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**The Road to Reality - A Complete Guide to the Laws of the Universe**

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**Roger Penrose**

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THE ROAD TO REALITY

BY ROGER PENROSE

The Emperorâ€™s New Mind:

Concerning Computers, Minds,

and the Laws of Physics

Shadows of the Mind:

A Search for the Missing Science

of Consciousness

Roger Penrose

T H E R O A D TO

REALITY

A Complete Guide to the Laws

of the Universe

JONATHAN CAPE

LONDON

Published by Jonathan Cape 2004

2 4 6 8 10 9 7 5 3 1

Copyright ÃŸ Roger Penrose 2004

Roger Penrose has asserted his right under the Copyright, Designs

and Patents Act 1988 to be identified as the author of this work

This book is sold subject to the condition that it shall not,

by way of trade or otherwise, be lent, resold, hired out,

or otherwise circulated without the publisherâ€™s prior

consent in any form of binding or cover other than that

in which it is published and without a similar condition

including this condition being imposed on the

subsequent purchaser

First published in Great Britain in 2004 by

Jonathan Cape

Random House, 20 Vauxhall Bridge Road,

London SW1V 2SA

Random House Australia (Pty) Limited

20 Alfred Street, Milsons Point, Sydney,

New South Wales 2061, Australia

Random House New Zealand Limited

18 Poland Road, Glenfield,

Auckland 10, New Zealand

Random House South Africa (Pty) Limited

Endulini, 5A Jubilee Road, Parktown 2193, South Africa

The Random House Group Limited Reg. No. 954009

www.randomhouse.co.uk

A CIP catalogue record for this book

is available from the British Library

ISBN 0â€“224â€“04447â€“8

Papers used by The Random House Group Limited are natural,

recyclable products made from wood grown in sustainable forests;

the manufacturing processes conform to the environmental

regulations of the country of origin

Printed and bound in Great Britain by

William Clowes, Beccles, Suffolk

Contents

Preface

xv

Acknowledgements

xxiii

Notation

xxvi

Prologue

1

1 The roots of science

7

1.1

1.2

1.3

1.4

1.5

The quest for the forces that shape the world

Mathematical truth

Is Platoâ€™s mathematical world â€˜realâ€™?

Three worlds and three deep mysteries

The Good, the True, and the Beautiful

2 An ancient theorem and a modern question

2.1

2.2

2.3

2.4

2.5

2.6

2.7

The Pythagorean theorem

Euclidâ€™s postulates

Similar-areas proof of the Pythagorean theorem

Hyperbolic geometry: conformal picture

Other representations of hyperbolic geometry

Historical aspects of hyperbolic geometry

Relation to physical space

3 Kinds of number in the physical world

3.1

3.2

3.3

3.4

3.5

A Pythagorean catastrophe?

The real-number system

Real numbers in the physical world

Do natural numbers need the physical world?

Discrete numbers in the physical world

4 Magical complex numbers

4.1

4.2

7

9

12

17

22

25

25

28

31

33

37

42

46

51

51

54

59

63

65

71

The magic number â€˜iâ€™

Solving equations with complex numbers

v

71

74

Contents

4.3

4.4

4.5

Convergence of power series

Caspar Wesselâ€™s complex plane

How to construct the Mandelbrot set

5 Geometry of logarithms, powers, and roots

5.1

5.2

5.3

5.4

5.5

Geometry of complex algebra

The idea of the complex logarithm

Multiple valuedness, natural logarithms

Complex powers

Some relations to modern particle physics

6 Real-number calculus

6.1

6.2

6.3

6.4

6.5

6.6

What makes an honest function?

Slopes of functions

Higher derivatives; C1 -smooth functions

The â€˜Eulerianâ€™ notion of a function?

The rules of diVerentiation

Integration

7 Complex-number calculus

7.1

7.2

7.3

7.4

Complex smoothness; holomorphic functions

Contour integration

Power series from complex smoothness

Analytic continuation

8 Riemann surfaces and complex mappings

8.1

8.2

8.3

8.4

8.5

76

81

83

86

86

90

92

96

100

103

103

105

107

112

114

116

122

122

123

127

129

135

The idea of a Riemann surface

Conformal mappings

The Riemann sphere

The genus of a compact Riemann surface

The Riemann mapping theorem

135

138

142

145

148

9 Fourier decomposition and hyperfunctions

153

9.1

9.2

9.3

9.4

9.5

9.6

9.7

vi

Fourier series

Functions on a circle

Frequency splitting on the Riemann sphere

The Fourier transform

Frequency splitting from the Fourier transform

What kind of function is appropriate?

Hyperfunctions

153

157

161

164

166

168

172

Contents

10 Surfaces

179

10.1

10.2

10.3

10.4

10.5

179

181

185

190

193

Complex dimensions and real dimensions

Smoothness, partial derivatives

Vector Welds and 1-forms

Components, scalar products

The Cauchyâ€“Riemann equations

11 Hypercomplex numbers

11.1

11.2

11.3

11.4

11.5

11.6

The algebra of quaternions

The physical role of quaternions?

Geometry of quaternions

How to compose rotations

CliVord algebras

Grassmann algebras

12 Manifolds of n dimensions

12.1

12.2

12.3

12.4

12.5

12.6

12.7

12.8

12.9

Why study higher-dimensional manifolds?

Manifolds and coordinate patches

Scalars, vectors, and covectors

Grassmann products

Integrals of forms

Exterior derivative

Volume element; summation convention

Tensors; abstract-index and diagrammatic notation

Complex manifolds

13 Symmetry groups

13.1

13.2

13.3

13.4

13.5

13.6

13.7

13.8

13.9

13.10

Groups of transformations

Subgroups and simple groups

Linear transformations and matrices

Determinants and traces

Eigenvalues and eigenvectors

Representation theory and Lie algebras

Tensor representation spaces; reducibility

Orthogonal groups

Unitary groups

Symplectic groups

14 Calculus on manifolds

14.1

14.2

14.3

14.4

DiVerentiation on a manifold?

Parallel transport

Covariant derivative

Curvature and torsion

198

198

200

203

206

208

211

217

217

221

223

227

229

231

237

239

243

247

247

250

254

260

263

266

270

275

281

286

292

292

294

298

301

vii

Contents

14.5

14.6

14.7

14.8

Geodesics, parallelograms, and curvature

Lie derivative

What a metric can do for you

Symplectic manifolds

15 Fibre bundles and gauge connections

15.1

15.2

15.3

15.4

15.5

15.6

15.7

15.8

Some physical motivations for Wbre bundles

The mathematical idea of a bundle

Cross-sections of bundles

The CliVord bundle

Complex vector bundles, (co)tangent bundles

Projective spaces

Non-triviality in a bundle connection

Bundle curvature

16 The ladder of inWnity

16.1

16.2

16.3

16.4

16.5

16.6

16.7

Finite Welds

A Wnite or inWnite geometry for physics?

DiVerent sizes of inWnity

Cantorâ€™s diagonal slash

Puzzles in the foundations of mathematics

Turing machines and GoÂ¨delâ€™s theorem

Sizes of inWnity in physics

17 Spacetime

17.1

17.2

17.3

17.4

17.5

17.6

17.7

17.8

17.9

The spacetime of Aristotelian physics

Spacetime for Galilean relativity

Newtonian dynamics in spacetime terms

The principle of equivalence

Cartanâ€™s â€˜Newtonian spacetimeâ€™

The Wxed Wnite speed of light

Light cones

The abandonment of absolute time

The spacetime for Einsteinâ€™s general relativity

18 Minkowskian geometry

18.1

18.2

18.3

18.4

18.5

18.6

18.7

viii

Euclidean and Minkowskian 4-space

The symmetry groups of Minkowski space

Lorentzian orthogonality; the â€˜clock paradoxâ€™

Hyperbolic geometry in Minkowski space

The celestial sphere as a Riemann sphere

Newtonian energy and (angular) momentum

Relativistic energy and (angular) momentum

303

309

317

321

325

325

328

331

334

338

341

345

349

357

357

359

364

367

371

374

378

383

383

385

388

390

394

399

401

404

408

412

412

415

417

422

428

431

434

Contents

19 The classical Welds of Maxwell and Einstein

19.1

19.2

19.3

19.4

19.5

19.6

19.7

19.8

Evolution away from Newtonian dynamics

Maxwellâ€™s electromagnetic theory

Conservation and Xux laws in Maxwell theory

The Maxwell Weld as gauge curvature

The energyâ€“momentum tensor

Einsteinâ€™s Weld equation

Further issues: cosmological constant; Weyl tensor

Gravitational Weld energy

20 Lagrangians and Hamiltonians

20.1

20.2

20.3

20.4

20.5

20.6

The magical Lagrangian formalism

The more symmetrical Hamiltonian picture

Small oscillations

Hamiltonian dynamics as symplectic geometry

Lagrangian treatment of Welds

How Lagrangians drive modern theory

21 The quantum particle

21.1

21.2

21.3

21.4

21.5

21.6

21.7

21.8

21.9

21.10

21.11

Non-commuting variables

Quantum Hamiltonians

SchroÂ¨dingerâ€™s equation

Quantum theoryâ€™s experimental background

Understanding waveâ€“particle duality

What is quantum â€˜realityâ€™?

The â€˜holisticâ€™ nature of a wavefunction

The mysterious â€˜quantum jumpsâ€™

Probability distribution in a wavefunction

Position states

Momentum-space description

22 Quantum algebra, geometry, and spin

22.1

22.2

22.3

22.4

22.5

22.6

22.7

22.8

22.9

22.10

22.11

The quantum procedures U and R

The linearity of U and its problems for R

Unitary structure, Hilbert space, Dirac notation

Unitary evolution: SchroÂ¨dinger and Heisenberg

Quantum â€˜observablesâ€™

yes/no measurements; projectors

Null measurements; helicity

Spin and spinors

The Riemann sphere of two-state systems

Higher spin: Majorana picture

Spherical harmonics

440

440

442

446

449

455

458

462

464

471

471

475

478

483

486

489

493

493

496

498

500

505

507

511

516

517

520

521

527

527

530

533

535

538

542

544

549

553

559

562

ix

Contents

22.12

22.13

Relativistic quantum angular momentum

The general isolated quantum object

23 The entangled quantum world

23.1

23.2

23.3

23.4

23.5

23.6

23.7

23.8

23.9

23.10

Quantum mechanics of many-particle systems

Hugeness of many-particle state space

Quantum entanglement; Bell inequalities

Bohm-type EPR experiments

Hardyâ€™s EPR example: almost probability-free

Two mysteries of quantum entanglement

Bosons and fermions

The quantum states of bosons and fermions

Quantum teleportation

Quanglement

24 Diracâ€™s electron and antiparticles

24.1

24.2

24.3

24.4

24.5

24.6

24.7

24.8

Tension between quantum theory and relativity

Why do antiparticles imply quantum Welds?

Energy positivity in quantum mechanics

DiYculties with the relativistic energy formula

The non-invariance of ]=]t

CliVordâ€“Dirac square root of wave operator

The Dirac equation

Diracâ€™s route to the positron

25 The standard model of particle physics

25.1

25.2

25.3

25.4

25.5

25.6

25.7

25.8

The origins of modern particle physics

The zigzag picture of the electron

Electroweak interactions; reXection asymmetry

Charge conjugation, parity, and time reversal

The electroweak symmetry group

Strongly interacting particles

â€˜Coloured quarksâ€™

Beyond the standard model?

26 Quantum Weld theory

26.1

26.2

26.3

26.4

26.5

26.6

26.7

26.8

26.9

x

Fundamental status of QFT in modern theory

Creation and annihilation operators

InWnite-dimensional algebras

Antiparticles in QFT

Alternative vacua

Interactions: Lagrangians and path integrals

Divergent path integrals: Feynmanâ€™s response

Constructing Feynman graphs; the S-matrix

Renormalization

566

570

578

578

580

582

585

589

591

594

596

598

603

609

609

610

612

614

616

618

620

622

627

627

628

632

638

640

645

648

651

655

655

657

660

662

664

665

670

672

675

Contents

26.10

26.11

Feynman graphs from Lagrangians

Feynman graphs and the choice of vacuum

27 The Big Bang and its thermodynamic legacy

27.1

27.2

27.3

27.4

27.5

27.6

27.7

27.8

27.9

27.10

27.11

27.12

27.13

Time symmetry in dynamical evolution

Submicroscopic ingredients

Entropy

The robustness of the entropy concept

Derivation of the second lawâ€”or not?

Is the whole universe an â€˜isolated systemâ€™?

The role of the Big Bang

Black holes

Event horizons and spacetime singularities

Black-hole entropy

Cosmology

Conformal diagrams

Our extraordinarily special Big Bang

28 Speculative theories of the early universe

28.1

28.2

28.3

28.4

28.5

28.6

28.7

28.8

28.9

28.10

Early-universe spontaneous symmetry breaking

Cosmic topological defects

Problems for early-universe symmetry breaking

InXationary cosmology

Are the motivations for inXation valid?

The anthropic principle

The Big Bangâ€™s special nature: an anthropic key?

The Weyl curvature hypothesis

The Hartleâ€“Hawking â€˜no-boundaryâ€™ proposal

Cosmological parameters: observational status?

29 The measurement paradox

29.1

29.2

29.3

29.4

29.5

29.6

29.7

29.8

29.9

The conventional ontologies of quantum theory

Unconventional ontologies for quantum theory

The density matrix

Density matrices for spin 12: the Bloch sphere

The density matrix in EPR situations

FAPP philosophy of environmental decoherence

SchroÂ¨dingerâ€™s cat with â€˜Copenhagenâ€™ ontology

Can other conventional ontologies resolve the â€˜catâ€™?

Which unconventional ontologies may help?

30 Gravityâ€™s role in quantum state reduction

30.1

30.2

Is todayâ€™s quantum theory here to stay?

Clues from cosmological time asymmetry

680

681

686

686

688

690

692

696

699

702

707

712

714

717

723

726

735

735

739

742

746

753

757

762

765

769

772

782

782

785

791

793

797

802

804

806

810

816

816

817

xi

Contents

30.3

30.4

30.5

30.6

30.7

30.8

30.9

30.10

30.11

30.12

30.13

30.14

Time-asymmetry in quantum state reduction

Hawkingâ€™s black-hole temperature

Black-hole temperature from complex periodicity

Killing vectors, energy Xowâ€”and time travel!

Energy outXow from negative-energy orbits

Hawking explosions

A more radical perspective

SchroÂ¨dingerâ€™s lump

Fundamental conXict with Einsteinâ€™s principles

Preferred SchroÂ¨dingerâ€“Newton states?

FELIX and related proposals

Origin of Xuctuations in the early universe

31 Supersymmetry, supra-dimensionality, and strings

31.1

31.2

31.3

31.4

31.5

31.6

31.7

31.8

31.9

31.10

31.11

31.12

31.13

31.14

31.15

31.16

31.17

31.18

Unexplained parameters

Supersymmetry

The algebra and geometry of supersymmetry

Higher-dimensional spacetime

The original hadronic string theory

Towards a string theory of the world

String motivation for extra spacetime dimensions

String theory as quantum gravity?

String dynamics

Why donâ€™t we see the extra space dimensions?

Should we accept the quantum-stability argument?

Classical instability of extra dimensions

Is string QFT Wnite?

The magical Calabiâ€“Yau spaces; M-theory

Strings and black-hole entropy

The â€˜holographic principleâ€™

The D-brane perspective

The physical status of string theory?

32 Einsteinâ€™s narrower path; loop variables

32.1

32.2

32.3

32.4

32.5

32.6

32.7

Canonical quantum gravity

The chiral input to Ashtekarâ€™s variables

The form of Ashtekarâ€™s variables

Loop variables

The mathematics of knots and links

Spin networks

Status of loop quantum gravity?

33 More radical perspectives; twistor theory

33.1

33.2

xii

Theories where geometry has discrete elements

Twistors as light rays

819

823

827

833

836

838

842

846

849

853

856

861

869

869

873

877

880

884

887

890

892

895

897

902

905

907

910

916

920

923

926

934

934

935

938

941

943

946

952

958

958

962

Contents

33.3

33.4

33.5

33.6

33.7

33.8

33.9

33.10

33.11

33.12

33.13

33.14

Conformal group; compactiWed Minkowski space

Twistors as higher-dimensional spinors

Basic twistor geometry and coordinates

Geometry of twistors as spinning massless particles

Twistor quantum theory

Twistor description of massless Welds

Twistor sheaf cohomology

Twistors and positive/negative frequency splitting

The non-linear graviton

Twistors and general relativity

Towards a twistor theory of particle physics

The future of twistor theory?

34 Where lies the road to reality?

34.1

34.2

34.3

34.4

34.5

34.6

34.7

34.8

34.9

34.10

Great theories of 20th century physicsâ€”and beyond?

Mathematically driven fundamental physics

The role of fashion in physical theory

Can a wrong theory be experimentally refuted?

Whence may we expect our next physical revolution?

What is reality?

The roles of mentality in physical theory

Our long mathematical road to reality

Beauty and miracles

Deep questions answered, deeper questions posed

968

972

974

978

982

985

987

993

995

1000

1001

1003

1010

1010

1014

1017

1020

1024

1027

1030

1033

1038

1043

Epilogue

1048

Bibliography

1050

Index

1081

xiii

I dedicate this book to the memory of

DENNIS SCIAMA

who showed me the excitement of physics

Preface

The purpose of this book is to convey to the reader some feeling for

what is surely one of the most important and exciting voyages of discovery

that humanity has embarked upon. This is the search for the underlying

principles that govern the behaviour of our universe. It is a voyage that

has lasted for more than two-and-a-half millennia, so it should not surprise us that substantial progress has at last been made. But this journey

has proved to be a profoundly diYcult one, and real understanding has,

for the most part, come but slowly. This inherent diYculty has led us

in many false directions; hence we should learn caution. Yet the 20th

century has delivered us extraordinary new insightsâ€”some so impressive

that many scientists of today have voiced the opinion that we may be

close to a basic understanding of all the underlying principles of physics.

In my descriptions of the current fundamental theories, the 20th century

having now drawn to its close, I shall try to take a more sober view.

Not all my opinions may be welcomed by these â€˜optimistsâ€™, but I expect

further changes of direction greater even than those of the last century.

The reader will Wnd that in this book I have not shied away from

presenting mathematical formulae, despite dire warnings of the severe

reduction in readership that this will entail. I have thought seriously

about this question, and have come to the conclusion that what I have

to say cannot reasonably be conveyed without a certain amount of

mathematical notation and the exploration of genuine mathematical

concepts. The understanding that we have of the principles that actually

underlie the behaviour of our physical world indeed depends upon some

appreciation of its mathematics. Some people might take this as a cause

for despair, as they will have formed the belief that they have no

capacity for mathematics, no matter at how elementary a level. How

could it be possible, they might well argue, for them to comprehend the

research going on at the cutting edge of physical theory if they cannot

even master the manipulation of fractions? Well, I certainly see the

diYculty.

xv

Preface

Yet I am an optimist in matters of conveying understanding. Perhaps I

am an incurable optimist. I wonder whether those readers who cannot

manipulate fractionsâ€”or those who claim that they cannot manipulate

fractionsâ€”are not deluding themselves at least a little, and that a good

proportion of them actually have a potential in this direction that they are

not aware of. No doubt there are some who, when confronted with a line

of mathematical symbols, however simply presented, can see only the stern

face of a parent or teacher who tried to force into them a non-comprehending parrot-like apparent competenceâ€”a duty, and a duty aloneâ€”and

no hint of the magic or beauty of the subject might be allowed to come

through. Perhaps for some it is too late; but, as I say, I am an optimist and

I believe that there are many out there, even among those who could never

master the manipulation of fractions, who have the capacity to catch some

glimpse of a wonderful world that I believe must be, to a signiWcant degree,

genuinely accessible to them.

One of my motherâ€™s closest friends, when she was a young girl, was

among those who could not grasp fractions. This lady once told me so

herself after she had retired from a successful career as a ballet dancer. I

was still young, not yet fully launched in my activities as a mathematician,

but was recognized as someone who enjoyed working in that subject. â€˜Itâ€™s

all that cancellingâ€™, she said to me, â€˜I could just never get the hang of

cancelling.â€™ She was an elegant and highly intelligent woman, and there is

no doubt in my mind that the mental qualities that are required in

comprehending the sophisticated choreography that is central to ballet

are in no way inferior to those which must be brought to bear on a

mathematical problem. So, grossly overestimating my expositional abilities, I attempted, as others had done before, to explain to her the simplicity and logical nature of the procedure of â€˜cancellingâ€™.

I believe that my eVorts were as unsuccessful as were those of others.

(Incidentally, her father had been a prominent scientist, and a Fellow of

the Royal Society, so she must have had a background adequate for the

comprehension of scientiWc matters. Perhaps the â€˜stern faceâ€™ could have

been a factor here, I do not know.) But on reXection, I now wonder

whether she, and many others like her, did not have a more rational

hang-upâ€”one that with all my mathematical glibness I had not noticed.

There is, indeed, a profound issue that one comes up against again and

again in mathematics and in mathematical physics, which one Wrst encounters in the seemingly innocent operation of cancelling a common

factor from the numerator and denominator of an ordinary numerical

fraction.

Those for whom the action of cancelling has become second nature,

because of repeated familiarity with such operations, may Wnd themselves

insensitive to a diYculty that actually lurks behind this seemingly simple

xvi

Preface

procedure. Perhaps many of those who Wnd cancelling mysterious are

seeing a certain profound issue more deeply than those of us who press

onwards in a cavalier way, seeming to ignore it. What issue is this? It

concerns the very way in which mathematicians can provide an existence

to their mathematical entities and how such entities may relate to physical

reality.

I recall that when at school, at the age of about 11, I was somewhat

taken aback when the teacher asked the class what a fraction (such as 38)

actually is! Various suggestions came forth concerning the dividing up of

pieces of pie and the like, but these were rejected by the teacher on the

(valid) grounds that they merely referred to imprecise physical situations

to which the precise mathematical notion of a fraction was to be applied;

they did not tell us what that clear-cut mathematical notion actually is.

Other suggestions came forward, such as 38 is â€˜something with a 3 at the top

and an 8 at the bottom with a horizontal line in betweenâ€™ and I was

distinctly surprised to Wnd that the teacher seemed to be taking these

suggestions seriously! I do not clearly recall how the matter was Wnally

resolved, but with the hindsight gained from my much later experiences as

a mathematics undergraduate, I guess my schoolteacher was making a

brave attempt at telling us the deWnition of a fraction in terms of the

ubiquitous mathematical notion of an equivalence class.

What is this notion? How can it be applied in the case of a fraction and

tell us what a fraction actually is? Let us start with my classmateâ€™s â€˜something with a 3 at the top and an 8 on the bottomâ€™. Basically, this is

suggesting to us that a fraction is speciWed by an ordered pair of whole

numbers, in this case the numbers 3 and 8. But we clearly cannot regard the

6

fraction as being such an ordered pair because, for example, the fraction 16

3

is the same number as the fraction 8, whereas the pair (6, 16) is certainly not

the same as the pair (3, 8). This is only an issue of cancelling; for we can

6

3

write 16

as 3 2

8 2 and then cancel the 2 from the top and the bottom to get 8.

Why are we allowed to do this and thereby, in some sense, â€˜equateâ€™ the pair

(6, 16) with the pair (3, 8)? The mathematicianâ€™s answerâ€”which may well

sound like a cop-outâ€”has the cancelling rule just built in to the deWnition of

a fraction: a pair of whole numbers (a n, b n) is deemed to represent the

same fraction as the pair (a, b) whenever n is any non-zero whole number

(and where we should not allow b to be zero either).

But even this does not tell us what a fraction is; it merely tells us

something about the way in which we represent fractions. What is a

fraction, then? According to the mathematicianâ€™s â€˜â€˜equivalence classâ€™â€™

notion, the fraction 38, for example, simply is the inWnite collection of all

pairs

(3, 8), ( 3, 8), (6, 16), ( 6, 16), (9, 24), ( 9, 24), (12, 32), . . . ,

xvii

Preface

where each pair can be obtained from each of the other pairs in the list by

repeated application of the above cancellation rule.* We also need deWnitions telling us how to add, subtract, and multiply such inWnite collections

of pairs of whole numbers, where the normal rules of algebra hold, and

how to identify the whole numbers themselves as particular types of

fraction.

This deWnition covers all that we mathematically need of fractions (such

as 12 being a number that, when added to itself, gives the number 1, etc.), and

the operation of cancelling is, as we have seen, built into the deWnition. Yet it

seems all very formal and we may indeed wonder whether it really captures

the intuitive notion of what a fraction is. Although this ubiquitous equivalence class procedure, of which the above illustration is just a particular

instance, is very powerful as a pure-mathematical tool for establishing

consistency and mathematical existence, it can provide us with very topheavy-looking entities. It hardly conveys to us the intuitive notion of what 38

is, for example! No wonder my motherâ€™s friend was confused.

In my descriptions of mathematical notions, I shall try to avoid, as far

as I can, the kind of mathematical pedantry that leads us to deWne a

fraction in terms of an â€˜inWnite class of pairsâ€™ even though it certainly

has its value in mathematical rigour and precision. In my descriptions here

I shall be more concerned with conveying the ideaâ€”and the beauty and

the magicâ€”inherent in many important mathematical notions. The idea of

a fraction such as 38 is simply that it is some kind of an entity which has the

property that, when added to itself 8 times in all, gives 3. The magic is that

the idea of a fraction actually works despite the fact that we do not really

directly experience things in the physical world that are exactly quantiWed

by fractionsâ€”pieces of pie leading only to approximations. (This is quite

unlike the case of natural numbers, such as 1, 2, 3, which do precisely

quantify numerous entities of our direct experience.) One way to see that

fractions do make consistent sense is, indeed, to use the â€˜deWnitionâ€™ in

terms of inWnite collections of pairs of integers (whole numbers), as

indicated above. But that does not mean that 38 actually is such a collection.

It is better to think of 38 as being an entity with some kind of (Platonic)

existence of its own, and that the inWnite collection of pairs is merely one

way of our coming to terms with the consistency of this type of entity.

With familiarity, we begin to believe that we can easily grasp a notion like 38

as something that has its own kind of existence, and the idea of an â€˜inWnite

collection of pairsâ€™ is merely a pedantic deviceâ€”a device that quickly

recedes from our imaginations once we have grasped it. Much of mathematics is like that.

* This is called an â€˜equivalence classâ€™ because it actually is a class of entities (the entities, in this

particular case, being pairs of whole numbers), each member of which is deemed to be equivalent,

in a speciWed sense, to each of the other members.

xviii

Preface

To mathematicians (at least to most of them, as far as I can make out),

mathematics is not just a cultural activity that we have ourselves created,

but it has a life of its own, and much of it Wnds an amazing harmony with

the physical universe. We cannot get any deep understanding of the laws

that govern the physical world without entering the world of mathematics.

In particular, the above notion of an equivalence class is relevant not only

to a great deal of important (but confusing) mathematics, but a great deal

of important (and confusing) physics as well, such as Einsteinâ€™s general

theory of relativity and the â€˜gauge theoryâ€™ principles that describe the

forces of Nature according to modern particle physics. In modern physics,

one cannot avoid facing up to the subtleties of much sophisticated mathematics. It is for this reason that I have spent the Wrst 16 chapters of this

work directly on the description of mathematical ideas.

What words of advice can I give to the reader for coping with this?

There are four diVerent levels at which this book can be read. Perhaps you

are a reader, at one end of the scale, who simply turns oV whenever a

mathematical formula presents itself (and some such readers may have

diYculty with coming to terms with fractions). If so, I believe that there is

still a good deal that you can gain from this book by simply skipping all

the formulae and just reading the words. I guess this would be much like

the way I sometimes used to browse through the chess magazines lying

scattered in our home when I was growing up. Chess was a big part of the

lives of my brothers and parents, but I took very little interest, except that

I enjoyed reading about the exploits of those exceptional and often strange

characters who devoted themselves to this game. I gained something from

reading about the brilliance of moves that they frequently made, even

though I did not understand them, and I made no attempt to follow

through the notations for the various positions. Yet I found this to be

an enjoyable and illuminating activity that could hold my attention.

Likewise, I hope that the mathematical accounts I give here may convey

something of interest even to some profoundly non-mathematical readers

if they, through bravery or curiosity, choose to join me in my journey of

investigation of the mathematical and physical ideas that appear to underlie our physical universe. Do not be afraid to skip equations (I do this

frequently myself) and, if you wish, whole chapters or parts of chapters,

when they begin to get a mite too turgid! There is a great variety in the

diYculty and technicality of the material, and something elsewhere may be

more to your liking. You may choose merely to dip in and browse. My

hope is that the extensive cross-referencing may suYciently illuminate

unfamiliar notions, so it should be possible to track down needed concepts

and notation by turning back to earlier unread sections for clariWcation.

At a second level, you may be a reader who is prepared to peruse

mathematical formulae, whenever such is presented, but you may not

xix

Preface

have the inclination (or the time) to verify for yourself the assertions that

I shall be making. The conWrmations of many of these assertions constitute the solutions of the exercises that I have scattered about the mathematical portions of the book. I have indicated three levels of difficulty by the

icons â€“

very straight forward

needs a bit of thought

not to be undertaken lightly.

It is perfectly reasonable to take these on trust, if you wish, and there is no

loss of continuity if you choose to take this position.

If, on the other hand, you are a reader who does wish to gain a facility

with these various (important) mathematical notions, but for whom the

ideas that I am describing are not all familiar, I hope that working through

these exercises will provide a signiWcant aid towards accumulating such

skills. It is always the case, with mathematics, that a little direct experience

of thinking over things on your own can provide a much deeper understanding than merely reading about them. (If you need the solutions, see

the website www.roadsolutions.ox.ac.uk.)

Finally, perhaps you are already an expert, in which case you should

have no diYculty with the mathematics (most of which will be very

familiar to you) and you may have no wish to waste time with the

exercises. Yet you may Wnd that there is something to be gained from

my own perspective on a number of topics, which are likely to be somewhat diVerent (sometimes very diVerent) from the usual ones. You may

have some curiosity as to my opinions relating to a number of modern

theories (e.g. supersymmetry, inXationary cosmology, the nature of the Big

Bang, black holes, string theory or M-theory, loop variables in quantum

gravity, twistor theory, and even the very foundations of quantum theory).

No doubt you will Wnd much to disagree with me on many of these topics.

But controversy is an important part of the development of science, so I

have no regrets about presenting views that may be taken to be partly

at odds with some of the mainstream activities of modern theoretical

physics.

It may be said that this book is really about the relation between

mathematics and physics, and how the interplay between the two strongly

inXuences those drives that underlie our searches for a better theory of the

universe. In many modern developments, an essential ingredient of these

drives comes from the judgement of mathematical beauty, depth, and

sophistication. It is clear that such mathematical inXuences can be vitally

important, as with some of the most impressively successful achievements

xx

Preface

of 20th-century physics: Diracâ€™s equation for the electron, the general

framework of quantum mechanics, and Einsteinâ€™s general relativity. But

in all these cases, physical considerationsâ€”ultimately observational

onesâ€”have provided the overriding criteria for acceptance. In many of

the modern ideas for fundamentally advancing our understanding of the

laws of the universe, adequate physical criteriaâ€”i.e. experimental data, or

even the possibility of experimental investigationâ€”are not available. Thus

we may question whether the accessible mathematical desiderata are suYcient to enable us to estimate the chances of success of these ideas. The

question is a delicate one, and I shall try to raise issues here that I do not

believe have been suYciently discussed elsewhere.

Although, in places, I shall present opinions that may be regarded as

contentious, I have taken pains to make it clear to the reader when I am

actually taking such liberties. Accordingly, this book may indeed be used

as a genuine guide to the central ideas (and wonders) of modern physics. It

is appropriate to use it in educational classes as an honest introduction to

modern physicsâ€”as that subject is understood, as we move forward into

the early years of the third millennium.

xxi

Acknowledgements

It is inevitable, for a book of this length, which has taken me about eight

years to complete, that there will be a great many to whom I owe my thanks.

It is almost as inevitable that there will be a number among them, whose

valuable contributions will go unattributed, owing to congenital disorganization and forgetfulness on my part. Let me Wrst express my special

thanksâ€”and also apologiesâ€”to such people: who have given me their

generous help but whose names do not now come to mind. But for various

speciWc pieces of information and assistance that I can more clearly

pinpoint, I thank Michael Atiyah, John Baez, Michael Berry, Dorje

Brody, Robert Bryant, Hong-Mo Chan, Joy Christian, Andrew Duggins,

Maciej Dunajski, Freeman Dyson, Artur Ekert, David Fowler, Margaret

Gleason, Jeremy Gray, Stuart HameroV, Keith Hannabuss, Lucien Hardy,

Jim Hartle, Tom Hawkins, Nigel Hitchin, Andrew Hodges, Dipankar

Home, Jim Howie, Chris Isham, Ted Jacobson, Bernard Kay, William

Marshall, Lionel Mason, Charles Misner, Tristan Needham, Stelios Negrepontis, Sarah Jones Nelson, Ezra (Ted) Newman, Charles Oakley, Daniel

Oi, Robert Osserman, Don Page, Oliver Penrose, Alan Rendall, Wolfgang

Rindler, Engelbert SchuÂ¨cking, Bernard Schutz, Joseph Silk, Christoph

Simon, George Sparling, John Stachel, Henry Stapp, Richard Thomas,

Gerard tâ€™Hooft, Paul Tod, James Vickers, Robert Wald, Rainer Weiss,

Ronny Wells, Gerald Westheimer, John Wheeler, Nick Woodhouse, and

Anton Zeilinger. Particular thanks go to Lee Smolin, Kelly Stelle, and Lane

Hughston for numerous and varied points of assistance. I am especially

indebted to Florence Tsou (Sheung Tsun) for immense help on matters of

particle physics, to Fay Dowker for her assistance and judgement concerning various matters, most notably the presentation of certain quantummechanical issues, to Subir Sarkar for valuable information concerning

cosmological data and the interpretation thereof, to Vahe Gurzadyan

likewise, and for some advance information about his cosmological

Wndings concerning the overall geometry of the universe, and particularly

to Abhay Ashtekar, for his comprehensive information about loopvariable theory and also various detailed matters concerning string theory.

xxiii

Acknowledgements

I thank the National Science Foundation for support under grants PHY

93-96246 and 00-90091, and the Leverhulme Foundation for the award of

a two-year Leverhulme Emeritus Fellowship, during 2000â€“2002. Part-time

appointments at Gresham College, London (1998â€“2001) and The Center

for Gravitational Physics and Geometry at Penn State University, Pennsylvania, USA have been immensely valuable to me in the writing of this

book, as has the secretarial assistance (most particularly Ruth Preston)

and oYce space at the Mathematical Institute, Oxford University.

Special assistance on the editorial side has also been invaluable, under

diYcult timetabling constraints, and with an author of erratic working

habits. Eddie Mizziâ€™s early editorial help was vital in initiating the process

of converting my chaotic writings into an actual book, and Richard

Lawrence, with his expert eYciency and his patient, sensitive persistence,

has been a crucial factor in bringing this project to completion. Having to

Wt in with such complicated reworking, John Holmes has done sterling

work in providing a Wne index. And I am particularly grateful to William

Shaw for coming to our assistance at a late stage to produce excellent

computer graphics (Figs. 1.2 and 2.19, and the implementation of the

transformation involved in Figs. 2.16 and 2.19), used here for the Mandelbrot set and the hyperbolic plane. But all the thanks that I can give to

Jacob Foster, for his Herculean achievement in sorting out and obtaining

references for me and for checking over the entire manuscript in a remarkably brief time and Wlling in innumerable holes, can in no way do justice to

the magnitude of his assistance. His personal imprint on a huge number of

the end-notes gives those a special quality. Of course, none of the people

I thank here are to blame for the errors and omissions that remain, the sole

responsibility for that lying with me.

Special gratitude is expressed to The M.C. Escher Company, Holland

for permission to reproduce Escher works in Figs. 2.11, 2.12, 2.16, and

2.22, and particularly to allow the modiWcations of Fig. 2.11 that are used

in Figs. 2.12 and 2.16, the latter being an explicit mathematical transformation. All the Escher works used in this book are copyright (2004) The

M.C. Escher Company. Thanks go also to the Institute of Theoretical

Physics, University of Heidelberg and to Charles H. Lineweaver for permission to reproduce the respective graphs in Figs. 27.19 and 28.19.

Finally, my unbounded gratitude goes to my beloved wife Vanessa, not

merely for supplying computer graphics for me on instant demand (Figs.

4.1, 4.2, 5.7, 6.2â€“6.8, 8.15, 9.1, 9.2, 9.8, 9.12, 21.3b, 21.10, 27.5, 27.14,

27.15, and the polyhedra in Fig. 1.1), but for her continued love and care,

and her deep understanding and sensitivity, despite the seemingly endless

years of having a husband who is mentally only half present. And Max,

also, who in his entire life has had the chance to know me only in such a

distracted state, gets my warmest gratitudeâ€”not just for slowing down the

xxiv

Acknowledgements

writing of this book (so that it could stretch its life, so as to contain at least

two important pieces of information that it would not have done otherwise)â€”but for the continual good cheer and optimism that he exudes,

which has helped to keep me going in good spirits. After all, it is through

the renewal of life, such as he himself represents, that the new sources of

ideas and insights needed for genuine future progress will come, in the

search for those deeper laws that actually govern the universe in which

we live.

xxv

Notation

(Not to be read until you are familiar with the concepts, but perhaps Wnd

the fonts confusing!)

I have tried to be reasonably consistent in the use of particular fonts in

this book, but as not all of this is standard, it may be helpful to the reader

to have the major usage that I have adopted made explicit.

Italic lightface (Greek or Latin) letters, such as in w2 , pn , log z,

cos y, eiy , or ex are used in the conventional way for mathematical variables which are numerical or scalar quantities; but established numerical

constants, such as e, i, or p or established functions such as sin, cos, or log

are denoted by upright letters. Standard physical constants such as c, G, h,

h, g, or k are italic, however.

A vector or tensor quantity, when being thought of in its (abstract)

entirety, is denoted by a boldface italic letter, such as R for the Riemann

curvature tensor, while its set of components might be written with italic

letters (both for the kernel symbol its indices) as Rabcd . In accordance with

the abstract-index notation, introduced here in Â§12.8, the quantity Rabcd

may alternatively stand for the entire tensor R, if this interpretation is

appropriate, and this should be made clear in the text. Abstract linear

transformations are kinds of tensors, and boldface italic letters such as T

are used for such entities also. The abstract-index form T a b is also used

here for an abstract linear transformation, where appropriate, the staggering of the indices making clear the precise connection with the ordering of

matrix multiplication. Thus, the (abstract-)index expression S a b T b c stands

for the product ST of linear transformations. As with general tensors, the

symbols S a b and T b c could alternatively (according to context or explicit

speciWcation in the text) stand for the corresoponding arrays of componentsâ€”these being matricesâ€”for which the corresponding bold upright

letters S and T can also be used. In that case, ST denotes the corresponding matrix product. This â€˜ambivalentâ€™ interpretation of symbols such as

Rabcd or S a b (either standing for the array of components or for the

abstract tensor itself) should not cause confusion, as the algebraic (or

diVerential) relations that these symbols are subject to are identical for

xxvi

Notation

both interpretations. A third notation for such quantitiesâ€”the diagrammatic notationâ€”is also sometimes used here, and is described in Figs.

12.17, 12.18, 14.6, 14.7, 14.21, 19.1 and elsewhere in the book.

There are places in this book where I need to distinguish the 4-dimensional spacetime entities of relativity theory from the corresponding ordinary 3-dimensional purely spatial entities. Thus, while a boldface italic

notation might be used, as above, such as p or x, for the 4-momentum or

4-position, respectively, the corresponding 3-dimensional purely spatial

entities would be denoted by the corresponding upright bold letters p or x.

By analogy with the notation T for a matrix, above, as opposed to T for an

abstract linear transformation, the quantities p and x would tend to be

thought of as â€˜standing forâ€™ the three spatial components, in each case,

whereas p and x might be viewed as having a more abstract componentfree interpretation (although I shall not be particularly strict about this).

The Euclidean â€˜lengthâ€™ of a 3-vector quantity a Â¼ (a1 ,a2 ,a3 ) may be written

a, where a2 Â¼ a21 Ã¾ a22 Ã¾ a23 , and the scalar product of a with b Â¼ (b1 ,b2 ,b3 ),

written a . b Â¼ a1 b1 Ã¾ a2 b2 Ã¾ a3 b3 . This â€˜dotâ€™ notation for scalar products

applies also in the general n-dimensional context, for the scalar (or inner)

product a . j of an abstract covector a with a vector j.

A notational complication arises with quantum mechanics, however,

since physical quantities, in that subject, tend to be represented as linear

operators. I do not adopt what is a quite standard procedure in this

context, of putting â€˜hatsâ€™ (circumXexes) on the letters representing the

quantum-operator versions of the familiar classical quantities, as I believe

that this leads to an unnecessary cluttering of symbols. (Instead, I shall

tend to adopt a philosophical standpoint that the classical and quantum

entities are really the â€˜sameâ€™â€”and so it is fair to use the same symbols for

eachâ€”except that in the classical case one is justiWed in ignoring quantities

of the order of h, so that the classical commutation properties ab Â¼ ba can

hold, whereas in quantum mechanics, ab might diVer from ba by something of order h.) For consistency with the above, such linear operators

would seem to have to be denoted by italic bold letters (like T), but that

would nullify the philosophy and the distinctions called for in the preceding paragraph. Accordingly, with regard to speciWc quantities, such as the

momentum p or p, or the position x or x, I shall tend to use the same

notation as in the classical case, in line with what has been said earlier in

this paragraph. But for less speciWc quantum operators, bold italic letters

such as Q will tend to be used.

The shell letters N, Z, R, C, and Fq , respectively, for the system of

natural numbers (i.e. non-negative integers), integers, real numbers, complex numbers, and the Wnite Weld with q elements (q being some power of a

prime number, see Â§16.1), are now standard in mathematics, as are the

corresponding Nn , Zn , Rn , Cn , Fnq , for the systems of ordered n-tuples

xxvii

Notation

of such numbers. These are canonical mathematical entities in standard

use. In this book (as is not all that uncommon), this notation is extended

to some other standard mathematical structures such as Euclidean 3-space

E3 or, more generally, Euclidean n-space En . In frequent use in this book

is the standard Xat 4-dimensional Minkowski spacetime, which is itself a

kind of â€˜pseudo-â€™ Euclidean space, so I use the shell letter M for this space

(with Mn to denote the n-dimensional versionâ€”a â€˜Lorentzianâ€™ spacetime

with 1 time and (n 1) space dimensions). Sometimes I use C as an

adjective, to denote â€˜complexiWedâ€™, so that we might consider the complex

Euclidean 4-space, for example, denoted by CEn . The shell letter P can

also be used as an adjective, to denote â€˜projectiveâ€™ (see Â§15.6), or as a noun,

with Pn denoting projective n-space (or I use RPn or CPn if it is to be

made clear that we are concerned with real or complex projective n-space,

respectively). In twistor theory (Chapter 33), there is the complex 4-space

T, which is related to M (or its complexiWcation CM) in a canonical

way, and there is also the projective version PT. In this theory, there is

also a space N of null twistors (the double duty that this letter serves

causing no conXict here), and its projective version PN.

The adjectival role of the shell letter C should not be confused with that

of the lightface sans serif C, which here stands for â€˜complex conjugate ofâ€™

(as used in Â§13.1,2). This is basically similar to another use of C in particle

physics, namely charge conjugation, which is the operation which interchanges each particle with its antiparticle (see Chapters 24, 25). This

operation is usually considered in conjunction with two other basic particle-physics operations, namely P for parity which refers to the operation

of reXection in a mirror, and T, which refers to time-reveral. Sans serif

letters which are bold serve a diVerent purpose here, labelling vector

spaces, the letters V, W, and H, being most frequently used for this

purpose. The use of H, is speciWc to the Hilbert spaces of quantum

mechanics, and Hn would stand for a Hilbert space of n complex dimensions. Vector spaces are, in a clear sense, Xat. Spaces which are (or could

be) curved are denoted by script letters, such as M, S, or T , where there is

a special use for the particular script font I to denote null inWnity. In

addition, I follow a fairly common convention to use script letters for

Lagrangians (L) and Hamiltonians (H), in view of their very special status

in physical theory.

xxviii

Prologue

Am-tep was the Kingâ€™s chief craftsman, an artist of consummate skills. It

was night, and he lay sleeping on his workshop couch, tired after a

handsomely productive eveningâ€™s work. But his sleep was restlessâ€”perhaps from an intangible tension that had seemed to be in the air. Indeed,

he was not certain that he was asleep at all when it happened. Daytime had

comeâ€”quite suddenlyâ€”when his bones told him that surely it must still be

night.

He stood up abruptly. Something was odd. The dawnâ€™s light could not

be in the north; yet the red light shone alarmingly through his broad

window that looked out northwards over the sea. He moved to the

window and stared out, incredulous in amazement. The Sun had never

before risen in the north! In his dazed state, it took him a few moments to

realize that this could not possibly be the Sun. It was a distant shaft of a

deep Wery red light that beamed vertically upwards from the water into the

heavens.

As he stood there, a dark cloud became apparent at the head of the

beam, giving the whole structure the appearance of a distant giant parasol,

glowing evilly, with a smoky Xaming staV. The parasolâ€™s hood began to

spread and darkenâ€”a daemon from the underworld. The night had been

clear, but now the stars disappeared one by one, swallowed up behind this

advancing monstrous creature from Hell.

Though terror must have been his natural reaction, he did not move,

transWxed for several minutes by the sceneâ€™s perfect symmetry and awesome beauty. But then the terrible cloud began to bend slightly to the east,

caught up by the prevailing winds. Perhaps he gained some comfort from

this and the spell was momentarily broken. But apprehension at once

returned to him as he seemed to sense a strange disturbance in the ground

beneath, accompanied by ominous-sounding rumblings of a nature quite

unfamiliar to him. He began to wonder what it was that could have

caused this fury. Never before had he witnessed a Godâ€™s anger of such

magnitude.

1

Prologue

His Wrst reaction was to blame himself for the design on the sacriWcial

cup that he had just completedâ€”he had worried about it at the time. Had

his depiction of the Bull-God not been suYciently fearsome? Had that god

been oVended? But the absurdity of this thought soon struck him. The fury

he had just witnessed could not have been the result of such a trivial

action, and was surely not aimed at him speciWcally. But he knew that

there would be trouble at the Great Palace. The Priest-King would waste

no time in attempting to appease this Daemon-God. There would be

sacriWces. The traditional oVerings of fruits or even animals would not

suYce to pacify an anger of this magnitude. The sacriWces would have to

be human.

Quite suddenly, and to his utter surprise, he was blown backwards

across the room by an impulsive blast of air followed by a violent wind.

The noise was so extreme that he was momentarily deafened. Many of his

beautifully adorned pots were whisked from their shelves and smashed

to pieces against the wall behind. As he lay on the Xoor in a far corner of

the room where he had been swept away by the blast, he began to recover

his senses, and saw that the room was in turmoil. He was horriWed to see

one of his favourite great urns shattered to small pieces, and the wonderfully detailed designs, which he had so carefully crafted, reduced to

nothing.

Am-tep arose unsteadily from the Xoor and after a while again approached the window, this time with considerable trepidation, to re-examine that terrible scene across the sea. Now he thought he saw a

disturbance, illuminated by that far-oV furnace, coming towards him.

This appeared to be a vast trough in the water, moving rapidly towards

the shore, followed by a cliVlike wall of wave. He again became transWxed,

watching the approaching wave begin to acquire gigantic proportions.

Eventually the disturbance reached the shore and the sea immediately

before him drained away, leaving many ships stranded on the newly

formed beach. Then the cliV-wave entered the vacated region and struck

with a terrible violence. Without exception the ships were shattered, and

many nearby houses instantly destroyed. Though the water rose to great

heights in the air before him, his own house was spared, for it sat on high

ground a good way from the sea.

The Great Palace too was spared. But Am-tep feared that worse might

come, and he was rightâ€”though he knew not how right he was. He did

know, however, that no ordinary human sacriWce of a slave could now be

suYcient. Something more would be needed to pacify the tempestuous

anger of this terrible God. His thoughts turned to his sons and daughters,

and to his newly born grandson. Even they might not be safe.

Am-tep had been right to fear new human sacriWces. A young girl and a

youth of good birth had been soon apprehended and taken to a nearby

2

Prologue

temple, high on the slopes of a mountain. The ensuing ritual was well

under way when yet another catastrophe struck. The ground shook with

devastating violence, whence the temple roof fell in, instantly killing all the

priests and their intended sacriWcial victims. As it happened, they would lie

there in mid-ritualâ€”entombed for over three-and-a-half millennia!

The devastation was frightful, but not Wnal. Many on the island where

Am-tep and his people lived survived the terrible earthquake, though the

Great Palace was itself almost totally destroyed. Much would be rebuilt

over the years. Even the Palace would recover much of its original splendour, constructed on the ruins of the old. Yet Am-tep had vowed to leave

the island. His world had now changed irreparably.

In the world he knew, there had been a thousand years of peace,

prosperity, and culture where the Earth-Goddess had reigned. Wonderful

art had been allowed to Xourish. There was much trade with neighbouring

lands. The magniWcent Great Palace was a huge luxurious labyrinth, a

virtual city in itself, adorned by superb frescoes of animals and Xowers.

There was running water, excellent drainage, and Xushed sewers. War was

almost unknown and defences unnecessary. Now, Am-tep perceived the

Earth-Goddess overthrown by a Being with entirely diVerent values.

It was some years before Am-tep actually left the island, accompanied

by his surviving family, on a ship rebuilt by his youngest son, who was a

skilled carpenter and seaman. Am-tepâ€™s grandson had developed into an

alert child, with an interest in everything in the world around. The voyage

took some days, but the weather had been supremely calm. One clear

night, Am-tep was explaining to his grandson about the patterns in the

stars, when an odd thought overtook him: The patterns of stars had been

disturbed not one iota from what they were before the Catastrophe of the

emergence of the terrible daemon.

Am-tep knew these patterns well, for he had a keen artistâ€™s eye. Surely,

he thought, those tiny candles of light in the sky should have been blown

at least a little from their positions by the violence of that night, just as his

pots had been smashed and his great urn shattered. The Moon also had

kept her face, just as before, and her route across the star-Wlled heavens

had changed not one whit, as far as Am-tep could tell. For many moons

after the Catastrophe, the skies had appeared diVerent. There had been

darkness and strange clouds, and the Moon and Sun had sometimes worn

unusual colours. But this had now passed, and their motions seemed

utterly undisturbed. The tiny stars, likewise, had been quite unmoved.

If the heavens had shown such little concern for the Catastrophe, having

a stature far greater even than that terrible Daemon, Am-tep reasoned,

why should the forces controlling the Daemon itself show concern for

what the little people on the island had been doing, with their foolish

rituals and human sacriWce? He felt embarrassed by his own foolish

3

Prologue

thoughts at the time, that the daemon might be concerned by the mere

patterns on his pots.

Yet Am-tep was still troubled by the question â€˜why?â€™ What deep forces

control the behaviour of the world, and why do they sometimes burst forth

in violent and seemingly incomprehensible ways? He shared his questions

with his grandson, but there were no answers.

...

A century passed by, and then a millennium, and still there were no

answers.

...

Amphos the craftsman had lived all his life in the same small town as his

father and his father before him, and his fatherâ€™s father before that. He

made his living constructing beautifully decorated gold bracelets, earrings,

ceremonial cups, and other Wne products of his artistic skills. Such work

had been the family trade for some forty generationsâ€”a line unbroken

since Am-tep had settled there eleven hundred years before.

But it was not just artistic skills that had been passed down from

generation to generation. Am-tepâ€™s questions troubled Amphos just as

they had troubled Am-tep earlier. The great story of the Catastrophe

that destroyed an ancient peaceful civilization had been handed down

from father to son. Am-tepâ€™s perception of the Catastrophe had also

survived with his descendants. Amphos, too, understood that the heavens

had a magnitude and stature so great as to be quite unconcerned by that

terrible event. Nevertheless, the event had had a catastrophic eVect on the

little people with their cities and their human sacriWces and insigniWcant

religious rituals. Thus, by comparison, the event itself must have been the

result of enormous forces quite unconcerned by those trivial actions of

human beings. Yet the nature of those forces was as unknown in

Amphosâ€™s day as it was to Am-tep.

Amphos had studied the structure of plants, insects and other small

animals, and crystalline rocks. His keen eye for observation had served

him well in his decorative designs. He took an interest in agriculture and

was fascinated by the growth of wheat and other plants from grain. But

none of this told him â€˜why?â€™, and he felt unsatisWed. He believed that there

was indeed reason underlying Natureâ€™s patterns, but he was in no way

equipped to unravel those reasons.

One clear night, Amphos looked up at the heavens, and tried to make

out from the patterns of stars the shapes of those heroes and heroines who

formed constellations in the sky. To his humble artistâ€™s eye, those shapes

made poor resemblances. He could himself have arranged the stars far

more convincingly. He puzzled over why the gods had not organized the

4

Prologue

stars in a more appropriate way? As they were, the arrangements seemed

more like scattered grains randomly sowed by a farmer, rather than the

deliberate design of a god. Then an odd thought overtook him: Do not seek

for reasons in the speciWc patterns of stars, or of other scattered arrangements of objects; look, instead, for a deeper universal order in the way that

things behave.

Amphos reasoned that we Wnd order, after all, not in the patterns that

scattered seeds form when they fall to the ground, but in the miraculous

way that each of those seeds develops into a living plant having a superb

structure, similar in great detail to one another. We would not try to seek

the meaning in the precise arrangement of seeds sprinkled on the soil; yet,

there must be meaning in the hidden mystery of the inner forces controlling the growth of each seed individually, so that each one follows essentially the same wonderful course. Natureâ€™s laws must indeed have a

superbly organized precision for this to be possible.

Amphos became convinced that without precision in the underlying

laws, there could be no order in the world, whereas much order is indeed

perceived in the way that things behave. Moreover, there must be precision

in our ways of thinking about these matters if we are not to be led seriously

astray.

It so happened that word had reached Amphos of a sage who lived in

another part of the land, and whose beliefs appeared to be in sympathy

with those of Amphos. According to this sage, one could not rely on the

teachings and traditions of the past. To be certain of oneâ€™s beliefs, it was

necessary to form precise conclusions by the use of unchallengeable

reason. The nature of this precision had to be mathematicalâ€”ultimately

dependent on the notion of number and its application to geometric forms.

Accordingly, it must be number and geometry, not myth and superstition,

that governed the behaviour of the world.

As Am-tep had done a century and a millennium before, Amphos took

to the sea. He found his way to the city of Croton, where the sage and his

brotherhood of 571 wise men and 28 wise women were in search of truth.

After some time, Amphos was accepted into the brotherhood. The name

of the sage was Pythagoras.

5

1

The roots of science

1.1 The quest for the forces that shape the world

What laws govern our universe? How shall we know them? How

may this knowledge help us to comprehend the world and hence guide

its actions to our advantage?

Since the dawn of humanity, people have been deeply concerned by

questions like these. At Wrst, they had tried to make sense of those

inXuences that do control the world by referring to the kind of understanding that was available from their own lives. They had imagined that

whatever or whoever it was that controlled their surroundings would do

so as they would themselves strive to control things: originally they had

considered their destiny to be under the inXuence of beings acting very

much in accordance with their own various familiar human drives. Such

driving forces might be pride, love, ambition, anger, fear, revenge, passion,

retribution, loyalty, or artistry. Accordingly, the course of natural

eventsâ€”such as sunshine, rain, storms, famine, illness, or pestilenceâ€”

was to be understood in terms of the whims of gods or goddesses motivated by such human urges. And the only action perceived as inXuencing

these events would be appeasement of the god-Wgures.

But gradually patterns of a diVerent kind began to establish their reliability. The precision of the Sunâ€™s motion through the sky and its clear

relation to the alternation of day with night provided the most obvious

example; but also the Sunâ€™s positioning in relation to the heavenly orb of

stars was seen to be closely associated with the change and relentless

regularity of the seasons, and with the attendant clear-cut inXuence on

the weather, and consequently on vegetation and animal behaviour. The

motion of the Moon, also, appeared to be tightly controlled, and its phases

determined by its geometrical relation to the Sun. At those locations on

Earth where open oceans meet land, the tides were noticed to have a

regularity closely governed by the position (and phase) of the Moon.

Eventually, even the much more complicated apparent motions of the

planets began to yield up their secrets, revealing an immense underlying

precision and regularity. If the heavens were indeed controlled by the

7

Â§1.1

CHAPTER 1

whims of gods, then these gods themselves seemed under the spell of exact

mathematical laws.

Likewise, the laws controlling earthly phenomenaâ€”such as the daily

and yearly changes in temperature, the ebb and Xow of the oceans, and the

growth of plantsâ€”being seen to be inXuenced by the heavens in this

respect at least, shared the mathematical regularity that appeared to

guide the gods. But this kind of relationship between heavenly bodies

and earthly behaviour would sometimes be exaggerated or misunderstood

and would assume an inappropriate importance, leading to the occult and

mystical connotations of astrology. It took many centuries before the

rigour of scientiWc understanding enabled the true inXuences of the

heavens to be disentangled from purely suppositional and mystical ones.

Yet it had been clear from the earliest times that such inXuences did indeed

exist and that, accordingly, the mathematical laws of the heavens must

have relevance also here on Earth.

Seemingly independently of this, there were perceived to be other regularities in the behaviour of earthly objects. One of these was the tendency

for all things in one vicinity to move in the same downward direction,

according to the inXuence that we now call gravity. Matter was observed

to transform, sometimes, from one form into another, such as with the

melting of ice or the dissolving of salt, but the total quantity of that matter

appeared never to change, which reXects the law that we now refer to as

conservation of mass. In addition, it was noticed that there are many

material bodies with the important property that they retain their shapes,

whence the idea of rigid spatial motion arose; and it became possible to

understand spatial relationships in terms of a precise, well-deWned geometryâ€”the 3-dimensional geometry that we now call Euclidean. Moreover,

the notion of a â€˜straight lineâ€™ in this geometry turned out to be the same as

that provided by rays of light (or lines of sight). There was a remarkable

precision and beauty to these ideas, which held a considerable fascination

for the ancients, just as it does for us today.

Yet, with regard to our everyday lives, the implications of this mathematical precision for the actions of the world often appeared unexciting

and limited, despite the fact that the mathematics itself seemed to represent a deep truth. Accordingly, many people in ancient times would allow

their imaginations to be carried away by their fascination with the subject

and to take them far beyond the scope of what was appropriate. In

astrology, for example, geometrical Wgures also often engendered mystical

and occult connotations, such as with the supposed magical powers of

pentagrams and heptagrams. And there was an entirely suppositional

attempted association between Platonic solids and the basic elementary

states of matter (see Fig. 1.1). It would not be for many centuries that the

deeper understanding that we presently have, concerning the actual

8

The roots of science

Â§1.2

Fig. 1.1 A fanciful association, made by the ancient Greeks, between the Wve

Platonic solids and the four â€˜elementsâ€™ (Wre, air, water, and earth), together with

the heavenly Wrmament represented by the dodecahedron.

relationships between mass, gravity, geometry, planetary motion, and the

behaviour of light, could come about.

1.2 Mathematical truth

The Wrst steps towards an understanding of the real inXuences controlling Nature required a disentangling of the true from the purely suppositional. But the ancients needed to achieve something else Wrst, before

they would be in any position to do this reliably for their understanding of

Nature. What they had to do Wrst was to discover how to disentangle the

true from the suppositional in mathematics. A procedure was required for

telling whether a given mathematical assertion is or is not to be trusted as

true. Until that preliminary issue could be settled in a reasonable way, there

would be little hope of seriously addressing those more diYcult problems

concerning forces that control the behaviour of the world and whatever

their relations might be to mathematical truth. This realization that the key

to the understanding of Nature lay within an unassailable mathematics was

perhaps the Wrst major breakthrough in science.

Although mathematical truths of various kinds had been surmised

since ancient Egyptian and Babylonian times, it was not until the

great Greek philosophers Thales of Miletus (c.625â€“547 bc) and

9

Â§1.2

CHAPTER 1

Pythagoras1* of Samos (c.572â€“497 bc) began to introduce the notion of

mathematical proof that the Wrst Wrm foundation stone of mathematical

understandingâ€”and therefore of science itselfâ€”was laid. Thales may have

been the Wrst to introduce this notion of proof, but it seems to have been the

Pythagoreans who Wrst made important use of it to establish things that

were not otherwise obvious. Pythagoras also appeared to have a strong

vision of the importance of number, and of arithmetical concepts, in

governing the actions of the physical world. It is said that a big factor in

this realization was his noticing that the most beautiful harmonies produced

by lyres or Xutes corresponded to the simplest fractional ratios between

the lengths of vibrating strings or pipes. He is said to have introduced the

â€˜Pythagorean scaleâ€™, the numerical ratios of what we now know to be

frequencies determining the principal intervals on which Western music is

essentially based.2 The famous Pythagorean theorem, asserting that the

square on the hypotenuse of a right-angled triangle is equal to the sum of

the squares on the other two sides, perhaps more than anything else, showed

that indeed there is a precise relationship between the arithmetic of numbers

and the geometry of physical space (see Chapter 2).

He had a considerable band of followersâ€”the Pythagoreansâ€”situated

in the city of Croton, in what is now southern Italy, but their inXuence on

the outside world was hindered by the fact that the members of the

Pythagorean brotherhood were all sworn to secrecy. Accordingly, almost

all of their detailed conclusions have been lost. Nonetheless, some of these

conclusions were leaked out, with unfortunate consequences for the

â€˜molesâ€™â€”on at least one occasion, death by drowning!

In the long run, the inXuence of the Pythagoreans on the progress of

human thought has been enormous. For the Wrst time, with mathematical

proof, it was possible to make signiWcant assertions of an unassailable

nature, so that they would hold just as true even today as at the time that

they were made, no matter how our knowledge of the world has progressed since then. The truly timeless nature of mathematics was beginning

to be revealed.

But what is a mathematical proof? A proof, in mathematics, is an

impeccable argument, using only the methods of pure logical reasoning,

which enables one to infer the validity of a given mathematical assertion

from the pre-established validity of other mathematical assertions, or from

some particular primitive assertionsâ€”the axiomsâ€”whose validity is taken

to be self-evident. Once such a mathematical assertion has been established in this way, it is referred to as a theorem.

Many of the theorems that the Pythagoreans were concerned with were

geometrical in nature; others were assertions simply about numbers. Those

*Notes, indicated in the text by superscript numbers, are gathered at the ends of the chapter

(in this case on p. 23).

10

The roots of science

Â§1.2

that were concerned merely with numbers have a perfectly unambiguous

validity today, just as they did in the time of Pythagoras. What about the

geometrical theorems that the Pythagoreans had obtained using their

procedures of mathematical proof? They too have a clear validity today,

but now there is a complicating issue. It is an issue whose nature is more

obvious to us from our modern vantage point than it was at that time of

Pythagoras. The ancients knew of only one kind of geometry, namely that

which we now refer to as Euclidean geometry, but now we know of many

other types. Thus, in considering the geometrical theorems of ancient

Greek times, it becomes important to specify that the notion of geometry

being referred to is indeed Euclidâ€™s geometry. (I shall be more explicit

about these issues in Â§2.4, where an important example of non-Euclidean

geometry will be given.)

Euclidean geometry is a speciWc mathematical structure, with its own

speciWc axioms (including some less assured assertions referred to as postulates), which provided an excellent approximation to a particular aspect of

the physical world. That was the aspect of reality, well familiar to the ancient

Greeks, which referred to the laws governing the geometry of rigid objects

and their relations to other rigid objects, as they are moved around in 3dimensional space. Certain of these properties were so familiar and selfconsistent that they tended to become regarded as â€˜self-evidentâ€™ mathematical truths and were taken as axioms (or postulates). As we shall be seeing in

Chapters 17â€“19 and Â§Â§27.8,11, Einsteinâ€™s general relativityâ€”and even the

Minkowskian spacetime of special relativityâ€”provides geometries for the

physical universe that are diVerent from, and yet more accurate than, the

geometry of Euclid, despite the fact that the Euclidean geometry of the

ancients was already extraordinarily accurate. Thus, we must be careful,

when considering geometrical assertions, whether to trust the â€˜axiomsâ€™ as

being, in any sense, actually true.

But what does â€˜trueâ€™ mean, in this context? The diYculty was well

appreciated by the great ancient Greek philosopher Plato, who lived in

Athens from c.429 to 347 bc, about a century after Pythagoras. Plato

made it clear that the mathematical propositionsâ€”the things that could be

regarded as unassailably trueâ€”referred not to actual physical objects (like

the approximate squares, triangles, circles, spheres, and cubes that might

be constructed from marks in the sand, or from wood or stone) but to

certain idealized entities. He envisaged that these ideal entities inhabited a

diVerent world, distinct from the physical world. Today, we might refer to

this world as the Platonic world of mathematical forms. Physical structures,

such as squares, circles, or triangles cut from papyrus, or marked on a Xat

surface, or perhaps cubes, tetrahedra, or spheres carved from marble,

might conform to these ideals very closely, but only approximately. The

actual mathematical squares, cubes, circles, spheres, triangles, etc., would

11

Â§1.3

CHAPTER 1

not be part of the physical world, but would be inhabitants of Platoâ€™s

idealized mathematical world of forms.

1.3 Is Platoâ€™s mathematical world â€˜realâ€™?

This was an extraordinary idea for its time, and it has turned out to be a

very powerful one. But does the Platonic mathematical world actually

exist, in any meaningful sense? Many people, including philosophers,

might regard such a â€˜worldâ€™ as a complete Wctionâ€”a product merely of

our unrestrained imaginations. Yet the Platonic viewpoint is indeed an

immensely valuable one. It tells us to be careful to distinguish the precise

mathematical entities from the approximations that we see around us in

the world of physical things. Moreover, it provides us with the blueprint

according to which modern science has proceeded ever since. Scientists will

put forward models of the worldâ€”or, rather, of certain aspects of the

worldâ€”and these models may be tested against previous observation and

against the results of carefully designed experiment. The models are

deemed to be appropriate if they survive such rigorous examination and

if, in addition, they are internally consistent structures. The important

point about these models, for our present discussion, is that they are

basically purely abstract mathematical models. The very question of the

internal consistency of a scientiWc model, in particular, is one that requires

that the model be precisely speciWed. The required precision demands that

the model be a mathematical one, for otherwise one cannot be sure that

these questions have well-deWned answers.

If the model itself is to be assigned any kind of â€˜existenceâ€™, then this

existence is located within the Platonic world of mathematical forms. Of

course, one might take a contrary viewpoint: namely that the model is

itself to have existence only within our various minds, rather than to take

Platoâ€™s world to be in any sense absolute and â€˜realâ€™. Yet, there is something

important to be gained in regarding mathematical structures as having a

reality of their own. For our individual minds are notoriously imprecise,

unreliable, and inconsistent in their judgements. The precision, reliability,

and consistency that are required by our scientiWc theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we Wnd a far greater robustness than can be located in any

particular mind. Does this not point to something outside ourselves,

with a reality that lies beyond what each individual can achieve?

Nevertheless, one might still take the alternative view that the mathematical world has no independent existence, and consists merely of

certain ideas which have been distilled from our various minds and

which have been found to be totally trustworthy and are agreed by all.

12

The roots of science

Â§1.3

Yet even this viewpoint seems to leave us far short of what is required. Do

we mean â€˜agreed by allâ€™, for example, or â€˜agreed by those who are in their

right mindsâ€™, or â€˜agreed by all those who have a Ph.D. in mathematicsâ€™

(not much use in Platoâ€™s day) and who have a right to venture an â€˜authoritativeâ€™ opinion? There seems to be a danger of circularity here; for to judge

whether or not someone is â€˜in his or her right mindâ€™ requires some external

standard. So also does the meaning of â€˜authoritativeâ€™, unless some standard of an unscientiWc nature such as â€˜majority opinionâ€™ were to be adopted

(and it should be made clear that majority opinion, no matter how

important it may be for democratic government, should in no way be

used as the criterion for scientiWc acceptability). Mathematics itself indeed

seems to have a robustness that goes far beyond what any individual

mathematician is capable of perceiving. Those who work in this subject,

whether they are actively engaged in mathematical research or just using

results that have been obtained by others, usually feel that they are merely

explorers in a world that lies far beyond themselvesâ€”a world which

possesses an objectivity that transcends mere opinion, be that opinion

their own or the surmise of others, no matter how expert those others

might be.

It may be helpful if I put the case for the actual existence of the Platonic

world in a diVerent form. What I mean by this â€˜existenceâ€™ is really just the

objectivity of mathematical truth. Platonic existence, as I see it, refers to

the existence of an objective external standard that is not dependent upon

our individual opinions nor upon our particular culture. Such â€˜existenceâ€™

could also refer to things other than mathematics, such as to morality or

aesthetics (cf. Â§1.5), but I am here concerned just with mathematical

objectivity, which seems to be a much clearer issue.

Let me illustrate this issue by considering one famous example of a

mathematical truth, and relate it to the question of â€˜objectivityâ€™. In 1637,

Pierre de Fermat made his famous assertion now known as â€˜Fermatâ€™s Last

Theoremâ€™ (that no positive nth power3 of an integer, i.e. of a whole

number, can be the sum of two other positive nth powers if n is an integer

greater than 2), which he wrote down in the margin of his copy of the

Arithmetica, a book written by the 3rd-century Greek mathematician

Diophantos. In this margin, Fermat also noted: â€˜I have discovered a

truly marvellous proof of this, which this margin is too narrow to contain.â€™

Fermatâ€™s mathematical assertion remained unconWrmed for over 350

years, despite concerted eVorts by numerous outstanding mathematicians.

A proof was Wnally published in 1995 by Andrew Wiles (depending on the

earlier work of various other mathematicians), and this proof has now

been accepted as a valid argument by the mathematical community.

Now, do we take the view that Fermatâ€™s assertion was always true, long

before Fermat actually made it, or is its validity a purely cultural matter,

13

Â§1.3

CHAPTER 1

dependent upon whatever might be the subjective standards of the community of human mathematicians? Let us try to suppose that the validity

of the Fermat assertion is in fact a subjective matter. Then it would not be

an absurdity for some other mathematician X to have come up with an

actual and speciWc counter-example to the Fermat assertion, so long as X

had done this before the date of 1995.4 In such a circumstance, the

mathematical community would have to accept the correctness of Xâ€™s

counter-example. From then on, any eVort on the part of Wiles to prove

the Fermat assertion would have to be fruitless, for the reason that X had

got his argument in Wrst and, as a result, the Fermat assertion would now

be false! Moreover, we could ask the further question as to whether,

consequent upon the correctness of Xâ€™s forthcoming counter-example,

Fermat himself would necessarily have been mistaken in believing in the

soundness of his â€˜truly marvellous proofâ€™, at the time that he wrote his

marginal note. On the subjective view of mathematical truth, it could

possibly have been the case that Fermat had a valid proof (which would

have been accepted as such by his peers at the time, had he revealed it) and

that it was Fermatâ€™s secretiveness that allowed the possibility of X later

obtaining a counter-example! I think that virtually all mathematicians,

irrespective of their professed attitudes to â€˜Platonismâ€™, would regard such

possibilities as patently absurd.

Of course, it might still be the case that Wilesâ€™s argument in fact

contains an error and that the Fermat assertion is indeed false. Or there

could be a fundamental error in Wilesâ€™s argument but the Fermat assertion

is true nevertheless. Or it might be that Wilesâ€™s argument is correct in its

essentials while containing â€˜non-rigorous stepsâ€™ that would not be up to the

standard of some future rules of mathematical acceptability. But these

issues do not address the point that I am getting at here. The issue is the

objectivity of the Fermat assertion itself, not whether anyoneâ€™s particular

demonstration of it (or of its negation) might happen to be convincing to

the mathematical community of any particular time.

It should perhaps be mentioned that, from the point of view of mathematical logic, the Fermat assertion is actually a mathematical statement

of a particularly simple kind,5 whose objectivity is especially apparent.

Only a tiny minority6 of mathematicians would regard the truth of such

assertions as being in any way â€˜subjectiveâ€™â€”although there might be some

subjectivity about the types of argument that would be regarded as being

convincing. However, there are other kinds of mathematical assertion

whose truth could plausibly be regarded as being a â€˜matter of opinionâ€™.

Perhaps the best known of such assertions is the axiom of choice. It is not

important for us, now, to know what the axiom of choice is. (I shall

describe it in Â§16.3.) It is cited here only as an example. Most mathematicians would probably regard the axiom of choice as â€˜obviously trueâ€™, while

14

The roots of science

Â§1.3

others may regard it as a somewhat questionable assertion which might

even be false (and I am myself inclined, to some extent, towards this

second viewpoint). Still others would take it as an assertion whose

â€˜truthâ€™ is a mere matter of opinion or, rather, as something which can be

taken one way or the other, depending upon which system of axioms and

rules of procedure (a â€˜formal systemâ€™; see Â§16.6) one chooses to adhere to.

Mathematicians who support this Wnal viewpoint (but who accept the

objectivity of the truth of particularly clear-cut mathematical statements,

like the Fermat assertion discussed above) would be relatively weak Platonists. Those who adhere to objectivity with regard to the truth of the

axiom of choice would be stronger Platonists.

I shall come back to the axiom of choice in Â§16.3, since it has some

relevance to the mathematics underlying the behaviour of the physical

world, despite the fact that it is not addressed much in physical theory. For

the moment, it will be appropriate not to worry overly about this issue. If

the axiom of choice can be settled one way or the other by some appropriate form of unassailable mathematical reasoning,7 then its truth is indeed

an entirely objective matter, and either it belongs to the Platonic world or

its negation does, in the sense that I am interpreting this term â€˜Platonic

worldâ€™. If the axiom of choice is, on the other hand, a mere matter of

opinion or of arbitrary decision, then the Platonic world of absolute

mathematical forms contains neither the axiom of choice nor its negation

(although it could contain assertions of the form â€˜such-and-such follows

from the axiom of choiceâ€™ or â€˜the axiom of choice is a theorem according

to the rules of such-and-such mathematical systemâ€™).

The mathematical assertions that can belong to Platoâ€™s world are precisely those that are objectively true. Indeed, I would regard mathematical

objectivity as really what mathematical Platonism is all about. To say that

some mathematical assertion has a Platonic existence is merely to say that

it is true in an objective sense. A similar comment applies to mathematical

notionsâ€”such as the concept of the number 7, for example, or the rule of

multiplication of integers, or the idea that some set contains inWnitely

many elementsâ€”all of which have a Platonic existence because they are

objective notions. To my way of thinking, Platonic existence is simply a

matter of objectivity and, accordingly, should certainly not be viewed as

something â€˜mysticalâ€™ or â€˜unscientiWcâ€™, despite the fact that some people

regard it that way.

As with the axiom of choice, however, questions as to whether some

particular proposal for a mathematical entity is or is not to be regarded as

having objective existence can be delicate and sometimes technical. Despite this, we certainly need not be mathematicians to appreciate the

general robustness of many mathematical concepts. In Fig. 1.2, I have

depicted various small portions of that famous mathematical entity known

15

Â§1.3

CHAPTER 1

b

c

d

(a)

(b)

(c)

(d)

Fig. 1.2 (a) The Mandelbrot set. (b), (c), and (d) Some details, illustrating blowups of those regions correspondingly marked in Fig. 1.2a, magniWed by respective

linear factors 11.6, 168.9, and 1042.

as the Mandelbrot set. The set has an extraordinarily elaborate structure,

but it is not of any human design. Remarkably, this structure is deWned by

a mathematical rule of particular simplicity. We shall come to this explicitly in Â§4.5, but it would distract us from our present purposes if I were to

try to provide this rule in detail now.

The point that I wish to make is that no one, not even Benoit Mandelbrot himself when he Wrst caught sight of the incredible complications in

the Wne details of the set, had any real preconception of the setâ€™s extraordinary richness. The Mandelbrot set was certainly no invention of any

human mind. The set is just objectively there in the mathematics itself. If it

has meaning to assign an actual existence to the Mandelbrot set, then that

existence is not within our minds, for no one can fully comprehend the setâ€™s

16

The roots of science

Â§1.4

endless variety and unlimited complication. Nor can its existence lie within

the multitude of computer printouts that begin to capture some of its

incredible sophistication and detail, for at best those printouts capture

but a shadow of an approximation to the set itself. Yet it has a robustness

that is beyond any doubt; for the same structure is revealedâ€”in all its

perceivable details, to greater and greater Wneness the more closely it is

examinedâ€”independently of the mathematician or computer that examines

it. Its existence can only be within the Platonic world of mathematical

forms.

I am aware that there will still be many readers who Wnd diYculty with

assigning any kind of actual existence to mathematical structures. Let me

make the request of such readers that they merely broaden their notion of

what the term â€˜existenceâ€™ can mean to them. The mathematical forms of

Platoâ€™s world clearly do not have the same kind of existence as do ordinary

physical objects such as tables and chairs. They do not have spatial

locations; nor do they exist in time. Objective mathematical notions

must be thought of as timeless entities and are not to be regarded as

being conjured into existence at the moment that they are Wrst humanly

perceived. The particular swirls of the Mandelbrot set that are depicted

in Fig. 1.2c or 1.2d did not attain their existence at the moment that they

were Wrst seen on a computer screen or printout. Nor did they come about

when the general idea behind the Mandelbrot set was Wrst humanly put

forthâ€”not actually Wrst by Mandelbrot, as it happened, but by R. Brooks

and J. P. Matelski, in 1981, or perhaps earlier. For certainly neither

Brooks nor Matelski, nor initially even Mandelbrot himself, had any

real conception of the elaborate detailed designs that we see in Fig. 1.2c

and 1.2d. Those designs were already â€˜in existenceâ€™ since the beginning of

time, in the potential timeless sense that they would necessarily be revealed

precisely in the form that we perceive them today, no matter at what time

or in what location some perceiving being might have chosen to examine

them.

1.4 Three worlds and three deep mysteries

Thus, mathematical existence is diVerent not only from physical existence

but also from an existence that is assigned by our mental perceptions. Yet

there is a deep and mysterious connection with each of those other two

forms of existence: the physical and the mental. In Fig. 1.3, I have

schematically indicated all of these three forms of existenceâ€”the physical,

the mental, and the Platonic mathematicalâ€”as entities belonging to three

separate â€˜worldsâ€™, drawn schematically as spheres. The mysterious connections between the worlds are also indicated, where in drawing the diagram

17

Â§1.4

CHAPTER 1

Platonic

mathematical

world

3

1

2

Mental

world

Physical

world

Fig. 1.3 Three â€˜worldsâ€™â€”

the Platonic mathematical,

the physical, and the

mentalâ€”and the three

profound mysteries in the

connections between them.

I have imposed upon the reader some of my beliefs, or prejudices, concerning these mysteries.

It may be noted, with regard to the Wrst of these mysteriesâ€”relating the

Platonic mathematical world to the physical worldâ€”that I am allowing

that only a small part of the world of mathematics need have relevance to

the workings of the physical world. It is certainly the case that the vast

preponderance of the activities of pure mathematicians today has no

obvious connection with physics, nor with any other science (cf. Â§34.9),

although we may be frequently surprised by unexpected important applications. Likewise, in relation to the second mystery, whereby mentality

comes about in association with certain physical structures (most speciWcally, healthy, wakeful human brains), I am not insisting that the majority

of physical structures need induce mentality. While the brain of a cat may

indeed evoke mental qualities, I am not requiring the same for a rock.

Finally, for the third mystery, I regard it as self-evident that only a small

fraction of our mental activity need be concerned with absolute mathematical truth! (More likely we are concerned with the multifarious irritations,

pleasures, worries, excitements, and the like, that Wll our daily lives.) These

three facts are represented in the smallness of the base of the connection of

each world with the next, the worlds being taken in a clockwise sense in the

diagram. However, it is in the encompassing of each entire world within

the scope of its connection with the world preceding it that I am revealing

my prejudices.

Thus, according to Fig. 1.3, the entire physical world is depicted as

being governed according to mathematical laws. We shall be seeing in later

chapters that there is powerful (but incomplete) evidence in support of this

contention. On this view, everything in the physical universe is indeed

18

The roots of science

Â§1.4

governed in completely precise detail by mathematical principlesâ€”

perhaps by equations, such as those we shall be learning about in chapters

to follow, or perhaps by some future mathematical notions fundamentally diVerent from those which we would today label by the term â€˜equationsâ€™. If this is right, then even our own physical actions would be entirely

subject to such ultimate mathematical control, where â€˜controlâ€™ might still

allow for some random behaviour governed by strict probabilistic

principles.

Many people feel uncomfortable with contentions of this kind, and I

must confess to having some unease with it myself. Nonetheless, my

personal prejudices are indeed to favour a viewpoint of this general nature,

since it is hard to see how any line can be drawn to separate physical

actions under mathematical control from those which might lie beyond it.

In my own view, the unease that many readers may share with me on this

issue partly arises from a very limited notion of what â€˜mathematical

controlâ€™ might entail. Part of the purpose of this book is to touch upon,

and to reveal to the reader, some of the extraordinary richness, power, and

beauty that can spring forth once the right mathematical notions are hit

upon.

In the Mandelbrot set alone, as illustrated in Fig. 1.2, we can begin to

catch a glimpse of the scope and beauty inherent in such things. But even

these structures inhabit a very limited corner of mathematics as a whole,

where behaviour is governed by strict computational control. Beyond this

corner is an incredible potential richness. How do I really feel about the

possibility that all my actions, and those of my friends, are ultimately

governed by mathematical principles of this kind? I can live with that. I

would, indeed, prefer to have these actions controlled by something residing in some such aspect of Platoâ€™s fabulous mathematical world than to

have them be subject to the kind of simplistic base motives, such as

pleasure-seeking, personal greed, or aggressive violence, that many

would argue to be the implications of a strictly scientiWc standpoint.

Yet, I can well imagine that a good many readers will still have diYculty

in accepting that all actions in the universe could be entirely subject to

mathematical laws. Likewise, many might object to two other prejudices

of mine that are implicit in Fig. 1.3. They might feel, for example, that I

am taking too hard-boiled a scientiWc attitude by drawing my diagram in a

way that implies that all of mentality has its roots in physicality. This is

indeed a prejudice, for while it is true that we have no reasonable scientiWc

evidence for the existence of â€˜mindsâ€™ that do not have a physical basis, we

cannot be completely sure. Moreover, many of a religious persuasion

would argue strongly for the possibility of physically independent minds

and might appeal to what they regard as powerful evidence of a diVerent

kind from that which is revealed by ordinary science.

19

Â§1.4

CHAPTER 1

A further prejudice of mine is reXected in the fact that in Fig. 1.3 I have

represented the entire Platonic world to be within the compass of mentality. This is intended to indicate thatâ€”at least in principleâ€”there are no

mathematical truths that are beyond the scope of reason. Of course, there

are mathematical statements (even straightforward arithmetical addition

sums) that are so vastly complicated that no one could have the mental

fortitude to carry out the necessary reasoning. However, such things

would be potentially within the scope of (human) mentality and would

be consistent with the meaning of Fig. 1.3 as I have intended to represent

it. One must, nevertheless, consider that there might be other mathematical statements that lie outside even the potential compass of reason, and

these would violate the intention behind Fig. 1.3. (This matter will be

considered at greater length in Â§16.6, where its relation to GoÂ¨delâ€™s famous

incompleteness theorem will be discussed.)8

In Fig. 1.4, as a concession to those who do not share all my personal

prejudices on these matters, I have redrawn the connections between the

three worlds in order to allow for all three of these possible violations of

my prejudices. Accordingly, the possibility of physical action beyond the

scope of mathematical control is now taken into account. The diagram

also allows for the belief that there might be mentality that is not rooted in

physical structures. Finally, it permits the existence of true mathematical

assertions whose truth is in principle inaccessible to reason and insight.

This extended picture presents further potential mysteries that lie even

beyond those which I have allowed for in my own preferred picture of the

world, as depicted in Fig. 1.3. In my opinion, the more tightly organized

scientiWc viewpoint of Fig. 1.3 has mysteries enough. These mysteries are

not removed by passing to the more relaxed scheme of Fig. 1.4. For it

Platonic

mathematical

world

Mental

world

20

Physical

world

Fig. 1.4 A redrawing of

Fig. 1.3 in which violations

of three of the prejudices of

the author are allowed for.