New Theories of Everything

also by john d. barrow The Left Hand of Creation (with Joseph Silk) L’Homme et le Cosmos (with Frank J. Tipler) The Anthropic Cosmological Principle (with Frank J. Tipler) The World Within the World The Artful Universe Pi in the Sky Perchè il mondo è matematico? Impossibility The Origin of the Universe Between Inner Space and Outer Space The Universe that Discovered Itself The Book of Nothing The Constants of Nature: From Alpha to Omega The Inﬁnite Book The Artful Universe Expanded

john d. barrow

New Theories of Everything the quest for ultimate explanation

‘I am very interested in the Universe—I am specialising in the Universe and all that surrounds it’ — peter cook

Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With oﬃces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © John D. Barrow 2007 Published in the United States by Oxford University Press Inc., New York The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2007 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by Clays Ltd., St Ives plc ISBN 978–0–19–280721–2 1 3 5 7 9 10 8 6 4 2

to roger Who still believes there should always be something rather than nothing

Preface to the Second Edition

I was pleased to take the opportunity, provided by the Press, to prepare a new edition of Theories of Everything. Interest in this subject has continued unabated since my ﬁrst attempts to explain their scope and limitations, and to place them in a broader cultural context than that of mathematical physics. Many new possibilities have emerged in the pursuit of a ﬁnal string theory, and cosmology has taken an unexpected path into a realm populated by many other possible universes. Both developments have undermined the naïve expectations of many, that a Theory of Everythng would uniquely and completely specify all the deﬁning quantities of the Universe that make it a possible home for life. There is a long way to go before the physicists’ Theory of Everything is formulated and decisively tested. In the meantime, I hope that this extended survey of the newest developments will help point readers in the right direction and illuminate the way. John D. Barrow Cambridge, February 2007

Preface to the First Edition

‘Everything’ is a big subject. Yet modern scientists believe they have stumbled upon a key which unlocks the mathematical secret at the heart of the Universe: a discovery that points them towards a monumental ‘Theory of Everything’ which will unite all the laws of Nature into a single statement that reveals the inevitability of everything that was, is, and is to come in the physical world. Such dreams are not new; Einstein wasted the latter part of his life in a fruitless and isolated quest for just this Theory of Everything. But today such schemes are not to be found only on the desks of a few maverick thinkers and unconstrained speculators; they have entered the mainstream of theoretical physics and are worked upon by a growing population of the world’s brightest young thinkers. This turn of events raises many deep questions. Can their quest really succeed? Can our understanding of the logic underlying physical reality be completed? Do we forsee a day when fundamental physics will be complete, leaving only the complex details latent within those laws to be unravelled? Is this truly the new frontier of abstract thought? This book is an attempt to describe what the challenge facing Theories of Everything really is; to pick out those aspects of things which must be understood before we can have any right to claim that we understand them. We shall try to show that while Theories of Everything, as currently conceived, may well prove necessary if we are to understand the Universe around and within us, they are far from suﬃcient. We shall introduce the reader to those extra ingredients which we need to complete our understanding of what is, and in so doing we aim to display many new ideas and speculations which transcend traditional thinking about the scope and structure of scientiﬁc inquiry. Numerous people have helped this book come to completion. The Senatus of the University of Glasgow invited the author to deliver a series of Giﬀord Lectures at the University of Glasgow in January of 1988 and this book elaborates upon the content of some of those lectures. I am particularly indebted to Neil Spurway for his gracious help with everything associated with those lectures. For advertent or inadvertent comments and discussions which have helped in the writing of this book I am grateful to David Bailin, Margaret Boden, Danko Bosanac, Gregory Chaitin, Paul Davies, Bernard d’Espagnat,

viii

preface to the first edition

Jeﬀrey Friedman, Michael Green, Chris Isham, John Manger, Bill McCrea, Leon Mestel, John Polkinghorne, Aaron Sloman, John Maynard Smith, Neil Spurway, Euan Squires, René Thom, Frank Tipler, John Wheeler, Denys Wilkinson, Peter Williams, and Tom Willmore. Writing a book can be a miserable business, not only for the author, but for all those in his immediate orbit. The most perceptive reﬂection upon this situation was one made by the late Sir Peter Medawar. It applies not only to the activities of authors, but to obsessives of many sorts: ‘. . . it is a proceeding that makes one rather inhuman, selﬁshly guarding every second of one’s time and becoming inattentive about personal relationships; one soon formed the opinion that anyone who used three words where two would have done was a bore of insuﬀerable prolixity whose company must at all times be shunned. A danger sign that fellow-obsessionals will at once recognize is the tendency to regard the happiest moments of your life as those that occur when someone who has an appointment to see you is prevented from coming.’ Because of the danger of such distortions, family members require special thanks for their patience and forbearance in the face of frequent neglect. Elizabeth has supplied her constant support in innumerable ways; without it this work would never have begun. Finally, our children, David, Roger, and Louise, have shown a keen and unnerving interest in the progress of the manuscript without which the book would undoubtedly have been ﬁnished in half the time. J.D.B. Brighton, September 1990

acknowledgements Figure 7.3 is reprinted by permission of the publishers from Mind Children by Hans Moravec, Cambridge, Mass.: Harvard University Press, Copyright © 1988 by the President and Fellows of Harvard College. Figs 7.2 and 7.4 are adapted from the same source. Figure 7.6 is copyright © R. V. Solé, reproduced by permission of the artist.

Contents

Ultimate explanation

An eightfold way Myths Creation myths Algorithmic compressibility

1 4 8 10

Laws

The legacy of law The quest for unity Roger Boscovich Symmetries Inﬁnities—to be or not to be? From strings to ‘M’ A ﬂight of rationalistic fancy Goodbye to all that

14 17 19 22 26 32 36 43

Initial conditions

At the edge of things Axioms Mathematical Jujitsu Initial conditions and time symmetry Time without time Cosmological time The problem of time Absolute space and time How far is far enough? The quantum mystery of time Quantum initial conditions The great divide

44 45 51 61 62 66 76 78 83 85 88 90

Forces and particles

The stuﬀ of the Universe The copy-cat principle Elementarity

93 95 100

x contents

8 9

The atom and the vortex A world beside itself

102 104

Constants of Nature

110

The importance of being constant Fundamentalism What do constants tell us? Varying constants The cosmological constant

110 112 117 124 128

Broken symmetries

136

The never-ending story Broken symmetry Natural theology: A tale of two tales The ﬂaws of nature Chaos Chance The unpredictability of sex Symmetry-breaking in the Universe

136 138 140 143 145 148 152 154

Organizing principles

160

Where the wild things are Big AL Time Being and becoming organized The arrow of time Far from equilibrium The sands of time The way of the world

160 169 173 176 180 182 185 188

Selection effects

192

Ubiquitous bias

192

Is ‘pi’ really in the sky?

202

In the centre of immensities The number of the rose Philosophies of mathematics What is mathematics? Mathematics and physics: An eternal golden braid The intelligibility of the world Algorithmic compressibility rides again Continuity—a bridge too far?

202 204 206 212 219 224 231 233

contents The secret of the Universe Is the Universe a computer? The unknowable

236 238 242

Select Bibliography Index

247 256

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chapter 1

Ultimate explanation I have yet to see any problem, however complicated, which when you looked at it in the right way, did not become still more complicated. — poul anderson

an eightfold way It seemed to me a superlative thing—to know the explanation of everything, why it comes to be, why it perishes, why it is. — socrates

How, when, and why did the Universe come into being? Such ultimate questions have been out of fashion for centuries. Scientists grew wary of them; theologians and philosophers grew weary of them. But suddenly scientists are asking such questions in all seriousness and theologians ﬁnd their thinking pre-empted and guided by the mathematical speculations of a new generation of scientists. Ironically, few theologians have an adequate training in physics to keep abreast of the details, and few physicists have a suﬃcient appreciation of the wider questions to make a fruitful dialogue easy. The theologians think they know the questions but cannot understand the answers. The physicists think they know the answers but don’t know the questions. An optimist might thus regard a dialogue as a recipe for enlightenment, whilst the pessimist might predict the likely outcome to be a state in which we ﬁnd ourselves knowing neither the questions nor the answers. Modern physicists believe they have stumbled upon a key which leads to the mathematical secret at the heart of the Universe—a discovery that points towards a ‘Theory of Everything’, a single all-embracing picture of all the laws of Nature from which the inevitability of all things seen must follow with

2 ultimate explanation

unimpeachable logic. With possession of this cosmic Rosetta Stone, we could read the book of Nature in all tenses: we could understand all that was, is, and is to come. Of such a prospect, there has always been speculation but never conﬁdence. But is this conﬁdence now misplaced? This is one of the questions that the reader will be in a position to answer after turning the ﬁnal page of this book. It is our intention to spell out the diﬀerent ingredients that must comprise any scientiﬁc understanding of the Universe in which we live. These we shall ﬁnd to be more diverse and slippery than has fondly been imagined by the purveyors of Theories of Everything. Of course, we must be circ*mspect in our use of such a loaded term as ‘Everything’. Does it really mean everything: the works of Shakespeare, the Taj Mahal, the Mona Lisa? No, it doesn’t. And the way in which such particulars of the world ﬁt into the general scheme of things we shall discuss at some length in the pages to come. It is a vital distinction that needs to be made in our approach to the study of Nature. For we might like to know if there are things which cannot be straitjacketed into the mathematically determined world of science. We shall see that there are, and we will attempt to explain how they may be distinguished from the codiﬁable and predictable ingredients of the scientiﬁc world that will populate any Theory of Everything. Scanning the past millennia of human achievement reveals just how much has been achieved during the last three hundred years since Newton set in motion the eﬀective mathematization of Nature. We have found that the world is curiously adapted to a simple mathematical description. It is enigma enough that the world is described by mathematics; but by simple mathematics, of the sort that a few years energetic study now produces familiarity with, this is a mystery within an enigma. Several are the reactions to this state of aﬀairs. We could regard the Newtonian revolution as the discovery of a master key which opens doors faster with constant use. And although the pace of discovery has quickened dramatically in recent times, it will none the less continue to do so indeﬁnitely. Our present pace of discovery of truths about seemingly fundamental things does not necessarily indicate that we are about to converge upon the spot where all the treasure lies buried. The process of discovery could continue indeﬁnitely either because the complexity of Nature is truly bottomless or because we have chosen a particular way of describing Nature which, while being as accurate as we desire, is none the less at best always but an asymptotic approximation that only an inﬁnite number of reﬁnements could make correspond exactly to reality. More pessimistically, our human frame and its eventful evolutionary past may place real limits upon the concepts that we can accommodate. Why should our cognitive processes have tuned themselves to such an extravagant

ultimate explanation

quest as the understanding of the entire Universe? Is it not more likely that the Universe is, in Haldane’s words, ‘queerer than we can ever know’? Whatever our speculations about our own position in the history of scientiﬁc discovery, we surely regard with a Copernican suspicion any idea that our human mental powers should be adequate to handle an understanding of Nature at its ultimate level. Why should it be us? None of the sophisticated ideas involved appear to oﬀer any selective advantage to be exploited during the preconscious period of our evolution. Alternatively, we might take the optimistic view that our recent sucess is indicative of a golden age of discovery which will near completion during the early years of the next century. Thereafter, fundamental science will be more or less complete. True, there will be things left to discover, but they will be matters of detail, applications of known principles, polishing, elegant reformulation, or metaphysical rumination. Historians of science will look back at this and neighbouring centuries as the time when we discovered the laws of Nature. We have been this way before. Perhaps there is a psychological desire to bring things to a successful completion as the end of each century approaches. Near the end of the last century, many also felt the work of science to be all but done. The Prussian patent oﬃce was closed down in the belief that there were no more inventions to be made. But some work carried out by a junior at another patent oﬃce in Berne changed all that and opened up all the vistas of twentieth-century physics. Can we hope to give ultimate explanations of the Universe? Is there a Theory of Everything and what could it tell us? And just what would such a theory actually encompass? By their very nature, scientiﬁc investigations do not know their end from their beginning. We cannot tell how much of what at present we might be loath even to call science will need to be included in such an all-embracing picture of the world. Indeed, history teaches some interesting lessons in this respect. Today, physicists accept the atomistic viewpoint that material bodies are at root composed of identical elementary particles, as well supported by evidence. It is taught in every university in the world. Yet, this theory of physics began amongst the early Greeks as a philosophical, or even mystical, religion without any supporting observational evidence whatsoever. Thousands of years would pass before we even had the means to gather this evidence. Atomism began life as a philosophical idea that would fail virtually every contemporary test of what should be regarded as ‘scientiﬁc’; yet, eventually, it became the cornerstone of physical science. One suspects that there are ideas of a similar groundless status by today’s standards that will in the future take their place within the accepted ‘scientiﬁc’ picture of reality.

4 ultimate explanation

In the chapters ahead, we shall take a look at this quest for ultimate explanation and inquire a little into its ancient and modern precedents. We shall stress, unlike many other commentators, that, while knowledge of such a Theory of Everything, if it exists, is necessary in order to understand the physical universe we see about us, it is far from suﬃcient to achieve that goal. Other essential ingredients are required. Without them, our knowledge will always remain incomplete and partial, and our quest for ultimate explanation will remain unfulﬁlled. We shall see how our understanding of the Universe is inﬂuenced by eight essential ingredients:

r laws of Nature, r initial conditions, r the identity of forces and particles, r constants of Nature, r broken symmetries, r organizing principles, r selection biases, and r categories of thought. As our story develops, we shall enlarge upon the nature and contribution of these ingredients to the search for ultimate explanation. It is the author’s naïve hope that some of the ideas that we shall encounter along the way may be of wider interest than merely as support for a cautious attitude towards the likely scope of any Theory of Everything. But before we begin to follow this eightfold way, let us begin at the beginning and look back at some of the ﬁrst Theories of Everything and how their motivations have matured into those of the twentieth-century enquirers into the nature of things.

myths When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. — st paul

If you browse through the ancient mythological accounts of the origin of the world and the situation of its inhabitants, the overwhelming impression one obtains is of having wandered into a Theory of Everything. All around there is completeness, conﬁdence, and certainty. There is a place for everything, and

ultimate explanation

everything is in its proper place. Nothing happens by chance. There are neither gaps nor uncertainties. No room for progress; no room for doubt. All things are interwoven into a tapestry of meaning pulled taut by the cords of certainty. Surely these were the ﬁrst Theories of Everything. The term ‘myth’ has taken upon itself a meaning in everyday English usage that betrays its real content. It is a much maligned word. To call something ‘a myth’, to label a politician’s assurances as ‘mythical’, is now just the journalese for saying these things are false or unreliable. Alternatively, we may simply bundle up myths with legends, fairy stories, and all manner of other fantastic or imaginative literature. But to do so is to miss a layer of meaning that is crucial for our enquiry. A myth is a story imbued with a meaning. The message it contains transcends the naïve medium of the story and allows the hearer to understand why things are as they are. By studying the myths of a particular culture, we do not learn anything terribly interesting about the origin of the Universe or of mankind in the way that their original hearers did; rather, we appreciate how they deﬁne the outer boundaries of the imagination of their authors. They reveal what things they have thought about, how far they have followed them, those things they see as important enough to merit explanation, and the extent to which they regard the world as a unity. Once we start asking what the details of these myths mean we have removed ourselves from the mindset of the original hearers. It is like asking the meaning of Little Red Riding Hood. No nursery child would dream of asking such a question: if they did so, they would cease to be a child. Like fairy tales, myths are meaningful at many unconscious levels. Too precise an analysis of their message and meaning would remove this multiplicity of layers and reduce the number of hearers who could be inﬂuenced by its messages. Myths do not arise from data or as solutions to practical problems. They emerge as antidotes for mankind’s psychological suspicion of smallness and insigniﬁcance in the face of things he cannot understand. Our modern attempts to explain everything within some all-encompassing scientiﬁc picture diﬀer in certain subtle respects when compared with ancient speculative explanations. For the ancients, it was breadth alone that was the hallmark of success for their Theories of Everything. For us, it is breadth and depth that count. If we claim to explain everything that is found in the world by a system of thought which proposes that the whole Universe came into being one hundred years ago with all its complex components ready-made, but bearing all the features of having already existed for millennia, then we do indeed attain a breadth of ‘explanation’ but our explanation possesses no depth whatsoever. We can extract no more from our theory save what we put

6 ultimate explanation

into it. A similar theory to the one just proposed was actually considered in the nineteenth century by Philip Gosse in an attempt to reconcile the conﬂict over fossil evidence for the Earth’s great antiquity and widespread public belief in special creation having occurred only a few thousand years ago. Gosse proposed that the rocks appeared with the pre-aged fossils already present, bearing (false) witness to past generations of evolution. A deep theory, by contrast, is one which is able to provide explanations for a wide range of things with a minimal contribution being made to the conclusion by the number of input assumptions. The depth of a particular consequence could be characterized by the eﬀort expended in performing the shortest chain of logical reasoning from the assumptions to the conclusion: the amount of waste heat that a computer would have to generate in the process of computing the answer from scratch. The weakness of mythological Theories of Everything played a key role in their structure and evolution. If one has a weak explanation, then it lacks real explanatory power. As a result, each fresh fact that is discovered requires a new ingredient in order to weave it into the pre-existing tapestry. We see this displayed most clearly by the proliferation of deities in most ancient cultures. Each time a short chain of explanations (‘Why is it raining?’—‘Because the rain-god is crying’) ends, it tends to end at a deity. In any attempt at ultimate explanation—whether it be mythological or mathematical—there are psychologically acceptable bottom lines. In most mythological stories, the entry of an overseeing deity marks an acceptable end to the backtrack of ‘why’ questions. The more arbitrary and disparate one’s explanations for the events of Nature, so the more deities one will tend to invent. At ﬁrst, myths must have been simple and focused upon a single question. With the passage of time, they became intricate and unwieldy, bound only by the laws of poetic form. A new fantasy, a new god: one by one they can be added to the patchwork. There was no sense of the need for economy in the multiplication of arbitrary causes and explanations. All that mattered was that they ﬁtted together in some plausible way. Today such patterns of explanation are not acceptable. Ultimate explanation no longer means only a story that encompasses everything. An indiscriminate multiplication of deities creates other problems. It implies a conﬂict of legislation in the natural world. A picture of universal laws imposed upon the world by a Supreme Being will not easily emerge. Indeed, even when we look at the relatively sophisticated society of the Greek gods, we do not ﬁnd the notion of an all-powerful cosmic lawgiver very evident. Events are decided by negotiation, deception, or argument, rather than by

ultimate explanation

omnipotent decree. Creation proceeds by committee rather than by ﬁat. In the end, any appeal to such a moody collection of initial causes leads to the multiplication of ad hoc explanations, a spawning of unnecessary complexity that is going to require more of the same to keep it going in the future. There is no plausible route towards simplicity. By interlinking causes, by searching always for unity in the face of superﬁcial diversity, modern scientiﬁc explanations prize depth above breadth. A deep and narrow theory can, and often does, graduate to become a deep and broad one. A broad and shallow theory never does. It is not clear how we should regard the originators of the ﬁrst mythological Theories of Everything. We tend to assume they were realists and hence at worst foolish, at best wrong, in their description of the world. But although most of their hearers undoubtedly did take such stories literally—indeed many people hold somewhat similar views today—there may well have been others who thought of them only as images of some unreachable truth, or cynics who saw them as useful fables or devices for maintaining the status quo. Lest we relegate the myth-makers and their objectives to the miasmal mists of the past, we should remind ourselves of the way in which the desire for completeness of explanation continued down the centuries. The most striking example is that of the medievals with their bookish desire to codify and order everything that we know or ever could know of Heaven and Earth. Great systems like the Summa of Aquinas or Dante’s Divine Comedy sought to unify all existing knowledge into a labyrinthine unity. Everything had a place; everything had a meaning. As C. S. Lewis observes, it was altogether a little too stiﬂing: The human imagination has seldom had before it an object so sublimely ordered as the medieval cosmos. If it has an aesthetic fault, it is perhaps, for us who have known romanticism, a shade too ordered. For all its vast spaces it might in the end aﬄict us with a kind of claustrophobia. Is there nowhere any vagueness. No undiscovered by-ways? No twilight? Can we never really get out of doors?

And, just as primitive peoples found that unity and completeness led to a vast and unwieldy patchwork of uneasy alliances in order that everything could ﬁnd a place, so the medievals’ desire to harmonize all knowledge into a Theory of Everything became unmanageably complicated. Where the primitive mind met the challenge of completeness by imaginative invention and was then faced with the problem of ﬁtting all these imaginings together, the medieval mind was fettered by its respect for existing books and authorities. It regarded the inherited written words of the ancient philosophers with the same ultimate

8 ultimate explanation

authority that modern physicists attach to experimental evidence. But the sheer volume of these written authorities ensured that any uniﬁcation of their philosophical thinking was a vast enterprise. The twentieth century is not immune to such desires either. We have only to look at the problems that had to be faced over the deﬁnition and meaning of mathematics near the turn of the century. The formalists wished to protect mathematics from paradox by making it a closed shop: they deﬁned it to be the sum total of all the logical deductions made using all possible rules of inference from all possible starting assumptions. As we shall see in a later chapter, this attempt to trammel up all possible mathematical consequences proved impossible. The desire for completeness could not be realized even here, in the most formalized and controllable human empire of knowledge. This modern urge for completeness had developed hand-in-hand with the desire for a uniﬁed picture of the world. Where the ancients were content to create many minor deities, each of whom had a hand in explaining the origins of particular things, but might often be in conﬂict with one another, the legacy of the great monotheistic religions is the expectation of a single over-arching explanation for the Universe. The unity of the Universe is a deep-rooted expectation. A description of the Universe that was not uniﬁed in its mode of description, but fragmented into pieces, would invite our minds to look for a further principle which related them to a single source. Again, we notice that this motivation is essentially religious. There is no logical reason why the Universe should not contain surds or arbitrary elements that do not relate to the rest.

creation myths It is necessary to recognise that with respect to unity and coherence, mythical explanation carries one much further than scientiﬁc explanation. For science does not, as its primary objective, seek a complete and deﬁnitive explanation of the Universe . . . It satisﬁes itself with partial and conditional responses. Whether they be magical, mythical or religious, the other systems of explanation include everything. They are applied to all domains. They answer all questions. They account for the origin, for the present and even for the evolution of the universe. — françois jacob

We are so familiar with myths and scientiﬁc explanations for everything around us that it is no easy task to place ourselves in the prehistoric mindset that existed before any such abstractions were commonplace. We might think that the alternatives available were simply to rely on reason or sight, or upon

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faith in some invisible personalities or spirits. But this is a false dichotomy. At such a primitive stage, it is very much an act of faith to seek any parallel between our thoughts and the way things are in the outside world. It is by no means obvious that the great impersonal forces of the natural world are amenable to discussion or explanation, far less to prediction. Indeed, so awesome and devastating are many of their eﬀects that they might more persuasively appear to be an enemy or, worse still, the irrational forces of chaos and darkness. It is with such scales lifted from our eyes that we should approach the ideas that evolved concerning the origins of the world that we ﬁnd in the mythology and traditions of every culture. These stories are often exhibited as illustrating the prescience of a few ancients for some favourite modern idea like the creation of the Universe out of nothing or its inﬁnite age; but there should be no serious intent behind such juxtaposition of ancient and modern. It is merely that distorted perspective on the past that ﬁnds it to be signiﬁcant solely where it presages our present thinking. Ancient cosmology was not scientiﬁc. Its raison d’être was neither to explain observations nor make predictions. Rather, it was to embroider a tapestry of meaning within which its authors could represent themselves, and with respect to which they could evaluate the status of the unknown and the mysterious. The organization of their local society could be justiﬁed and reinforced by making it commensurate with the story of the world’s origin and form. The starkness of the contrast between their aims and ours is strikingly captured by Frances Yates: The basic diﬀerence between the attitude of the magician to the world and the attitude of the scientist towards the world is that the former wants to draw the world into himself, whilst the scientist does just the opposite, he externalizes and impersonalizes the world by a movement of will in an entirely opposite direction.

The primitive belief in order and in the sequence of cause and eﬀect displayed by myths is consistent with the belief that it is necessary to have some reason for the existence of everything—a reason that pays due respect for the natural forces which hold life and death in their hands. If one’s view of Nature involves a personiﬁcation of natural forces, then this search for reason reduces to the attribution of blame. Such generalized assumptions by no means lead to a unique collection of ideas about how the Universe came into being. But if one scans all the known myths concerning the origins of the Universe, they reveal a surprisingly small collection of cosmogonical notions. We ﬁnd rather rarely, and then somewhat ambiguously, a belief in creation of the world out of nothing, but we ﬁnd also a belief in the restructuring of the world out of

10 ultimate explanation

pre-existent chaos. Often it suﬃces for a story to explain the ordered world which we now see. The notion of explaining some pre-existent state from which the world was fashioned either is not called for or is recognized for the cul-de-sac that it will turn out to be. Occasionally, we ﬁnd adherence to the notion of a cyclic pattern of history taking its cue from the diurnal and seasonal periodicities of the natural world or, more adventurously, to a world that had no beginning. Elsewhere, we encounter the picturesque idea that the world hatched from a ‘cosmic egg’ or appeared as the progeny of the embrace of two world-parents. In the same vein, we ﬁnd a collection of traditions in which the world emerges from some primeval womb or is ﬁshed from the primordial waters of chaos by a heroic diver. Finally, there is a mythological pattern which embroiders the theme of some titanic ﬁgure engaged in a cataclysmic battle against opposing forces of chaos and darkness. Out of the heroic victory of light over darkness is born our own Cosmos. All of these formulae for dealing with the existence of the world are happy to establish some initial cause beyond which explanations will not be sought. The cause is simple in that it is singular, whereas the world of experience is bewilderingly plural. These fantastic speculations diﬀer from any modern scientiﬁc approach to the origin of things because they look to an ultimate purpose as part of the motivation or the initial mode of creation. Yet they share one aspect with modern attempts to understand the Universe. All begin as attempts to explain everything we see about us and ﬁnd this quest leads inexorably back to the ultimate question: how did the Universe originate? Today, the real goal of the search for a Theory of Everything is not just to understand the structure of all the forms of matter that we ﬁnd around us but to understand why there is any matter at all, to attempt to show that both the existence and the particular structure of the physical Universe can be understood, to discover whether, in Einstein’s words, ‘God could have made the Universe in a diﬀerent way; that is, whether the necessity of logical simplicity leaves any freedom at all’.

algorithmic compressibility Irrationality is the square root of all evil. — douglas hofstadter

The goal of science is to make sense of the diversity of Nature. It is not based upon observation alone. It employs observation to gather information about the world and to test predictions about how the world will react to

ultimate explanation

new circ*mstances, but in between these two procedures lies the heart of the scientiﬁc process. This is nothing more than the transformation of lists of observational data into abbreviated form by the recognition of patterns. The recognition of such a pattern allows the information content of the observed sequence of events to be replaced by a shorthand formula which possesses the same, or almost the same, information content. As the scientiﬁc method has matured, so we have become aware of more sophisticated types of pattern, new forms of symmetry and new types of algorithm that can miraculously condense vast arrays of observational data into compact formulae. Newton discovered that all the information he could possibly record about the motion of bodies in the heavens or on Earth could be encapsulated in the simple rules that he called the ‘three laws of motion’ together with his law of gravitation. We can extend this image of science in a manner that sharpens its focus. Suppose we are presented with any string of symbols. They do not have to be numbers but let us assume for the sake of illustration that they are. We say that the string is ‘random’ if there is no other representation of the string which is shorter than itself. But we will say that it is ‘non-random’ if there does exist such an abbreviated representation. So, for example, if we take the string of numbers 2, 4, 6, 8, . . . , and so on ad inﬁnitum, then we can represent it more succinctly by recognizing it to be just the list of positive even numbers. It is clearly non-random. A short computer program could instruct the machine to generate the entire inﬁnite sequence. In general, the shorter the possible representation of a string of numbers, the less random it is. If there is no abbreviated representation at all, then the string is random in the real sense that it contains no discernible order that can be exploited to code its information content more concisely. It has no representation short of a full listing of itself. Any string of symbols that can be given an abbreviated representation is called algorithmically compressible. On this view, we recognize science to be the search for algorithmic compressions. We list sequences of observed data. We try to formulate algorithms that compactly represent the information content of those sequences. Then we test the correctness of our hypothetical abbreviations by using them to predict the next terms in the string. These predictions can then be compared with the future direction of the data sequence. Without the development of algorithmic compressions of data all science would be replaced by mindless stamp collecting—the indiscriminate accumulation of every available fact. Science is predicated upon the belief that the Universe is algorithmically compressible and the modern search for a Theory of Everything is the ultimate expression of that belief, a belief that there is an abbreviated representation of the logic

12 ultimate explanation

behind the Universe’s properties that can be written down in ﬁnite form by human beings. This reﬂection on the compressibility of Nature also nudges us towards an understanding of why mathematics is so useful in practice. Our scientiﬁc theories always seemed to be described by mathematics, and physicists seem only interested in Theories of Everything that are couched in the language of mathematics. Is this telling us something profound about the nature of the Universe or the nature of mathematics? It is simplest to think of mathematics simply as the catalogue of all possible patterns. Some of those patterns are especially attractive and are used for decorative purposes, others are patterns in time or in chains of cause and eﬀect. Some are described solely in abstract terms, while others are made manifest on paper or in stone. When viewed in this way, it is inevitable that the world is described by mathematics. We could not exist in a universe in which there was no pattern or order of any sort. Some order is inevitable for us, and the description of that order (and all the other sorts that we can imagine) is what we call mathematics. So, the fact that mathematics describes the world is not a mystery, but the exceptional utility of mathematics is. It could have been that the patterns behind the world were of exceptional complexity which allowed no algorithms to be developed which approximated them in simple ways. Such a universe would ‘be’ mathematical but we would not ﬁnd mathematics terribly useful in practice. We could prove all sorts of ‘existence’ theorems about what structures exist but we would not be able to predict the future in detail using mathematics in the way that mission control at NASA does. Seeing it in this light, we recognize that the great mystery about mathematics and the world is that such simple mathematics is so far-reaching. Very simple patterns, described by mathematics that is easily within our grasp, allow us to explain and understand a huge part of the Universe and the happenings within it. This is another way of saying that the Universe is extremely compressible in the algorithmic sense. An awful lot of its observed complexity can be reduced to the presence of very simple patterns, described by short formulae and small equations. In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality, that the ultimate patterns that give the Universe its shape and feel will also be ‘simple’ in the sense that we can understand them and discover them. It relies on the complexity of our minds, and the reach of our technologies, being suﬃcient to understand and ﬁnd those ultimate patterns. All things being equal, the most likely state of aﬀairs would be that our capabilities are vastly more or vastly less than those required for the task. A situation in which we are just able to understand the

ultimate explanation

ultimate patterns behind the Universe using contemporary mathematics has a suspiciously un-Copernican element to it—why are we so closely matched in complexity to the Universe. The human mind is the device that allows us to abbreviate the information content of reality in this way. The brain is the most eﬀective algorithmic compressor of information that we have so far encountered in Nature. It reduces complex sequences of sense data to simple abbreviated forms which permit the existence of thought and memory. The natural limits that nature imposes upon the sensitivity of our eyes and ears prevents us from being overloaded with information about the world. They ensure that the brain receives a manageable amount of information when we look at a picture. If we could see everything down to sub-atomic scales then the informationprocessing capacity of our brains would need to be prohibitively large. The processing speed would need to be far larger than it now is in order for bodily responses to occur quickly enough to evade dangerous natural processes. This we shall have more to say about in the ﬁnal chapter of our story, when we come to discuss the mathematical aspects of our mental processing. This simple picture of the process of scientiﬁc enquiry as the search for algorithmic compressions is a compelling one, but it is also a naïve one in many ways. In the chapters to follow, we shall see why this is so and explore the eight ingredients which we have already highlighted as being necessary for our understanding of the physical world, to show what role each plays in the modern quest for an all-encompassing picture of the world. We shall start with the oldest notion: that of the laws of Nature.

chapter 2

Laws Search well another world; who studies this. — henry vaughan

the legacy of law We are the music-makers And we are the dreamers of dreams Wandering by lone sea-breakers And sitting by desolate streams; World-losers and world-forsakers, On whom the pale moon gleams: Yet we are the movers and shakers Of the world forever, it seems. — arthur o’shaughnessy

Many threads entwined to form our concept of a law of Nature. At ﬁrst, primitive societies and groups were impressed primarily by the irregularities of Nature: mishap, plague, and pestilence. In time, emphasis refocused upon the regularities of the environment and the means by which they could be most fruitfully exploited for advantage. Sense began to emerge from the welter of disparate natural phenomena. The irregularities became exceptions to, rather than conceptions of, the natural state of the world. It emerged that some degree of organization might lurk behind the ordered facets of the world just as it lay behind the ordered results of mankind’s interventions in Nature. Social and religious views coloured early ideas about the organization of the world. There were many paradigms. For some, the world was a living organism growing and maturing towards some great purposeful culmination. All its

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constituents contained innate imperatives which moved them to trace out the ways predestined for them. They followed not the rules of some external diktat but the manifestations of their immanent properties. The meanings of things were to be found in their ends, not in their present or past states. For others, the world was a cosmic city, ordered by transcendent laws and rules imposed by a Supreme Being. Moreover, it was a walled city within which order was preserved for our beneﬁt. Beyond its borders lay chaos and evil. In other cultures, quite diﬀerent ideas held sway. No outside lawgiver was imagined. No outside lawgiver was necessary. Instead, all things seemed to work together in harmony to compose the common good by mutual consent and interaction. The order in the world was seen as that of the ant colony, wherein every individual plays its part to produce a coherent selfintereacting whole. It is a spontaneous response to the requirements of the system as a whole, not the inﬂexible result of eternal and unchangeable laws of Nature. Diﬀerent modern cultures have been variously inﬂuenced by their religious heritage in coming to a satisfying picture of natural laws. In the JudaeoChristian West, the inﬂuence of the divine lawgiver has been paramount. The laws of Nature are the dictates of a transcendent God. They enshrine faith in the existence of an underlying order to things. They sanction the investigation of Nature as a secular activity. They outlaw Nature gods and the potential conﬂicts of polygamous legislation in the Universe. Farther East, in cultures like that of the early Chinese, the dominant picture was more liberal in style, with Nature operating holistically to produce a harmonious equilibrium in which every ingredient interacts with its fellows to produce a whole that is more than the sum of its parts. It is not hard to see why the Eastern holistic perspective made scientiﬁc progress so diﬃcult. It denies the intuition that one can study parts of the world in isolation from the rest—that one can analyse the world— and understand a part without knowing the whole. In modern terms, the Western perspective has regarded Nature as a linear phenomenon in which what happens at a given place and time is determined exclusively by what has occurred at nearby places immediately beforehand. The holistic view assumes nature to be intrinsically non-linear so that non-local inﬂuences predominate and interact with one another to form a complicated whole. It is not that the Eastern approach was misguided. It was simply premature. Only very recently, aided by versatile computer graphics, have scientists come to terms with the description of intrinsically complex non-linear systems. A successful study of natural laws needs to start with the simple linear problems

16 laws

if it is ever to graduate successfully to the holistic complexities created by non-linearity. Having drawn with broad brush-strokes the inter-relationship between religious beliefs and the wider philosophy of nature that it engenders within a society it is important to inject a note of caution. It is common for apologists to press the argument further and claim that modern science has emerged because of, or even from, the West’s Christian religious roots. There is undoubtedly some grain of truth in this claim, rightly interpreted; but its uncritical acceptance is as mistaken as the common notion that religion and science have always been at war like the forces of darkness and light. The monotheistic basis for the concept of universal laws of Nature contains an element of the truth because modern science is something that has developed to fruition after the early events which shape religious history. Moreover, many great scientists were overtly religious and brought to their scientiﬁc work an explicit religious justiﬁcation and motivation. While these facts cannot be denied, it is a giant leap to infer from this summary of events that modern science is therefore a necessary consequence of our Christian past which would not otherwise have arisen. Here, the apologist is seeking to persuade that the practice of science or the concept of universal laws is a logical outcome of a certain range of religious beliefs rather than merely something that has been fostered by them. Religious scientists, like Boyle, Newton, or Maxwell, undoubtedly existed in profusion, but they inevitably stressed those aspects of their religion which accorded well with their scientiﬁc intuitions and activities. They were satisﬁed that their work was in tune with a Christian view of the world in an age when the public face of religion was a far greater factor in people’s lives than it is today. There were always other strands of Christian doctrine, less obviously convivial to the pursuit of theoretical science, which the very same scientists would subconsciously downplay or simply ignore. Others, who found science distasteful, materialistic, or even blasphemous, could always be found amongst the ranks of the theologians and philosophers. The virtues necessary for the successful pursuit of science are neither speciﬁcally nor exclusively those engendered by our Judaeo-Christian heritage, nor, indeed, by any other. To believe that science has necessary rather than actual religious precursors is to subscribe to a deterministic theory of history with unique eﬀects and causes. The real world is immeasurably more complicated: it is a skein of many strands, knotted and tangled, whose beginning is out of reach and whose end we cannot know.

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the quest for unity Man shall not join what God has torn asunder. — wolfgang pauli

As we have become more demanding of our explanations and pictures of the Universe, so we have found the scale of what we must explain to be far greater in extent than our predecessors could ever have imagined. As complexity has grown, so has physics fragmented into specializations, which in turn have found themselves partitioned into manageable pieces. Each has enjoyed its own successes in building up mathematical theories of the diﬀerent fundamental forces of Nature and has endowed us with eﬀective descriptions of each of the diﬀerent interactions between particles of matter and light. The most striking aspect of these theories, beyond that of their huge success, is that until only recently they have been distinct in form and content, each compartmentalized from the others as though bearing witness to some curious paranoia in Nature. This goes against the grain of our belief in the unity of Nature. Only very rarely have ambitious scientists attempted to construct a theory of physics which would unite all the disparate and successful theories of the diﬀerent forces of Nature into a single coherent framework from which all things could in principle be derived. One of the earliest with a distinctly modern perspective was Bernhard Riemann, the nineteenth-century creator of the systematic study of non-Euclidean geometries. He envisaged a ‘total theory of physics’ united by mathematics, and wrote to Richard Dedekind of his belief that one can set up a completely self-contained mathematical theory, which proceeds from the elementary laws that are valid for individual points to processes in the actually given continuously ﬁlled space, without distinguishing whether it is gravity, electricity, magnetism, or the equilibrium of heat that is being treated.

The most famous modern attempts to implement it were those of Eddington and Einstein. They failed for many reasons. In retrospect, we recognize that knowledge of the elementary-particle world was then so seriously incomplete that neither Eddington nor Einstein were in a position even to see what needed to be uniﬁed, let alone how to do it. However, the ﬂame they ﬁrst ignited has remained glowing faintly in the background, often overshadowed by the ﬁreworks provided by the latest advances in the understanding of particular pieces of nature, until being fanned into prominence by the most recent attempts by

18 laws

theoretical physicists to illuminate our picture of the Universe. Whereas past uniﬁers were regarded as lone eccentrics by their colleagues, tolerated because of the brilliance of their other contributions to physics, the uniﬁers of today populate the mainstream of physics and continually add to their number the most gifted young students. This is what distinguishes the physics of the 1980s from any that has gone before. The current breed of candidates for the title of a ‘Theory of Everything’ hope to provide an encapsulation of all the laws of nature into a simple and single representation. The fact that such a uniﬁcation is even sought tells us something important about our expectations regarding the Universe. These we must have derived from an amalgam of our previous experience of the world and our inherited religious beliefs about its ultimate nature and signiﬁcance. Our monotheistic traditions reinforce the assumption that the Universe is at root a unity, that it is not governed by diﬀerent legislation in diﬀerent places, neither the residue of some clash of the Titans wrestling to impose their arbitrary wills upon the nature of things, nor the compromise of some cosmic committee. Our Western religious tradition also endows us with the assumption that things are governed by a logic that exists independently of those things, that laws are externally imposed as though they were the decrees of a transcendent divine legislator. In other respects, our prejudices reﬂect a mixture of diﬀerent traditions. Some feel the force of the Greek imperative that the structure of the Universe is a necessary and inﬂexible truth that could not be otherwise, while others inherit the feeling that the Universe is contingent. In this connection, it is interesting to recall the commentary supplied by Charles Babbage the eccentric nineteenth-century pioneer of computing devices who was much exercised by the concept of the laws of Nature. He was the ﬁrst to liken the Universe to a computer whose program (as we would now call it) comprised the laws of Nature; but this image provided him more readily with the conception of a diﬀerent program or one which might turn up irregularities and novelty very occasionally: The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an inﬁnite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws.

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Our attraction to that quality which we have come to call ‘beauty’, and which we associate with the detection of innate unity and harmony in the face of superﬁcial diversity, has led us to expect that the unity of the Universe should be expressed in certain particular ways. If we are physicists we might often hear talk of the ‘beauty’ or ‘elegance’ of particular ideas or theories to such an extent that, like Dirac, ∗ we make aesthetic quality a guide or even a prerequisite for the formulation of correct mathematical theories of Nature. The aesthetic imperative of Dirac strikes the life scientist as strange, the more so when he discovers how ineﬀective physicists, for all their mathematical powers, so often prove to be when they stray into his menagerie. For physicists are used to dealing with the pristine symmetries and fundamental laws of Nature. This habit conditions them to seek and expect symmetry and mathematical elegance everywhere they look. But the living world is not a marble palace. It is the higgledy-piggledy outcome of natural selection and the competition between many interacting factors. The outcome is often neither elegant nor symmetrical.

roger boscovich Dear Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. — roger boscovich

Our picture of the physical world has expanded so rapidly during this century that it requires some eﬀort to put oneself in the shoes of the scientist of a past century. For Newton, there was no classiﬁcation of the diﬀerent forces of Nature. Radioactivity and nuclear forces were unknown; electricity and magnetism were diﬀerent observed phenomena. Until Newton united them, the terrestrial and celestial inﬂuences of gravity were conceptually quite distinct. Newton simpliﬁed our apprehension of the world by explaining all gravitational phenomena within a simple scheme which attributed the observed eﬀects to the action of a single attractive force acting between all massive bodies. Despite the success of this programme, and the other areas of thermodynamics and optics in which Newton was able to bring logical simplicity to a plethora of confusing observations, he knew that there were ∗

On being asked what he meant by the beauty of a mathematical theory of physics, Dirac replied that if the questioner was a mathematician then he did not need to be told, but were he not a mathematician then nothing would be able to convince him of it.

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areas still shrouded in mystery. He speculated that there must exist other forces of Nature—‘very strong attractions’—which hold material bodies together, but he could take that intuition no further. One of the most remarkable and neglected ﬁgures in the history of modern European science was Roger Boscovich. A Dalmatian Jesuit, at once a poet and architectural advisor to Popes, cosmopolitan diplomat and man of aﬀairs, socialite and theologian, conﬁdant of governments and Fellow of the Royal Society, but most of all a mathematician and scientist, Boscovich was a passionate Newtonian who was the ﬁrst to have a scientiﬁc vision of a Theory of Everything. His most famous work the Theoria Philosophiae Naturalis, was ﬁrst published in Vienna in 1758. After several editions, it culminated in the enlarged and revised Venetian edition of 1763. Its inﬂuence was wide and deep, especially in Britain, where Faraday, Maxwell, and Kelvin would record their indebtedness to its inspiration. Boscovich aimed to extend Newton’s overall picture of Nature in several important ways. In particular, he sought to ‘derive all observed physical phenomena from a single law’. In so doing, he introduced a number of new concepts which still form part of the intuition of scientists. He emphasized the atomistic notion that Nature was composed of identical elementary particles and then aimed to show that the existence in Nature of larger objects with ﬁnite sizes was a consequence of the way their elementary constituents interact one with another. The resulting structures were equilibrium states between opposing forces of attraction and repulsion. This was the ﬁrst serious attempt to understand the existence of solid objects in Nature. He saw that Newton’s inverse-square law of gravitation alone was insuﬃcient to explain the existence of structures with particular sizes because it endowed gravity with no characteristic scale of length over which its eﬀects were especially manifest. The inverse-square law singles out no particular scale of length as special and has an inﬁnite range. To explain objects of particular sizes requires a balance between gravity and some other force. Boscovich proposed a grand uniﬁed force law which included all known physical eﬀects. This was his ‘Theory’, as he called it. It approached the inversesquare law of Newtonian gravitation at large distances as required by observations of the lunar motions. But on smaller length scales, it is alternately attractive and repulsive and so gives rise to equilibrium structures whose sizes are dictated by the characteristic length scales built into the force law. The ‘Law of Forces’ he proposed is shown in Figure 2.1. Boscovich lays great stress upon the fact that this law is not merely a ‘haphazard’ aggregate of forces but needs to be a ‘single continuous curve’, which, he argues, witnesses to

Dⴕ

laws

o v

B D g

r H

n Cⴕ

Gⴕ

Eⴕ

Fⴕ

A ab

l h F

t I

c z

x T p

y K

e fq

Figure 2.1 Boscovich’s original universal force law, reproduced from his Theory of Natural Philosophy, ﬁrst published in 1758. The variation of the force between two ‘points of matter’ as the distance between them changes is traced by the undulatory curve passing through the sequence of points DFHKMOQSTV. The distance between them is given along the abscissa AC; the strength of the force along the ordinate AB. The force is repulsive when this curve lies above the line AC and attractive when it lies below it. At very large distances (at and beyond V), it is attractive and approaches Newton’s inverse-square law of force produced by gravity. The repulsive nature of the force as the separation of the two points tends to zero prevents all matter collapsing to zero size. Regarding this picture, Boscovich remarks: ‘A Law of this kind will seem at ﬁrst sight to be very complicated, and to be the result of combining together several different laws in a haphazard sort of way; but it can be of the simplest kind and not complicated in the slightest degree; it can be represented for instance by a single continuous curve . . . It is sufﬁcient merely to glance at it.’

the uniﬁed all-encompassing nature of the theory. In addition to the pictorial representation of his force law illustrated here, Boscovich also introduced the idea of expressing his law as a convergent series of mathematical terms in powers of inverse distance, each smaller than its predecessor but the longer the sum is extended, the better becomes its approximation to the true force law. There are many other innovations in Boscovich’s detailed treatise, but we are interested here in drawing attention to just this one point: that he was the ﬁrst to envisage, seek, and propose a uniﬁed mathematical theory of all the forces of Nature. His continuous force law was the ﬁrst scientiﬁc Theory of Everything. Perhaps, in the eighteenth century, only a generalist like Boscovich, who successfully uniﬁed intellectual and administrative activities in every area of thought and practice would have the presumption that Nature herself was no less multicultural.

22 laws

symmetries But you see, I can believe a thing without understanding it. It’s all a matter of training. — (Lord Peter Wimsey in Have His Carcase) dorothy sayers

For the early Greeks, the most perfect laws of Nature were its static harmonies. In the last two hundred years, the concept of a law of Nature has come to mean a set of rules which tell us how things change in space and in time. Thus, knowing the state of a system here and now, we seek a device for predicting its state at future times and in other places. But curiously, such laws of change can always be recast into completely equivalent statements which assert that something must not change: such unchanging quantities are known as invariances. During the nineteenth century, mathematicians invested much time in classifying all the possible types of change and associated invariance that could exist, in both concrete and abstract terms. This classiﬁcation gave rise to the branch of mathematics which we now call group theory. A ‘group’ is simply a collection of changes which possess three simple properties: there must be the possibility of no change, there must exist the possibility of undoing or reversing each change to restore its original state, and any two consecutive changes must give a result that could equally well be attained by another single change. Each of the most basic physical laws that we know of corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. The symmetry group describes all the variations that can be formed from an initial seed pattern whilst still leaving some underlying theme unchanged. Thus, for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time (that is, the result of an experiment should not depend on the time at which it was carried out, all other factors being identical); the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to the directional orientation of your laboratory in space. Other conserved quantities in physics, which arise as the constants of integration of the laws of change, turn out to be equivalent to other less intuitive invariances of the laws of Nature. It is interesting to note that the conservation of energy was not used by Newton. Moreover, in the post-Newtonian discussions regarding the theological relevance of Newton’s successful description of the world, the

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existence of conservation laws appears to have played some role in the growth of atheism amongst scientists. Some, like Newton himself, felt that there was need within the Newtonian dynamical model of the known universe (the solar system) for the sustaining and regulating hand of the Deity, but the subsequent discovery of conservation laws indicated that Nature possessed built-in sustaining principles which stopped the world from just ceasing to be. There were fewer roles for the Deity to play than had been believed. It was in this context that Laplace made his famous admission that ‘nous n’avons pas besoin de cette hypothèse-là’ with regard to the sustaining role of the Deity in maintaining the motions within the solar system. Later the pendulum would swing back and the need to violate a conservation law of Nature in order to bring the Universe into being out of nothing persuaded many of the need for supernatural intervention. Moreover, the evident success of the concept of laws of Nature led to a reformulation of the Design Argument for the existence of God. We shall refrain from elaborating upon it here, but later, in Chapter 6, we shall return to highlight its special signiﬁcance. Even today there persists amongst many a feeling that the creation of the Universe out of nothing must violate some basic conservation law that stops one getting something for nothing. Nevertheless, there is actually no evidence that the Universe as a whole possesses a non-zero value of any such conserved quantity. The total mass-energy of all the constituents of a ﬁnite Universe appears to be always equal in magnitude but opposite in sign to the total gravitational potential energies of those particles. It could suddenly thus appear spontaneously without violating the conservation of mass-energy. Similarly, there is no evidence that the Universe possesses any overall net rotation or electric charge. It may well transpire that we discover some other conserved attribute that is non-zero for the Universe as a whole or obtain evidence that the Universe does indeed possess a non-zero electric charge or rotation. These ideas are based upon the supposition that the Universe is ﬁnite in size. Not only do we not know whether this is the case, but we cannot know because the ﬁnite speed of light ensures that we can only ever see a ﬁnite portion of the entire Universe. If the Universe were inﬁnite in extent, then it is not known how one should associate conserved quantities with it and the question of whether it can appear out of ‘nothing’ without violating the conservation of charge, rotational momentum, and energy is a far subtler, unanswered question. The fact that laws of change can be represented as invariances of the world under all possible changes that respect a particular innate pattern struck a resonant chord with physicists’ expectations regarding the presence of symmetry

24 laws

and harmony in Nature. Symmetry has become the dominant theme in fundamental physics. Elementary-particle physics is singularly Platonic in this respect. Mathematicians of the past have catalogued all the distinct patterns of change that exist and have diligently encoded their essential ingredients into that branch of mathematics now known as group theory. By searching through its kaleidoscope of all possible patterns, the particle physicist can extract candidate symmetries to impose upon the world. The candidates need to pass some initial screening to ensure that they can accommodate all the necessary ingredients of the elementary-particle world and do not have some obvious consequence at variance with reality. The successfully vetted candidates then graduate to a more detailed mathematical outworking, which results in a gamut of predictions as to how particles should interact in a world governed by the imposed symmetry. Thus, a blind faith in symmetry provides an eﬃcient recipe for generating candidate theories of elementaryparticle interactions. No such machinery exists to generate candidate theories to explain the workings of less basic entities like economies or weather systems. The stronghold of symmetry is the unseen world of the smallest things. Each of the four forces of Nature is accurately described by a theory that derives from the assumption of a particular invariance under all possible changes. The quest for uniﬁcation proceeds by seeking to embed the separate patterns preserved by the several forces of Nature within a single ‘Grand Uniﬁed’ pattern into which the sub-patterns ﬁt uniquely and completely. Such schemes are not easy to ﬁnd and until recently carried with them unfortunate defects which came to light when the resulting pattern of invariance was used to compute observable quantities. Inﬁnite answers were obtained which had to be dealt with in particular ways in order to produce sensible predictions. So far, this ﬂaw has been found to be absent only in a narrow class of unusual physical theories which have been proposed as the most complete laws of Nature by Michael Green, John Schwarz, and Edward Witten. These are known as ‘superstring’ theories. The preﬁx ‘super’ alludes to a powerful symmetry that they respect. This ‘supersymmetry’ has been proposed as a symmetry between otherwise distinct classes of elementary particles called fermions and bosons. In most situations, this amounts to a symmetry between matter and radiation. This idea was prevalent long before Green, Schwarz, and Witten. What they were able to do was wed it to the powerful concept of a ‘string’. Earlier theories of elementary particles had regarded the most elementary entities of Nature as point particles having no ﬁnite extent (they can be arbitrarily localized and they would oﬀer no evidence of any internal structure

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when bombarded by other high-energy particles). They were described by quantum ﬁeld theories in which the most basic elements are points of zero size. For the most part, they worked satisfactorily, but they were invariably beset by the disease of the inﬁnities, and rather ad hoc mathematical remedies have had to be employed to suppress them. As time went on, they also became rather cumbersome: more and more quantum ﬁelds had to be introduced for each variety of elementary particle required to complete the picture. Strings tie things up more neatly. If the most elementary entities in Nature are regarded as strings (lines) rather than points, then all the unpleasant divergences in calculated quantities magically disappear for some very special universal symmetries. This reversal of fortunes arises because of the intrinsic diﬀerences in the way points and lines interact. In Figure 2.2 is displayed the schematic picture of a point particle and a string interaction in space and time. The particle interaction has obvious sharp corners that translate into mathematical inﬁnities, whereas the smooth tube-like picture of the string interaction creates no such hiatuses. In eﬀect, a certain collection of possible laws of Nature, and these only, may be ﬁnite and self-consistent. Time Space

C D E 1

B (a)

(b)

Figure 2.2 Diagrammatic representations of (a) interactions between two point particles A and B, mediated by the exchange of E, which results in the production of C and D; and (b) the interaction between two string loops which leads to two resultant strings. The diagrams represent the interactions in space and time with all the dimensions of space idealized to one for ease of presentation. As the point moves through space and time it traces out a line, whereas as the loops move through space and time they trace out tubes. The mathematical inﬁnities associated with the point interaction arise because of the sharp corners at points 1 and 2 in (a). By contrast, the string interaction has no sharp corners at all and its smooth continuous character is a consequence of the absence of mathematical inﬁnities in its calculation.

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These fundamental strings possess a tension that varies with the energy of the environment in which they reside and this tension becomes large enough to shrink the loops of string to approximate points at the low energies we witness in the Universe today. But, in the extremities of the Big Bang, the essential stringiness of things should be manifest. It is theories of this sort that have aroused talk of ﬁnding a ‘Theory of Everything’. If we had the correct version, it should in principle contain all the laws of radioactivity, gravity, electromagnetism, and nuclear physics. The enticing aspect of the string theories has been the unexpected discovery that the requirement of ﬁniteness and consistency alone should prove to be so constraining. Our attitude towards the laws of Nature and some ultimate codiﬁcation of them into a possibly unique and self-consistently speciﬁed ‘Theory of Everything’ is a search for an ultimate symmetry of the world from within whose straightjacket there follow all the allowed causal laws of change governing the forces and particles of Nature. Our approach to such an apparent panacea must be tempered by an appreciation of how the laws of Nature—the Theory of Everything—might be related to the Universe.

infinities—to be or not to be? Take it to the limit one more time. — the eagles

The boundless, timeless, and endless have attracted and confused human minds for thousands of years. From East to West, sophisticated cultures devised words to express the concept of inﬁnity as well as arguments to include it or banish it from their models of the world. Aristotle ﬁrst distinguished clearly between ‘actual’ and ‘potential’ inﬁnities in the fourth century bc. The ‘actual’ variety, which involved inﬁnite values of observable or measurable quantities here and now, were outlawed. On the other hand, he permitted the less threatening notion of a potentially inﬁnite sequence, such as the unending list of positive whole numbers or an eternity of future time, where the inﬁnity was neither achieved nor reached. It couldn’t hurt you. In many ways ancient attitudes towards actual inﬁnities mirrored that towards the existence of a vacuum. For Aristotle, the two were intimately connected because in an empty space there would be no resistance to motion and bodies would eventually move with inﬁnite speed. Therefore, no

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perfect vacuum could exist in the physical universe. Medieval scientists devised ingenious arguments to avoid either an inﬁnity or a perfect vacuum ever occurring. A ‘celestial agent’ was imagined to act as a cosmic censor, ensuring that any opportunity to create a real vacuum or an inﬁnity, even if permitted by the laws of mechanics, was never taken advantage of by Nature. Much has happened to change our conceptions of the inﬁnite since those path-breaking arguments emerged. But it still challenges theologians, philosophers, and scientists to understand it, cut it down to size, ﬁnd out if it comes in diﬀerent shapes and sizes, and to decide whether we want to outlaw it or welcome it with open arms into our descriptions of the Universe. Inﬁnity is also very much a live issue. We have seen that physicists have spent the past twentyﬁve years searching for a Theory of Everything that unites all the known laws of Nature into a single mathematical statement. That search has been signiﬁcantly guided by an attitude towards the existence of actual physical inﬁnities. In theories of particle physics, the appearance of an inﬁnite answer to a question about the magnitude of a measurable quantity was always taken as a warning that you had made a wrong turn. For decades, the inevitable appearance of an inﬁnity in the calculations was managed by a strange subtraction procedure that removed the divergent part from the calculation to leave only a ﬁnite residue to compare with observations. Although the results of this so-called ‘renormalization’ process gave spectacularly good agreement with experiments, there was always deep unease that this ugliness could not be part of Nature’s economy. The true theory must be ﬁnite. This all changed in 1984, when Green and Schwarz showed that a particular type of physics theory—a ‘superstring’ theory—could indeed be wholly ﬁnite. The enthusiasm with which the new theories were embraced by physicists was a consequence of their ingenious banishment of inﬁnities, a problem that had plagued their predecessors. The path towards superstring theories awaits experimental endorsem*nt. But the energy with which they have been pursued reﬂects the philosophy of scientists who believe that the appearance of an actual inﬁnity in a physical theory is a signal that it is being stretched beyond its domain of applicability. The usual response is to upgrade the theory until the inﬁnities are smoothed into large, but ﬁnite, quantities. Engineers, for example, know this well. You can exorcise the appearance of inﬁnities in simple models of rapid aerodynamic ﬂows by simply including more realism in the description of the friction of the air. The crack of a whip is the sonic boom from the tip travelling faster than the speed of sound. A simple calculation that ignored the friction of air would say that this involved something changing inﬁnitely quickly. But a more

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detailed modelling of the air-ﬂow properties turns this inﬁnity into a very rapid but ﬁnite change. Despite the general adoption of this ‘inﬁnities-mean-you-must-try-harder’ dictum in relation to physical theories, one area of science has been willing to take predictions of actual inﬁnities more seriously. Cosmologists see there is room for a lot of inﬁnities in the Universe. Many are of the ‘potential’ variety— the Universe might be inﬁnite in size, face an inﬁnite future lifetime, or contain an inﬁnite number of atoms or stars. These are all potential inﬁnities in Aristotle’s sense. But there is one aspect of them that seems alarming to our common sense. While potential inﬁnities pose no local threat to the fabric of reality, we do have to face up to Nietzsche’s inﬁnite replication paradox: if the universe is inﬁnite in extent and exhaustively random, then any event that has a ﬁnite probability of occurring here and now (such as you reading this book) must be occurring inﬁnitely often elsewhere at this very moment. Moreover, for every history we have pursued here, all possible alternatives are acted out, wrong choices made simultaneously with right choices. This is a grave challenge to ethics and to the theology of almost every religion. Some ﬁnd it so alarming that they regard it as a powerful argument for a ﬁnite universe. However, it should be remembered that the ﬁniteness of the speed of light insulates us from contact with our doubles. We can only see and receive signals from a ﬁnite part of the universe. The distance that we would have to travel before we should expect to encounter a copy of ourselves 28 is 1010 metres whereas the greatest distance that light has had time to reach us from is a mere 1027 metres. For all practical purposes, we experience a ﬁnite part of the Universe. The challenge to cosmologists does not end there, though. They also have to worry about ‘actual’ inﬁnities. For decades, cosmologists have been happy to live with the notion that the universe of space and time began expanding from an initial big bang ‘singularity’, where temperature, density and just about everything else, was inﬁnite at some ﬁnite time in the past. Furthermore, when large stars exhaust their nuclear fuel and implode as a result of their own gravity, they appear doomed to reach a state of inﬁnite density in ﬁnite time. But this is all neatly kept out of reach. Black holes are believed to be always shrouded by an ‘event horizon’—a surface of no-return through which things can fall in but not pass out—so that we can neither see the inﬁnite density at the black hole’s centre, nor feel its eﬀects, from the outside. Roger Penrose, of Oxford University, believes that actual inﬁnities do occur both at the start of the Universe and at the centre of black holes. He once proposed that the laws of Nature provide a form of cosmic censorship that ensures that such naked physical inﬁnities are always enclosed by event horizons. This

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is reminiscent of the medievals’ celestial agent invoked to avoid the creation of a perfect vacuum. It means that the eﬀects of the physical inﬁnity are conﬁned to the inside of the black hole and cannot inﬂuence the outside world. The cosmological inﬁnity at the beginning of the Universe is the one that inﬂuences us and, on this picture, could determine everything about the Universe that we see today. By contrast, cosmologists with a particle-physics perspective tend to see these black-hole and cosmological inﬁnities merely as a signal that the theory has overextended itself and needs to be improved to exorcise these inﬁnities. As a result, we ﬁnd much interest in the prospect of universes that bounce back into expansion if run backwards in time towards their apparent beginning. Our presently expanding Universe is suspected by some cosmologists of having arisen from the rebound, at ﬁnite density and temperature, of a previously contracting phase in its history. From the outside, we cannot see what is happening inside a black hole. But if we fell in, we would be facing an uncertain fate as we approached the centre. Is there a real physical inﬁnity waiting there, or does energy slip away into another dimension of space, or simply disappear into nothing, or get soaked by exciting a never-ending sequence of vibrations of the superstrings at the core of all matter and energy? We just do not know. But again the issue of ﬁnite versus inﬁnite is a crucial guiding principle. Do we treat the appearance of an inﬁnity as a signal to update our theory, or do we treat it more seriously as an indication that new types of law govern inﬁnite physical quantities, laws that could dictate how our Universe began and how matter meets its end under the relentless implosion of gravity. Cosmologists have another strange potential inﬁnity to contemplate: the possibility of an inﬁnite future. Is the Universe on course to last forever? Its contemplation leads quickly to philosophy, for what does ‘forever’ mean? And to biology and computer science—or can life, in any form, continue forever? And to the social sciences—and what would it mean socially, personally, mentally, legally, materially, and psychologically for us to live forever? The last question, at least, is one to which we can all think of answers. In the long run, living forever might not prove as attractive as it seems at ﬁrst. You might even welcome the televangelist who oﬀers you the promise of ﬁnite life. Mathematicians have also had to face up to the reality of inﬁnity. Twice in the last 120 years, mathematics has faced civil war over the matter, leaving many a casualty and much bitterness. Some wished to outlaw actual inﬁnities and redeﬁne boundaries to forbid all treatment of them as real ‘things’. Journals were compromised and mathematicians ostracized. As we shall see in the next chapter, the nineteenth-century German mathematician Georg Cantor ﬁrst showed how to make sense of the paradoxes of

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inﬁnity. He elegantly deﬁned inﬁnite collections as those that could be put in one-to-one correspondence with subsets of themselves. This enabled him to go on and answer deeper questions: Can one inﬁnity be bigger than another? Is there an ultimate inﬁnity beyond which nothing bigger can be constructed or conceived, or do inﬁnities go on forever? Cantor answered all these questions in precise ways, but did not live long enough to see the fruits of his genius form part of the acknowledged body of mathematics. He was sidelined and undermined by inﬂuential ﬁnitist opponents, and for long periods he turned instead to the study of history and theology and suﬀered bouts of depression before his death in 1918. Remarkably, theologians were the ﬁrst to seize on the importance of Cantor’s work. They had long struggled to make sense of the inﬁnities lurking within their doctrines. Is God alone inﬁnite? Must he not be ‘bigger’ than other more mundane inﬁnities? Many investigations of the inﬁnite had been unpopular because they seemed to be challenging the belief that only God was inﬁnite. Cantor’s work changed all that. He revealed that there is a neverending hierarchy of inﬁnities, each unambiguously bigger than the last. This enabled us to distinguish between three diﬀerent types of inﬁnity: the mathematical, the physical, and the transcendental. Some thinkers accept them all, some accept only some, some accept none. The ancients, beginning with Zeno, were challenged by the paradoxes of inﬁnities on many fronts. But what about philosophers today? What sort of problems do they worry about? There are live issues on the interface between science and philosophy that are concerned with whether it is possible to build an ‘inﬁnity machine’ that can perform an inﬁnite number of tasks in a ﬁnite time. Of course, this simple question needs some clariﬁcation: What exactly is meant by ‘possible’, ‘tasks’, ‘number’, ‘inﬁnite’, ‘ﬁnite’ and, by no means least, by ‘time’? Classical physics appears to impose few physical limits on the functioning of inﬁnity machines because there is no limit to the speed at which signals can travel or switches can move. Newton’s laws allow an inﬁnity machine. This can be seen by exploiting a discovery about Newtonian dynamics made in 1971 by the US mathematician Jeﬀ Xia. First take four particles of equal mass and arrange them in two binary pairs orbiting with equal but oppositely directed spins in two separate parallel planes. Now introduce a ﬁfth much lighter particle that oscillates back and forth along a perpendicular line joining the mass centres of the two orbiting binary pairs. Xia showed that such a system of ﬁve particles will expand to inﬁnite size in a ﬁnite time! How does this happen? The little oscillating particle runs back and forth between the binary pairs, each time creating an unstable meeting of three bodies. The lighter particle then gets kicked back, and the binary pair recoils outwards to conserve momentum. The lighter particle then

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travels across to the other binary and the same ménage à trois is repeated there. This continues without end, accelerating the binary pairs apart so strongly that they become inﬁnitely separated while the lighter particle undergoes an inﬁnite number of oscillations in the process. Unfortunately (or perhaps fortunately), this behaviour is not possible when relativity is taken into account. No information can be transmitted faster than the speed of light and gravitational forces cannot become arbitrarily strong in Einstein’s theory of motion and gravitation; nor can masses get arbitrarily close to each other and recoil—there is a limit to how close separation can get, after which an ‘event horizon’ surface encloses the particles to form a black hole. Their fate is then literally sealed—no such inﬁnity machine could send information to the outside world. But this does not mean that all relativistic inﬁnity machines are forbidden. Indeed, Einstein’s relativity of time that is a requirement of all observers, no matter what their motion, opens up some interesting new possibilities for completing inﬁnite tasks in ﬁnite time. Could it be that one observer could move fast enough to see an inﬁnite number of computations occurring in a ﬁnite amount of their lifetime? The famous motivating example of this sort is the so-called twin paradox. Two identical twins are given diﬀerent future careers. Tweedlehome stays at home while Tweedleaway goes away on a space ﬂight at a speed approaching that of light. When they are eventually reunited, relativity predicts that Tweedleaway will ﬁnd Tweedlehome to be much older. The twins have experienced diﬀerent careers in space and time because of the acceleration and deceleration that Tweedleaway underwent on his round trip. Time passes more slowly on the accelerated and decelerated trip. So can we ever send a computer on a journey so extreme that it could accomplish an inﬁnite number of operations by the time it returns to its stay-at-home owner? Itamar Pitowsky, a philosopher of science at the Hebrew University of Jerusalem, argued that if Tweedleaway could accelerate his spaceship suﬃciently strongly, then he could record a ﬁnite amount of the Universe’s history on his own clock while his twin records an inﬁnite amount of time. Does this, he wondered, permit the existence of a ‘Platonist computer’—one that could carry out an inﬁnite number of operations along some trajectory through space and time and print out answers that we could see back home. Alas, there is a problem— for the receiver to stay in contact with the computer, it also has to accelerate dramatically to maintain the ﬂow of information. Eventually the gravitational forces become stupendous and it is always torn apart if it has any ﬁnite size. Notwithstanding these problems a check-list of properties has been compiled for universes that can allow an inﬁnite number of tasks to be completed in ﬁnite time, or ‘supertasks’ as they have become known. These are called

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Malament–Hogarth (MH) universes after David Malament, a University of Chicago philosopher, and Mark Hogarth, a former Cambridge University research student, who in 1992 investigated the conditions under which supertasks were theoretically possible. Supertasks open the fascinating prospect of ﬁnding or creating conditions under which an inﬁnite number of things can be seen to be accomplished in a ﬁnite time. This has all sorts of consequences for computer science and mathematics because it would remove the distinction between computable and uncomputable operations. It is something of a surprise that MH universes are possibilities but, unfortunately, they have properties that suggest they are not realistic unless we embrace some disturbing notions, such as the prospect of things happening without causes, and travel through time. The most serious by-product of being allowed to build an inﬁnity machine is rather more alarming though. Observers who stray into bad parts of these universes will ﬁnd that being able to perform an inﬁnite number of computations in a ﬁnite time means that any amount of radiation, no matter how small, gets compressed to zero wavelength and ampliﬁed to inﬁnite energy along the inﬁnite computational trail. Thus any attempt to transmit the output from an inﬁnite number of computations will zap the receivers and destroy them. So far, these dire problems seem to rule out the practicality of engineering a relativistic inﬁnity machine in such a way that we could safely receive and store the information. But the universes in which inﬁnite tasks are possible in ﬁnite time include a type of space (called ‘anti-de Sitter space’) that plays a key role in the structure of the very superstring theories that looked so appealingly ﬁnite. Perhaps inﬁnity still lurks in the wings ready to play a new and unexpected role in the drama of the Universe.

from strings to ‘m’ ‘But do you really mean, sir,’ said Peter, ‘that there could be other worlds—all over the place, just around the corner—like that?’ ‘Nothing is more probable’, said the Professor, taking off his spectacles and beginning to polish them, while he muttered to himself, ‘I wonder what they do teach them at these schools.’ – c. s. lewis, The Lion, The Witch, and The Wardrobe

After the initial excitement that followed the proofs that string theories are ﬁnite, many years of detailed study followed with hundreds of young mathematicians and physicists ﬂocking to join this research area at the world’s

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leading physics departments. Progress was slow and diﬃcult. It emerged that there were ﬁve varieties of string theory available to consider as a Theory of Everything, all ﬁnite and logically self-consistent, but all diﬀerent. This was a little disconcerting. You wait nearly a century for a Theory of Everything then, suddenly, ﬁve come along all at once. They had exotic sounding names that described aspects of the mathematical patterns they contained— type I, type IIA, and type IIB superstring theories, SO(32) and E8 heterotic string theories, and eleven-dimensional supergravity. These theories are all unusual in that they have ten dimensions of space and time, with the exception of the last one, which has eleven. Although it is not demanded by the ﬁniteness of the theory, it is generally assumed that just one of these ten or eleven dimensions is a ‘time’ and the others are spatial dimensions. Of course, we do not live in a ten- or eleven-dimensional space so in order to reconcile such a world with what we see it must be assumed that only three of the dimensions of space in these theories became large and the others remain ‘trapped’ with (so far) unobservably small sizes. It is remarkable that in order to achieve a ﬁnite theory we seem to need many more dimensions of space than those that we are aware of. This might be regarded as a prediction of the theory. It is a consequence of the amount of ‘room’ that is needed to accommodate the patterns governing the four known forces of Nature inside a single one without them being able to hive themselves oﬀ into sub-patterns that only talk to themselves rather than to everything else. Nobody knows why three dimensions (rather than one or four or eight, say) became large nor whether the number of large dimensions is something that arises at random (and so could be diﬀerent— and may be diﬀerent elsewhere in the Universe) or is an inevitable consequence of the laws of physics that could not be otherwise without destroying the logical self-consistency of the theory. One thing that we do know is that only in spaces with three large dimensions can things bind together to form structures like atoms, molecules, planets, and stars. No complexity and no life is possible except in spaces with three large dimensions. So, even if the number of large dimensions is diﬀerent in diﬀerent parts of the Universe, or separate universes are possible with diﬀerent numbers of large dimensions, we would have to ﬁnd ourselves living in one with three large dimensions no matter how improbable that might be, because we could exist in no other. At ﬁrst, it was hoped that one of these theories would turn out to be special and attention would then narrow in to reveal it to be the true Theory of Everything. Unfortunately, things were not so simple and progress was slow and unremarkable until Edward Witten, at Princeton, discovered that these diﬀerent string theories are not really diﬀerent. They are linked

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to one another by mathematical transformations that amount to exchanging large distances for small ones, and vice versa in a particular way. But this revealed that the ﬁve string theories were not the fundamental things that physicists had been searching for. Instead, they were each limiting situations of another deeper, but as yet unfound, Theory of Everything, which lives in eleven dimensions of space and time. That theory became known as ‘M Theory’, where M has been said to be an abbreviation for Mystery, Matrix, or Millennium, just as you like. We can think of M theory as the ball in Figure 2.3 and parts of its surface reveal the ﬁve string theories as limiting situations, cast like shadows upon it. The presence of the eleven-dimensional supergravity theory on the surface means that it might be that the hidden M theory is also eleven-dimensional but looks ten-dimensional at some places on its surface.

Figure 2.3 The known string theories appear to be limiting cases of a deeper underlying ‘M’ theory that has yet to be found. Each theory is described by a mathematical symmetry it displays. All of them exist in ten dimensions of space and time only, except for IID supergravity, which exists only in eleven dimensions.

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These mathematical discoveries launched an intensive search for the underlying M theory. But so far it has not been found. Other possibilities have emerged along the way, with the arguments of Lisa Randall and Raman Sundrum that the three-dimensional space that we inhabit may be thought of as the surface of a higher-dimensional space in which the strong, weak, and electromagnetic forces act only in that three-dimensional surface while the force of gravity reaches out into all the other dimensions as well. This is why it is so much weaker than the other three forces of Nature in this picture. Do these ‘extra’ dimensions of space really exist? This is a key question for all these new Theories of Everything. In most versions, the other dimensions are so small (10–33 cm) that no direct experiment will ever see them. But, in some variants, they can be much bigger. The interesting feature is that only the force of gravity will ‘feel’ these extra dimensions and be modiﬁed by their presence. In these cases the extra dimensions could be up to one hundredth of a millimetre in extent and they would alter the form of the law of gravity over these and smaller distances. Big changes in Newton’s inverse-square law of gravitational attraction between masses would occur, changing to an inversefourth power of the separation between masses, for example. This sounds like a major change but unfortunately it is very diﬃcult to test. Gravity is so weak that the form of the law of gravity is untested at these tiny distances. It is too diﬃcult to isolate the gravitational forces from all the overwhelmingly larger forces of adhesion, friction, magnetism, and so forth that dominate on small scales—look at that ﬂy walking on the ceiling, gravity is too weak to beat the forces of surface adhesion that hold his feet to the paintwork. This gives experimental physicists a wonderful challenge: test the form of the law of gravity on submillimetre scales. There is another fascinating consequence of extra dimensions that we shall have more to say about in Chapter 5. It involves changes to the constants of Nature. One of the changes to our picture of the world that results from accepting that we live in a nine- or ten-dimensional space is that the true constants of physics live in that number of dimensions too. The ones that we measure in the laboratory and have been in the habit of calling constants are not the truly fundamental constants of Nature at all, they are merely shadows of the higher-dimensional reality cast on our three large dimensions. Indeed, there is no reason why they need be constant at all and we ﬁnd that if the ‘other’ dimensions were to be slowly changing then we would see that because our three-dimensional ‘constants’ would change at the same rate as the change in the average size of the other dimensions. This means that observational searches for tiny variations in the traditional constants of Nature might reveal

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eﬀects caused by wobbles or steady changes in other dimensions of space. Although we can’t see them directly, in this case we can still see the eﬀects of their existence in our own three-dimensional space.

a flight of rationalistic fancy Why can’t somebody give us a list of things that everybody thinks and nobody says, and another list of things that everybody says and nobody thinks. — oliver wendell holmes

Let us examine some simple options that we can take with regard to the status of the laws of Nature. They provide a modern version of some ancient paradigms. Suppose, for simplicity, we restrict ourselves to three concepts: that of God (G) in the traditional omniscient and omnipotent sense; that of the Universe (U), taken to encompass the entire material world of space and time; and that of the laws of Nature (L), which prescribe its workings. The inter-relationships assumed between these three concepts rather succinctly encapsulate a number of diﬀerent philosophies of Nature. With regard to the pair U and L, we might choose one of ﬁve simple positions: 1. 2. 3. 4. 5.

U is a subset of L; L is a subset of U; L is the same as U; L is non-existent; U is non-existent.

These are illustrated schematically in Figure 2.4. The ﬁrst option takes the laws of Nature to be something that transcends the physical Universe. The Universe is one of its particular manifestations. There may be others either in possibility or in actuality. It is important to notice that the recent direction of research in cosmology which has sought to provide a mathematical account of the creation of the Universe out of ‘nothing’ implicitly assumes the situation (1). It must assume that there pre-exist laws of Nature and other primitive notions like logic prior to the creation of the material Universe. If such a research programme were to be successful and come up with a self-consistent picture of the appearance of the physical Universe which made predictions repeatedly borne out by experiment, then the next research

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L U L

(1)

(2)

(3)

(4)

(5)

Figure 2.4 The possible relationships between the concepts of U, the material Universe, and L, the laws of Nature, investigated in the text.

programme would seek to understand why the laws of Nature, which allowed that, and no other, Universe to appear, do themselves exist and whether they could be diﬀerent. This ultimate quest lies far in the future, but it is interesting to consider that if the Universe as a whole is described by a law of Nature like that enshrined in Einstein’s general theory of relativity then there must exist a logical structure larger than the physical Universe. Certainly such an assumption is made implicitly in most cosmological studies. For one considers diﬀerent possible mathematical models of the Universe each obeying the same laws of Nature but diﬀering in their choice of starting states. Unfortunately, no observations could tell us whether any cosmological theory described by a set of mathematical equations really did describe the entire Universe, if only because we can only ever see a ﬁnite part of it. If we subscribe to option (2), then we are nudged towards the view that the laws of Nature really possess some spatial or temporal dependence within the Universe. Elsewhere there may exist diﬀerent laws or even none at all. There may exist islands of rationality within a possibly inﬁnite universe. Since we know that the existence of observers like ourselves, and indeed observers considerably unlike ourselves, requires certain regularities to exist, we should not be surprised to ﬁnd ourselves inhabiting one of the rational suburbs of such a chaotic universe. Attempts have been made to demonstrate that it

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is plausible to start the evolution of the Universe in a state which does not possess an exact adherence of things to some of its familiar laws and yet show that, as it expands, ages, and cools, behaviours at variance with what we have come to call laws of Nature will become rarer and rarer, so that in our lowenergy world, ﬁfteen billion years after the beginning of things, we observe an approximate adherence to certain patterns of behaviour that is so close to perfect that we assume it to be perfect. Taken to its logical conclusion, this philosophy would aim to show that all, or almost all, the observed laws of Nature are consequences of the late epoch of cosmic history at which we have come upon the scene. Back in the earliest moments of the Big Bang the situation would be largely lawless and very, very diﬀerent. Another, more sceptical, interpretation of the second alternative is to regard the laws of Nature as an invention of human minds, which themselves have emerged from the stuﬀ of the Universe by natural processes. In diﬀerent parts of the Universe, the historical process that led to this would necessarily be diﬀerent: the environmental pressures would demand diﬀerent responses and distinct evolutionary pathways would no doubt be followed. On this view, the laws of Nature are a creation, either in whole or in part, of minds and will thus vary from galaxy to galaxy in line with the distribution of sentient beings in the Universe. This view, whilst common enough amongst philosophers, has little to commend itself to scientists, because it does not lead to any future research programme which might test it, falsify it, or enlarge upon its content. It is something of a speculative dead-end. All one can do is await contact with hypothetical extraterrestrials and compare their ‘laws’ with our ‘laws’. The perspective (3) equates the Universe and the laws of Nature in a spirit that goes back at least as far as St Augustine and Philo of Alexandria, who avoided the problem of deciding what God was doing before the creation of the world by pointing out that there was no ‘before’ because time was part of the created order. Such a deduction involves the perception that time is not just measured by natural phenomena like the swinging of a pendulum but may in some deep sense always be associated with physical events within the Universe rather than imposed upon it as a transcendental back-drop. This leads to the natural conclusion that the Universe is coeval with time itself. In Philo’s submission, Time began either simultaneously with the world or after it. For since time is a measured space determined by the world’s movement, and since movement could not be prior to the object moving, but must of necessity arise either after

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it or simultaneously with it, it follows of necessity that time also is either coeval with or later born than the world.

A similar perspective had been forced upon modern cosmologists up until recent years. Before attempts to understand quantum cosmology began in earnest, one was faced with the conclusion that our Universe must have experienced a space-time singularity at some ﬁnite moment in the past. Before this singularity, the Universe did not exist; afterwards, it did. The mathematical description of space and time predicts that both concepts must cease to exist at this singularity. It is the boundary of the Universe. Conversely, we are forced to regard universes which possess a past singularity as having an origin out of literally nothing at some past moment. At that moment, the material Universe, the laws of Nature, and the very fabric of space and time must come into being together. It is important to stress that, although Einstein’s general theory of relativity predicts that there can exist such a singularity in our past, it provides no reason why such a creation out of nothing should occur. If one does not want to come to terms with such a stark beginning to things, then there are ways of avoiding the conclusion that there existed a past singularity. If gravity were ever to become a repulsive, rather than an attractive, force in the distant past (and this seems rather likely given our present understanding of how matter could behave at very high energies), then the Universe need not have experienced a singular beginning. We oﬀer this merely as an illustration of the perspective (3). We might also point out that this alternative may accommodate wider possibilities because the Universe is expanding and changing in time. Does this mean that we should, on this view, expect the laws of Nature to possess a reciprocal time variation? In fact, it is not logically possible for all the laws of Nature to be changing. Either there are no laws at all or there are invariant laws. Any changing law can always be traced back to the invariance of some more basic quantity which governs the rule of change. The alternative, that there exists no invariant bedrock, would mean that there could exist no laws of Nature at all. This leads us to our next option. The fourth possibility, that there are no laws of Nature, is an extreme one. It might be defended in two ways. On the one hand, those of a more philosophical persuasion might seek to persuade us that what we choose to call the laws of Nature may be nothing more than the mental categories that our brains are forced to adopt in order to make sense of our experience. For all we know, there may exist no deep reality governed by true laws of Nature. Alternatively, a more realist perspective might be to imagine, as some physicists have done,

40 laws

that, as the Universe expands and ages from a state of chaos created by the simultaneous presence of all possible orders, some of these forms of order become predominant, so that after billions of years they dominate aﬀairs so eﬀectively that they pass for preordained laws of Nature rather than merely the stubbornest of possibilities. This possibility we have already mooted above. This situation also includes one in which there are many universes governed by a single set of underlying laws. Each universe is an outcome of the laws for which some things can fall out diﬀerently. Until fairly recently, this scenario was a philosophically possible one with little scientiﬁc basis. But investigations of the wider consequences of the theory of the inﬂationary universe have provided the ﬁrst reasons to take it a little more seriously. There is now good observational evidence that supports the idea that our visible universe underwent a surge of accelerated expansion, which we call ‘inﬂation’, in its very early stages. Observations of the small temperature variations in the microwave radiation left over from the early stages of the Universe display the same characteristic pattern of statistical variations that are predicted to result if we live in the vastly inﬂated image of a tiny primordial ﬂuctuation. But this same theory makes other predictions that are not amenable to observational test. It predicts that the little ﬂuctuations that inﬂate in the early Universe should continue producing further inﬂation from tiny parts of themselves over and over again. The process that results is an eternal self-reproduction process that creates a universe which is very diﬀerent from place to place and at diﬀerent times. We ﬁnd ourselves living in a local ‘bubble’ that, like any of the others, may have had a beginning and may have an end. But the whole ‘multiverse’ of bubbles need have no beginning and no end. Each of the bubbles can diﬀer in many respects. In the simplest versions of these theories the diﬀerences are in age and density. In other versions the diﬀerences are more fundamental: some bubbles have diﬀerent numbers of fundamental forces of Nature and diﬀerent numbers of dimensions of space. In these cases, the diﬀerent bubbles are like diﬀerent universes with some diﬀerent laws even though they are all part of the same universal space. In eﬀect, each of the bubbles is an outcome of the underlying laws of Nature which endow each bubble with some common features but many diﬀerent ones. Some of those diﬀering features are things that we have long regarded as so fundamental—like the number of dimensions of space—that they must be programmed into the Universe irrevocably, now turn out to be things that can fall out diﬀerently as outcomes of the laws of gravity and particle physics. All this is possible inside one universe or, in other theories still, in metaphysically separate ‘other’ universes governed by other laws that are logically self-consistent modiﬁcations of the ones that we

laws

know already. In all these situations the Universe—whether it is unique or one of many—is an outcome of the laws of Nature and, in a metaphysical sense, contained within them. The last of our choices, that there is no Universe, is a peculiar form of nihilism that no earnest philosopher has ever taken seriously. However, it is of interest because if the quantum cosmological models which seek to create the Universe out of nothing are considered then this view encapsulates their ‘pre-initial’ state. One cannot therefore argue that such a position is logically impossible or self-contradictory, since it is an admitted precursor to the present state within this cosmological description. It may be unstable in some peculiar sense, but it is hard to see why it should be impossible. To argue in this direction would seem to take us perilously close to resurrecting the infamous Ontological Argument of Anselm and others, that there can exist concepts like that of a Supreme Being whose very conception necessitates their existence. This seems particularly dubious when one tries to conceive of how there could exist some entity whose non-existence would imply a logical contradiction. For others, there is a tension primarily between the concepts of God and the Universe, rather than between the laws of Nature and the Universe. Indeed the concept of a Supreme Being is in all cultures a more primitive and natural notion than that of laws of Nature. It could well be argued that no culture arrived at a robust concept of the latter without a preliminary concept of the former. Again, it is convenient to list the naïve possibilities as follows: (i) (ii) (iii) (iv) (v)

U is a subset of G; G is a subset of U; G is the same as U; G is non-existent; U is non-existent.

The ﬁrst option, that the Universe is part of God, is called panentheism in the terminology adopted by the eighteenth-century German philosopher Krause. Theologians distinguish this view from simple theism by associating the latter with a view that God is wholly other than the Universe, both above it and becond it. The panentheist believes that God is in all things but not identical to them. The situation (ii) would be consistent with the sceptical attitude that the notion of ‘God’ is a creation solely of the human mind and hence of the purely material processes that gave rise to it. Alternatively, if the Deity were of the non-traditional sort and in some way limited to the role of a Superbeing within the Universe, this situation would be approximated. There are

42 laws

many science ﬁction stories that have explored this paternalistic Superbeing scenario. The semi-religious vision of advanced forms of intelligent extraterrestrial life discussed by many enthusiasts for the search for extraterrestrial radio signals might also ﬁt into this category. The third possible relationship is associated with the doctrine of pantheism which regards God and the natural Universe as one and the same thing. This is a common view to be found in many non-personal Eastern religions and also amongst agnostic scientists; it is also a view with which Einstein professed some sympathy. It is what he means when he talks of his God being that of Spinoza, the philosopher most associated with the pantheistic view. Our last two possibilities are easily dealt with. The option (iv) is the position of the atheist, whilst (v) has already been discussed as possibility (5) above. The third side of our triangle of relationships consists of the possible interrelation of the laws of Nature and a Deity: (a) (b) (c) (d) (e)

L is a subset of G; G is a subset of L; G is the same as L; L is non-existent; G is non-existent.

The ﬁrst case is in line with a Judaeo-Christian tradition that views the laws of Nature as constraints that God imposes upon the Universe. This was, for instance the view of Newton, who consequently maintained that the laws of Nature could have been diﬀerent and could be suspended arbitrarily according to divine ﬁat. The second possibility is somewhat akin to the schools of Process Theology that propose an evolving Deity. In this case, God is constrained by some higher-order logic. Although this would be diﬃcult to reconcile with many pictures of an omnipotent Deity, it is diﬃcult to draw the line between this position and what is generally assumed implicitly even in these pictures, that God’s actions are bound by certain constraints of logic and related to such concepts as ‘good’ and ‘evil’. Alternatively, this option may be interpreted nontheistically as one in which the concept of God is an inevitable outworking of the laws of Nature in the minds of certain species of complex biocomputers like ourselves. The third case, which equates the laws of Nature with God, is similar to the impersonal picture of God adopted by some pantheists. But it also resembles the view of the Deists which emerged as a lowest common denominator in response to the labyrinth that seventeenth-century theologians of all creeds

laws

found themselves within. It reduced the number of attributes which the Deity was expected to display in the Universe and reduced Him to the role of initial cause and sustainer of the laws of Nature who thereafter maintained all things in harmonious development. The cases (d) and (e) we have already met as (4) and (iv) above.

goodbye to all that Every dogma must have its day. — h. g. wells

Our exploration of the laws of Nature has been rather cursory. ∗ The reason is deliberate: to most minds the issue of a Theory of Everything is about nothing more than the laws of Nature. It is a quest for the most basic and most comprehensive versions of those universal laws. From these, it is assumed that everything one might want to know or explain regarding the nature of the observed universe would follow with a little work. In the chapters to follow, we hope to undermine this dogma and reveal what other aspects of the physical world, distinct from the traditional image of laws of Nature, are needed to understand its overall structure. To come to terms with them will require either additional facets of the Universe to be uncovered or the concept of a law of Nature to be considerably deepened and widened to unify it to other concepts that are at present logically disjoint from it. From Boscovich to superstrings, the searchers for a uniﬁed Theory of Everything have focused upon ﬁnding the all-encompassing laws of Nature to the exclusion of all else. At root, this prejudice has grown from an implicit subservience to the Platonic emphasis upon timeless universals as more important in the nature of things than the world of particulars that we observe and experience. In the chapters to follow, we shall examine the challenges to this view that are oﬀered by our latest ideas about the physical world. The ﬁrst is almost familiar. Since science pays homage to the gods of change, it needs to know how things began if it is to know anything at all. ∗

A more extensive study of this subject can be found in the author’s earlier volume The World within the World.

chapter 3

Initial conditions Once upon a time and a very good time it was. — james joyce

at the edge of things Science is a differential equation. Religion is a boundary condition. — alan turing

Laws of Nature tell us how things change. Yet behind them we believe there to lurk invariances that straitjacket reality. Nature can do whatever she pleases so long as these charmed quantities stay the same throughout the change. The Theory of Everything seeks to provide us with the ultimate directory of all possible changes. The guiding principle in the search for this all-controlling formula is that it must be a single law, not a collection of diﬀerent pieces. The logical unity of the Universe demands a single invariance that remains unchanged in the face of all the complexity and transience we see about us from the smallest sub-atomic scales to the farthest reaches of outer space. Identifying this over-arching symmetry, if it does indeed exist and is manifest in a form that is intelligible to us, may be the nearest thing we could get to discovering the ‘secret of the Universe’. Yet this is still not enough. Even if we knew the rules which govern how all things change, then we can only understand the present structure of things if we know how they began. This is a legacy of our belief in the rule of cause and eﬀect in the Universe and our representation of laws of Nature as diﬀerential equations or algorithms in which output is determined uniquely by input. Diﬀerential equations are mathematical ‘machines’ which allow us to predict

initial conditions

the future from the present. Equally, they enable us to use the present to reconstruct the past.

axioms Set theory can be viewed as a form of exact theology. — rudy rucker

In mathematics, the role of initial conditions is played by axioms. These are the initial postulates that are made before we start employing any deductive reasoning. The classic example of an axiomatic system is that of plane geometry formulated by Euclid in about 300 bc. It forms the model of all rigorous mathematical schemes. The axioms are initial assumptions which are taken as self-evidently true. From them, logical deductions can proceed under stipulated rules of reasoning. These rules of logical reasoning are analogous to the scientists’ laws of Nature, whilst the axioms play the role of initial conditions. We are not free to pick any axioms we might care to choose. They must be logically consistent. But there is no limit to their number although the number of axioms that we introduce will determine the size and richness of the logical deductions that can follow from them. Whereas Euclid and most other pre-nineteenth-century mathematicians knew that logical consistency was essential in any choice of axioms, they were also strongly biased towards picking axioms which mirrored the way the world was observed to work. Thus Euclid’s axioms—for example, that parallel straight lines never meet, or that there is only one straight line joining any two points on a ﬂat surface—are the self-evident fruits of one’s experience of drawing lines on a ﬂat surface. Later mathematicians did not feel so encumbered and have required only consistency from their lists of axioms. They need have no correspondence with anything we can see or abstract from experience. It remains to be seen whether the initial conditions appropriate to the deepest physical problems, like the cosmological problem which we shall discuss below, will have speciﬁcations which are directly related to visualizable physical things, or whether they will be abstract mathematical or logical notions that enforce only self-consistency. Even if the latter situation prevails, it may transpire that the requirement of self-consistency in a system as self-evidently complex as the physical universe is adequate to ﬁx those initial conditions uniquely and completely. Another important lesson we have learnt from the mathematicians’ approach to axiomatic systems is that one can quantify the amount of

46 initial conditions

information that is contained in a collection of axioms. None of the possible deductions that can be proved from these axioms using the allowed rules of reasoning can possess more information than was contained in the axioms. In essence, this is the reason for the famous limits to the power of logical deduction expressed by Gödel’s incompleteness theorem. The axioms of ordinary arithmetic (and any axiomatic system rich enough to contain the whole of arithmetic) contain less information than some arithmetical statements and hence those axioms and their associated rules of reasoning cannot determine whether these statements are true or false. Note, however, that an axiomatic system which is not as large as the whole of arithmetic does not suﬀer from Gödel’s incompleteness. For example, the so-called Presburger arithmetic, which consists of the operation of addition upon zero and the positive whole numbers (but not subtraction) has the property that all its statements are decidable. Its reduced set of axioms contain suﬃcient information to ascertain the truth or falsity of all the statements that can be framed using its vocabulary. In the ﬁrst chapter, we introduced the notion of algorithmic compressibility as a criterion for determining the degree of randomness of mathematical expressions. We can make use of this concept again here to sharpen our discussion. If presented with a particular sequence then we cannot prove it to be random, although we can prove it to be non-random simply by ﬁnding a compression. The minimum compression that is possible for a logical system corresponds to the axioms of the system. Thus we see why there can be no theorem of the system which possesses a larger information content than the axioms of the system. Axioms are not therefore quite as straightforward as one might have hoped. It is often a rather subtle question to decide whether diﬀerent proposed axioms are truly independent of each other. There is one classic case of this sort which enshrouds one of the most diﬃcult unsolved problems in mathematics. It is called the continuum hypothesis. Prior to the work of Georg Cantor in the mid-nineteenth century, mathematicians had denied the existence of real inﬁnities. Indeed, inﬁnities were an ‘abomination’ in the words of one famous mathematician. Gauss’s views on the matter are that he would protest against using inﬁnite magnitude as something consummated; such a use is never admissible in mathematics. The inﬁnite is only a façon de parler: one has in mind limits which certain ratios approach as closely as is desirable, while other ratios may increase indeﬁnitely.

Here we see spelt out the notion that the inﬁnite can never be an actuality; it is merely a shorthand for something that can be as large as one wishes. But Cantor turned the world upside-down by treating inﬁnities like other

initial conditions

mathematical quantities and creating an entire series of inﬁnities of diﬀerent sizes. The smallest was the set of natural numbers {1, 2, 3, 4, 5, . . . }, which was labelled ℵ0 (aleph-nought). Another inﬁnite set is said to have the same size (or cardinality) as ℵ0 if its members can be put into a direct one-to-one correspondence with the natural numbers; that is, if they can be systematically counted. For example, the inﬁnite set of all the even numbers {2, 4, 6, 8, 10, . . . } can be counted in this way by the correspondences displayed by the sequence of arrows in Figure 3.1(a). The arrowed path in Figure 3.1(b) then shows how all the rational fractions laid out in an inﬁnite array can be counted one by one without any being omitted. This shows there to exist a direct one to one correspondence between ℵ0 and all the rational fractions through the sequence 11 , 21 , 12 , 13 , 22 , 31 , 41 , 32 , 23 , 14 , 15 , 24 , 33 , 42 , 51 , 61 , 52 , 43 , . . . , and so on ad inﬁnitum. Hence, in this precise sense, the rational fractions are an inﬁnite set of the same size as the natural numbers. At ﬁrst sight, this is a surprising result, since natural numbers are rather sparsely distributed whereas there seem to be rational fractions densely packed everywhere in-between them so a counting process ought to ﬁnd many more fractions that integers. But this intuition focuses too much upon the order in which the numbers appear, whereas the one-to-one correspondence that we have set up does not need to follow the order in size with which the fractions occur in between the integers. A fraction

1 1

2 1

3 1

4 1

5 1

6 1

7 1

1 2

2 2

3 2

4 2

5 2

6 2

7 2

1 3

2 3

3 3

4 3

5 3

6 3

7 3

1 4

2 4

1 5

1 6

(a)

(b)

Figure 3.1 (a) The positive even numbers form an inﬁnite set of the same ‘size’ as all the positive integers because the two sets can be put into the direct one-on-one relationship shown here. This means that they can be systematically counted. (b) The set of all rational fractions can also be put in a one-on-one relationship with the positive integers and hence be systematically counted if they are listed in the pattern shown and then counted in the order marked by the sequence of arrows, ad inﬁnitum.

48 initial conditions

is just speciﬁed by a pair of numbers and there are as many inﬁnite pairs of numbers as there are numbers. If we now try and count not just all the fractions but the decimals as well, then something qualitatively diﬀerent happens because there are so many more decimals than fractions. The jump in size that marks the step from the natural numbers to the decimals is comparable to the step one would have to take from just the numbers zero and one to the larger ones. To take such a step, further information is required, because the only way that we can make 2 from 0 and 1 is to add two 1’s together, but such a move requires us to be in possession of the concept of ‘two’ already. Cantor showed that if we try to count the number of inﬁnite decimals (the so-called ‘real numbers’), then we fail. They are of a higher cardinality than the natural numbers and so cannot be placed in a one-to-one correspondence with them. This he showed by an ingenious and very powerful new form of argument. It involves the notion of a diagonal number. For illustration, suppose we have four numbers of four digits in length: 1234 5678 9012 3456 Then the diagonal number 1616 is not one of the four numbers listed. What Cantor showed was that if we make this array of numbers inﬁnitely large then there is always a way of concocting a diagonal number that is not one of the inﬁnite list of numbers lined up to make the array. Suppose we just look at the real numbers between zero and 1 (it does not make any diﬀerence to the basic argument if we add all the others as well) and suppose that we can count all the inﬁnite decimals. This, Cantor showed, leads to a contradiction. Suppose we could write down all possible inﬁnite decimals and align them one-to-one with the natural numbers. Let us suppose the list begins as follows: 1 2 3 4 5 6

0.234566789 0.575603737 0.463214516 0.846216388 0.562194632 0.466732271

... ... ... ... ... ...

and so on to inﬁnity. Now take the diagonal number with decimal part composed of the highlighted digits: 0.273292 . . .

initial conditions

Next, alter each digit by adding one to it to get the new decimal 0.384303

...

Then this new number cannot appear anywhere on the original list because it diﬀers by one digit from every single horizontal entry. ∗ It must at least diﬀer from the ﬁrst entry in the ﬁrst digit and from the second entry in the second digit and so on. So, contrary to our original supposition, the list could not have contained all the possible decimals. Hence, our original assumption that all the inﬁnite decimals can be systematically counted was false. The real numbers possess a higher cardinality than the natural numbers and it is denoted by the symbol ℵ1 (‘aleph-one’). Cantor raised the intriguing question of whether there exist inﬁnite sets which are intermediate in size between the natural numbers and the real numbers. Cantor thought that there could not be, but was unable to prove it. This is called the continuum hypothesis. Indeed, Cantor appears to have suﬀered a mental breakdown brought about by the intellectual eﬀort he expended upon this question. The problem remains unsolved to this day. Nevertheless Kurt Gödel and his young American colleague Paul Cohen demonstrated some deep and unusual things about it. Gödel showed that if we merely treat the continuum hypothesis as an additional axiom and add it to the conventional axioms of set theory † then no logical contradiction can result. But then, in 1963, Cohen showed that the continuum hypothesis is independent of the ∗

As a technicality note that we have to remove any ambiguity about decimals that end with recurring 9s because 0.2399999999 . . . , say, is the same as 0.240000000. . . . So, by cutting out those decimal expansions that end in a run of zeros we identify the rational number 24/100 by 0.23999 . . . and not by 0.240000. . . .

†

The seven axioms of standard set theory which are intended to be suﬃcient for the deduction of all of mathematics (and hence for the mathematical representation of physics) are as follows. (1) Extensionality: two sets are equal if and only if they contain the same members. (2) Subsets: given a set S and some meaningful property, there exists a set containing members of S, and only those members of S which possess this property. (3) Pairing: given any two diﬀerent sets, there exist another set that contains just the members of these two sets. (4) Sum-set: if there is a set S whose members are themselves sets, then there exists a set (called the sum set of S) whose members are just the members of the members of S. (5) Inﬁnity: there exists at least one inﬁnite set (i.e. the natural numbers 1, 2, 3, . . . ). (6) Power set: for any set S, there exists another set whose members are the subsets of S. (7) Choice: if S is a set of sets that is not empty and no two distinct members of S have an element in common, then there exists a set which consists solely of a single element taken from each set of S. It is of these axioms that Kurt Gödel said: Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less conﬁdence in this kind of perception, i.e., in mathematical intuition, than in sense perception . . . They, too, may represent an aspect of objective reality.

50 initial conditions

other axioms of set theory (just as Euclid’s parallel postulate was eventually shown to be independent of the other axioms of plane geometry) and therefore could be neither proved nor refuted from those axioms. The lesson we learn here is that mathematical axioms are more like initial conditions for natural laws than we might have suspected. Indeed, it is the hope of some that they may turn out to be the same: that the ultimate assumptions one has to make about the input assumptions for the Theory of Everything are those required for logical consistency. But we have also learned that their nature and inter-relationship is extremely subtle. We are at liberty to choose whichever collection of them suits our purpose. Lacking an obvious intuitive guide as to the appropriateness of highly abstruse axioms (like the continuum hypothesis, for example), how do we know whether they should be included or not? Motivated by this experience with the continuum hypothesis problem Alonzo Church remarked that . . . if a choice must in some sense be made among the rival set theories, rather than merely and neutrally to develop the mathematical consequences of alternate theories, it seems that the only basis for it can be the same informal criterion of simplicity that governs the choice among rival physical theories when both or all of them equally explain the experimental facts.

Cohen’s demonstration that the continuum hypothesis is independent of the other axioms of set theory means that we are equally at liberty either to add it or its negation to the existing axioms of set theory. In each case, we could create a diﬀerent enlargement of set theory, just as we can retain Euclid’s parallel postulate or replace it by its negation to create logically consistent non-Euclidean geometries. If one is a mathematical Platonist who believes that mathematical entities really exist then only one of those two mutually exclusive set theories really exists, but if one is a constructivist or formalist then each are equally valid intellectual creations. There is one area of interaction between fundamental physics and these foundational questions regarding inﬁnity. It concerns the issue of whether a true continuum exists in reality or not. Most fundamental pictures of the physical world assume that the basic notions—ﬁelds, space, and time—are continuous entities rather than discrete bits. This issue of discreteness versus continuity is an ancient tension in natural philosophy that re-emerges in every era in new dress. The most important point about it for the structure of any theory of inﬁnity is the vast diﬀerence in complexity that would exist between a continuous and a discrete Theory of Everything. The reason for this is that the number of continuous transformations that exist between one set of real

initial conditions

numbers and another is a whole order of inﬁnity lower than the total number of possible transformations which are not continuous. The requirement of continuity produces a vast and surprising reduction in scope. Since these continuous transformations include the catalogue of possible relationships from which we draw that class of transformations (or ‘equations’) called the laws of physics, we see that a discontinuous world will be inﬁnitely more complex in its potentiality. It is less constrained in what it is allowed to do. At present, physicists are enamoured of symmetry and search only for continuous pictures of fundamental physics. Maybe, one day, they will be motivated to look at possible structures of a fundamentally discrete world. In Chapter 9, we shall look at some of the ideas that might provoke them to do so. What statements can be proved or disproved depends crucially upon the information content of the axioms at hand. Some philosophers of science have used Gödel’s theorems regarding the incompleteness of arithmetic (and hence of any logical system containing arithmetic) to argue that we can never know everything about the physical universe in terms of mathematical laws of Nature.

mathematical jujitsu No mathematical theorem has aroused as much interest among non-mathematicians as Gödel’s incompleteness theorem . . . One ﬁnds invocations not only in discussion groups dedicated to logic, mathematics, computing, or philosophy, where one might expect them, but also in groups dedicated to politics, religion, atheism, poetry, evolution, hip-hop, dating, and what have you. — torkel franzén

Gödel’s monumental demonstration, that complicated systems of mathematics have self-imposed limits on what they can prove, gradually changed the way in which philosophers and scientists viewed the world and our quest to understand it. Superﬁcially, it appears that all human investigations of the Universe must be limited. Science is based on mathematics; mathematics cannot discover all truths; therefore science cannot discover all truths. This is an argument that is often heard. One of Gödel’s contemporaries, the famous mathematician Hermann Weyl, described Gödel’s discovery as exercising a ‘constant drain on the enthusiasm’ with which he pursued his scientiﬁc research. In more recent times, a frequent writer on theology and science,

52 initial conditions

Stanley Jaki, has argued that Gödel’s theorem prevents us from gaining an understanding of the cosmos as a necessary truth, Clearly then no scientiﬁc cosmology, which of necessity must be highly mathematical, can have its proof of consistency within itself as far as mathematics goes. In the absence of such consistency, all mathematical models, all theories of elementary particles, including the theory of quarks and gluons . . . fall inherently short of being that theory which shows in virtue of its a priori truth that the world can only be what it is and nothing else. This is true even if the theory happened to account with perfect accuracy for all phenomena of the physical world known at a particular time.

and so is a fundamental barrier to human understanding of the Universe: It seems on the strength of Gödel’s theorem that the ultimate foundations of the bold symbolic constructions of mathematical physics will remain embedded forever in that deeper level of thinking characterized both by the wisdom and by the haziness of analogies and intuitions. For the speculative physicist this implies that there are limits to the precision of certainty, that even in the pure thinking of theoretical physics there is a boundary . . . An integral part of this boundary is the scientist himself, as a thinker.

Intriguingly, and just to show the important role human psychology plays in assessing the signiﬁcance of limits, some other scientists, like Freeman Dyson, acknowledge that Gödel places limits on our ability to discover the truths of mathematics and science, but interpret this as ensuring that science will go on forever. Dyson, who had some contact with Gödel during his time at Princeton’s Institute for Advanced Study, sees the incompleteness theorem as an insurance policy against the scientiﬁc enterprise, which he admires so much, coming to a self-satisﬁed end; for Gödel proved that the world of pure mathematics is inexhaustible; no ﬁnite set of axioms and rules of inference can ever encompass the whole of mathematics; given any set of axioms, we can ﬁnd meaningful mathematical questions which the axioms leave unanswered. I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.

In these two quite diﬀerent statements, we see the optimistic and the pessimistic responses to Gödel. The optimists, like Dyson, see his result as a guarantor of the never-ending character of human investigation. They see

initial conditions

scientiﬁc research as an essential part of the human spirit whose ﬁnal completion would have a disastrous demotivating eﬀect upon us. The pessimists, like Jaki, interpret Gödel as establishing that the human mind cannot know all of the secrets of Nature. They place more emphasis upon the possession and application of knowledge than on the process of acquiring it. The pessimist does not see the principal human beneﬁt of science as arising from the quest for knowledge itself. On reﬂection we should not be too surprised at such diametrically opposed responses. Many things in life create the same hiatus. It all depends whether you think your glass is half empty or half full. Gödel’s own view was as unexpected as ever. He thought that intuition, by which we can ‘see’ truths of mathematics and science, was a tool that would one day be valued just as formally and reverently as logic itself, I don’t see any reason why we should have less conﬁdence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in the future.

Gödel himself was not minded to draw any strong conclusions for the scope of physics from his incompleteness theorems. He made no connections with the Uncertainty Principle of quantum mechanics, which was another great deduction which limited our ability to know, and which was discovered by Heisenberg just a few years before Gödel proved his ﬁrst theorem. In fact, Gödel was rather hostile to any consideration of quantum mechanics at all. Those who worked at the same Institute (no one really worked with him) believed that this was a result of his frequent discussions with Einstein who, in the words of John Wheeler (who knew them both) ‘brainwashed Gödel’ into disbelieving quantum mechanics and the Uncertainty Principle. Greg Chaitin records this account of Wheeler’s attempt to draw Gödel out on the question of whether there is a connection between Gödel incompleteness and Heisenberg Uncertainty, Well, one day I was at the Institute for Advanced Study, and I went to Gödel’s oﬃce, and there was Gödel. It was winter and Gödel had an electric heater and had his legs wrapped in a blanket. I said ‘Professor Gödel, what connection do you see between your incompleteness theorem and Heisenberg’s uncertainty principle?’ And Gödel got angry and threw me out of his oﬃce!

The argument that mathematics contains unprovable statements, physics is based on mathematics, therefore physics will not be able to discover everything

54 initial conditions

that is true, has been around for a long time. With these worries in mind, let us look a little more closely at what Gödel’s result might have to say about the course of physics. The situation is not so clear-cut as the commentators would have us believe. It is useful to lay out the precise assumptions that underlie Gödel’s deduction of incompleteness. Gödel’s theorem says that if a formal system is 1. ﬁnitely speciﬁed 2. large enough to include arithmetic 3. consistent then it is incomplete. Condition 1 means that there is not an uncomputable inﬁnity of axioms. We could not, for instance, choose our system to consist of all the true statements about arithmetic because this collection cannot be ﬁnitely listed in the required sense. Condition 2 means that the formal system includes all the symbols and axioms used in arithmetic. The symbols are 0, ‘zero’, S, ‘successor of ’, +, ×, and =. Hence, the number two is the successor of the successor of zero, written as the term SS0, and two and plus two equals four is expressed as SS0 + SS0 = SSSS0. The structure of arithmetic plays a central role in the proof of Gödel’s theorem. Special properties of numbers, like their primeness and the fact that any number can be expressed in only one way as the product of the prime numbers that divide it, were used by Gödel to establish the vital correspondence between statements of mathematics and statements about mathematics. Thereby, linguistic paradoxes like that of the ‘liar’ could be embedded, like Trojan horses, within the structure of mathematics itself. Only logical systems which are rich enough to include arithmetic allow these incestuous encodings of statements about themselves to be made within their own language. Again, it is instructive to see how these requirements might fail to be met. Pick a theory that consists of references to (and relations between) only the ﬁrst ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) with base-10 arithmetic, then Condition 2 fails and such a mini-arithmetic is complete. Arithmetic makes statements about individual numbers, or terms (like SS0, above). If a system does not have individual terms like this but, like Euclidean geometry, only makes statements about a continuum of points, circles, and lines, in general, then it cannot satisfy Condition 2. And so, as Alfred Tarski ﬁrst showed, Euclidean geometry is complete. There is nothing magical about the ﬂat, Euclidean nature of the geometry either: the non-Euclidean geometries on

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curved surfaces are also complete. Similarly, if we had a logical theory dealing with numbers that only used the concept of ‘greater than’ and ‘less than’ without referring to any speciﬁc numbers then it would be complete: we can determine the truth or falsity of any statement about real numbers involving the ‘greater than’ relationship. Another example of a system that is smaller than arithmetic is arithmetic without the multiplication, ×, operation. This is called Presburger arithmetic (the full arithmetic is called Peano arithmetic after the mathematician who ﬁrst expressed it axiomatically, in 1889). At ﬁrst, this sounds strange—in our everyday encounters with multiplication it is nothing more than a shorthand way of doing addition (e.g., 2 + 2 + 2 + 2 + 2 + 2 = 2 × 6)—but in the full logical system of arithmetic, in the presence of logical quantiﬁers like ‘there exists’ or ‘for any’, multiplication permits constructions which are not merely equivalent to a succession of additions. Gödel showed, as part of his doctoral thesis work, that Presburger arithmetic is complete: all statements about the addition of natural numbers can be proved or disproved; all truths can be reached from the axioms. Similarly, if we create another truncated version of arithmetic, which does not have addition, but retains multiplication, this is also complete. It is only when addition and multiplication are simultaneously present that incompleteness emerges. Arithmetic is the watershed in complexity for incompleteness to appear. The use of Gödel’s theorem to place limits on what a mathematical theory of physics (or anything else) can ultimately tell us seems at ﬁrst to be a fairly straightforward consequence. But as one looks more carefully into the question, things are not quite so simple. Suppose, for the moment, that all the conditions required for Gödel’s theorem to hold are in place. What would incompleteness look like in practice? We are familiar with the situation of having a physical theory which makes accurate predictions about a wide range of observed phenomena: we might call it ‘the standard model’ or ‘string theory’. One day, we may be surprised by an observation about which it has nothing to say. It cannot be accommodated within its framework. Examples are provided by some so called ‘grand uniﬁed theories’ in particle physics. Some early editions of these theories had the property that all neutrinos must have zero mass. When neutrinos were observed to have a non-zero mass then we know that the new situation cannot be accommodated within our original theory. What do we do? We have encountered a certain sort of incompleteness, but we respond to it by extending or modifying the theory to include the new possibilities. Thus, in practice incompleteness looks very much like inadequacy in a theory.

56 initial conditions

In the case of arithmetic, if some statement about arithmetic is known to be undecidable (there are known statements of this sort, it means that both their truth and falsity are consistent with the axioms of arithmetic) then we have two ways of extending the structure. We can create two new arithmetics: one which adds the undecidable statement as an extra axiom, the other which adds its negation as a new axiom. Of course, the new arithmetics will still be incomplete, but they can always be extended to accommodate any incompleteness. Thus, in practice, a physical theory can always be enlarged by adding new principles which force all the undecidability into the part of the mathematical realm which has no physical manifestation. Incompleteness would then always be very hard, if not impossible, to distinguish from incorrectness or inadequacy. An interesting example of this dilemma is provided by the history of mathematics. During the sixteenth century, mathematicians started to explore what happened when they added together inﬁnite lists of numbers. If the quantities in the list get larger then the sum will ‘diverge’, that is, as the number of terms approaches inﬁnity so does the sum. An example is the sum 1 + 2 + 3 + 4 + 5 + . . . = inﬁnity. However, if the individual terms get smaller and smaller suﬃciently rapidly ∗ then the sum of an inﬁnite number of terms can get closer and closer to a ﬁnite limiting value which we shall call the sum of the series; for example 1+

1 1 1 2 1 + + + + ... = = 1.2337005. 9 25 36 49 8

This left mathematicians to worry about a most peculiar type of unending sum, 1 – 1 + 1 – 1 + 1 – 1 + 1 . . . = ????? If you divide up the series into pairs of terms it looks like (1 – 1) + (1 – 1) + . . . and so on. This is just 0 + 0 + 0 + . . . = 0 and the sum is zero. But think of the series as 1 – {1 – 1 + 1 – 1 + 1 – . . .} and it looks like 1 – {0} = 1. We seem to have proved that 0 = 1. Mathematicians had a variety of choices when faced with ambiguous sums like this. They could reject inﬁnities in mathematics and deal only with ﬁnite sums of numbers, or, as Cauchy showed in the early nineteenth century, the sum of a series like the last one must be deﬁned by specifying more closely ∗

That the terms in the sum get progressively smaller is a necessary but not a suﬃcient condition for an inﬁnite sum to be ﬁnite. For example, the sum 1 + 12 + 13 + 14 + 15 + . . . is inﬁnite.

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what is meant by its sum. The limiting value of the sum must be speciﬁed together with the procedure used to calculate it. The contradiction 0 = 1 arises only when one omits to specify the procedure used to work out the sum. In both cases it is diﬀerent and so the two answers are not the same. Thus, here we see a simple example of how a limit is side-stepped by enlarging the concept which seems to create limitations. Divergent series can be dealt with consistently so long as the concept of a sum for a series is suitably extended. Another possibility, which appears very likely to be true, is that the laws of Nature only use the decidable part of mathematics. We know that mathematics is an inﬁnite sea of possible structures. Only some of those structures and patterns appear to ﬁnd existence and application in the physical world. Very few of them are used to describe the laws of Nature. It may be that they are all from the subset of decidable truths. It is also possible that the conditions required to prove Gödel’s incompleteness do not apply to physical theories. Condition 1 requires the axioms of the theory to be listable. It might be that the laws of physics are not listable in this special sense. This would be a radical departure from the situation that we think exists, where the number of fundamental laws is believed to be not just listable, but ﬁnite (and very small). Yet, it is always possible that we are just scratching the surface of a bottomless tower of laws, only the top of which has signiﬁcant eﬀects upon our experience. However, if there were an unlistable inﬁnity of physical laws then we would face a more formidable problem than that of incompleteness. An equally interesting issue is that of ﬁniteness. It may be that the universe of physical possibilities is ﬁnite, although astronomically large. However, no matter how large the number of primitive quantities to which the laws refer, so long as they are ﬁnite the resulting system of interrelationships will be logically complete. We should stress that although we habitually assume that there is a continuum of points of space and time this is just an assumption that is very convenient for the use of simple mathematics. There is no deep reason to believe that space and time are continuous, rather than discrete, at their most fundamental microscopic level; in fact, there are some theories of quantum gravity that assume that they are not. Quantum theory has introduced discreteness and ﬁniteness in a number of places where once we believed in a continuum of possibilities. Curiously, if we give up this continuity, so that there is not necessarily another point in between any two suﬃciently close points you care to choose, space-time structure becomes vastly more complicated. Many more complicated things can happen. This question of ﬁniteness might also be bound up with the question of whether the

58 initial conditions

Universe is ﬁnite in volume and whether the number of elementary particles (or whatever the most elementary entities might be) of Nature are ﬁnite or inﬁnite in number. Thus there might only exist a ﬁnite number of terms to which the ultimate logical theory of the physical world applies. Hence, it would be complete. An interesting possibility with regard to the application of Gödel to the laws of physics is that Condition 2 of the incompleteness theorem might not be met. How could this be? Although we seem to make wide use of arithmetic, and much larger mathematical structures, when we carry out scientiﬁc investigations of the laws of Nature, this does not mean that the inner logic of the physical Universe needs to employ such a large structure. It is undoubtedly convenient for us to use large mathematical structures together with concepts like inﬁnity but this may be an anthropomorphism. The deep structure of the Universe may be rooted in a much simpler logic than that of full arithmetic, and hence be complete. All this would require would be for the underlying structure to contain either addition or multiplication but not both. Recall that all the sums that you have ever done have used multiplication simply as a shorthand for addition. They would be possible in Presburger arithmetic as well. Alternatively, a basic structure of reality that made use of simple relationships of a geometrical variety, or which derived from ‘greater than’ or ‘less than’ relationships, or subtle combinations of them all could also remain complete although the proofs needed to demonstrate them become very long. The fact that Einstein’s theory of general relativity replaces many physical notions like force and weight by geometrical distortions in the fabric of spacetime may well hold some clue about what is possible here. The laws of physics might be fully expressible in terms of a mathematical system that is complete, but in practice we would always be far more concerned with making sure that we had got the correct system than a complete system. Tarski showed that, unlike arithmetic of natural numbers, the ﬁrstorder theory of real numbers under addition and multiplication is decidable. This is rather surprising and may give some hope that theories of physics based on the real or complex numbers will evade undecidability. Many mathematical systems used in physics, like lattice theory, projective geometry, and Abelian group theory are also decidable, although others, notably non-Abelian groups are not. There is another important aspect of the situation to be keep in view. Even if a logical system is complete, it always contains unprovable ‘truths’. These are the axioms which are chosen to deﬁne the system and they are assumed to be independent of each other and consistent. And after they are chosen, all

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the logical system can do is deduce conclusions from them. In simple logical systems, like Peano arithmetic, the axioms seem reasonably obvious because we are thinking backwards—formalizing something that we have been doing intuitively for thousands of years. When we look at a subject like physics, there are parallels and diﬀerences. The axioms, or laws, of physics are the prime target of physics research. They are by no means intuitively obvious, because they govern regimes that can lie far outside our experience. The outcomes of those laws are unpredictable in certain circ*mstances because they involve symmetry breakings. Trying to deduce the laws from the outcomes is not something that we can ever do uniquely and completely by means of a computer program. Thus, we detect a completely diﬀerent emphasis in the study of formal systems and in physical science. In mathematics and logic, we start by deﬁning a system of axioms and laws of deduction. Then, we might try to show that the system is complete or incomplete, and deduce as many theorems as we can from the axioms. In science, we are not at liberty to pick any logical system of laws that we choose. We are trying to ﬁnd the system of laws and axioms (assuming there is one—or more than one perhaps) that will give rise to the outcomes that we see. It is always possible to ﬁnd a system of laws which will give rise to any set of observed outcomes. But it is the very set of unprovable statements that the logicians and the mathematicians ignore—the axioms and laws of deduction—that the scientist is most interested in discovering, rather than simply assuming. The only hope of proceeding as the logicians do, would be if for some reason there is only one possible set of axioms or laws of physics which could include all the forces that we know of. So, in summary so far, we have argued that there is no reason to believe that Gödel’s incompleteness theorem places any restriction on our ability to ﬁnd the ultimate laws of Nature—the Theory of Everything. Physics uses only a part of mathematics and that part can lie in the decidable area of mathematics. In fact, the mathematics used in expressing the known laws of Nature uses only simple patterns and the process of ﬁnding them is not beset by undecidability. However, whereas the laws of Nature are simple, their outcomes are not. They are complicated and asymmetrical and we have often been in a situation where we have a law of Nature in the form of a system of equations but we don’t know how to solve them to determine the outcomes of the laws. It is in this realm of the complicated outcomes of the known laws that we expect Gödel incompleteness to rear its head. We already know of a number of questions that we could ask about the Universe that cannot be answered because of incompleteness. They are not restrictions on determining the laws

60 initial conditions

of Nature but they do stop us using them to answer some simple questions that we might have expected to have accessible answers. Speciﬁc examples have been given of physical problems which are undecidable. As one might expect from what has just been said, they do not involve an inability to determine something fundamental about the nature of the laws of physics or the most elementary particles of matter. Rather, they involve an inability to perform some speciﬁc mathematical calculation, which inhibits our ability to determine the course of events in a well-deﬁned physical problem. However, although the problem may be mathematically well deﬁned, this does not mean that it is possible to create the precise conditions required for the undecidability to exist. An interesting series of examples of this sort have been created by the Brazilian mathematicians Francisco Doria and Newton da Costa. Responding to a challenge problem posed by the Russian mathematician Vladimir Arnold, they investigated whether it was possible to have a general mathematical criterion which would decide whether or not any equilibrium was stable. A stable equilibrium is a situation like a ball sitting in the bottom of a basin—displace it slightly and it returns to the bottom; an unstable equilibrium is like a needle balanced vertically—displace it slightly and it moves away from the vertical. When the equilibrium is of a simple nature this problem is very elementary; ﬁrst-year science students learn about it. But, when the equilibrium exists in the face of more complicated couplings between the diﬀerent competing inﬂuences, the problem soon becomes more complicated than the situation studied by science students. So long as there are only a few competing inﬂuences the stability of the equilibrium can still be decided by inspecting the equations that govern the situation. Arnold’s challenge was to discover an algorithm which tells us if this can always be done, no matter how many competing inﬂuences there are, and no matter how complex their interrelationships. By ‘discover’ he meant ﬁnd a formula into which you can feed the equations which govern the equilibrium along with your deﬁnition of stability, and out of which will pop the answer ‘stable’ or ‘unstable’. Strikingly, da Costa and Doria discovered that there can exist no such algorithm. There exist equilibria characterized by special solutions of mathematical equations whose stability is undecidable. In order for this undecidability to have an impact on problems of real interest in mathematical physics, the equilibria have to involve the interplay of very large numbers of diﬀerent forces. While such equilibria cannot be ruled out, they have not arisen yet in real physical problems. Da Costa and Doria went on to identify similar

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problems where the answer to a simple question, like ‘will the orbit of a particle become chaotic?’, is undecidable. The tentative conclusion we should draw from this discussion is that, just because physics makes use of mathematics, it is by no means obvious that Gödel places any straightforward limit upon the overall scope of physics to understand the laws of Nature of the Universe, but it will limit the sorts of questions we can answer about the details of their outcomes in practice.

initial conditions and time symmetry The historian is a prophet looking backwards. — august von schlegel

Sometimes initial conditions can exert such an all-pervasive inﬂuence that they create the impression that a new type of law is acting. The most familiar case is that of the so-called ‘second law of thermodynamics’ which stipulates that the entropy, or level of disorder, of a conﬁned physical system cannot decrease with the passage of time. Thus, we see coﬀee cups breaking accidentally into pieces, but we never see a cup re-form from the fragments. Our desks naturally degenerate from order to disorder but never vice versa. However, the laws of mechanics that govern the manner in which changes can occur allow the time-reverse of each of these common motions. Thus a world in which china fragments coalesce into Staﬀordshire china cups and untidy desks evolve steadily into tidy ones violates no law of Nature. The reason that things are invariably seen to proceed from bad to worse in closed systems is because the starting conditions necessary to manifest order-increase are fantastically unusual and the probability that they arise in practice is tiny. The fragments of china would all need to be moving at precisely the right speeds and in just the right directions so as to convene to form a cup. In practice there are vastly more ways for a desk to go from order to disorder than from disorder to order. Thus, it is the high probability of realizing the rather ‘typical’ conditions from which disorder is more likely to ensue that is responsible for the illusion of a disorder-creating law of Nature. This example of the second law of thermodynamics alerts us to the importance of understanding initial conditions, particularly in unfamiliar situations. For without that understanding we may be misled into seeking from a Theory of Everything explanations for things that it has no business explaining.

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Moreover, we see how the choice (or accident) of initial conditions creates a sense of time directionality in a physical environment. The ‘arrow’ of entropy increase is a reﬂection of the improbability of those initial conditions which are entropy-decreasing in a closed physical system. Everywhere we look in the Universe, we discern that closed physical systems evolve in the same sense from ordered states towards a state of complete disorder called thermal equilibrium. This cannot be a consequence of known laws of change, since at their most fundamental level these laws are timesymmetric—they permit the time-reverse of any allowed sequence of events. The initial conditions play a decisive role in endowing the world with its sense of temporal direction. In our later discussion of quantum cosmology, we shall explore some of the dramatic consequences of initial conditions for the entire Universe. It will become clear that some prescription for initial conditions is crucial if we are to understand the observed universe. A Theory of Everything needs to be complemented by some such independent prescription which appeals to simplicity, naturalness, economy, or some other equally metaphysical notion to underpin its credibility. The only radically diﬀerent alternative would seem to lie in a belief that the type of mathematical description of Nature that we have come to know and love—that of causal equations with starting conditions—is just an artefact of our own preferred categories of thought and merely an approximation to the true nature of things. At a deeper level, a sharp divide between those aspects of reality that we habitually call ‘laws’ and those which we have come to know as ‘initial conditions’ may simply not exist.

time without time There is nothing new under the sun. — ecclesiastes

Leibniz and Laplace both recognized a puzzling consequence of perfect determinism. If all our laws of motion are in the form of equations which determine the future uniquely and completely from the present, then by a perfect knowledge of the starting state it would be possible for a superbeing to predict the entire future history of the Universe from this raw material. Although the statements to this eﬀect by Laplace are often quoted and the concept of determinism in classical physics has assumed the title ‘Laplacian determinism’,

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there is an earlier and more explicit statement of the idea in Boscovich’s remarkable book of 1758 which we introduced in the last chapter. On the subject of determinism and continuity of motion, he writes: Any point of matter, setting aside free motions that arise from the action of arbitrary will, must describe some continuous curved line, the determination of which can be reduced to the following general problem. Given a number of points of matter and given, for each of them, the point of space that it occupies at any given instant of time; also given the direction and velocity of the initial motion if they were projected, or the tangential motion if they were already in motion; and given the law of forces expressed by some continuous curve [like his force law shown in Figure 2.1 of the last chapter] . . . it is required to ﬁnd the path of each of the points . . . Now, although a problem of such a kind surpasses all the powers of the human intellect, yet any geometer can easily see thus far, that the problem is determinate . . . a mind which had the powers requisite to deal with such a problem in a proper manner and was brilliant enough to perceive the solutions of it (and such a mind might even be ﬁnite, provided the number of points were ﬁnite, and the notion of the curve representing the law of forces were given by a ﬁnite representation), such a mind, I say, could from a continuous arc described in an interval of time, no matter how small, by all points of matter, derive the law of forces itself . . . Now, if the law of forces were known, and the position, velocity and direction of all the points at any given instant, it would be possible for a mind of this type to forsee all the necessary subsequent motions and states, and to predict all the phenomena that necessarily followed from them.

Later the practicalities of attaining such perfect knowledge would be addressed by scientists, and then, in the twentieth century, the quantum theory would question the principle of whether such knowledge could be acquired by any observer, and indeed of whether it even exists in any meaningful sense. But let us leave aside these important developments and examine one of the striking consequences of the rigidly deterministic world of Boscovich, Laplace, and Leibniz that underpins the majority of the day-to-day concerns of physical scientists whose work is not directly aﬀected by the ambiguities of quantum mechanics. In a completely deterministic world, all the information about its structure is implicit in the initial conditions. The existence of time is a mystery. There is no use for it. Nothing really needs to ‘happen’ it all lies latent in the laws and initial conditions. A ﬁrst reaction to this statement is to point to the laws of Nature as being algorithms that predict the future from the past, but we have seen that laws are equivalent to invariance principles, that is, statements to the eﬀect that some entity does not change. The deterministic straitjacket makes

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time appear superﬂuous. Everything that is ever going to happen is implicit in the starting state. Our present state contains all the information necessary to reconstruct the past and predict the future. In Joseph Conrad’s disquieting words from the Heart of Darkness, The mind of man is capable of anything—because everything is in it, all the past as well as all the future.

This situation always presented scientists of the pre-quantum era with a dilemma. During the nineteenth-century debates about the likelihood of Darwinian evolution as opposed to a special creation of the living world in its present wondrously adapted form, several scientiﬁc commentators remarked upon the essential convergence of these two views since the present state of the evolved world can be nothing more nor less than a precise mirror of particular initial conditions. Others fretted over the problem of free will in a world of rigid determinism. A consideration of this problem led James Clerk Maxwell to appreciate the world of diﬀerence between determinism in principle and determinism in practice. There exist a vast number of physical situations, from the weather to a beating heart, where the slightest uncertainty in our knowledge of the state of the system at one moment results in total loss of information about its exact state after a very short period of time. Almost identical presents lead to very diﬀerent futures. Such systems are called ‘chaotic’. Their prevalence is responsible for many of the complexities of life: the economy, money-market ﬂuctuations, or climatic variations. In these situations, it does not matter how precisely we may know the rules governing how changes occur because we cannot ascertain the present state of things with perfect accuracy. Our capacity to predict rapidly becomes empty. It is curious how long it took scientists to recognize the overwhelming inﬂuence of such sensitivity to starting conditions in the real world. So blinkered were they by the deterministic clockwork of the Newtonian world-view and the technological advances that grew out of it that the ‘laws that never shall be broken’ stood out as the dominant aspect of the world’s character. Only the deepest thinkers of the nineteenth century, like Maxwell and Poincaré, recognized the true nature of things, which so often leaves us unable to predict the actual future even if we had the precise laws of Nature in our hands. Maxwell’s thoughts about the problem of free will in practice led him to recognize that many sequences of natural events possess an extremely sensitive dependence upon their precise starting conditions. Later, it was Henri Poincaré’s attempts to understand the sensitive dynamics of the planetary motions in our solar system that led him also to appreciate that

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a very small cause which escapes our notice determines a considerable eﬀect that we cannot fail to see, and then we say that the eﬀect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that is, governed by laws. But it is not always so; it may happen that small diﬀerences in the initial conditions produce very great ones in the ﬁnal phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.

Here, Poincaré points out that this extreme sensitivity that the evolution possesses to the actual state of the motion leads to very complicated and erratic behaviour that cannot be uniquely traced back through its antecedents in practice. Hence, it is regarded as a ‘random’ phenomenon by those who observe it. There is nothing intrinsically indeterminate about the motions involved. If we could have perfectly accurate knowledge of the starting conditions, we could predict the future behaviour perfectly. What we now know that Poincaré did not is that quantum aspects of reality forbid the acquisition of such error-free knowledge of the initial conditions in principle, not merely in practice. Nor are these quantum restrictions far removed from experience. If we were to strike a snooker ball as accurately as the quantum uncertainty of Nature permits, then it would take merely a dozen collisions with the sides of the table and other balls for this uncertainty to have ampliﬁed to encompass the extent of the entire snooker table. Laws of motion would henceforth tell us nothing about the individual trajectory of the ball. Before leaving these prescient remarks of Maxwell and Poincaré, it is intriguing to search in Boscovich’s work to ﬁnd his thoughts about the practicalities of some ‘mind’ grasping the content of all motions. He seems to recognize the inevitability of perturbing inﬂuences in reality, although not their unstable character. And, of any aspiration to exploit determinism to obtain complete knowledge, he cautions: We cannot aspire to this, not only because our human intellect is not equal to the task, but also because we do not know the number, or the position and motion of each of these points . . . and there is another reason namely that the free motions produced by spiritual substances aﬀect these curves . . .

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The ubiquity of chaotic phenomena raises a further problem for our dreams of omniscience through the medium of a Theory of Everything. Even if we can overcome the problem of initial conditions to determine the most natural or uniquely consistent starting state, we may have to face the reality that there is inevitable uncertainty surrounding the prescription of the initial state which makes the prediction of the exact future state of the Universe impossible. Only statistical statements will be possible.

cosmological time Time is God’s way of keeping things from happening all at once. — anonymous texan graffiti

In most scientiﬁc problems the initial conditions are rather mundane. We prepare them in a particular way in order better to watch a certain type of eﬀect which we suspect will ensue. But in cosmology—the study of the structure and evolution of the Universe as a whole—the situation is altogether more interesting. For, without some knowledge of those cosmic initial conditions, our knowledge of the Universe remains seriously incomplete. It would appear that even knowledge of the Theory of Everything would prevent us understanding why the Universe began in a particular way. Given a sequence of numbers, we might guess the pattern between them which allows the next one to be predicted and the whole sequence to be algorithmically compressed, but be unable to say why it begins at the particular point that it does. Yet, what really singles out the problem of cosmological initial conditions is that it has metaphysical consequences. If there are special initial conditions which start the evolution of the Universe upon the course that leads to the present, what is it that selects those rather than any other starting conditions? Initial conditions determine the coarse-grained structure of the Universe over its largest dimensions. They will play a role in determining the size of the Universe, its shape, its temperature, and its composition. From what we have already said the situation appears clear-cut. There will be particular initial conditions which lead to the present observed state. All we can hope for is to discover what they were. But we shall see that the situation is more interesting than that, and for over twenty-ﬁve years the attitude of cosmologists to the issue of initial conditions has fuelled almost all our ideas about the structure of the Universe. And, because those initial conditions were set up more than ten billion years ago when the Universe resembled a vast experiment

initial conditions

in high-energy physics, their consideration brings cosmology into collision with our thinking about the ultimate structure of the elementary particles of matter. The question of why the Universe is as it is, is inextricably linked to that of why fundamental physics is the way that it is. Let us begin by exploring the ramiﬁcations and options of the traditional cosmological pictures in which there is a fundamental distinction between the laws of Nature and initial conditions. After Hubble’s discovery in the late 1920s that the Universe is in a state of overall expansion it was appreciated that this implies that the Universe must have had a ‘beginning’ in the sense that the present state of expansion could not be extended indeﬁnitely into the past. We appear to encounter a moment in our ﬁnite past when the density was inﬁnite and all matter was squashed to zero size. Later, in the mid-1960s, this ‘Big Bang’ picture was reinforced by the discovery of a cosmic heat radiation ﬁeld, greatly cooled by the expansion, which had been predicted should exist as a remnant of the early hot state. Subsequently, the careful study of the expanding universe models supplied by Einstein’s theory of general relativity has conﬁrmed further detailed predictions based upon what the Universe must have been like when it was just one second old. It is generally agreed by modern cosmologists that we have established the general framework of how the Universe behaved from when it was a second old until the present, some ﬁfteen billion years later. This is not to claim that we understand everything that occurred. We do not understand the detailed processes by which galaxies formed, but such processes actually exert a negligible inﬂuence upon the course of the overall expansion. Prior to one second after the apparent beginning, we are on altogether shakier ground. We no longer have direct fossil remnants from the early universe against which to check the accuracy of our reconstruction of its history. In order to reconstruct the history of the Universe in these ﬁrst instants, we require knowledge of the behaviour of matter at far higher energies than are accessible to us by terrestrial experiments. Indeed, the study of the very early stages of the Universe’s history may be the only way in which we can test our theories about the behaviour of matter at very high temperatures. For we might ﬁnd that if a certain hypothetical elementary particle were really to exist then it would survive the Big Bang in such profusion that the strength of its gravitational pull today would have caused the Universe’s expansion to decelerate at a rate far in excess of what is observed. We are therefore caught in a double bind. We need to know the behaviour of the elementary particles of matter in order to understand the very early universe, but we need to know what the early universe was like in order to discover the behaviour of elementary particles.

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With this warning taken on board, let us none the less continue to extrapolate our successful picture of the Universe into the ﬁrst second of its history using the latest ideas in elementary-particle physics as a guide to what is possible or probable during the dim and distant past. Traditionally (and currently) there are three distinct attitudes towards the problem of cosmological initial conditions:

r Show there are none. r Show that their inﬂuence is minimal. r Show that they have a special form. The ﬁrst option springs from a belief that the universe did not have a beginning—that there was no initial state. This stance was taken most adamantly by Hermann Bondi, Fred Hoyle, and Thomas Gold, who introduced the ‘steady-state’ theory of the Universe in 1948. The speciﬁc theory they proposed fell into conﬂict with observation long ago and its speciﬁc details are not important for our present discussion. What is most interesting is their motivation to avoid any special times occurring during the history of the Universe, just as Copernicus cautioned us against endowing special signiﬁcance upon any places in the Universe. Clearly, if the Universe begins expanding (or existing) at some ﬁnite past moment or ceases to expand (or exist) at some future moment, then these moments are special times for any observer. The ‘steady-statesmen’ called the extension of the Copernican Principle from spatial location to spatial and temporal location, the Perfect Cosmological Principle (a title which provoked Herbert Dingle into remarking that this was like ‘calling a spade a perfect agricultural instrument’ and some Americans to suggest that the stipulation that the Universe be the same at all times was merely a device by which its proposers could ensure that there would always be an England). Although the steady-state universe expands, it maintains a constant density at all times by the assumption that matter is being continuously created at a rate that exactly counterbalances the rarefaction that would otherwise result from the expansion. This continuous creation contrasts with the once and for all creation that was envisaged in the Big Bang cosmological models of that time. The fact that the creation rate exactly balances the eﬀects of the expansion was automatically ensured and the creation rate is so tiny, less than one atom in a cubic metre every ten billion years, that it could not be detected directly. Yet, despite the fact that there is no actual beginning to the Universe in this theory—it always has, and always will expand, on the average, at the same constant rate—it still requires its deﬁning parameters to be speciﬁed: there is

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no unique steady-state universe. The value of its constant universal density of matter, or, equivalently, its constant creation rate or the universal rate of expansion, needs to be explained. We must specify certain conditions at some moment of time to deﬁne this model. A Theory of Everything might tell us that the Universe has no beginning in time and expands in a fashion similar to the steady-state universe (at least until about ten billion years ago), but this would leave many things unexplained: the expansion rate of the Universe, the origin of galaxies, the heat content of the Universe, its imbalance between matter and antimatter. This logical incompleteness characterizes any cosmological model that is hypothesized to have existed from a past inﬁnity of time. It still requires extra speciﬁcations that play the role of ‘initial’ conditions, even if there is strictly no ‘initial’ moment in the temporal sense. In an inﬁnitely old universe, initial conditions are required at past temporal inﬁnity. It is interesting to reﬂect that for centuries philosophers and theologians have attempted to settle by pure thought the issue of whether the Universe could or could not be inﬁnitely old. That is, some have attempted to show that there is some logical contradiction inherent in the notion of a past inﬁnity of time. And some still do. Such ideas have some association with cosmological arguments for the existence of God, which not only seek to demonstrate that there must have been an origin to the Universe in time but go further in showing (or, in practice, assuming) that this requires there to have been an originator. This is a slippery argument, notwithstanding our ultimate ignorance about such overwhelming questions. A common form of this argument points to the fact that everything that we see has a cause, and hence the Universe must have a cause. But this argument has a dangerous bend in the middle of it. The Universe is not a ‘thing’ in the sense of all the other examples that are being cited. It is a collection of things, or as Wittgenstein put it ‘the world is the totality of the facts’. Our argument is thus seen to be analogous to arguing that all members of clubs have mothers, and therefore all clubs have mothers. One might also take issue with the claim that all events have causes. In the shadowy world of quantum theory, this need no longer be the case. We cannot tie individual observations to speciﬁc causes according to some interpretations of the quantum theory, and indeed this is one of the reasons why a quantum description of the whole Universe can in principle give a description of the creation of the material Universe without any direct initial cause being invoked. When discussing those features of the steady-state universe which would have to be speciﬁed to complement what the laws of Nature tell us, we

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mentioned the expansion rate, but not the shape, of the Universe. One of the unusual features of the steady-state model was that it was stable against any inﬂuences that might distort it away from possessing the same rate of expansion in every direction of the sky. If some violent event suddenly occurred somewhere in the Universe or the Deity temporarily intervened at one moment to make it expand faster in one direction than another, then with the passage of time these deviations from the state demanded by the Perfect Cosmological Principle would soon fade away and the expansion settle back into a perfectly symmetrical state. Such a property is a very attractive feature of any cosmological model because our Universe is observed to expand at the same rate in every direction to within one part in a thousand. The wider quest for a natural explanation of this surprising fact brings us to the second of the three general approaches that have been made to understanding the initial conditions of the Universe. It is evident that the most awkward feature about the inﬂuence of initial conditions in cosmology is the fact that they are the most uncertain aspect of our knowledge. It may well be that we can never know how (or if) the Universe began. Therefore there has always been a lobby of cosmological opinion that has seen it as expedient to seek an explanation for the present structure of the Universe that places the minimum onus for that structure upon those unknowable initial conditions. But how could this be done? There are many physical systems which rapidly lose memory of their initial conditions. By this we mean that their future states are to very high accuracy pretty much the same regardless of how they started out. Stir a large pot of treacle in a vigorous way and it will quickly settle down to the same placid state no matter how you stirred it. Drop a rock in air from a suﬃciently great height and it will hit the ground at essentially the same speed no matter how hard you threw it initially because the competing eﬀects of gravity accelerating the stone and air resistance slowing it down always act to create a situation where they have an equal and opposite eﬀect, and thereafter the stone feels no net force at all and falls at constant speed. The Universe could be like this. Cosmologists spent much of the 1970s looking for natural physical processes which might emerge during the early stages of the Universe and render its present state quite inevitable irrespective of the details of how it started. In particular, they hoped to explain why the visible universe possesses the remarkable property of expanding at the same rate in every direction to within one part in ten thousand. If it could be shown that no matter how disparate were the expansion rates in diﬀerent directions when the Universe began, so long as we wait long enough (and life takes a long time to evolve), we will

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always ﬁnd almost identical expansion rates in diﬀerent directions because physical processes always arise to transport energy from place to place and iron out disparities in expansion energy between one direction and another. This sounds like an attractive scenario. Unfortunately, the early attempts to implement it were largely unsuccessful. The main problem is that the smoothing of irregularities is one of those processes that is governed by the second law of thermodynamics. Irregularity in the expansion can only be reduced if this partial reduction in disorder (or ‘entropy’ as it is called) is paid for by an even larger production of entropy in another form. In practice, this compensating entropy appears in the form of heat radiation. Thus, if we build a chair out of disordered pieces of wood, we do not violate the second law, because we put a lot of physical and mental eﬀort into it, which is manifested as the production of heat and sound by our bodies. Yet we ﬁnd that the Universe does not contain very much heat radiation today and therefore very little smoothing of irregularities can have occurred in the past. Moreover, even if smoothing were to occur, there exists a vast array of cosmological models in which the irregularities could not become smooth by the present day. The smoothing eﬀects are not strong enough to overcome a tendency to become increasingly distorted that is latent in the starting conditions of some possible universes. As a result of these negative discoveries, cosmologists had become somewhat disenchanted with this route to explaining the large-scale regularity of the Universe by the end of the 1970s. But then a new idea emerged. Alan Guth pointed out that if the expansion rate of the Universe could be greatly increased for a short period during its early stages then one could explain the present structure of the Universe with only a minimal appeal to initial conditions, without having to worry about producing excessive heat. The inﬂationary universe is a recipe for doing just this. It is based upon the expectation that there exist certain types of matter in the realm of elementary particles which, in eﬀect, behave as though they exhibit gravitational repulsion rather than attraction. This is possible because they possess a negative pressure, or tension, and in the theory of relativity all forms of energy—and pressure is one of them—feel the force of gravity since they are all equivalent to a mass (via Einstein’s famous E = mc 2 formula relating energy E to mass m and the speed of light c ). If such a tension can appear during the earliest moments of the Universe’s expansion, then gravity no longer pulls matter back and decelerates the expansion of the Universe. Instead, it acts to accelerate the expansion. The period of acceleration is called the inﬂation of the Universe. It causes all distorting inﬂuences to diminish extremely quickly and the Universe rapidly assumes a highly symmetrical state of expansion, which explains the

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residual state of extreme regularity that we still witness today. If the period of inﬂation lasts for only a very brief period, it is suﬃcient to reduce all irregularities that might have been present initially to an inﬁnitesimally small level. It wipes the slate clean. Thus, it claims to explain the regular expansion that we presently observe, irrespective of the initial conditions. This is actually not quite true. There are always some maliciously chosen initial conditions that will not be damped down suﬃciently by a pre-speciﬁed period of inﬂation, but, conversely, if the initial conditions are chosen ﬁrst, then there always exists an amount of inﬂation that will suﬃce. It is something of a chicken-andegg problem. If you are allowed to pick the period of inﬂation after you have picked the initial conditions then you can always explain what we see, but if the period of inﬂation is ﬁxed ﬁrst by the laws and constants of Nature then there are always initial conditions whose inﬂuence cannot be made innocuous by the present. The answer to the question ‘What should be chosen ﬁrst?’ depends in a deep way upon one’s view of initial conditions and their relationship with the laws of Nature. If we retain the traditional classical view that initial conditions are independent of the laws of physics, then, in the absence of other information to the contrary, we should regard the initial conditions of the Universe as being freely speciﬁable, but they would then possess a secondary status with respect to the laws and the constants of physics. We can envisage diﬀerent initial conditions quite easily and are accustomed to specify them at will every time we employ laws of physics in the laboratory, but to alter a law of physics or the value of a fundamental constant is altogether more radical. Thus, it seems most reasonable to regard the constants and laws of physics, and hence the duration of any period of inﬂation, as having been ﬁxed before we specify initial conditions. With this choice, inﬂation cannot always deliver the observed Universe irrespective of initial conditions. It might still turn out that the unsuccessful starting states are in some sense ‘unlikely’ ones, but the question of what distinguishes a probable from an improbable initial state is still an open one. In the traditional Big Bang picture of the expanding universe, the relative sameness of the observed universe from place to place is something of a mystery. To understand the mystery, we must ﬁrst distinguish between the entire Universe, which might be inﬁnite in extent, and the ‘visible universe’, which is that part of it from which light has had time to travel since the expansion began. The visible universe can be thought of as a sphere of radius approximately equal to ﬁfteen billion light years centred upon us. Fifteen billion light years is the distance that light can have travelled in the ﬁfteen billion years we shall use as a good estimate of the time that has passed since the

initial conditions

expansion apparently began. (It is a reasonable average of the diﬀerent pieces of observational evidence which point to a 13 to 18 billion year age range for the universal expansion.) We know nothing about the Universe except from what we observe of the ﬁnite visible portion of it. For instance, no observations of the visible universe can ever tell us whether the entire Universe is ﬁnite or inﬁnite. It is our visible universe which displays remarkable large-scale uniformity. Yet, if we extrapolate this visible region backwards in time, we can determine how much smaller it would have been at earlier times in the Universe’s history. For example, when the Universe was one second old, our present visible universe would have been crammed into a region only one and a half light years in size. When the Universe was 10–35 of a second old, it would have been squeezed into a region a mere centimetre across. This sounds staggeringly small, but for the cosmologist it is unacceptably large. It is 3 × 1025 times larger than the size of the regions whose contents are in causal contact at that early time: for at that time the latter distance is simply 10–35 seconds multiplied by the speed of light (3 × 1010 centimetres per second), which gives 3 × 10–25 centimetres. The upshot of this state of aﬀairs is that the region which grows into the entire visible universe today is composed of a vast number of totally independent regions that cannot even ‘know’ of each other’s existence at very early times (see Figure 3.2). The root of this ‘horizon problem’, as it is called, is apparent from our description. The Universe expands too slowly early on, so that part of it E

C Time

Space

Zero of time

Figure 3.2 Signals sent out from two separate points, A and B, when the Universe begins expanding cannot reach each other until the time D. The interior of the wedges CAD and DBE represent the parts of space and time that can be contacted by signals emanating from A and B, respectively. This restriction upon communication arises because signals cannot travel faster than the speed of light; this means that communication is conﬁned to the interior of the wedges. Notice that A cannot predict the future. Conditions at D are not determined solely by the signal that A transmits, but also by that transmitted from B.

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which will encompass our visible universe today needs to have grown from a relatively large region at early times, a region far larger than any that can be kept smooth and regular at those times by physical processes that are limited in the extent of their inﬂuence by the speed of light. However, the period of accelerated expansion that characterizes the early evolution of the inﬂationaryuniverse models enables our entire visible universe to have evolved from a much smaller region at an early time like 10–35 of a second. In fact, if the inﬂation lasted for just a ﬂeeting moment—from 10–35 to 10–33 of a second—then our entire visible universe can have emerged from a region that was within the range of light signals at these very early times. The gross uniformity of the observed universe now has a plausible explanation. It is the expanded image of a minute region that was small enough to have been smoothed by physical processes obeying the restrictions on their scope imposed by relativity. In the standard Big Bang theory in which inﬂation does not occur, the observed universe cannot have arisen from any such causally correlated and coherent region. Instead, it is the coming together of a myriad of completely unrelated regions that would be expected to be very diﬀerent from one another and hence result in a visible universe that was wildly diﬀerent from place to place. This new picture of the early evolution of the Universe radically diminishes the role of initial conditions, because, although the entire visible universe partially reﬂects the structure of some ‘initial’ conditions that deﬁne the structure of the Universe prior to the onset of the inﬂation, the particular initial conditions that play that role are only a minute part of the entire map of initial conditions for the whole (possibly inﬁnite) Universe (see Figure 3.3). Time

Our visible universe today

Space

To infinity

Region of space which expands to become our visible universe today

To infinity

Figure 3.3 The structure of the visible universe is determined by conditions over only a tiny part of the ‘initial’ conditions of the Universe. If the Universe is inﬁnite in size, then both our visible part of it and the part of the initial conditions which determine it are but inﬁnitesimal parts of the whole.

initial conditions

This is disturbing to the scientist. It means that our observations of the structure of the visible universe can at best give us information about only a minute part of the initial conditions characterizing the ﬁrst moments of the expanding universe. We can never know about the structure of the whole of the initial conditions for the Universe by observational science. They are condemned to remain always partially within the realm of philosophy and theology. It also makes any test of this theory extraordinarily diﬃcult. Even if some varieties of initial condition do not allow any or enough inﬂation to take place, there will always be some part of the entire Universe initially where acceptable conditions will exist and that is all we require. As we shall see in a later chapter, we need to explore in some detail how our own existence plays a role in evaluating such theories. The picture of initial conditions that inﬂation presents us with is therefore of a possibly chaotic or random initial state for the Universe as we look from place to place—rather like the surface of the sea. Each minute local region will inﬂate independently of all the others by an amount determined by its local conditions. We will ﬁnd ourselves living inside one of these regions after it has greatly expanded. The inside of this region should look very smooth and expand uniformly, but beyond its boundary there are regions whose light rays have not yet had time to reach us. And these regions beyond our ken will in all probability be utterly diﬀerent in structure. We have a picture which can explain why our visible part of the Universe is smooth even though the entire Universe would not be expected to be. The inﬂationary period of expansion does not smooth out irregularity by entropy-producing processes like those explored by the cosmologists of the seventies. Rather, it sweeps the irregularity out beyond the horizon of our visible universe, where we cannot see it. The entire universe of stars and galaxies on view to us, on this hypothesis, is but the reﬂection of a minute, perhaps inﬁnitesimal, portion of the Universe’s initial conditions, whose ultimate extent and structure must remain forever unknowable to us. A Theory of Everything does not help here. The information contained in the observable part of the Universe derives from the evolution of a tiny part of the initial conditions for the entire Universe. The sum total of all the observations we could possibly make can only tell us about a minuscule portion of the whole. It is possible that the rigid divide between laws and initial conditions that we have just assumed does not exist; that for some laws only one type of initial condition is allowed. This is a possibility that we shall now explore a little further.

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The last of our options regarding the initial conditions of the Universe that we have left to consider is that there exists some special type of initial condition—eﬀectively, a ‘meta-law’ governing initial conditions. The inﬂationary philosophy chooses to regard the initial conditions as being freely speciﬁable and inﬂation is a means by which we can show that their precise form has little bearing on what we will see today, so long as the laws of physics, the nature of elementary particles, and the constants of Nature permit this magical phenomenon of inﬂation to occur. By contrast, the lobby for special initial conditions searches for a fundamental link between the notion of laws and initial conditions that transcends our normal experience in classical physics. Traditionally, initial conditions are not constrained by the form of the laws of change except in a very weak fashion. If a solution of an equation of change also ﬁxes the starting conditions uniquely, then this invariably means that the solution in question is extremely special, and hence unlikely to be realized in practice. To ﬁnd a deep connection between the form of laws of Nature and their permitted starting conditions, we need therefore to look to a situation where there exists some probabilistic element regarding the possible form of evolutionary behaviour. This is something that can be found in any quantum description of things. For the most part, these attempts to link laws with initial conditions have focused upon the rapidly growing, embryonic subject of quantum cosmology. In so doing, they ﬁnd themselves embroiled in other deep problems of a fundamental nature regarding the interpretation of quantum theory, about which much has been written elsewhere, and the less frequently discussed problem of time.

the problem of time The English are not a very spiritual people. So they invented cricket to give them some idea of eternity. — george bernard shaw

There is a long-standing philosophical puzzle regarding the nature of time that has emerged in the works of diﬀerent thinkers over millennia. It reduces to the question of whether time is an absolute background stage on which events are played out but yet remains unaﬀected by them, or whether it is a secondary concept wholly derivable from physical processes and hence aﬀected by them. If the former picture were adopted, then we could talk about the creation of

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the physical Universe of matter in time. It would be meaningful to discuss what occurred before the creation of the material universe and what might happen after it passed away. Here, time is a transcendent part of reality without a conceivable beginning or end. This idea lends itself readily to the Platonic notion that there exist certain eternal truths or blueprints from which the temporal realities derive their qualities. Indeed, time takes upon itself many of the qualities traditionally associated with a Deity. The alternative, an idea that emerges in Aristotle’s writings and more memorably in those of Augustine and Philo of Alexandria, before being elaborated by some of the early Islamic natural philosophers, is that time is something that comes into being with the Universe. Before the Universe was, there was no time, no concept of ‘before’. Such a device enabled the medieval Scholastics to evade diﬃcult conundrums about what took place before the creation of the world and what the Deity was doing in that period. In essence, this views time as a derived phenomenon, inextricably bound up with the contents of the Universe. The beginning of time is the moment when constants and laws of Nature must come into being ready-made and ready to go. In The City of God, St Augustine writes: Then assuredly the world was made, not in time, but simultaneously with time. For that which is made in time is made both after and before some time—after that which is past, before that which is future. But none could then be past, for there was no creature, by whose movements its duration could be measured. But simultaneously with time the world was made.

This is close to our common experience of time. We measure time using clocks, which are made of matter and which obey laws of Nature. We exploit the existence of periodic motions, whether they be revolutions of the Earth, oscillations of a pendulum, or vibrations of a caesium crystal; and the ‘ticks’ of these clocks deﬁne the passage of time for us. We have no everyday meaning to give to the notion of time aside from the process by which it is measured. We might thus defend an operationalist view, wherein time is deﬁned by its mode of measurement alone. Whereas, on the transcendental view of time, we might speak of bodies moving in time, the emphasis of the latter view is upon time being deﬁned by the motion of things. One of the advantages of the ﬁrst view is that one knows where one stands and what time is always going to look like: it is the same yesterday, today, and forever. By contrast the second picture promises to produce novel concepts of time—and might even do away with the concept altogether—as the material contents of the Universe alter their nature under varying conditions. We should be especially conscious of such a possibility as

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we backtrack towards those moments of extremis in the vicinity of the Big Bang. For any moment that appears to be the beginning of time inevitably exists where the very notion of time itself is likely to be most fragile. In an expanding and constantly changing universe, the operational view of time is likely to produce a subtle and variable conception of time’s place and meaning.

absolute space and time I do not deﬁne time, space, place and motion, as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices . . . — isaac newton

The image of a transcendent absolute time shadowing the march of events upon a cosmic billiard table of unending and unchanging space was the foundation of Newton’s monumental description of the world. Once the equations governing the change of the world in space and time are given then the whole future course of events is determined by the starting conditions. ∗ Time appears superﬂuous. Everything that is going to happen is programmed into the starting state. The Newtonian laws of motion could be applied to the description of the world and followed backwards in time. Our Universe is observed to be expanding, and hence a Newtonian description leads to the assertion that there must have been a past moment of time at which everything was compressed to zero size and inﬁnite density, the ‘Big Bang’ as it was ﬁrst termed by Fred Hoyle. However, because of the absolute nature of space and time in the Newtonian world-view, we cannot draw any conclusions about the Newtonian Big Bang constituting an origin to time, let alone the origin of the Universe. It is simply a past time at which known laws predict that some physical quantities become unboundedly large; we say they become inﬁnite in value there. But space and time go on regardless. ∗

This will not be true if other physical processes become involved. For example, in the archetypal situation of billiard balls moving according to Newton’s laws, their future behaviour after collisions depends upon the rigidity of the collisions and this involves knowledge of the behaviour of the materials out of which the balls are made. This information is beyond the scope of Newtonian mechanics.

initial conditions

The ﬁrst scientists to contemplate the signiﬁcance of places where things apparently cease to exist or become inﬁnite (‘singularities’, as we would now call them) in Newtonian theory were the eighteenth-century scientists Leonhard Euler and Roger Boscovich. They both considered the physical consequences of adopting force laws for gravitation other than Newton’s famous inverse-square law. They found some of the alternatives had the unpleasant feature that the solutions just cease to exist after some deﬁnite time in the future when one studied the behaviour of objects orbiting around a central sun. They cannot be continued forwards any further in a world governed by one of these maverick force laws. Boscovich thinks it absurd that the body must disappear from the Universe at the centre if the force law were inverse cube rather than inverse square. He draws attention to Euler’s earlier study of motion under the inﬂuence of gravity, where the master-mathematician asserts that the moving body on approaching the centre of forces is annihilated. How much more reasonable would it be to infer that this law of forces is an impossible one?

These appear to be the ﬁrst contemplations of such matters in the context of Newtonian mechanics. In fact, there are deep problems with attempting to apply Newton’s theory of gravity and motion to the Universe as a whole. It will not tolerate the consideration of an inﬁnite space distributed with matter: this leads to an inﬁnite aggregate of gravitational inﬂuences at any one point due to the inﬁnite number of gravitational attractions exerted by the others. Therefore a Newtonian universe must be ﬁnite in size and hence possess a boundary in space. If we think of Newtonian space stretching out straight in every direction, then this boundary must be a deﬁnite edge. For example, if the space is spherical about us at the centre, then the surface of the space is the surface of the sphere. Alternatively, the spatial universe could be a cube whose boundary was composed of the six faces of the cube. This prospect of a Universe with boundaries is a rather unattractive picture because we must specify how all physical quantities behave at these boundaries when the Universe is started at some time in the past. Thus, the Newtonian world requires the universe of matter to be a ﬁnite island of matter in an ocean of inﬁnite absolute space. Worse still, Newton’s theory is incomplete. It does not contain enough equations to tell us how all the allowed changes to the Universe actually occur. If the Universe expands or contracts at exactly the same rate in every direction then everything is indeed determined, but when any deviations from perfectly spherical expansion are allowed at the start then determinism breaks down

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for there are no Newtonian laws which dictate how the shape of the world will change with time. Clearly, Newton’s theory of absolute space and time is defective. The next step to take is to contemplate some coupling of the notions of space and time to the material contents of the world. The earliest and most intriguing speculation of this sort was made by William Cliﬀord, an English mathematician who contemplated just the type of situation that Einstein would build into the general theory of relativity. Cliﬀord was motivated by the mathematical investigations of Riemann who had formalized the geometric study of curved surfaces and spaces which possess non-Euclidean geometry (that is, the three interior angles of a triangle no longer add up to 180 degrees where the three corners of the triangle are formed by joining the shortest lines that can be drawn between them to form the sides of the triangle on the curved surface). Cliﬀord appreciated that the traditional space of Euclid is thus one of many and we can no longer assume that the geometry of the real world possesses the simple Euclidean form. The fact that it appears to be ﬂat locally is not persuasive because most curved surfaces appear ﬂat when viewed over small areas. After studying Riemann’s ideas, Cliﬀord proposed the following radical scenario in his paper of 1876: I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average ﬂat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

This prescience is rather remarkable. Although Einstein never seems to have been aware of these remarks, Cliﬀord’s intuitive idea became the central idea of the general theory of relativity. The geometry of space and the rate of ﬂow of time are no longer absolutely ﬁxed and independent of the material content of space and time. The matter content and its motion determine the geometry and the rate of ﬂow of time, and symbiotically this geometry dictates how matter is to move. Einstein’s elegant theory of gravitation possesses a set of

initial conditions

equations which dictate the connection between the matter content of the Universe and its space and time geometry. These are called ﬁeld equations and they generalize the Newtonian ﬁeld equation of Poisson, which encapsulates Newton’s inverse-square law of gravitation. In addition to this structure, there exist equations of motion which give the analogues of straight lines in the curved geometry. These generalize Newton’s laws of motion. One further erosion of time’s absolute Newtonian status occurs in Einstein’s theory. Einstein’s theory was built upon a premise that there are no preferred observers in the Universe, that is, there is no set of observers for whom all the laws of Nature look simpler. The laws of physics must have the same form for all observers no matter what their state of motion. In other words, however your laboratory is moving—whether it is accelerating or rotating with respect to that of your neighbour—you should both ﬁnd the same laws of physics to hold good. You may each measure observables to have diﬀerent values, but you will none the less ﬁnd them to be linked by the same invariant relationships. In Einstein’s world, there is no special class of observers for whom, by virtue of their motion and time-keeping arrangements, the laws of Nature look especially simple. This is not true in Newton’s formulation of motion. His famous laws of motion are found to hold only by experimenters moving in laboratories that are in uniform, non-rotating motion with respect to each other and with respect to the most distant stars, which he took to establish a state of absolute rest. Other observers who rotate or accelerate in unusual ways will observe the laws of motion to have a diﬀerent, more complicated form. In particular, and in violation to Newton’s famous ﬁrst law of motion, they will observe bodies acted upon by no forces to accelerate. This democracy of observers that Einstein built into the formulation of his general theory of relativity means that there is no preferred cosmic time. Whereas, in his special theory of relativity, there could exist no absolute standard of time—all time measurements are made relative to the state of motion of the observer—in the general theory of relativity, things are diﬀerent. There are many absolute times in general relativity. In fact, there appears to be an inﬁnite number of possible candidates. For instance, observers around the Universe could use the local mean density or expansion rate of the Universe to coordinate their time-keeping. Unfortunately, none of these absolute times has yet been found to possess a more fundamental status than the others. A good way to view an entire universe of space and time (a ‘space-time’) in Einstein’s theory is as a stack of spaces (imagine there to be only two dimensions of space rather than three for the sake of visualization), with each slice in the stack representing the whole universe of space at a diﬀerent time. The

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Figure 3.4 Each of the slices 1, 2, 3, 4, and 5 taken through space can be given a ‘time’ label that is calibrated by the radius of the circle (arrowed). As we progress up the curved surface, the increase in time is gauged by the increasing radii of the circles bounding the slices.

‘Time’ 3 2 1 0

time is a label identifying each slice in the stack. The discussion of the previous paragraph means that we can actually slice up the whole space-time block into a stack of ‘time-slices’ in many diﬀerent ways. We could slice through the solid stack at a variety of diﬀerent angles. This is why it is always more appropriate to talk about space-time rather than the somewhat ambiguous partners space and time. But the connection between matter and space-time geometry means that ‘time’ can be deﬁned internally by some geometrical property, like the curvature, of each slice and hence in terms of the gravitational ﬁeld of the matter on the slice which has distorted it from ﬂatness (see Figure 3.4 for a simple illustration). Thus we begin to see a glimmer of a possibility of associating time, including its beginning and its end, with some property of the contents of the Universe and the laws which govern how they change. The new picture of space-time rather than space and time considerably changes our attitude towards initial conditions and the possible beginning of the Universe. Because of the coupling that exists between the fabric of spacetime and matter, any singularity in the material content of space-time (for example, the inﬁnity in the density of matter which occurs in the traditional picture of the Big Bang) signals that space-time has come to an end as well. We now have singularities of space and time not merely singularities in space and time. Moreover, any space-time given by Einstein’s theory of general relativity is an entire Universe. Unlike in Newton’s theory, it can never merely describe some object sitting on an external stage of ﬁxed space. Thus the singularities of general relativity are features of the entire Universe, not just one place in it or one moment of its history. These singularities mark out the boundary of space and time.

initial conditions

If we study the expanding Universe according to this picture and trace its history backwards, then it is possible for it to begin at such a singularity. This prediction has been seized upon by many as proof that the Universe had a beginning in time. However, like any logical deduction, this conclusion follows from certain assumptions whose truth needs to be closely examined. The most shaky of these assumptions is that gravity is always attractive. Our modern theories of elementary particles contain many types of particle, and forms of matter, for which this assumption is not true. Indeed, the whole inﬂationaryuniverse picture which we introduced above is founded upon the requirement that it be not true, for only then can the brief period of accelerated ‘inﬂationary’ expansion arise. However, although the avoidance of a singularity might avoid a beginning to time, it would not save us from having to prescribe ‘initial’ conditions at some past moment to select our actual Universe from the inﬁnity of other possible worlds that begin at singularities. Even if there did exist a singularity, one must face the fact that there are diﬀerent types of singularity. The speciﬁcation of the properties of this singularity is an ‘initial’ condition to be speciﬁed on the boundary of our space and time. Some extra ingredient still needs to be found which could provide that speciﬁcation.

how far is far enough? There was a Door to which I found no key There was a Veil past which I could not see. — the rubaiyat of omar khayyam

General relativity (and any other relativistic theory of gravity which does not possess absolutely ﬁxed space or time) gives rise to another subtle property not present in simple Newtonian conceptions of space and time. There are actually many distinct space-times that can arise from the same initial conditions. Suppose that some space-time S has initial conditions set at some starting time zero which we shall label t0 . We can construct another space-time by removing all of that part of the ﬁrst space-time that lies to the future of some time t1 (later than t0 ) as well as the time t1 itself. The new space-time S is the same as S to the past of the moment t1 , but contains no space or time whatsoever to the future of t1 , as illustrated in Figure 3.5. But both S and S arise from the same initial state, and indeed we could have cut pieces oﬀ S in an inﬁnite number of diﬀerent ways to make other space-times which

84 initial conditions Time Space

S⬘

t0 (a)

(b)

Figure 3.5 Two space-times S and S which are determined by identical sets of initial conditions prescribed on the surface of initial time t0 . In the case (a), the space-time S is maximally extended, whereas in (b) it is brought to an end arbitrarily at some future time t1 , but no physical inﬁnity or other defect in the space-time structure arises then; the space-time S is therefore identical to S up to the time t1 , but does not exist to the future of that moment. In practice, it is always assumed that a given set of initial conditions leads to the maximally extended space-time and not one of the inﬁnite number of artiﬁcial alternatives which are identical up to some ﬁnite moment and then cease to exist for no physical reason.

start from the same initial conditions. Yet there is something unsavoury about S and its fellow neutered universes. It comes to an end at the allotted time t1 for no physical reason whatsoever. There is no singularity of any physical quantity. Indeed, we have not had to make mention of the material contents of the Universe at all. The equations that govern the behaviour of matter would still like to predict the future beyond t1 if only you would allow there to be a future. This arbitrary truncation of the future is regarded as unrealistically artiﬁcial, and cosmologists choose to exclude its possibility and specify the future evolution uniquely. To do so, it is necessary to introduce a further condition into the prescription of possible space-times, or universes, in theories like general relativity, in addition to the speciﬁcation of initial conditions and laws of Nature. One requires that the Universe should continue to exist until the laws of Nature governing the behaviour of mass and energy signal that time itself has come to an end at a real physical singularity. Under reasonable conditions, it transpires that there is a unique ‘biggest’ space-time which contains all the others starting from the same initial conditions and which is

initial conditions

obtained by letting time go forward until the equations predict a singularity. This maximally extended universe is the natural candidate for the space-time that actually arises from a particular set of initial conditions, although we should remember that in principle any of the other truncated realities could be the one that exists following the initial conditions of our Universe. If the maximally extended universe is not the extant one, then the end of the Universe of space and time could indeed come at any moment ‘like a thief in the night’, without any observable cause or warning. Despite all these subtleties regarding the nature of time, general relativity has failed to remove the traditional divide between laws and initial conditions. There is still always an initial slice to our space-time stack which determines what the others will look like to its future.

the quantum mystery of time It was a book to kill time for those who like it better dead. — rose macaulay

In quantum theory, the status of time is an even bigger mystery than it appeared to Newton and Einstein. If it exists in a transcendent way then it is not one of those quantities subject to the famous Uncertainty Principle of Heisenberg, but if it is deﬁned operationally by other intrinsic aspects of a physical system then it does suﬀer indirectly from the restrictions imposed by quantum uncertainty. Accordingly, when one attempts to produce a quantum description of the entire Universe, one might anticipate some unusual consequences for time. The most unusual has been the claim that a quantum cosmology permits us to interpret it as a description of a universe which has been created from nothing. The non-quantum cosmological models of general relativity may begin at a deﬁnite past moment of time deﬁned using certain types of clock. The initial conditions, which dictate the whole future behaviour of that universe, must be prescribed at that singularity. But, in quantum cosmology, the notion of time does not appear explicitly. Time is a construct of the matter ﬁelds and their conﬁgurations. Since we have equations which tell us something about how those conﬁgurations change as we look from one slice of space to another, it would be superﬂuous to have a ‘time’ as well. This is not altogether diﬀerent

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to the way in which a pendulum clock tells you time. The clock hands merely keep a record of how many swings the pendulum makes. There is no need to mention anything called ‘time’. Likewise, in the cosmological setting, we are labelling the slices in our ‘space-time’ stack by the matter conﬁguration which creates the intrinsic geometry of each slice. This information about the geometry and material conﬁguration is only available to us probabilistically in quantum theory and it is coded into something which has become known as the wave function of the Universe, which we shall henceforth call W. The generalization of Einstein’s equations to include quantum theory is one of the great problems of modern physics. One proposed route uses an equation ﬁrst found by the American physicists John A. Wheeler and Bryce De Witt. The Wheeler–De Witt equation describes the evolution of W. It is an adaptation of Schrödinger’s famous equation governing the wave function of ordinary quantum mechanics but with the curved space attributes of general relativity incorporated as well. If we knew the present form of W, it would tell us the probability that the observed universe would be found to possess certain large-scale features. It is hoped that these probabilities will turn out to be strongly concentrated around particular values in the same way that large everyday things have deﬁnite properties despite the microscopic uncertainties of quantum mechanics. If the greatly favoured values were similar to the values observed, then this would give an explanation of those features as a consequence of the fact that ours was one of the most ‘probable’ of all possible universes. However, to do this, one still requires some initial conditions for the Wheeler–De Witt equation—an initial form for the wave function of the Universe. The most useful quantity involved in the manipulation and study of W is the transition function T [x1 , t1 ; x2 , t2 ]. This gives the probability of ﬁnding the Universe in a state labelled by x2 at a time t2 if it was in a state x1 at an earlier time t1 , where the ‘times’ can be prescribed by some other attribute of the state of the Universe, for example its average density (see Figure 3.6). Of course, in non-quantum physics, the laws of Nature predict a deﬁnite future state will arise from a particular past one and we would not have use for such probabilistic notions. But, in quantum physics, a future state is determined only as an appropriately weighted sum over all the logically possible paths through space and time that the system could have taken. One of these paths might be the unique one that the non-quantum description would follow. We call this the classical path. In some situations, where there exists a conventional deterministic situation, its corresponding quantum description has a transition function that is principally determined by the classical path,

initial conditions Figure 3.6 Some paths for space-times whose boundary consists of two three-dimensional spaces with curvatures g 1 and g 2 where the matter ﬁelds are in the conﬁgurations m1 and m2 respectively.

g1,m1

g2,m2

leaving the others to combine so as to cancel each other out, rather like the peaks and troughs of waves that are out of phase. In fact, it is a deep question whether all possible starting conditions allowed for a quantum universe can give rise to a ‘classical’ universe when they expand to a large size. This may well turn out to be a very restrictive requirement, one necessary also for the existence of living observers, that marks our Universe out as unusual in the set of all possibilities. If this is true, then it would also have the interesting consequence that only by a study of its cosmological consequences could a complete appreciation of quantum mechanics be arrived at. In practice, W depends upon the conﬁguration of the matter in the Universe on a particular slice through the space stack and upon some internal geometrical property of the slice (like its curvature) which then eﬀectively labels its ‘time’ uniquely. Again, there is no special choice of geometrical quantity that is elevated above all others in labelling the slices in this way. There are many that will suﬃce and the Wheeler–De Witt equation then tells you how the wave function at one value of this internally deﬁned time is related to its form at another value of it. When we are close to the classical path, these developments of the wave function in internal time are straightforward to interpret as small ‘quantum corrections’ to ordinary classical physics. But this is not always the case, and, when the most probable path is far from the classical one, it becomes increasingly diﬃcult to interpret the quantum evolution as occurring ‘in’ time in any sense. That is, the collection of space slices that the Wheeler–De Witt equation gives us do not naturally stack to look like a space-time. None the less, the transition functions can still be found. The question of the initial conditions for the wave function now becomes the quantum analogue of the search for initial conditions. The transition function slots x 1 and t1 are where we could insert our candidates.

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quantum initial conditions There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. — a. n. whitehead

We have seen that the transition function T tells us about the transition from one conﬁguration of spatial geometry on which the matter has a particular arrangement to another. Let us think of it as T [m1 , g 1 ; m2 , g 2 ], where m labels the matter conﬁguration and g is some geometrical characteristic of space, like the curvature, which we are using as an internally deﬁned time at two values ‘1’ and ‘2’. We can envisage universes that begin at a single point rather than at an initial space, so that their development looks conical rather than cylindrical (as was the case in Figure 3.6). This is illustrated schematically in Figure 3.7. Yet this is no great advance in our attempt to transmogrify the idea of initial conditions, because the singularity of the non-quantum cosmological models always shows up as a feature of the classical quantum path, and in any case we just seem to be picking a particular initial condition, which happens to describe creation from an initial pre-existent point, for no good reason. We have not severed the dualism between laws (represented here by the WheelerDe Witt equation) and initial conditions. There is a radical path that may now be taken. One should stress that it may well turn out to be empty of any physical signiﬁcance. It is an article of faith. If we look at Figures 3.6 and 3.7, then we can see how the stipulation of an initial condition g 1 relates to the state of the space further up the tube or the cone at g 2 . Could the boundaries of the conﬁgurations at g 1 and g 2 be combined in some way so that they describe a single smooth space which contains no nasty singularities? We know of simple possibilities in two dimensions, like the surface of a sphere, which are smooth and free of any singular points. So we might try

Single point

Figure 3.7 A space-time path whose boundary consists of a curved three-dimensional space of curvature g 2 and a single initial point, rather than another threedimensional space. If there is a singularity in the curvature or the matter conﬁguration at the point, we cannot calculate the transition probability T from this point to the state with curvature g 2 . If this had been possible, it would give the probability of a particular type of universe arising from a ‘point’ rather than from ‘nothing’.

initial conditions

Figure 3.8 An appealing path is one whose boundary is smoothly rounded off so that it consists of just a single threedimensional space, with no conical ‘point’ at the base as there was in Figure 3.7. This admits an interpretation of the transition probability as creation out of ‘nothing’ because no initial state exists: there is a single boundary. This can be employed as the picture of the three-dimensional boundary of a fourdimensional space-time only if we suppose that time behaves like another dimension of space.

to conceive of the whole boundary of the four-dimensional space-time to be not g 1 and g 2 but a single smooth surface in three dimensions. This might be the surface of a sphere sitting in four space dimensions. One of the curious and attractive features of these smooth surfaces that mathematicians habitually consider regardless of their dimension, which we can visualize better by returning to the two-dimensional surface of an ordinary sphere, is that they are ﬁnite in size but nevertheless have no edge: the surface of the sphere has a ﬁnite area (it would only require a ﬁnite amount of paint to paint it), but however one moves one never runs into an unusual point like the apex of a cone. We might describe the sphere as being without boundary from the point of view of ﬂatlanders living on its surface. Interestingly, such a conﬁguration can be conceived for the initial state of the Universe (see Figure 3.8). However— and now comes the radical step—the sphere we are using as an example is a space of three dimensions with a two-dimensional surface as a bondary. But, for our quantum boundary, we need a three-dimensional space as a boundary. However, this requires the four-dimensional thing of which it is the boundary to be a four-dimensional space and not a four-dimensional space-time, which is what the real Universe has always been assumed to be. Therefore, it is proposed that our ordinary concept of time is transcended in this quantum cosmological setting and becomes like another dimension of space, so making three-plus-one dimensions of space and time into a four-dimensional space. This is not quite as mystical as it might sound because physicists have often carried out this ‘change time into space’ procedure as a useful trick for doing certain problems in ordinary quantum mechanics, although they did not imagine that time was really like space. At the end of the calculation, they just swop back into the usual interpretation of there being one dimension of time and three other qualitatively diﬀerent dimensions of what we call space. The radical character of this approach is that it regards time as being truly like space in the ultimate quantum gravitational environment of the Big Bang. As one moves far away from the beginning of the Universe, so the quantum eﬀects start to interfere in a destructive fashion and the Universe is expected to follow the classical path with greater and greater accuracy. When this happens,

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the conventional notion of time as a distinct concept to that of space begins to crystallize out. Conversely, as one approaches the beginning, so the conventional picture of time melts away and time becomes indistinguishable from space as the eﬀects of the boundary condition are felt. This ‘no boundary’ condition was proposed by James Hartle and Stephen Hawking for aesthetic reasons. It avoids singularities from the initial state and removes the conventional dualism between laws and initial conditions. This it can achieve if the distinction between space and time is lost. More precisely, the ‘no boundary’ proposal stipulates that, in order to work out the wave function of the Universe, we compute it as the weighted aggregate of paths which are restricted to those four-dimensional spaces which possess a single ﬁnite smooth boundary like the spherical one we have just discussed. The transition probability that this prescription provides for the production of a wave function with some other matter content m2 in a geometrical conﬁguration g 2 just has the form T [m2 , g 2 ]. Thus there are no slots corresponding to any ‘initial’ state characterized by m1 and g 1 . Hence, this is often described as giving a picture of ‘creation out of nothing’, in which T gives the probability of a certain type of universe having been created out of nothing. The eﬀect of the ‘time becomes space’ proposal is that there is no deﬁnite moment or point of creation. In more conventional quantum mechanical terms, we would say that the Universe is the result of a quantum mechanical tunnelling process, where it must be interpreted as having tunnelled from nothing at all. Quantum tunnelling processes, which are familiar to physicists and routinely observed, correspond to transitions which do not have a classical path.

the great divide I sometimes ask myself how it came about that I was the one to develop the theory of relativity. The reason, I think, is that a normal adult never stops to think about problems of space and time. These are things which he has thought of as a child. But my intellectual development was retarded, as a result of which I began to wonder about space and time only when I had already grown up. — albert einstein

The overall picture one gets of this type of quantum beginning is that the Wheeler–De Witt equation gives the law of Nature which describes how the wave function W changes. The geometry of the space can be used as a measure of time which looks essentially like the ordinary time of general relativity when

initial conditions

one is far from the Big Bang. But, as one looks back towards that instant which we would have called the zero of time, the notion of time fades away and ultimately ceases to exist. This type of quantum universe has not always existed; it comes into being just as the classical cosmologies could, but it does not start at a Big Bang where physical quantities are inﬁnite and where further initial conditions need to be speciﬁed. In neither case is there any information as to what it may have come into being from. We should stress again that this is a radical proposal (which we shall discover, in Chapter 5, can become even more radical). It has two ingredients: the ﬁrst is the ‘time becomes space’ proposal; the second is the addition of the ‘no boundary’ proposal—a single prescription for the state of the Universe, which subsumes the roles of both initial equations and laws of Nature in the traditional picture. Even if one subscribes to the ﬁrst ingredient, there are many choices one could have used instead of the second to specify the state of a Universe which tunnels into existence out of nothing. These would all have required some additional speciﬁcation of information. The study of the wave function of the Universe is in its infancy. It will undoubtedly change in many ways before it is done. The ‘no boundary’ condition leaves much to be desired. It probably contains too little information to describe all the observable features of a real universe containing irregularities like galaxies. It must be supplemented by additional information about the matter ﬁelds in the Universe and how they distribute themselves. Of course, it may also be complete nonsense. The important lesson for us to draw from it here is the extent to which our traditional dualism regarding initial conditions and laws might be mistaken. It might be an artefact of our experience of a realm of Nature in which quantum eﬀects are small. If a theory of Nature is truly uniﬁed, then we might expect that it would exploit the possibility of keeping time in terms of the material contents of the Universe so as to marry together the constituents of Nature with the laws governing their change and the nature of time itself. However, we are still left with a choice as to the boundary condition which should be imposed upon some entity like the wave function of the Universe. No matter how economical its prescription, it is an inescapable fact that the ‘no boundary’ condition and its various rivals are picked out only for aesthetic reasons. They are not demanded by the internal logical consistency of the quantum universe. The dualistic view that initial conditions are independent of laws of Nature must be reassessed in the case of the initial conditions for the Universe as a whole. If the Universe is unique—the only logically consistent possibility—then the initial conditions are unique and become in eﬀect a

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law of Nature themselves. This is the motivation of those who seek basic principles which might serve to delineate the initial conditions of the Universe. If this is truly the case, then it introduces another new ingredient into our thinking about the Universe, because it points to a fundamental asymmetry between the past and the future in the make-up of the laws of Nature. On the other hand, if we believe that there are many possible universes—indeed may actually be many possible universes ‘somewhere’—then initial conditions need have no special status. They could be just as in more mundane physical problems: those deﬁning characteristics that specify one particular actuality from a general class of possibilities. The traditional view that initial conditions are for the theologians and evolution equations for the physicists seems to have been overthrown—at least temporarily. Cosmologists now engage in the study of initial conditions to discover whether there exists a ‘law’ of initial conditions, of which the ‘no boundary’ proposal would be just one possible example. This is radical indeed, but perhaps it is not radical enough. It is worrying that so many of the concepts and ideas being used in the modern mathematical description— ‘creation out of nothing’, ‘time coming into being with the Universe’—are just reﬁned images of rather traditional human intuitions and categories of thought. Surely, it is these traditional notions that motivate many of the concepts that are searched for and even found within modern theories that are cast in mathematical form. The ‘time becomes space’ proposal is the one truly radical element that we cannot attribute to our inheritance of past generations of human thinking in philosophical theology. One suspects that a good many more habitual concepts may need to be transformed before the true picture begins to emerge.

chapter 4

Forces and particles A vacuum is a hell of a lot better than some of the stuff that nature replaces it with. — tennessee williams

the stuff of the universe The scenery in the play was beautiful, but the actors got in front of it. — alexander woollcott

Devices of all sorts, whether they be computers or milling machines, need to act upon suitable raw material. If you design a spanner on general mechanical principles with a little bit of symmetry thrown in for appearances, it will none the less be useless if it fails to ﬁt your particular bolt-head shapes. Likewise, a Theory of Everything needs information about what particles and forces actually exist. A knowledge of the laws of Nature is of little use unless one knows what it is that those laws govern. In this respect the contrast between the traditional classical physics of Newton and the elementary-particle world is striking. Newton emphasized the universality of his laws of motion: they apply without exception to ‘all bodies’ irrespective of their other idiosyncrasies. Yet it is this very universality that prevents the laws of classical physics having anything to say about what particles or bodies actually do exist. They focus upon certain universal attributes of particles, like their mass, to the exclusion of all others. To those of us who have grown up learning about Newtonian mechanics from schooldays, this seems a familiar and reasonable approach but how diﬃcult it must have been for the ﬁrst students of motion to identify the salient features of a real object which should be included in the laws of motion.

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A beautiful illustration of the dilemma is presented by the French scientist Moreau de Maupertuis, the originator of the Principle of Least Action during the eighteenth century. With regard to the laws of momentum conservation which govern the collision between bodies like snooker or pool balls, he observes: If someone who had never touched a body or seen how bodies collide, but who was experienced in mixing colours, saw a blue object move toward a yellow one, and were asked what would happen if these two bodies collide, he would probably say that the blue body would turn green as soon as it united with the yellow one.

It is not diﬃcult to appreciate why the number of properties of a body which can be important for its dynamics has to be minimal. Typical goodsized objects, like rocks, footballs, or cars have so many individual properties that if the laws governing their motion were closely associated with many of their deﬁning properties then it would be as good as having no laws at all. Every rock, car, or billiard ball is diﬀerent in a myriad of ways and each would respond to the same law in very diﬀerent ways. Such a situation is very similar to that found in many early Greek writings. They did not readily have the notion of an external Lawgiver in Nature who dictated external laws of Nature. Instead, they were partial to the notion that bodies contained immanent tendencies which dictated how they would move. Whereas Plato sought to understand what was observed in the world in terms of another world of perfect blueprints of which observed things were but an imperfect approximation, Aristotle believed that these ‘forms’ which dictated how things behaved were not inhabitants of some abstract other-world but were in the things themselves. Aristotle’s ideas held sway for thousands of years, until they were discarded because of a combination of religious and scientiﬁc considerations. Newton dismissed this tradition of innate tendencies rather forcefully in his correspondence with Richard Bentley: That gravity should be innate, inherent and essential to Matter . . . is to me so great an Absurdity, that I believe that no Man who has in philosophical matters a competent Faculty of thinking, can ever fall into it.

Newton saw that it was necessary to discard this view if one was to move forward and separate what we do know from what we do not. No universal laws could emerge if we regarded laws of Nature as innate to the particles they governed. The future course of physics until the early twentieth century therefore regarded the material content of the Universe as logically distinct from the laws that governed it. The former had to be discovered by observation

forces and particles

whilst the laws which governed the behaviour of particular things acted upon a very small number of attributes, like electric charge or mass, whose identities were revealed by our accumulated experience.

the copy-cat principle Repetition is the only form of permanence that nature can achieve. — george santayana

The laws of elementary-particle behaviour are diﬀerent precisely because the objects they govern are not diﬀerent. Whereas the rocks and billiard balls of classical physics are all diﬀerent the most elementary particles of matter fall into classes of identical particles: all electrons are the same, all muons the same, and so on, throughout the elementary-particle world. It is a world of clones. Once you have seen one electron, you have seen them all. But it is this copycat principle that makes it possible for the laws which govern the behaviour of electrons and muons to be closely linked to the intrinsic properties of electrons and muons without sacriﬁcing their universality. It also plays a crucial role in our human quest to understand the Universe, for it underpins our belief that by an exhaustive study of a small part of the Universe we can approach an understanding of the whole. The fact that Nature displays populations of identical elementary particles is its most remarkable property. It is the ‘ﬁne tuning’ that surpasses all others. Our experience of the Universe has never given us any reason to doubt the assumption that all electrons are the same, all photons are the same wherever and whenever they are. In the nineteenth century, James Clerk Maxwell highlighted the fact that the physical world was composed of identical atoms which were not subject to gradual mutation or evolution. Today, we look for some deeper explanation of the elementary particles of Nature in terms of a Theory of Everything like string theory. One of the perplexing features of the successful theories of the electromagnetic, weak and strong interactions of particle physics that provoked particle physicists to search for a deeper underlying theory was the profusion of elementary particles. There were so many of them that it suggested that there was a smaller and more basic population of entities inside them. Perhaps they were not elementary at all? Could they be composed of diﬀerent combinations of a far smaller number of elementary

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objects? Attempts to make theories of possible building blocks were never very compelling and led to no testable predictions. String theories oﬀer another route to solving this problem. Instead of a Theory of Everything containing a population of elementary point-like particles, string theories introduce basis entities that are lines or loops of energy which have a tension. As the temperature rises the loops shudder and vibrate in an increasingly stringy fashion, but as the temperature falls the tension increases and the loops become more and more point-like. So, at low energies the strings behave like points and allow the theory to make the same types of accurate prediction about what we should see as the intrinsically pointlike theories do. However, at high energies, things are diﬀerent. The hope is that it will be possible to determine the principal energies of vibration of the superstrings. All strings, even guitar strings, have a collection of special vibrational energies that they naturally take up when disturbed. It is hoped that, if we could calculate these special energies for the superstring, then they would (by virtue of Einstein’s famous formula of mass–energy equivalence: E = mc 2 ) correspond to a collection of masses that correspond in some way to the ‘particles’ that we call elementary. So far, these energies have proved too hard to calculate. However, one of them has been found: it corresponds to a particle with zero mass and two quantum units of spin. This spin ensures that it mediates attractions between all masses. It is the particle that we call the ‘graviton’ and shows that string theory necessarily includes the phenomenon of gravitation—a remarkable and compelling feature since earlier candidates for a Theory of Everything all failed miserably whenever they were challenged to ﬁnd a way to include gravity in the uniﬁcation story. It is this repeatability of things that is the hallmark of the most basic entities in Nature and at root it is the reason why there can be accuracy and reliability in the physical world, whether it be in DNA replication or in the stability of the properties of matter. But we shall ﬁnd it opens up the possibility that the strict divide between the laws of Nature and the entities that they govern may be compromised when we probe Nature more deeply, just as in the last chapter we found it possible to muddy the divide between laws and initial conditions in the quantum description of the Universe. If there exists a real divide between the constituents of the Universe and the laws that govern them, then any Theory of Everything would require additional information to restrict the identities of particles. This seems unsatisfactory to amateur universe builders like ourselves. We would expect that things could be perfectly uniﬁed in some sense, so that the laws and the ultimate particles of Nature that they govern

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are married together in a union of perfect and unique intercompatibility. The laws should decree what their subjects are in addition to what they may do. This symbiosis of laws and particles and forces has begun to come to pass in modern physics as a result of the discovery of a breed of physical theory called a gauge theory. All the best theories of the fundamental forces of Nature—of gravity, of electromagnetism, and of the weak and strong nuclear forces—are gauge theories. Let us dwell upon how this threefold union of particles, forces, and laws comes about within this jurisdiction. For the Newtonian physicist whose laws governed the behaviour of objects on an absolute space moving through unbending time, forces moved things in a mysterious way. Gravity acted instantaneously between masses by a process that Newton found it fruitless to enquire into any further. Gradually, throughout the twentieth century, the eﬀect of the cosmic speed limit for the transfer of information imposed by Einstein’s special theory of relativity has made its presence felt. Instantaneous gravitational eﬀects would violate that limit by allowing signals to be transmitted faster than the speed of light in a vacuum. As a result, we picture forces of Nature as mediated by the exchange of particles between the bodies which are in interaction. Thus the gravitational force is mediated by the exchange of gravitons, the electromagnetic force by the exchange of photons, the weak interaction by the exchange of massive W or Z particles, and the strong interaction between quarks by the exchange of gluons. In some cases, these exchange particles actually feel the force which they mediate. This is the case for gravitation and for the strong and weak interactions, although not for the electromagnetic interaction which acts between electrically charged elementary particles. The interactions between such particles are mediated by the exchange of a photon of light which is uncharged. Thus we see that the forces of Nature are deeply entwined with the elementary particles of Nature. They cannot be considered independently. The other arms of the golden triangle, the connection between forces and particles and the laws themselves exist only in these elegant creations called gauge theories. Their emergence has undermined a longstanding prejudice regarding the Galilean and Newtonian revolutions in the description of Nature that scientists ceased asking ‘why’ questions of Nature and were content to know only ‘how’ things were. Curiously, modern particle physicists are quite diﬀerent. Gauge theories show that physicists need not be content to possess theories that are perfectly accurate in their description of how particles move and interact. They can know something of why those particles exist and why they interact in the manner seen.

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The most successful fundamental theories of physics—general relativity (the theory of gravity), quantum chromodynamics (the theory of the strong sub-nuclear forces between quarks and gluons), and the Weinberg–Salam theory (the uniﬁed theory of the electromagnetic and weak interactions)—are all theories of a particular type known as local gauge theories. We have already seen in Chapter 2 how certain geometrical invariances of the laws of Nature are equivalent to the imposition of physical invariances. For each symmetry, there exists an associated conserved quantity. This correspondence is maintained even when the symmetries involved are more esoteric than simple rotations or translations in space. These additional invariances are called internal symmetries and correspond to invariances under various relabellings of the particles involved, for example swapping the identities of all the protons and neutrons in the Universe. Gauge symmetries are diﬀerent again. They do not lead to conserved quantities in Nature; rather, they impose powerful requirements upon the form and scope of the laws of Nature. In particular, they dictate what forces of Nature exist and the properties of the elementary particles which they govern. The simplest example is that of a global gauge symmetry. It demands that the world be invariant if we shift every point in the same way. Imagine such an operation performed upon an object like your hand. It would be transported in space but would look the same. But it is unnatural to suppose that the changes be the same everywhere. If a particle changes at this moment on the other side of the Universe, then a particle here and now cannot know this at least until a light signal has had time to pass between them. It would require instantaneous signalling to keep in step. Global gauge invariance is a somewhat unappealing restriction that retains echoes of Newton’s instantaneous action at a distance. This leads us to demand the more realistic, but much more stringent, requirement that things be invariant under local gauge symmetry, wherein every point can change in a diﬀerent way. Invariance in this case seems impossible. In our earlier example, every part of your hand would move oﬀ in diﬀerent directions. The only way in which things can be kept invariant under such general changes is if certain forces exist which constrain the allowed motions. Imagine some elastic bands taut around your hand which restrict the ways in which parts of it can move: the elementary-particle world is akin to having an inﬁnite network of entwined constraints like this which transform all possible changes into a small class of particular ones. In this way, the imposition of invariance under local gauge symmetry actually dictates what forces of Nature exist between the particles involved. They reveal why there must be electromagnetism as well as how it operates.

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Einstein’s general theory of relativity is a local gauge theory of this sort. Einstein wished to generalize the Principle of Special Relativity which maintained that the laws of physics be the same for all observers moving at constant relative velocities to the situation where they accelerate. The only way in which this is possible for all observers in arbitrary accelerated motion is for there to exist a gravitational ﬁeld. The Platonic faith in symmetry and the implementation of those symmetries as the basis for gauge theories is the foundation of our knowledge of elementary-particle interactions. Yet is does not tell us everything. It fails to tell us how many particles of a similar type there must be. Why are there three types of neutrino rather than just one. Why there is only one variety of photon. Nature appears to have used a ‘copy-cat’ principle in two ways. It has created populations of identical particles like electrons and electrontype neutrinos; but it has also created muons and muon-type neutrinos and tau particles and their associated neutrinos. These are similar to the electron and its neutrino in many ways. What one would like to know is why there exist these small variations on the same major theme and why there are just three of them and no more. The diﬀerent gauge theories have failed to tell us how many of these copies there must be. Moreover, to complete a fully uniﬁed picture of the Universe, we must do something about the fact that we have many diﬀerent gauge theories which must be uniﬁed into a single description by embedding the diﬀerent symmetries associated with individual theories into a bigger over-riding pattern, or grand uniﬁed theory. Grand uniﬁcation removes the problems of diﬀerent disjoint theories, but it still does not solve the problem of what limits the number of types of similar particles. Gauge theories, by their very nature, are built upon symmetry. These symmetries are built up by operation of a ﬁnite number of variations upon a single theme. There are only a ﬁnite number of basic generators of the possible patterns that span all the possibilities compatible with the maintenance of a particular symmetry. The greater the number of basic generators so the larger the range of patterns. Furthermore, the basic generators of the set of patterns consistent with any underlying symmetry deﬁne that symmetry and correspond to the ‘carrier’ particles which mediate the forces of Nature. Thus, in Maxwell’s theory of electromagnetism, there is just one generator of the symmetry and this corresponds to the photon; the symmetry governing the weak force has three generators corresponding to the electrically neutral Z boson and the positively and negatively charged W bosons; the strong force between quarks has eight generators corresponding to the eight varieties of gluon carrying the three varieties of the type of charge called ‘colour’ which

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the strong force recognizes. As a result, we see that there is a certain element of ﬁniteness built into all of these theories. The ﬁniteness of the symmetry is associated with the ﬁnite number of the elementary particles that are the basic generators of the symmetry. A world which had a bottomless inﬁnity of elementary particles would be well on the way to anarchy. Its symmetries would need to be so large that their inﬂuences would be tantalizingly weak.

elementarity We spoke of the ‘Properties of Things’, and of the degree to which these properties could be investigated. As an extreme thought, the following question was proposed: Supposing it were possible to discover all the properties of a grain of sand, would we then have gained a complete knowledge of the whole universe? Would there then remain no unsolved component of our comprehension of the universe? — a. moszkowski

The most topical aspect of the identiﬁcation of the forces and particles of Nature is to know the identity of the most elementary entities in Nature. Until only a few years ago, they were invariably imagined to be idealized ‘points’ of zero size. Quarks and leptons were taken to be particles of this sort, exhibiting no evidence of internal structure in any particle scattering experiment. If a particle physicist were asked how many angels can dance on a quark, he could answer none without a moment’s hesitation. However, theories in which the most basic entities are points—quantum ﬁeld theories as they are known—possess unpleasant mathematical properties. They lead to mathematical inﬁnities that must be ignored in the process of calculating observable quantities. This can usually be done by following a systematic recipe which amounts to ignoring the inﬁnite part of any answer, but the procedure is aesthetically rather unappealing. It has only been tolerated in practice because the ﬁnite parts that remain in these calculations after the inﬁnite parts have been removed produce predictions of observed quantities that are correct to fantastic precision. There is clearly a deep truth somewhere close to the heart of this picture. It has now been recognized that theories in which the most elementary objects are lines or loops (‘strings’), rather than points, can avoid these defects. Moreover, whereas the point particle schemes require a separate point endowed with characteristics like mass to be speciﬁed for each elementary particle separately, a single string possesses an inﬁnite number of modes of

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vibrations, just like the harmonics of a violin string, and the energy of each diﬀerent mode will correspond to a diﬀerent elementary-particle mass (via the mass–energy equivalence E = mc2 ). Most of this collection of particle masses will be concentrated around unobservably high energies, but the others should include the masses of the known elementary particles. Furthermore, the number of copies of each type of particle appears to be tied to the underlying symmetrical structure of these theories. They have the scope to tell us why there are three varieties of neutrino at low energy. Whereas earlier elementary-particle theories could provide no explanation for this aspect of things, string theories link it to the laws of Nature in a deep way and turn it into an answerable ‘why’ question. This explanatory potential is the great hope of string theories and is the hallmark of their claim to be a Theory of Everything. They should contain within them the deep connection between the symmetries or laws of Nature and the entities which those laws govern, but as yet the diﬃculty of extracting that information from the theory has proved insurmountable. It is one thing to have the Theory of Everything; quite another to solve it. One day it is hoped that deﬁnite predictions of the masses of the elementary particles of Nature will be extracted from this theory and compared with observation. Strings aim to explain all the properties of the elementary particles of Nature. But in terms of what will they explain them? What are the properties of strings themselves? Strings possess one deﬁning property which is their tension. This quantity plays a crucial role in the overall picture of how strings can be reconciled with the miraculous experimental success of the point-like quantum ﬁeld theories in explaining the observed features of the world at lower energies. For the strings possess a tension that varies with the energy of the environment, so that at low energies the tension is high and pulls the strings taut into points and we recover the favourable features of a world of point-like elementary particles. At high energies, where the string tension is low, their essential stringiness becomes evident and creates behaviour that is qualitatively diﬀerent from that of the point-particle theories. Unfortunately, at present, the mathematical expertise required to reveal these properties is somewhat beyond us. For the ﬁrst time, modern physicists have found that oﬀ-the-shelf mathematics is insuﬃcient to extract the physical content of their theories. But, in time, suitable techniques will no doubt emerge, or perhaps a better way to look at the theory will be found: one that is conceptually and technically simpler. In summary, we have seen that we need to know the identity of the forces and particles of Nature. At present, we believe, perhaps mistakenly, that we have identiﬁed all the fundamental forces. We have working gauge theories,

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References