S.G. Shanker is Special Associate Professor in Philosophy at York University, Ontario.

"Shanker presents a new and interesting interpretation of Wittgenstein's philosophy of mathematics. I am persuaded that he is reading Wittgenstein basically correctly. Wittgenstein's views seem even more radical and bizarre on Shanker's reading than they did before. Shanker also argues for the correctness of Wittgenstein's position, but I find that completely implausible. Nevertheless, his argument is vigorous and shows an excellent, up-to-date grasp of the relevant literature." — Nicholas D. Goodman

• Preface List of Abbreviations 1. Wittgenstein's Turning-Point 1. Philosophy's Debt to Schlick2. Sceptical Confusions about Rule-following3. Dispersing the Clouds of Epistemological Confusion 2. The Strains in the Realist/Anti-realist Framework 1. Wittgenstein's Verificationism2. Wittgenstein's 'Conversion"3. The Objectivity of Mathematical Knowledge 3. The Nature of Proof 1. The Burden of Proof2. Observing the Law of Excluded Middle3. Mathematical 'Stimuli' 4. Surveyability 1. The Bounds of Perspicuity2. 'Probabilistic Proofs'3. The Appel-Haken Solution of the Four-Colour Problem 5. The Perils of Prose 1. The Nature of Infinity2. Coming to Grips with the Irrational3. 'Prose': The Meeting-point Between Mathematics and Philosophy 6. Consistency 1. Hilbert's Programme2. Mathematical Totems and Tabus3. The Application of an 'Inconsistent System' 7. The Recovery of Certainty 1. The Foundations of the Foundations of Mathematics2. Equations are Rules of Syntax3. Unconventional Conventionalism 8. Freedom and Necessity 1. Discovering, Creating, Inventing2. Anarchy versus Autonomy3. The Interrelatedness of Mathematical Truths Bibliography Index

S.G. Shanker is Special Associate Professor in Philosophy at York University, Ontario.

"Shanker presents a new and interesting interpretation of Wittgenstein's philosophy of mathematics. I am persuaded that he is reading Wittgenstein basically correctly. Wittgenstein's views seem even more radical and bizarre on Shanker's reading than they did before. Shanker also argues for the correctness of Wittgenstein's position, but I find that completely implausible. Nevertheless, his argument is vigorous and shows an excellent, up-to-date grasp of the relevant literature." — Nicholas D. Goodman

• Preface List of Abbreviations 1. Wittgenstein's Turning-Point 1. Philosophy's Debt to Schlick2. Sceptical Confusions about Rule-following3. Dispersing the Clouds of Epistemological Confusion 2. The Strains in the Realist/Anti-realist Framework 1. Wittgenstein's Verificationism2. Wittgenstein's 'Conversion"3. The Objectivity of Mathematical Knowledge 3. The Nature of Proof 1. The Burden of Proof2. Observing the Law of Excluded Middle3. Mathematical 'Stimuli' 4. Surveyability 1. The Bounds of Perspicuity2. 'Probabilistic Proofs'3. The Appel-Haken Solution of the Four-Colour Problem 5. The Perils of Prose 1. The Nature of Infinity2. Coming to Grips with the Irrational3. 'Prose': The Meeting-point Between Mathematics and Philosophy 6. Consistency 1. Hilbert's Programme2. Mathematical Totems and Tabus3. The Application of an 'Inconsistent System' 7. The Recovery of Certainty 1. The Foundations of the Foundations of Mathematics2. Equations are Rules of Syntax3. Unconventional Conventionalism 8. Freedom and Necessity 1. Discovering, Creating, Inventing2. Anarchy versus Autonomy3. The Interrelatedness of Mathematical Truths Bibliography Index

S.G. Shanker is Special Associate Professor in Philosophy at York University, Ontario.

"Shanker presents a new and interesting interpretation of Wittgenstein's philosophy of mathematics. I am persuaded that he is reading Wittgenstein basically correctly. Wittgenstein's views seem even more radical and bizarre on Shanker's reading than they did before. Shanker also argues for the correctness of Wittgenstein's position, but I find that completely implausible. Nevertheless, his argument is vigorous and shows an excellent, up-to-date grasp of the relevant literature." — Nicholas D. Goodman

• Preface List of Abbreviations 1. Wittgenstein's Turning-Point 1. Philosophy's Debt to Schlick2. Sceptical Confusions about Rule-following3. Dispersing the Clouds of Epistemological Confusion 2. The Strains in the Realist/Anti-realist Framework 1. Wittgenstein's Verificationism2. Wittgenstein's 'Conversion"3. The Objectivity of Mathematical Knowledge 3. The Nature of Proof 1. The Burden of Proof2. Observing the Law of Excluded Middle3. Mathematical 'Stimuli' 4. Surveyability 1. The Bounds of Perspicuity2. 'Probabilistic Proofs'3. The Appel-Haken Solution of the Four-Colour Problem 5. The Perils of Prose 1. The Nature of Infinity2. Coming to Grips with the Irrational3. 'Prose': The Meeting-point Between Mathematics and Philosophy 6. Consistency 1. Hilbert's Programme2. Mathematical Totems and Tabus3. The Application of an 'Inconsistent System' 7. The Recovery of Certainty 1. The Foundations of the Foundations of Mathematics2. Equations are Rules of Syntax3. Unconventional Conventionalism 8. Freedom and Necessity 1. Discovering, Creating, Inventing2. Anarchy versus Autonomy3. The Interrelatedness of Mathematical Truths Bibliography Index