Mehr zum Titel: | I Origins of Formal Structure1. The Natural Numbers -- 2. Infinite Sets -- 3. Permutations -- 4. Time and Order -- 5. Space and Motion -- 6. Symmetry -- 7. Transformation Groups -- 8. Groups -- 9. Boolean Algebra -- 10. Calculus, Continuity, and Topology -- 11. Human Activity and Ideas -- 12. Mathematical Activities -- 13. Axiomatic Structure -- II From Whole Numbers to Rational Numbers -- 1. Properties of Natural Numbers -- 2. The Peano Postulates -- 3. Natural Numbers Described by Recursion -- 4. Number Theory -- 5. Integers -- 6. Rational Numbers -- 7. Congruence -- 8. Cardinal Numbers -- 9. Ordinal Numbers -- 10. What Are Numbers? -- III Geometry -- 1. Spatial Activities -- 2. Proofs without Figures -- 3. The Parallel Axiom -- 4. Hyperbolic Geometry -- 5. Elliptic Geometry -- 6. Geometric Magnitude -- 7. Geometry by Motion -- 8. Orientation -- 9. Groups in Geometry -- 10. Geometry by Groups -- 11. Solid Geometry -- 12. Is Geometry a Science? -- IV Real Numbers -- 1. Measures of Magnitude -- 2. Magnitude as a Geometric Measure -- 3. Manipulations of Magnitudes -- 4. Comparison of Magnitudes -- 5. Axioms for the Reals -- 6. Arithmetic Construction of the Reals -- 7. Vector Geometry -- 8. Analytic Geometry -- 9. Trigonometry -- 10. Complex Numbers -- 11. Stereographic Projection and Infinity -- 12. Are Imaginary Numbers Real? -- 13. Abstract Algebra Revealed -- 14. The Quaternions-and Beyond -- 15. Summary -- V Functions, Transformations, and Groups -- 1. Types of Functions -- 2. Maps -- 3. What Is a Function? -- 4. Functions as Sets of Pairs -- 5. Transformation Groups -- 6. Groups -- 7. Galois Theory -- 8. Constructions of Groups -- 9. Simple Groups -- 10. Summary: Ideas of Image and Composition -- VI Concepts of Calculus -- 1. Origins -- 2. Integration -- 3. Derivatives -- 4. The Fundamental Theorem of the Integral Calculus -- 5. Keplers Laws and Newtons Laws -- 6. Differential Equations -- 7. Foundations of Calculus -- 8. Approximations and Taylors Series -- 9. Partial Derivatives -- 10. Differential Forms -- 11. Calculus Becomes Analysis -- 12. Interconnections of the Concepts -- VII Linear Algebra -- 1. Sources of Linearity -- 2. Transformations versus Matrices -- 3. Eigenvalues -- 4. Dual Spaces -- 5. Inner Product Spaces -- 6. Orthogonal Matrices -- 7. Adjoints -- 8. The Principal Axis Theorem -- 9. Bilinearity and Tensor Products -- 10. Collapse by Quotients -- 11. Exterior Algebra and Differential Forms -- 12. Similarity and Sums -- 13. Summary -- VIII Forms of Space -- 1. Curvature -- 2. Gaussian Curvature for Surfaces -- 3. Arc Length and Intrinsic Geometry -- 4. Many-Valued Functions and Riemann Surfaces -- 5. Examples of Manifolds -- 6. Intrinsic Surfaces and Topological Spaces -- 7. Manifolds -- 8. Smooth Manifolds -- 9. Paths and Quantities -- 10. Riemann Metrics -- 11. Sheaves -- 12. What Is Geometry? -- IX Mechanics -- 1. Keplers Laws -- 2. Momentum, Work, and Energy -- 3. Lagranges Equations -- 4. Velocities and Tangent Bundles -- 5. Mechanics in Mathematics -- 6. Hamiltons Principle -- 7. Hamiltons Equations -- 8. Tricks versus Ideas -- 9. The Principal Function -- 10. The Hamilton-Jacobi Equation -- 11. The Spinning Top -- 12. The Form of Mechanics -- 13. Quantum Mechanics -- X Complex Analysis and Topology -- 1. Functions of a Complex Variable -- 2. Pathological Functions -- 3. Complex Derivatives -- 4. Complex Integration -- 5. Paths in the Plane -- 6. The Cauchy Theorem -- 7. Uniform Convergence -- 8. Power Series -- 9. The Cauchy Integral Formula -- 10. Singularities -- 11. Riemann Surfaces -- 12. Germs and Sheaves -- 13. Analysis, Geometry, and Topology -- XI Sets, Logic, and Categories -- 1. The Hierarchy of Sets -- 2. Axiomatic Set Theory -- 3. The Propositional Calculus -- 4. First Order Language -- 5. The Predicate Calculus -- 6. Precision and Understanding -- 7. Gödel Incompleteness Theorems -- 8. Independence Results -- 9. Categories and Functions -- 10. Natural Transformations -- 11. Universals -- 12. Axioms on Functions -- 13. Intuitionistic Logic -- 14. Independence by Means of Sheaves -- 15. Foundation or Organization? -- XII The Mathematical Network -- 1. The Formal -- 2. Ideas -- 3. The Network -- 4. Subjects, Specialties, and Subdivisions -- 5. Problems -- 6. Understanding Mathematics -- 7. Generalization and Abstraction -- 8. Novelty -- 9. Is Mathematics True? -- 10. Platonism -- 11. Preferred Directions for Research -- 12. Summary -- List of Symbols. |
