Mathematics Form and Function

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.