Preface ix 0 Introduction 1PART I: History and Philosophy of Mathematics 5 1 Egyptian Mathematics 7 2 Scales of Notation 11 3 Prime Numbers 15 4 Sumerian-Babylonian Mathematics 21 5 More about Mesopotamian Mathematics 25 6 The Dawn of Greek Mathematics 29 7 PVthagoras and His School 33 8 Perfect Numbers 37 9 Regular Polyhedra 41 10 The Crisis of incommensurables 47 11 Whom Heraclitus to Democritus 53 12 Mathematics in Athens 59 13 Plato and Aristotle on Mathematics 67 14 Constructions with Ruler and Compass 71 15 The Impossibility of Solving the Classical Problems 79 16 Euclid 83 17 Non-Euclidean Geometry and Hilbert's Axioms 89 18 Alexandria from 3o0 BC to 20O BC 93 19 Archimedes 97 ZO Alexandria from 200 BC to 500 AD 103 21 Mathematics in China and india 111 22 Mathematics in Islamic Countries 117 23 New Beginnings in Europe 121 24 Mathematics in the Renaissance 125 25 The Cubic and Quartic Equations 133 26 Renaissance Mathematics Continued 139 27 The Seventeenth Century in dance 145 28 The Seventeenth Century Continued 153 29 Leibniz 159 30 The Eighteenth Century 163 31 The Law of Quadratic Reciprocity 169PART II: FOundations of Mathematics 173 1 The Number System 175 2 Natural Numbers (Peano's Approach) 179 3 The integers 183 4 The auctionals 187 5 The Real Numbers 191 6 Complex Numbers 195 7 The Fundamental Theorem of Algebra 199 8 Quaternions 2O3 9 Quaternions Applied to Number Theory 207 10 Quaternions Applied to Physics 211 11 Quaternions in Quantum Mechanics 215 12 Cardinal Numbers 219 13 Cardinal Arithmetic 223 14 Continued "actions 227 15 The Fundamental Theorem of Arithmetic 231 16 Linear Diophantine Equations 233 17 Quadratic Surds 237 18 Pythagorean triangles and Fermat's Last Theorem 241 19 What is a Calculation? 245 20 Recursive and Recursively Enumerable Sets 251 21 Hilbert's Tenth Problem 255 22 Lambda Calculus 259 23 Logic from Aristotle to Russell 265 24 Intuitionistic Propositional Calculus 271 25 How to interpret intuitionistic Logic 277 26 Intuitionistic Predicate Calculus 281 27 Intuitionistic Type Theory 285 28 Godel's Theorems 289 29 Proof of G5del's incompleteness Theorem 291 30 More about G5del's Theorems 293 31 Concrete Categories 295 32 Graphs and Categories 297 33 Functors 299 34 Natural transformations 303 35 A Natural transformation between Vector Spaces 307References 311Index 321
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