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[Roger Penrose] The road to reality a complete gu(BookFi.org) .pdf



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Titolo: The Road to Reality - A Complete Guide to the Laws of the Universe
Autore: Roger Penrose

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THE ROAD TO REALITY

BY ROGER PENROSE

The Emperor’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 ß 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 publisher’s prior
consent in any form of binding or cover other than that
in which it is published and without a similar condition
including this 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 0–224–04447–8
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 Plato’s mathematical world ‘real’?
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
Euclid’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 ‘i’
Solving equations with complex numbers

v

71
74

Contents

4.3
4.4
4.5

Convergence of power series
Caspar Wessel’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 ‘Eulerian’ 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 Cauchy–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
Cantor’s diagonal slash
Puzzles in the foundations of mathematics
Turing machines and Go¨del’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
Cartan’s ‘Newtonian spacetime’
The Wxed Wnite speed of light
Light cones
The abandonment of absolute time
The spacetime for Einstein’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 ‘clock paradox’
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
Maxwell’s electromagnetic theory
Conservation and Xux laws in Maxwell theory
The Maxwell Weld as gauge curvature
The energy–momentum tensor
Einstein’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
Schro¨dinger’s equation
Quantum theory’s experimental background
Understanding wave–particle duality
What is quantum ‘reality’?
The ‘holistic’ nature of a wavefunction
The mysterious ‘quantum jumps’
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: Schro¨dinger and Heisenberg
Quantum ‘observables’
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
Hardy’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 Dirac’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
CliVord–Dirac square root of wave operator
The Dirac equation
Dirac’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
‘Coloured quarks’
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: Feynman’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 law—or not?
Is the whole universe an ‘isolated system’?
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 Bang’s special nature: an anthropic key?
The Weyl curvature hypothesis
The Hartle–Hawking ‘no-boundary’ 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
Schro¨dinger’s cat with ‘Copenhagen’ ontology
Can other conventional ontologies resolve the ‘cat’?
Which unconventional ontologies may help?

30 Gravity’s role in quantum state reduction
30.1
30.2

Is today’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
Hawking’s black-hole temperature
Black-hole temperature from complex periodicity
Killing vectors, energy Xow—and time travel!
Energy outXow from negative-energy orbits
Hawking explosions
A more radical perspective
Schro¨dinger’s lump
Fundamental conXict with Einstein’s principles
Preferred Schro¨dinger–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 don’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 Calabi–Yau spaces; M-theory
Strings and black-hole entropy
The ‘holographic principle’
The D-brane perspective
The physical status of string theory?

32 Einstein’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 Ashtekar’s variables
The form of Ashtekar’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 physics—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 insights—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 ‘optimists’, 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 fractions—or those who claim that they cannot manipulate
fractions—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 competence—a duty, and a duty alone—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 mother’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. ‘It’s
all that cancelling’, she said to me, ‘I could just never get the hang of
cancelling.’ 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 ‘cancelling’.
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 ‘stern face’ 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-up—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 ‘something with a 3 at the top
and an 8 at the bottom with a horizontal line in between’ 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 classmate’s ‘something with a 3 at the top and an 8 on the bottom’. 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, ‘equate’ the pair
(6, 16) with the pair (3, 8)? The mathematician’s answer—which may well
sound like a cop-out—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 mathematician’s ‘‘equivalence class’’
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 mother’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 ‘inWnite class of pairs’ even though it certainly
has its value in mathematical rigour and precision. In my descriptions here
I shall be more concerned with conveying the idea—and the beauty and
the magic—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 fractions—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 ‘deWnition’ 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 ‘inWnite
collection of pairs’ is merely a pedantic device—a device that quickly
recedes from our imaginations once we have grasped it. Much of mathematics is like that.
* This is called an ‘equivalence class’ 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 Einstein’s general
theory of relativity and the ‘gauge theory’ 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 –
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: Dirac’s equation for the electron, the general
framework of quantum mechanics, and Einstein’s general relativity. But
in all these cases, physical considerations—ultimately observational
ones—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 criteria—i.e. experimental data, or
even the possibility of experimental investigation—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 physics—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
thanks—and also apologies—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 Schu¨cking, Bernard Schutz, Joseph Silk, Christoph
Simon, George Sparling, John Stachel, Henry Stapp, Richard Thomas,
Gerard t’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 2000–2002. Part-time
appointments at Gresham College, London (1998–2001) 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 Mizzi’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, 6.2–6.8, 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 gratitude—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)—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 §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 components—these being matrices—for which the corresponding bold upright
letters S and T can also be used. In that case, ST denotes the corresponding matrix product. This ‘ambivalent’ 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 quantities—the diagrammatic notation—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 ‘standing for’ 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 ‘length’ of a 3-vector quantity a ¼ (a1 ,a2 ,a3 ) may be written
a, where a2 ¼ a21 þ a22 þ a23 , and the scalar product of a with b ¼ (b1 ,b2 ,b3 ),
written a . b ¼ a1 b1 þ a2 b2 þ a3 b3 . This ‘dot’ 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 ‘hats’ (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 ‘same’—and so it is fair to use the same symbols for
each—except that in the classical case one is justiWed in ignoring quantities
of the order of h, so that the classical commutation properties ab ¼ 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 §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 ‘pseudo-’ Euclidean space, so I use the shell letter M for this space
(with Mn to denote the n-dimensional version—a ‘Lorentzian’ spacetime
with 1 time and (n 1) space dimensions). Sometimes I use C as an
adjective, to denote ‘complexiWed’, 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 ‘projective’ (see §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 ‘complex conjugate of’
(as used in §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 King’s chief craftsman, an artist of consummate skills. It
was night, and he lay sleeping on his workshop couch, tired after a
handsomely productive evening’s work. But his sleep was restless—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
come—quite suddenly—when his bones told him that surely it must still be
night.
He stood up abruptly. Something was odd. The dawn’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 parasol’s hood began to
spread and darken—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 scene’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 God’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 completed—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 right—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-ritual—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-tep’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 artist’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 ‘why?’ 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 father’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 generations—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-tep’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-tep’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
Amphos’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 ‘why?’, and he felt unsatisWed. He believed that there
was indeed reason underlying Nature’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 artist’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. Nature’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 one’s beliefs, it was
necessary to form precise conclusions by the use of unchallengeable
reason. The nature of this precision had to be mathematical—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
events—such as sunshine, rain, storms, famine, illness, or pestilence—
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 Sun’s motion through the sky and its clear
relation to the alternation of day with night provided the most obvious
example; but also the Sun’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

§1.1

CHAPTER 1

whims of gods, then these gods themselves seemed under the spell of exact
mathematical laws.
Likewise, the laws controlling earthly phenomena—such as the daily
and yearly changes in temperature, the ebb and Xow of the oceans, and the
growth of plants—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 geometry—the 3-dimensional geometry that we now call Euclidean. Moreover,
the notion of a ‘straight line’ 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

§1.2

Fig. 1.1 A fanciful association, made by the ancient Greeks, between the Wve
Platonic solids and the four ‘elements’ (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.625–547 bc) and
9

§1.2

CHAPTER 1

Pythagoras1* of Samos (c.572–497 bc) began to introduce the notion of
mathematical proof that the Wrst Wrm foundation stone of mathematical
understanding—and therefore of science itself—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
‘Pythagorean scale’, 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 followers—the Pythagoreans—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
‘moles’—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 assertions—the axioms—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

§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 Euclid’s geometry. (I shall be more explicit
about these issues in §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 ‘self-evident’ mathematical truths and were taken as axioms (or postulates). As we shall be seeing in
Chapters 17–19 and §§27.8,11, Einstein’s general relativity—and even the
Minkowskian spacetime of special relativity—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 ‘axioms’ as
being, in any sense, actually true.
But what does ‘true’ 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 propositions—the things that could be
regarded as unassailably true—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

§1.3

CHAPTER 1

not be part of the physical world, but would be inhabitants of Plato’s
idealized mathematical world of forms.

1.3 Is Plato’s mathematical world ‘real’?
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 ‘world’ as a complete Wction—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 world—or, rather, of certain aspects of the
world—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 ‘existence’, 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
Plato’s world to be in any sense absolute and ‘real’. 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

§1.3

Yet even this viewpoint seems to leave us far short of what is required. Do
we mean ‘agreed by all’, for example, or ‘agreed by those who are in their
right minds’, or ‘agreed by all those who have a Ph.D. in mathematics’
(not much use in Plato’s day) and who have a right to venture an ‘authoritative’ opinion? There seems to be a danger of circularity here; for to judge
whether or not someone is ‘in his or her right mind’ requires some external
standard. So also does the meaning of ‘authoritative’, unless some standard of an unscientiWc nature such as ‘majority opinion’ 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 themselves—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 ‘existence’ 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 ‘existence’
could also refer to things other than mathematics, such as to morality or
aesthetics (cf. §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 ‘objectivity’. In 1637,
Pierre de Fermat made his famous assertion now known as ‘Fermat’s Last
Theorem’ (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: ‘I have discovered a
truly marvellous proof of this, which this margin is too narrow to contain.’
Fermat’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 Fermat’s assertion was always true, long
before Fermat actually made it, or is its validity a purely cultural matter,
13

§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 X’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 X’s forthcoming counter-example,
Fermat himself would necessarily have been mistaken in believing in the
soundness of his ‘truly marvellous proof’, 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 Fermat’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 ‘Platonism’, would regard such
possibilities as patently absurd.
Of course, it might still be the case that Wiles’s argument in fact
contains an error and that the Fermat assertion is indeed false. Or there
could be a fundamental error in Wiles’s argument but the Fermat assertion
is true nevertheless. Or it might be that Wiles’s argument is correct in its
essentials while containing ‘non-rigorous steps’ 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 anyone’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 ‘subjective’—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 ‘matter of opinion’.
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 §16.3.) It is cited here only as an example. Most mathematicians would probably regard the axiom of choice as ‘obviously true’, while
14

The roots of science

§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
‘truth’ 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 ‘formal system’; see §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 §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 ‘Platonic
world’. 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 ‘such-and-such follows
from the axiom of choice’ or ‘the axiom of choice is a theorem according
to the rules of such-and-such mathematical system’).
The mathematical assertions that can belong to Plato’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
notions—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 elements—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 ‘mystical’ or ‘unscientiWc’, 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

§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 §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 set’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 set’s
16

The roots of science

§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 revealed—in all its
perceivable details, to greater and greater Wneness the more closely it is
examined—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 ‘existence’ can mean to them. The mathematical forms of
Plato’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
forth—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 ‘in existence’ 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 existence—the physical,
the mental, and the Platonic mathematical—as entities belonging to three
separate ‘worlds’, drawn schematically as spheres. The mysterious connections between the worlds are also indicated, where in drawing the diagram
17

§1.4

CHAPTER 1

Platonic
mathematical
world

3

1

2
Mental
world
Physical
world

Fig. 1.3 Three ‘worlds’—
the Platonic mathematical,
the physical, and the
mental—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 mysteries—relating the
Platonic mathematical world to the physical world—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. §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

§1.4

governed in completely precise detail by mathematical principles—
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 ‘equations’. If this is right, then even our own physical actions would be entirely
subject to such ultimate mathematical control, where ‘control’ 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 ‘mathematical
control’ 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 Plato’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 ‘minds’ 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

§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 that—at least in principle—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 §16.6, where its relation to Go¨del’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.


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