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### Summary

### Summary

In this wide-ranging book, Brian Davies discusses the basis for scientists' claims to knowledge about the world. He looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. He rejects the claim that all scientificknowledge is provisional, by citing examples from chemistry, biology and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. A large number of examples are used to illustrate these points, and many of the deep issues in today's worlddiscussed - from psychology and evolution to quantum theory, consciousness and even religious belief. Disentangling knowledge from opinion and aspiration is a hard task, but this book provides a clear guide to the difficulties.

### Author Notes

Educated at the University of Oxford, Brian Davies is Professor of Mathematics at King's College, London and a Fellow of the Royal Society. He developed the theory of open quantum systems, writing a monograph on the subject, which became the standard text. He has published almost 300 articles and four books on subjects ranging from quantum theory to pure mathematics, and is currently working in both computational analysis and the philosophy of science. He has held visiting positions at a number of leading universities in Europe and the USA.

### Reviews 1

### Choice Review

Contemplating the nature of scientific knowledge as that knowledge is reflected in an unforgiving empiricist mirror, Davies (mathematics, King's College, London) has produced a wholly engrossing, highly readable, immensely wise, though at times curmudgeonly, study. But reader beware--Davies is anything if not opinionated, and he is forthright about those opinions as well. Perhaps surprising for an author mathematician, the theme that threads its way through the book--whether the chapter subject is mind, astronomy, quantum physics, physiology, evolution, or mathematics itself--is that there is no basis for seeing mathematics as anything other than a fallible, convenient, invented tool--end of mystery. Stating that opinion has Davies frankly chastising colleague Roger Penrose ("Penrose's idea ... is simply wrong"), Kurt Godel ("... can only serve to confuse"), and Steven Weinberg ("I have struggled to understand what he means ..."). Nor does Davies refrain from other bracing first-person interjections. Still, there is so much in this book that is brilliantly synthesized, eloquently written about, and, yes, persuasively argued as to make it a near indispensable contribution to the literature. A small drawback is an inadequate index, to be remedied, one hopes, in a second edition. ^BSumming Up: Highly recommended. General readers; upper-division undergraduates through faculty. M. Schiff College of Staten Island, CUNY

### Table of Contents

1 Perception and Language | p. 1 |

1.1 Preamble | p. 1 |

1.2 Light and Vision | p. 3 |

Introduction | p. 3 |

The Perception of Colour | p. 4 |

Interpretation and Illusion | p. 6 |

Disorders of the Brain | p. 13 |

The World of a Bat | p. 15 |

What Do We See? | p. 16 |

1.3 Language | p. 18 |

Physiological Aspects of Language | p. 18 |

Social Aspects of Language | p. 22 |

Objects, Concepts, and Existence | p. 24 |

Numbers as Social Constructs | p. 27 |

Notes and References | p. 31 |

2 Theories of the Mind | p. 33 |

2.1 Preamble | p. 33 |

2.2 Mind-Body Dualism | p. 34 |

Plato | p. 34 |

Mathematical Platonism | p. 37 |

The Rotation of Triangles | p. 41 |

Descartes and Dualism | p. 43 |

Dualism in Society | p. 46 |

2.3 Varieties of Consciousness | p. 49 |

Can Computers Be Conscious? | p. 50 |

Godel and Penrose | p. 52 |

Discussion | p. 54 |

Notes and References | p. 59 |

3 Arithmetic | p. 61 |

Introduction | p. 61 |

Whole Numbers | p. 62 |

Small Numbers | p. 62 |

Medium Numbers | p. 64 |

Large Numbers | p. 65 |

What Do Large Numbers Represent? | p. 66 |

Addition | p. 67 |

Multiplication | p. 68 |

Inaccessible and Huge Numbers | p. 71 |

Peano's Postulates | p. 75 |

Infinity | p. 78 |

Discussion | p. 80 |

Notes and References | p. 83 |

4 How Hard can Problems Get? | p. 85 |

Introduction | p. 85 |

The Four Colour Problem | p. 87 |

Goldbach's Conjecture | p. 88 |

Fermat's Last Theorem | p. 89 |

Finite Simple Groups | p. 90 |

A Practically Insoluble Problem | p. 91 |

Algorithms | p. 93 |

How to Handle Hard Problems | p. 96 |

Notes and References | p. 97 |

5 Pure Mathematics | p. 99 |

5.1 Introduction | p. 99 |

5.2 Origins | p. 100 |

Greek Mathematics | p. 100 |

The Invention of Algebra | p. 103 |

The Axiomatic Revolution | p. 103 |

Projective Geometry | p. 107 |

5.3 The Search for Foundations | p. 109 |

5.4 Against Foundations | p. 113 |

Empiricism in Mathematics | p. 116 |

From Babbage to Turing | p. 117 |

Finite Computing Machines | p. 123 |

Passage to the Infinite | p. 125 |

Are Humans Logical? | p. 127 |

5.5 The Real Number System | p. 130 |

A Brief History | p. 131 |

What is Equality? | p. 134 |

Constructive Analysis | p. 135 |

Non-standard Analysis | p. 137 |

5.6 The Computer Revolution | p. 138 |

Discussion | p. 139 |

Notes and References | p. 140 |

6 Mechanics and Astronomy | p. 143 |

6.1 Seventeenth Century Astronomy | p. 143 |

Galileo | p. 146 |

Kepler | p. 151 |

Newton | p. 153 |

The Law of Universal Gravitation | p. 154 |

6.2 Laplace and Determinism | p. 157 |

Chaos in the Solar System | p. 158 |

Hyperion | p. 160 |

Molecular Chaos | p. 161 |

A Trip to Infinity | p. 163 |

The Theory of Relativity | p. 164 |

6.3 Discussion | p. 166 |

Notes and References | p. 170 |

7 Probability and Quantum Theory | p. 171 |

7.1 The Theory of Probability | p. 171 |

Kolmogorov's Axioms | p. 172 |

Disaster Planning | p. 174 |

The Paradox of the Children | p. 175 |

The Letter Paradox | p. 175 |

The Three Door Paradox | p. 176 |

The National Lottery | p. 177 |

Probabilistic Proofs | p. 178 |

What is a Random Number? | p. 179 |

Bubbles and Foams | p. 181 |

Kolmogorov Complexity | p. 182 |

7.2 Quantum Theory | p. 183 |

History of Atomic Theory | p. 184 |

The Key Enigma | p. 186 |

Quantum Probability | p. 188 |

Quantum Particles | p. 190 |

The Three Aspects of Quantum Theory | p. 192 |

Quantum Modelling | p. 193 |

Measuring Atomic Energy Levels | p. 195 |

The EPR Paradox | p. 196 |

Reflections | p. 198 |

Schrodinger's Cat | p. 199 |

Notes and References | p. 202 |

8 Is Evolution a Theory? | p. 203 |

Introduction | p. 203 |

The Public Perception | p. 204 |

The Geological Record | p. 205 |

Dating Techniques | p. 209 |

The Mechanisms of Inheritance | p. 213 |

Theories of Evolution | p. 217 |

Some Common Objections | p. 225 |

Discussion | p. 230 |

Notes and References | p. 232 |

9 Against Reductionism | p. 235 |

Introduction | p. 235 |

Biochemistry and Cell Physiology | p. 238 |

Prediction or Explanation | p. 240 |

Money | p. 242 |

Information and Complexity | p. 243 |

Subjective Consciousness | p. 245 |

The Chinese Room | p. 246 |

Zombies and Related Issues | p. 248 |

A Physicalist View | p. 250 |

Notes and References | p. 251 |

10 Some Final Thoughts | p. 253 |

Order and Chaos | p. 253 |

Anthropic Principles | p. 256 |

From Hume to Popper | p. 259 |

Empiricism versus Realism | p. 266 |

The Sociology of Science | p. 270 |

Science and Technology | p. 274 |

Conclusions | p. 276 |

Notes and References | p. 279 |

Bibliography | p. 281 |

Index | p. 289 |