Mathematics Form and Function

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?.
• 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.