Mathematics, Form and Function

This book records my efforts over the past four years to capture in words a description of the form and function of Mathematics, as a background for the Philosophy of Mathematics. My efforts have been encouraged by lec­ tures that I have given at Heidelberg under the auspices of the Alexander von Humboldt Stiftung, at the University of Chicago, and at the University of Minnesota, the latter under the auspices of the Institute for Mathematics and Its Applications. Jean Benabou has carefully read the entire manuscript and has offered incisive comments. George Glauberman, Car­ los Kenig, Christopher Mulvey, R. Narasimhan, and Dieter Puppe have provided similar comments on chosen chapters. Fred Linton has pointed out places requiring a more exact choice of wording. Many conversations with George Mackey have given me important insights on the nature of Mathematics. I have had similar help from Alfred Aeppli, John Gray, Jay Goldman, Peter Johnstone, Bill Lawvere, and Roger Lyndon. Over the years, I have profited from discussions of general issues with my colleagues Felix Browder and Melvin Rothenberg. Ideas from Tammo Tom Dieck, Albrecht Dold, Richard Lashof, and Ib Madsen have assisted in my study of geometry. Jerry Bona and B. L. Foster have helped with my examina­ tion of mechanics. My observations about logic have been subject to con­ structive scrutiny by Gert Miiller, Marian Boykan Pour-El, Ted Slaman, R. Voreadou, Volker Weispfennig, and Hugh Woodin.

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.