Mathematics: Form and Function: Form and Function

Main subject categories: • Philosophy of Mathematics • Development of Mathematics • History of Mathematics • Mathematics in General • Mathematics for NonmathematiciansChapter Headings: • 1. Origins of Formal Structure • 2. From Whole Numbers to Rational Numbers • 3. Geometry • 4. Real Numbers • 5. Functions, Transformations, and Groups • 6. Concepts of Calculus • 7. Linear Algebra • 8. Forms of Space • 9. Mechanics • 10. Complex Analysis and Topology • 11. Sets, Logic, and Categories • 12. The Mathematical Network • Bibliography • List of Symbols • IndexFrom the Preface 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 lectures 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, Carlos 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 examination of mechanics. My observations about logic have been subject to constructive scrutiny by Gert Miiller, Marian Boykan Pour-El, Ted Slaman, ... Cover......Page 1 Title page......Page 3 Copyright page......Page 4 Preface......Page 5 Contents......Page 7 Introduction......Page 13 CHAPTER I. Origins of Formal Structure......Page 18 1. The Natural Numbers......Page 19 2. Infinite Sets......Page 22 3. Permutations......Page 23 4. Time and Order......Page 25 5. Space and Motion......Page 28 6. Symmetry......Page 31 7. Transformation Groups......Page 33 8. Groups......Page 34 9. Boolean Algebra......Page 38 10. Calculus, Continuity, and Topology......Page 41 11. Human Activity and Ideas......Page 46 12. Mathematical Activities......Page 48 13. Axiomatic Structure......Page 52 1. Properties of Natural Numbers......Page 54 2. The Peano Postulates......Page 55 3. Natural Numbers Described by Recursion......Page 59 4. Number Theory......Page 60 5. Integers......Page 62 6. Rational Numbers......Page 63 7. Congruence......Page 64 8. Cardinal Numbers......Page 66 9. Ordinal Numbers......Page 68 10. What Are Numbers?......Page 70 1. Spatial Activities......Page 73 2. Proofs without Figures......Page 75 3. The Parallel Axiom......Page 79 4. Hyperbolic Geometry......Page 82 5. Elliptic Geometry......Page 85 6. Geometric Magnitude......Page 87 7. Geometry by Motion......Page 88 8. Orientation......Page 94 9. Groups in Geometry......Page 97 10. Geometry by Groups......Page 99 11. Solid Geometry......Page 101 12. Is Geometry a Science?......Page 103 1. Measures of Magnitude......Page 105 2. Magnitude as a Geometric Measure......Page 106 3. Manipulations of Magnitudes......Page 109 4. Comparison of Magnitudes......Page 110 5. Axioms for the Reals......Page 114 6. Arithmetic Construction of the Reals......Page 117 7. Vector Geometry......Page 119 8. Analytic Geometry......Page 121 9. Trigonometry......Page 122 10. Complex Numbers......Page 126 11. Stereographic Projection and Infinity......Page 128 12. Are Imaginary Numbers Real?......Page 130 13. Abstract Algebra Revealed......Page 131 14. The Quaternions—and Beyond......Page 132 15. Summary......Page 133 1. Types of Functions......Page 135 2. Maps......Page 137 3. What Is a Function?......Page 138 4. Functions as Sets of Pairs......Page 140 5. Transformation Groups......Page 145 6. Groups......Page 147 7. Galois Theory......Page 150 8. Constructions of Groups......Page 154 9. Simple Groups......Page 158 10. Summary: Ideas of Image and Composition......Page 159 1. Origins......Page 162 2. Integration......Page 164 3. Derivatives......Page 166 4. The Fundamental Theorem of the Integral Calculus......Page 167 5. Kepler's Laws and Newton's Laws......Page 170 6. Differential Equations......Page 173 7. Foundations of Calculus......Page 174 8. Approximations and Taylor's Series......Page 179 9. Partial Derivatives......Page 180 10. Differential Forms......Page 185 11. Calculus Becomes Analysis......Page 190 12. Interconnections of the Concepts......Page 195 1. Sources of Linearity......Page 197 2. Transformations versus Matrices......Page 200 3. Eigenvalues......Page 203 4. Dual Spaces......Page 205 5. Inner Product Spaces......Page 208 6. Orthogonal Matrices......Page 210 7. Adjoints......Page 212 8. The Principal Axis Theorem......Page 214 9. Bilinearity and Tensor Products......Page 216 10. Collapse by Quotients......Page 220 11. Exterior Algebra and Differential Forms......Page 222 12. Similarity and Sums......Page 225 13. Summary......Page 230 1. Curvature......Page 231 2. Gaussian Curvature for Surfaces......Page 234 3. Arc Length and Intrinsic Geometry......Page 238 4. Many-Valued Functions and Riemann Surfaces......Page 240 5. Examples of Manifolds......Page 245 6. Intrinsic Surfaces and Topological Spaces......Page 248 7. Manifolds......Page 251 8. Smooth Manifolds......Page 256 9. Paths and Quantities......Page 259 10. Riemann Metrics......Page 263 11. Sheaves......Page 264 12. What Is Geometry?......Page 268 1. Kepler's Laws......Page 271 2. Momentum, Work, and Energy......Page 276 3. Lagrange's Equations......Page 279 4. Velocities and Tangent Bundles......Page 286 5. Mechanics in Mathematics......Page 289 6. Hamilton's Principle......Page 290 7. Hamilton's Equations......Page 294 8. Tricks versus Ideas......Page 299 9. The Principal Function......Page 301 10. The Hamilton—Jacobi Equation......Page 304 11. The Spinning Top......Page 307 12. The Form of Mechanics......Page 313 13. Quantum Mechanics......Page 315 1. Functions of a Complex Variable......Page 319 2. Pathological Functions......Page 322 3. Complex Derivatives......Page 324 4. Complex Integration......Page 329 5. Paths in the Plane......Page 334 6. The Cauchy Theorem......Page 340 7. Uniform Convergence......Page 345 8. Power Series......Page 348 9. The Cauchy Integral Formula......Page 350 10. Singularities......Page 353 11. Riemann Surfaces......Page 356 12. Germs and Sheaves......Page 363 13. Analysis, Geometry, and Topology......Page 368 CHAPTER XI. Sets, Logic, and Categories......Page 370 1. The Hierarchy of Sets......Page 371 2. Axiomatic Set Theory......Page 374 3. The Propositional Calculus......Page 380 4. First Order Language......Page 382 5. The Predicate Calculus......Page 385 6. Precision and Understanding......Page 389 7. Goedel Incompleteness Theorems......Page 391 8. Independence Results......Page 395 9. Categories and Functions......Page 398 10. Natural Transformations......Page 402 11. Universals......Page 404 12. Axioms on Functions......Page 410 13. Intuitionistic Logic......Page 414 14. Independence by Means of Sheaves......Page 416 15. Foundation or Organization?......Page 418 CHAPTER XII. The Mathematical Network......Page 421 1. The Formal......Page 422 2. Ideas......Page 427 3. The Network......Page 429 4. Subjects, Specialties, and Subdivisions......Page 434 5. Problems......Page 440 6. Understanding Mathematics......Page 443 7. Generalization and Abstraction......Page 446 8. Novelty......Page 450 9. Is Mathematics True?......Page 452 10. Platonism......Page 459 11. Preferred Directions for Research......Page 461 12. Summary......Page 465 Bibliography......Page 469 List of Symbols......Page 473 Index......Page 475 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. This book is intended to describe the practical and conceptual origins of Mathematics and the character of its development - not in historical terms, but in intrinsic terms