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Inhaltsverzeichnis
I Origins of Formal Structure. - 1. 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. Kepler s Laws and Newton s Laws. - 6. Differential Equations. - 7. Foundations of Calculus. - 8. Approximations and Taylor s 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. Kepler s Laws. - 2. Momentum, Work, and Energy. - 3. Lagrange s Equations. - 4. Velocities and Tangent Bundles. - 5. Mechanics in Mathematics. - 6. Hamilton s Principle. - 7. Hamilton s 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? . - XIIThe 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.
