Discover and The
Sciences. He is mathematics consultant for New Scientist and writes the ‘Mathemati-
cal Recreations’ columns in Scientific American. He also writes science fiction stories
and has made numerous radio and television appearances. In 1995 the Royal Society
awarded himtheMichael Faraday Medal fortheyear'smostsignificant contribution to
the public understanding of science, and he has been selected for the 1997 Communi-
cator Award of the Joint Policy Board for Mathematics in the USA.
Does God Play Dice?
THE NEW MATHEMATICS OF CHAOS
Second edition
Ian Stewart
PENGUIN BOOKS
PENGUIN BOOKS
Published by the Penguin Group
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www.penguin.com
First published by Basil Blackwell 1989
Published in Penguin Books 1990
Second edition published 1997
12
Copyright © Ian Stewart, 1989, 1997
All rights reserved
The moral right of the author has been asserted
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of
trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any
form of binding or cover other than that in which it is published and without a similar condition including this con-
dition being imposed on the subsequent purchaser
ISBN: 978-0-14-192807-4
Contents
Preface to the Second Edition
Prologue Clockwork or Chaos?
1 Chaos from Order
2 Equations for Everything
3 The Laws of Error
4 The Last Universalist
5 One-way Pendulum
6 Strange Attractors
7 The Weather Factory
8 Recipe for Chaos
9 Sensitive Chaos
10 Fig-trees and Feigenvalues
11 The Texture of Reality
12 Return to Hyperion
13 The Imbalance of Nature
14 Beyond the Butterfly
15 Von Neumann's Dream
16 Chaos and the Quantum
17 Farewell, Deep Thought
Epilogue Dicing with the Deity
Further Reading
Illustration Acknowledgements
Index
Preface to the Second Edition
ThefirsteditionofDoesGodPlayDice?,publishedin1989,didn'thaveapreface.I
was going through a period when I didn't write prefaces because I thought that nobody
ever reads them, so it started with a prologue instead. The prologue remains, but now
there's a preface to go with it – after all, two prologues would be overkill. If you al-
readyownthefirsteditionandarewonderingwhetherthisoneisdifferentenoughtobe
worth buying, you should either read this bit or thumb through Chapters 14–17 where
mostofthenewstufflives.Ifnot,justbuyitnow,OK?Youcandecidewhethertoread
the preface when you get the book home.
‘Chaos’ is not just a trendy word for random. In the sense now prevalent in science,
it is an entirely new and different concept. Chaos occurs when a deterministic (that is,
non-random) system behaves inanapparently random manner.Itmay soundparadoxi-
cal,but‘apparently’hidesmanysins.Thebigdiscoveryofthelastdecadeisthatchaos
is just as common as traditional types of regular behaviour, such as steady states and
periodiccycles.Inretrospect,there'snothingverysurprisingaboutchaos.Fromtoday's
viewpoint, and with the benefit of 20/20 hindsight, it's very easy to understand how
chaos arises and why it often occurs. Despite this lots of people, many of them sci-
entists, still talk about chaos as if it is something weird and exotic. Sorry, but it isn't.
Chaos is just as down-to-earth as periodic cycles. But over several centuries we've got
used to periodic cycles, whereas we've only just stumbled upon the existence of chaos
andwehaven'tgotusedtoityet.That'snotasurpriseeither:chaosismuchmoresubtle
and intricate.
Chaos has come a long way since 1989. It has transmuted, especially in the popular
press,intosomethingcalled‘chaostheory’.Ithinkit'samistaketoviewchaosasathe-
oryinitsownright,butIappreciate thatjournalists needacatchyphrasetosumupthe
area, and what else would you call it? So I sometimes use that phrase myself, though
on the whole I use it to refer to the popular image of chaos, as distinct from that of
practising scientists. However, chaos isn't a theory. It's a concept, and one that cannot
sensibly be separated from the rest of dynamics. It's an idea that cuts across all of the
traditionalsubjectboundariesofscience.It'samissingpiecefromavastjigsawpuzzle.
It's a far-reaching unification of order and disorder. But whatever it is, chaos no more
deservestobeisolatedasatheoryinitsownrightthan‘skeletontheory’deservestobe
isolated from zoology.
There is a new theory, which goes by such names as nonlinear systems theory, dy-
namical systems theory, or nonlinear dynamics: new not in the sense that nothing like
it ever existed before, but in the sense that it has really ‘taken off’ and deserves to be
consideredasatheoryinitsownright.Oftenthisiswhatpeoplereallymeanwhenthey
say‘chaostheory’.Infactthereareatleasttwomeaningstotheword‘theory’Thereis
the sense usedinthe phrase ‘quantum theory’ or‘relativity theory’ –astatement about
hownaturebehaves.Theutilityofsuchatheorydependsuponitmatchingnaturesuffi-
cientlywell.Nonlinearsystemstheoryisatheoryintheothersense,acoherentbodyof
mathematical knowledge with a clear and consistent identity. As such, there is no seri-
ousquestionaboutitscorrectness:whenmathematicsiswrongthemistakesareusually
glaring. The big question is: does the concept of chaos afford new scientific insights?
(If not, then we'll have to scrub nonlinear dynamics: you can't divorce one from the
other.) Can chaos theory in the mathematical sense become the basis for new theories
in the scientific sense?
This is where chaos can become controversial, because it is no longer just a matter
ofcheckingthemathematicsandmakingsuretherearen'tanyerrors.Analogously,cal-
culus is a valid mathematical theory, but that doesn't imply that every application of
calculus to science must be right. If you set up the wrong differential equations when
you model the motion of the Moon, then no matter how correctly you apply calculus,
you'llgetnonsense.Thesameistrueifyourtheoreticalmodelgenerateschaos:thelink
fromthemodeltochaosmaybeimpeccable,butwhataboutthelinkfromrealitytothe
model?
Does God Play Dice? has two distinct themes. One is to explain the mathematical
concept of chaos, and why it is both natural and inevitable. The other is to ask: does
chaos occur in the real world? In order for that question to make sense, it has to be re-
formulated. No mathematics occurs in the real world. What it does is model the real
world in a useful manner. The geometry of the circle helps us understand why wheels
roll smoothly, but you won't find a genuine mathematical circle on a car. You can find
two sheep, two apples, or two bookends in the real world, but you'll never encounter
the number two as such. So the question should be: ‘does the mathematical concept of
chaos model the real world in a useful manner, and does it help us understand some of
the things we see?’
If you look at what's appearing in the scientific journals, it is absolutely clear that
the answer is ‘yes’. In 1995 I went to a conference in Utah on Applications of Dynam-
ical Systems, run by the Society for Industrial and Applied Mathematics. SIAM is the
premier professional body for the applied mathematicians of the most technologically
advancedcountryintheworld,notsomeMickeyMouseconglomerationonthelunatic
fringe. Some five hundred mathematicians attended over a period of four days, and
there were over two hundred research talks (in parallel sessions). About half of them
were either about chaos, or about issues that arise from it, such as new methods of da-
ta analysis. So if anyone tries to tell you that chaos is nothing but media hype, they're
wrong.It'sbeenaroundtoolongforthat,anditnowrunsfartoodeeplyinthescientific
consciousness. Of course this level of activity does not imply that every proposed ap-
plication of chaos is correct. The assumption that once chaos theory is ‘proved’ in one
area then you are automatically forced to swallow it in all areas – which I think is one
reason why some critics are so relentlessly negative – stems from the confusion about
the two meanings of ‘theory’ that I've just mentioned. Each application must prove its
worth in its own right and within its own area of science.
ThisneweditionofDoesGodPlayDice?differsfromitspredecessormainlybyin-
cluding new material on applications. I've left the original edition virtually untouched:
nothing has happened since it was published to require major surgery. There are three
completelynewchaptersinsertedneartheend.Thefirstisonpredictioninchaoticsys-
tems, which is perfectly possible, depending on what you want to predict; it also dis-
cusses various related issues. I've included several new applications, ranging from the
pulsationsofvariablestarstoqualitycontrolinthespring-makingindustry.Thesecond
new chapter is about the control of chaotic systems, a potential source of practical ap-
plicationsandacasestudyinwhatadvantagesaccruewhenyoulearnhowtousechaos
instead of trying to pretend that it doesn't exist. Among the applications here are ways
to steer artificial satellites more economically, and work heading in the direction of in-
telligent heart pacemakers.
The third new chapter is much more speculative. It is an attempt to explain how the
concept of chaos might lead to a new answer to Einstein's famous question, the title of
this book.Einstein wasworryingaboutquantum mechanics, which isgenerally heldto
be irreducibly probabilistic. Is it possible that the apparent randomness of the quantum
world is actually deterministic chaos? Would the course of physics have been different
if chaos had been discovered before quantum mechanics? In 1989 there wasn't a great
dealtosayaboutthesequestions,buttodaythereis.Thereisonequitespecificpropos-
al in the scientific literature: speculative, but based on solid discoveries, some of them
very new. It's a fascinating story, and all of the ingredients are good science: it is only
the overall mix that is speculative. And if you don't speculate, you don't accumulate.
I have also brought the earlier chapters up to date. That at least some instances of
turbulence in fluids are due to chaos is now absolutely certain. There are new results
onthedynamicsofthesolarsystem,whichitseemswillnotsurviveinitspresentform
formuchmorethananotherbillionyearsorso.Theuniverseisclumpieronevenlarger
scales than we thought. Chaos in at least some ecosystems is close to being an estab-
lishedfact.Fractalgeometryhasacquiredseriouscommercialuses.Mathematicaltech-
nique has advanced to the point that we can nowprove, in all rigour,that the model set
upbythemeteorologist EdwardLorenzdefinitely doesleadtochaos.Thisisbadnews
for those stalwarts of orthodoxy who maintained that the appearance of chaos was due
to computer error, but good news for the logical underpinnings of nonlinear dynamics.
Finally,acompaniontochaostheoryhasnowappeared,knownascomplexity theo-
ry. Chaos theory tells us that simple systems can exhibit complex behaviour; complex-
ity theory tells us that complex systems can exhibit simple ‘emergent’ behaviour. No
discussionofchaosnowadayswouldbecompletewithoutsomementionofcomplexity
theory, so I've put it in the final chapter. Complexity theory definitely is controversial,
but it brings a welcome breath of fresh air to a sackful of stuffy overblown old-fash-
ioned linear theories. I'm absolutely convinced that over the next few decades the kind
ofthinkingtowardswhichcomplexitytheoristsarecurrentlygropingwillturnouttobe
offundamentalimportanceinnearlyeveryareaofscientificactivity.Idon'tthinkcom-
plexitytheoryholdstheanswers–yet–butIdothinkitoffersamuchmoreinteresting
angle on the questions, which in turn suggests new ways to look for the answers.
I'mnottryingtosellyouchaos.I'mnotaprophetofanewreligionseekingconverts.
Idon'twantyourfaith–perishthethought.WhatI'mtryingtodoistosetbeforeyou,in
ascomprehensibleaformasIcanmanage,theinformationthatyouneedtomakeyour
own judgements about chaos's present achievements and future potential. I've tried to
make it clear when I'm being speculative. The rest of the time I'm presenting ideas or
results that have been published in the serious scientific and mathematical literature.
That doesn't mean that they are necessarily right, but it does mean that they are re-
spectable…
I'mnowbecominghorriblyawareofthereasonwhypeopledon'treadprefaces:they
dogoon,don'tthey?AndIhaven'ttoldyouaboutallthenewapplicationsofchaosthat
I had to leave out for lack of room – chaos in the Earth's molten interior, in the aurora
borealis, in the deep structure of spacetime, in ant colonies, in coding theory and com-
munication, in the voices of opera singers…
I think I'll stop.
Ian Stewart
Coventry
January 1996.
