Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.

List of Figures ix

List of Tables x

Introduction 1

Part I The Science of Quantity and Structure

1 The Aristotelian Realist Point of View 11

      The reality of universals 11

      Platonism and nominalism 12

      The reality of relations and structure 15

      'Unit-making' properties and sets 16

      Causality 17

      Aristotelian epistemology 18

2 Uninstantiated Universals and 'Semi-Platonist' Aristotelianism 21

      Determinables and determinates 22

      Uninstantiated shades of blue and huge numbers 23

      Possibles by recombination? 25

      Semi-Platonist Aristotelianism 26

3 Elementary Mathematics: The Science of Quantity 31

      Two realist theories of mathematics: quantity versus structure 31

      Continuous quantity and ratios 34

      Discrete quantity and numbers 36

      Discrete quantity and sets 38

      Discrete and continuous quantity compared 44

      Defining 'quantity' 45

4 Higher Mathematics: Science of the Purely Structural 48

      The rise of structure in mathematics 48

      Structuralism in recent philosophy of mathematics 49

      Abstract algebra, groups, and modern pure mathematics 51

      Structural commonality in applied mathematics 54

      Defining 'structure' 56

      The sufficiency of mereology and logic 59

      Is quantity a kind of structure? 63

5 Necessary Truths about Reality 67

      Examples of necessity 67

      Objections and replies 71

6 The Formal Sciences Discover the Philosophers' Stone 82

      A brief survey of the formal or mathematical sciences 83

      The formal sciences search for a place in the sun 89

      Real certainty: program verification 92

      Real certainty: the other formal sciences 95

      Experiment in the formal sciences 98

7 Comparisons and Objections 101

      Frege's limited options 101

      The Platonist/nominalist false dichotomy 104

      Nominalism 106

      Constructions in set theory 108

      Avoiding the question: what are sets? 110

      Overemphasis on the infinite 111

      Measurement and the applicability of mathematics 113

      The indispensability argument 114

      Modal and Platonist structuralism 117

      Epistemology and 'access' 121

      Naturalism: non-Platonist realisms 122

8 Infinity 129

      Infinity, who needs it? 130

      Paradoxes of infinity? 134

      'Potential' infinity? 136

      Knowing the infinite 140

9 Geometry: Mathematics or Empirical Science? 141

      What is geometry? Plan A: multidimensional quantities 143

      What is geometry? Plan B: the shapes of possible spaces 146

      The grit-or-gunk controversy: does space consist of points? 150

      Real non-spatial 'spaces' with geometric structure 153

      The space of colours 154

      Spaces of vectors 155

      The real space we live in 156

      Non-Euclidean geometry: the 'loss of certainty' in mathematics? 160

Part II Knowing Mathematical Reality

10 Knowing Mathematics: Pattern Recognition and Perception of Quantity and Structure 165

      The registering of mathematical properties by measurement devices and artificial intelligence 167

      Babies and animals: the simplest mathematical perception 172

      Animal and infant knowledge of quantity 173

      Perceptual knowledge of pattern and structure 176

11 Knowing Mathematics: Visualization and Understanding 180

      Imagination and the uninstantiated 180

      Visualization for understanding structure 181

      The return of visualization, and its neglect 185

      Why visualization has been persona non grata in the philosophy of mathematics 185

      The mind and structural properties: the mysteriousness of understanding 188

      The chiliagon and the limits of visualization 191

12 Knowing Mathematics: Proof and Certainty 192

      Proof: a chain of insights 192

      Symbolic proof 'versus' visualization: their respective advantages 196

      Proof: logicist, 'if-thenist' and formalist errors 197

      Axioms, formalization and understanding 200

      Counting 202

      Knowing the infinite 203

13 Explanation in Mathematics 207

      Explanation in pure mathematics 208

      How do pure mathematical explanations fit into accounts of explanation? 212

      Geometrical explanation in science 215

      Non-geometrical mathematical explanation in science 217

      Aristotelian realism for explanatory success 220

14 Idealization: An Aristotelian View 222

      Applied mathematics without idealization 224

      Approximation with simple structures, not idealization 225

      Negative and complex numbers, ideal points, and other extensions of ontology 229

      Zero 234

      The empty set 238

15 Non-Deductive Logic in Mathematics 241

      Estimating the probability of conjectures 242

      Evidence for (and against) the Riemann Hypothesis 245

      The classification of finite simple groups 250

      Probabilistic relations between necessary truths? 254

      The problem of induction in mathematics 257

      Epilogue: Mathematics, Last Bastion of Reason 260

Notes 263

Select Bibliography 302

Index 305

James Franklin is Professor of Mathematics at the University of New South Wales, Australia and founder of the 'Sydney School' in the philosophy of mathematics. His books include The Science of Conjecture: Evidence and Probability Before Pascal; Corrupting the Youth: A History of Philosophy in Australia; and What Science Knows.

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