Cover Page -- Title Page -- Copyright Page -- Introduction -- Note on Logical Symbolism -- Chapter 1: Plato's Philosophies of Mathematics -- 1.1: Meno -- 1.2: A Priori -- 1.3: Relevance -- 1.4: What Are We Talking About? -- 1.5: How Do We Know? -- 1.6: Modality -- 1.7: Cogency -- 1.8: Deduction -- 1.9: Whence the Premises? -- Chapter 2: Geometry -- 2.1: Euclid -- 2.2: The Fifth Postulate -- 2.3: Non-Euclidean Geometries -- 2.4: Formal and Physical Geometry -- 2.5: Conceptual Constraints -- 2.6: Which Geometry? -- 2.7: The Theory of Groups -- 2.8: Pythagorean Geometry Has a Better Metric -- 2.9: Desargues -- 2.10: Conclusions -- Chapter 3: Formalism -- 3.1: More Geometrico -- 3.2: Formalising -- 3.3: Maximum Cogency -- 3.4: The Theory of Formal Systems -- 3.5: Meaning and Interpretation -- 3.6: Epistemological Formalism -- 3.7: The Logicist Programme -- Chapter 4: Numbers: The Cardinal Approach -- 4.1: Etymology -- 4.2: The Uses of Numbers -- 4.3: How Many? -- 4.4: Nought -- 4.5: Quotifiers and Quotities -- 4.6: Frege's Extensions and Sets -- 4.7: Paradigm Sets -- Chapter 5: Numbers: The Ordinal Approach -- 5.1: The Superlative Approach -- 5.2: Dedekind's Successor -- 5.3: And So On -- 5.4: Grounding the Ordinals -- 5.5: How to Count -- 5.6: Ordinals and Cardinals -- 5.7: Conclusion -- Chapter 6: Numbers: The Abstract Approach -- 6.1: The Third Approach -- 6.2: Peano -- 6.3: Monomorphism and Non-standard Models -- 6.4: Immigration Control -- 6.5: The Fifth Postulate -- 6.6: Sorites Arithmetic -- 6.7: Dialogues -- 6.8: Recursive Reasoning -- 6.9: The Natural Numbers -- Chapter 7: The Infinite -- 7.1: In Defence of Doubt -- 7.2: Cardinality -- 7.3: The Mostest -- 7.4: Characterization of Intuitionism -- 7.5: Proofs and Dialogues -- 7.6: Verificationist Arguments for Intuitionism -- 7.7: Selective Scepticism -- 7.8: Ultrafinitism -- 7.9: Lax Finitism.

7.10: Actualising Potentiality -- 7.11: All -- Chapter 8: The Implications of Gödel's Theorem -- 8.1: Pons Asinorum -- 8.2: The Flavour of the Gödelian Argument -- 8.3: Gödel Numbering -- 8.4: Translation -- 8.5: Diagonalization -- 8.6: Conditions -- 8.7: Corollaries and Consequences -- 8.8: Church's Theorem and Turing's Theorem -- 8.9: Gödel's Second Theorem -- 8.10: Mechanism -- 8.11: Gödel's Theorem and Provability -- Chapter 9: Transitive Relations -- 9.1: Logic -- 9.2: Equivalence Relations -- 9.3: Functions -- 9.4: Identity in Difference -- 9.5: Ordering Relations -- 9.6: Macrostructure and Microstructure -- 9.7: The Continuum -- 9.8: The Marriage of Equivalence with Order -- 9.9: Converse Transitivity -- 9.10: Paradigm Partial Orderings -- 9.11: Lattices and Set Theory -- 9.12: Trees and Mereology -- Chapter 10: Prototopology0 -- 10.1: Togetherness -- 10.2: Axiomatic Approaches -- 10.3: Whitehead's Programme -- 10.4: Failure -- 10.5: Pointed and Linear Hopes -- 10.6: Alternatives -- 10.7: Mooreology -- 10.8: More Mereology -- Chapter 11: Magnitude and Measure -- 11.1: Quantum? and Quot? -- 11.2: The Point of Measuring -- 11.3: Equivalence Structures -- 11.4: Addition Rules -- 11.5: Limits and Zero -- 11.6: Zeno -- 11.7: Measures and Numbers -- Chapter 12: Down With Set Theory -- 12.1: Theory of Multitudes -- 12.2: Russell's Paradox -- 12.3: Responses to Paradox -- 12.4: Formal Approaches -- 12.5: The Skolem Paradox -- 12.6: The Axiom of Choice -- 12.7: The Continuum Hypotheses -- 12.8: Axioms and Existence -- 12.9: The Axiom of Extensionality -- 12.10: Conclusion -- Chapter 13: Chastened Logicism? -- 13.1: Logicism -- 13.2: What Is Logic? -- 13.3: Boolean Plus -- 13.4: Iterated Modalities -- 13.5: Completeness -- 13.6: Paradox -- 13.7: Second-order Logic -- 13.8: Analytic and A Priori Truth -- Chapter 14: Mathematical Knowledge.

14.1: Synthetic A Priori? -- 14.2: Not Seeing But Doing -- 14.3: Pattern Recognition -- 14.4: Lakatos -- 14.5: Cogency and Dialogue -- 14.6: On Behalf of the Fool -- 14.7: Hilbert -- 14.8: The Bed Theory of Truth -- 14.9: Mathematical Knowledge -- Chapter 15: Realism Revisited -- 15.1: Existence and Reality -- 15.2: Self-Subsistent Objects -- 15.3: Meaning and Impredicativity -- 15.4: Bivalence and Determinacy -- 15.5: Competing Truths -- 15.6: Contingency and Structure -- 15.7: Coherence and Depth -- 15.8: Laws of the Laws of Nature -- 15.9: Chastened Isms -- Envoi -- Summaries -- Numbers(Chapters Four, Five, Six and Eleven) -- I: Cardinal Approach -- II: Ordinal Approach -- III: Abstract Approach -- Schools -- I: Empiricism ( 1.1- 1.3, 2.4, 14.1) -- II: Platonism(Chapters One and Fifteen) -- III: Formalism(Chapter Three, 14.7) -- IV: Logicism ( 3.7, Chapters Four, Five, Six and Thirteen, 15.8, 15.9) -- V: Intuitionism ( 7.4- 7.7) -- Verdict.