Impossibility - The Limits of Science and the Science of - Angelfire

282 Pages·2000·3.82 MB·English

by  John D. Barrow

The meaning of the world is the separation of wish and fact.
KURT GÖDEL
In memory of
Roger Tayler
Impossibility
The Limits of Science and
the Science of Limits
JOHN D. BARROW
Astronomy Centre
University of Sussex
OXFORD UNIVERSITY PRESS • OXFORD
1998
Preface
The Preface is the most important part
of the book. Even reviewers read a preface.
PHILIP GUEDALLA
Both scientists and philosophers are much concerned with impossibilities.
Scientists like to show that things widely held to be impossible are in fact entire-
ly possible; philosophers, by contrast, are more inclined to demonstrate that
things widely regarded as perfectly feasible are in fact impossible. Yet, para-
doxically, science is only possible because some things are impossible.
The incontrovertible evidence that Nature is governed by reliable 'laws' allows
us to separate the possible from the impossible. Only those cultures for whom
there existed a belief that there was a distinction between the possible and the
impossible provided natural breeding grounds for scientific progress. But
'impossibility' is not only about science. In the pages that follow we shall look at
some of the ways in which the impossible in art, literature, politics, theology,
and logic has stimulated the human mind to take unexpected steps: revealing
how the concept of the impossible sheds new light on the nature and content of
the actual.
The idea of the impossible rings alarm bells in the minds of many. To some,
any suggestion that there might be limits to the scope of human understanding
of the Universe or to scientific progress is a dangerous meme that undermines
confidence in the scientific enterprise. Equally uncritical, are those who enthu-
siastically embrace any suggestion that science might be limited because they
suspect the motives and fear the dangers of unbridled investigation of the
unknown.
At the end of each century there seems to arise a stock-taking in science. We
shall see that at the end of the last century the issue of the limits of science
became a live one and attempts were made to pick out problems that could
never be solved. These problems still make interesting reading. But what will
people say about our concerns in a hundred years time? As we near the end of
the twentieth century we look back on an extraordinary century of progress. Yet
it is progress that possesses some extraordinary characteristics. A pattern has
emerged in many spheres of inquiry in which a scientific theory becomes so
successful in the quantity and quality of its accurate predictions that its practi-
VIII PREFACE
tioners start to wonder whether the end is in sight—whether their theory might
be able to explain everything within its encompass. But then something strange
happens. The theory predicts that it cannot predict. It turns out to be not simply
limited in scope, but self-limiting. This pattern is so strikingly recurrent that
it suggests to us that we can recognize mature scientific theories by their self-
limiting character. Such limits arise not merely because theories are inadequate,
inaccurate, or inappropriate: they tell us something profound about the nature
of knowledge and the implications of investigating the Universe from within.
Our study of the limits of science and the science of limits will take us from
the consideration of practical limits of cost, computability, and complexity to
the restrictions imposed on what we can know by our location in the middle of
the Nature's spectra of size, age, and complexity. We shall speculate about our
possible technological futures and locate our current abilities on the spectrum
of possibilities for the manipulation of Nature in the realms of the large, the
small, and the complex. But practicalities are not the only limits we face. There
may be limits imposed by the nature of our humanity. The human brain was not
evolved with science in mind. Scientific investigation, like our artistic senses, are
by-products of a mixed bag of attributes that survived preferentially because
they were better adapted to survive in the environments they faced in the far
distant past. Perhaps those ambiguous origins will compromise our quest for
an understanding of the Universe? Next, we shall start to pick at the edges of
possible knowledge. We shall learn that many of the great cosmological
questions about the beginning, the end, and the structure of our Universe are
unanswerable. Despite the confident exposition of the modern view of the
Universe by astronomers, these expositions are invariably simplified in ways
that disguise the reasons why we cannot know whether or not the Universe is
finite or infinite, open or closed, of finite age or eternal. Finally, we delve into
the mysteries of the famous theorems of Godel concerning the limitations of
mathematics. We know that there must exist statements of arithmetic whose
truth we can never confirm or deny. What does this really mean? What is the
fine print on this theorem? What are its implications for science? Does it mean
that there are scientific questions that we can never answer? We shall see that the
answers are unexpected and lead us to consider the possible meaning of incon-
sistency in Nature, of the paradoxes of time travel, the nature of freewill and the
workings of the mind. Finally, we shall explore some of the strange implications
of trying to pass from the consideration of individual choices to collective
choices. Whether it is the outcome of an election or the making up of one's
mind in the face of the brain's competing options, we find a deep impossibility
that may have ramifications throughout the domain of complex systems.
Here, in this strange world of fundamental limits we learn that worlds that are
complex enough for certain individualities to be manifest necessarily display an
PREFACE IX
open-endedness that defies capture within the confines of a single logical sys-
tem. Universes that are complex enough to give rise to consciousness impose
limits on what can be known about them from within.
By the end of our journey, I hope the reader will have come to see that there is
more to impossibility than first meets the eye. Its role in our understanding of
things is far from negative. Indeed, I believe that we will gradually come to
appreciate that the things that cannot be known, that cannot be done, and can-
not be seen, define our Universe more clearly, more completely, and more
sharply than those that can.
This book is dedicated to the memory of Roger Tayler, who sadly did not live
to see it finished. His selfless service to his colleagues at Sussex and to the wider
community of astronomers in Britain and around the world won him the
respect, admiration, and friendship ,of scientists everywhere. He is greatly
missed.
I would like to thank many people who helped me by their comments or
advice, or who provided pictures and references, especially David Bailin, Per
Bak, Margaret Boden, Michael Burt, Bernard Carr, John Casti, Greg Chaitin,
John Conway, Norman Dombey, George Ellis, Mike Hardiman, Susan Harrison,
Jim Hartle, Piet Hut, Janna Levin, Andrew Liddle, Andre Linde, Seth Lloyd,
Harold Morowitz, David Pringle, Martin Rees, Nicholas Rescher, Mark Ridley,
David Ruelle, John Maynard Smith, Lee Smolin, Debbie Sutcliffe, Karl Svozil,
Frank Tipler, Joseph Traub, and Wes Williams. My wife Elizabeth helped in
many practical ways, and accommodated innumerable new pieces of paper in
the house with surprising good humour, whilst the subject of this book merely
provoked our children, David, Roger, and Louise, to worry that there might
indeed be fundamental limits on the use of the telephone.
Brighton
November 1997 J.D.B.
Contents
Chapter 1: The art of the impossible 1
The power of negative thinking 1
Of faces and games 3
Those for whom all things are possible 7
Paradox 12
Visual paradox 13
Linguistic paradox 19
Limits to certainty 21
A cosmic speed limit 24
Summary 26
Chapter 2: The hope of progress 27
Over the rainbow 27
The voyage to Polynesia via Telegraph Avenue 31
Progress and prejudice 37
The big idea of unlimited knowledge 41
Negativism 45
Some nineteenth-century ideas of the impossible 48
Summary 55
Chapter 3: Back to the future 57
What do we mean by the limits of science? 47
Possible futures 58
Higgledy-piggledyology 65
Selective and absolute limits 68
Will we be builders or surgeons? 70
The futures market 72
How many discoveries are there still to be made? 83
Summary 84
Chapter 4: Being human 85
What are minds for? 85
Xii CONTENTS
Counting on words 91
Modern art and the death of a culture 95
Complexity matching: climbing Mount Improbable 96
Intractability 100
The frontier spirit 107
The end of diversity 109
Does science always bring about its own demise? Ill
Death and the death of science 113
The psychology of limits 114
Summary 116
Chapter 5: Technological limits 118
Is the Universe economically viable? 118
Why we are where we are 120
Some consequences of size 122
The forces of Nature 125
Manipulating the Universe 128
Criticality: the riddle of the sands 138
Demons: counting the cost 142
Two types of future 147
Is technological progress inevitable (or always desirable)?—a fable 150
Summary 153
Chapter 6: Cosmological limits 155
The last horizon 155
Inflation—still crazy after all these years 164
Chaotic inflation 169
Is the Universe open or closed? 170
Eternal inflation 171
The natural selection of universes 274
Topology 176
Did the Universe have a beginning? 178
Naked singularities: the final frontier 182
Dimensions 184
Symmetry-breaking 185
Summary 188
Chapter 7: Deep limits 190
Patterns in reality 190
Paradoxes 195
CONTENTS Xlll
Consistency 197
Time travel: is the Universe safe for historians? 199
Completeness 207
Impossible constructions 211
Metaphorical impossibilities 275
Summary 276
Chapter 8: Impossibility and us 218
Gödel's theorem and physics 218
Does Gödel stymie physics? 227
Gödel, logic, and the human mind 230
The problem of free will 232
The reaction game 236
Mathematics that comes alive 238
A stranger sort of impossibility 239
The Arrow Impossibility Theorem 242
Summary 246
Chapter 9: Impossibility: taking stock 248
Telling what is from what isn't 248
Notes 253
Index 275
CHAPTER 1
The art of the impossible
If an elderly but distinguished scientist says that something is possible he is
almost certainly right, but if he says that it is impossible he is very
probably wrong.
ARTHUR C.CLARKE
The power of negative thinking
That's what I like about Lord Young. While you all bring me problems,
he brings me solutions.
MARGARET THATCHER
Bookshelves are stuffed with volumes that expound the successes of the mind
and the silicon chip. We expect science to tell us what can be done and what is to
be done. Governments look to scientists to improve the quality of life and
safeguard us from earlier 'improvements'. Futurologists see no limit to human
inquiry, while social scientists see no end to the raft of problems it spawns. The
contemplation by our media of science's future path is dominated by our
expectations of great interventions: cracking the human genetic code, curing all
our bodily ills, manipulating the very atoms of the material universe, and,
ultimately, fabricating an intelligence that exceeds our own. Human progress
looks more and more like a race to manipulate the world around us on all scales,
great and small.
It would be easy to write such a scientific success story. But we have another
tale to tell: one that tells not of the known but of the unknown; of things
impossible; of limits and barriers which cannot be crossed. Perhaps this sounds
a little perverse. Surely there is little enough to say about the unknown without
dragging in the unknowable? But the impossible is a powerful and persistent
notion. Unnoticed, its influence upon our history has been deep and wide; its
place in our picture of what the Universe is like at its deepest levels is
undeniable. But its positive role has escaped the critics' attention. Our goal is to
uncover some of the limits of science: to see how our minds' awareness of the
impossible gives us a new perspective on reality.
When we are young we think we know everything. But if we grow wiser as we
2 THE ART OF THE IMPOSSIBLE
grow older we will gradually discover that we know less than we thought. The
poet W.H. Auden wrote of human development that
between the ages of twenty and forty we are engaged in the process of discovering
who we are, which involves learning the difference between accidental limitations
which it is our duty to outgrow and the necessary limitations of our nature beyond
which we cannot trespass with impunity.'
Our collective knowledge of the nuts and bolts of the Universe matures in a
similar way. Some knowledge is simply the accumulation of more facts, broader
theories, and better measurements by more powerful machines. Its rate of
growth is always limited by costs and practicalities that we steadily overcome by
attrition, little by little. But there is another form of knowledge. It is the
awareness that there are limits to one's theories even when they are right. While
the modest investigator might always suspect that there are things that will
remain beyond our reach, this is not quite what we have in mind. There is a path
of discovery that unveils limits that are an inevitable by-product of the knowing
process. Discovering what they are is a vital part of understanding the Universe.
This means that the investigation of the limits of our knowledge is more than a
delineation of the boundaries of the territory that science can hope to discover.
It becomes a crucial feature in our understanding of the nature of this collective
activity of discovery that we call science: a paradoxical revelation that we can
know what we cannot know. This is one of the most striking consequences of
human consciousness.
There is an intriguing pattern to many areas of deep human inquiry.
Observations of the world are made; patterns are discerned and described by
mathematical formulae. The formulae predict more and more of what is seen,
and our confidence in their explanatory and predictive power grows. Over a
long period of time the formulae seem to be infallible: everything they predict is
seen. Users of the magic formulae begin to argue that they will allow us to
understand everything. The end of some branch of human inquiry seems to be
in sight. Books start to be written, prizes begin to be awarded, and of the giving
of popular expositions there is no end. But then something unexpected
happens. It's not that the formulae are contradicted by Nature. It's not that
something is seen which takes the formulae by surprise. Something much more
unusual happens. The formulae fall victim of a form of civil war: they predict
that there are things which they cannot predict, observations which cannot be
made, statements whose truth they can neither affirm nor deny. The theory
proves to be limited, not merely in its sphere of applicability, but to be self-
limiting. Without ever revealing an internal inconsistency, or failing to account
for something we have seen in the world, the theory produces a 'no-go'
statement. We shall see that only unrealistically simple scientific theories avoid