The Conceptual Roots of Mathematics by Lucas, J.R. -- Read -- Imperial Library of Trantor

Index

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