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CONTENTS PREFACE V INTRODUCTION 3 PART I. ESSAYS 11 CHAPTER 1. From Myth to Mathematics to Knowledge 13 What is Mathematics? 13 Mathematics and Myths 14 Pre-Greek Mathematics 20 What is in a Name? 22 Universality of Mathematics 28 Some Contrasts between Greek and Present-Day Mathematics 30 Euclid's Elements 33 Mathematics on the Rise 37 Prefabricated Mathematics 42 Relation between Mathematics and Science 45 The Abstractness of Mathematics 48 CHAPTER 2. How History of Science Differs from Other History 59 Slowness of the Rise of History of Science 63 Trends and Individuality 73 Relations to Source Works 83 Documentation of Greek Science 85 Limitations of Subject Matter 96 The Quest for Spectacularity 101 Medievalism 106 Progress and Continuity 113 Conventionalism 120 The Mathematization of Science 126 CHAPTER 3. Revolutions in Physics and Crises in Mathematics 131 The Thesis of T. S. Kuhn 131 The Role of Max Planck 133 Crises in Mathematics 138 [vii] CONTENTS C HAP T E R 4. Aristotle's Physics and Today's Physics 143 Physics and Mathematical Physics 143 Physics and Philosophy 145 Time in Physics 147 Physics and Cosmology 148 Necessity and Chance 149 Chora ("Space") and Tapos ("Place") 152 Space and Matter 156 Gravitational Space 158 The Void (Kenon) 160 Processes-in-Nature 160 The Scope of Processes-in-Nature 162 Opposites 165 Motion of Motion (kineseos kinesis) 166 Simple and Composite Changes 170 Aristotle's Theory of Tapos 172 The Universe as One Physical System 175 Act and Potency 177 Conclusion 178 C HAP T E R 5. The Role of Mathematics in the Rise of Mechanics 179 Before the 17th Century 180 The 17th Century 187 The 18th Century 193 The Principle of Least Action 203 C HAP T E R 6. The Significance of Some Basic Mathematical Conceptions for Physics 209 I. The Significance of Multiplication 210 Greek Magnitudes 210 Physical Dimensions 211 Proportions 213 The Fundamental Theorem of Prime Numbers 214 II. The Significance of Functions 216 The Rise of Functions 216 [viii] CONTENTS The First Occurrence of Functions 222 Equations of State 224 III. The Significance of Real Numbers 226 Quality and Quantity 226 Vectors and Tensors 229 Importance of Scalars 232 IV. The Significance of Complex Numbers 234 The Complexification of Physics 234 Complex Numbers in the 18th Century 237 Complex Numbers in Optics 240 v. The Significance of Mathematical Space 243 The Space-Time Problem 243 Space before Aristotle 245 Space in Aristotle 246 Three-Dimensionality of Space 248 Mathematical Schema of Time 250 The Future of the Space-Time Problem 252 CHAPTER 7. The Essence of Mathematics 255 Relation to Science 255 Creative Formulas 257 Foundations: Mathematical Logic 260 Counterexamples 264 Constructiveness: Approximations 267 Space in Mathematics 270 CHA P TE R 8. The Essence of Analysis 275 I. Meaning of Analysis 275 Relation to Science 275 Derivatives 276 Real Numbers 277 Magnitudes 278 II. Stages of Development 279 Irrational Numbers 279 Areas and Volumes 280 Set Theory 280 Analytic Geometry 281 19th Century 282 [ix] CONTENTS 20th Century 282 πι. Functions 283 Higher Derivatives 283 General Functions 284 Power Series 285 Piecewise Analytic Functions 287 Fourier Series 288 Analytic Theory of Numbers 291 Least Square Approximation 292 Functions of Real Variables 293 iv. Analysis and Space 295 Conformal Mapping 295 Degrees of Freedom 297 Tensorial Functions 299 PART II. BIOGRAPHICAL SKETCHES 301 INDEX 373 [χ] THE ROLE OF MATHEMATICS IN THE RISE OF SCIENCE ...

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