Preface

List of Abbreviations

**1. Wittgenstein's Turning-Point**

1. Philosophy's Debt to Schlick

2. Sceptical Confusions about Rule-following

3. Dispersing the Clouds of Epistemological Confusion

**2. The Strains in the Realist/Anti-realist Framework**

1. Wittgenstein's Verificationism

2. Wittgenstein's 'Conversion"

3. The Objectivity of Mathematical Knowledge

**3. The Nature of Proof**

1. The Burden of Proof

2. Observing the Law of Excluded Middle

3. Mathematical 'Stimuli'

**4. Surveyability**

1. The Bounds of Perspicuity

2. 'Probabilistic Proofs'

3. The Appel-Haken Solution of the Four-Colour Problem

**5. The Perils of Prose**

1. The Nature of Infinity

2. Coming to Grips with the Irrational

3. 'Prose': The Meeting-point Between Mathematics and Philosophy

**6. Consistency**

1. Hilbert's Programme

2. Mathematical Totems and Tabus

3. The Application of an 'Inconsistent System'

**7. The Recovery of Certainty**

1. The Foundations of the Foundations of Mathematics

2. Equations are Rules of Syntax

3. Unconventional Conventionalism

**8. Freedom and Necessity**

1. Discovering, Creating, Inventing

2. Anarchy versus Autonomy

3. The Interrelatedness of Mathematical Truths

Bibliography

Index