Contents Preface xi A remark on notation xi Acknowledgments xii Chapter 1. Logic and foundations 1 §1.1. Material implication 1 §1.2. Errors in mathematical proofs 2 §1.3. Mathematical strength 4 §1.4. Stable implications 6 §1.5. Notational conventions 8 §1.6. Abstraction 9 §1.7. Circular arguments 11 §1.8. The classical number systems 12 §1.9. Round numbers 15 §1.10. The “no-self-defeating object” argument, revisited 16 §1.11. The “no-self-defeating object” argument, and the vagueness paradox 28 §1.12. A computational perspective on set theory 35 Chapter 2. Group theory 51 §2.1. Torsors 51 §2.2. Active and passive transformations 54 §2.3. Cayley graphs and the geometry of groups 56 §2.4. Group extensions 62 vii
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