Africa and Mathematics: From Colonial Findings Back to the Ishango Rods (Mathematics, Culture, and the Arts)

"This volume on ethnomathematics in Central Africa fills a gap in the current literature, focusing on a region rarely explored by other publications. It highlights the discovery of the Ishango rod, which was found to be the oldest mathematical tool in humanity's history, thereby shifting the origin of mathematics to the heart of Africa, and explores the different scientific hypotheses that emerged as a result. While it contains some high-level mathematics, the non-mathematical reader can easily skip these portions and enjoy the book's survey of African history, culture, and art." -- Prové de l'editor Foreword 6 Introduction 8 About the Contents 8 About the Language 8 Geographical Considerations 9 General Considerations 9 Acknowledgments 10 Contents 12 About the Author 15 Part I: Mathematics in the Heart of Africa 16 Chapter 1: Rationale and Sources 17 Is Mathematics Universal? 17 Teaching Geometry 18 Consequences for Textbooks 20 Ethnomathematical Studies 23 Collecting Data 25 Oral History 27 Recent Observations 30 Archeological Findings 30 Chapter 2: Storytelling and Music 33 Stories 33 The Moon and Lunar Months 34 Conceptions of the Universe in Traditional Rwanda 35 Names for Heavenly Phenomena in Rwanda 37 Science Fiction in Rwanda 40 Music 43 Chapter 3: Creative Counting 48 Number Bases 48 The Doubling Principle and Mixed Counting Forms 50 Counting Gestures 52 More Creativity 57 More Complicated Number Bases 58 Words for Larger Numbers 59 The Question of Base 12 61 Classical Explanations for Base 12 62 Duodecimal Hand Counting 64 Base 12 Vocabulary 65 Chapter 4: Drawings 67 Graphs, Mathematics, and Africa 67 Stories About African Graphs 68 Sand Drawings, Memory Aids, and Drawing Patterns 69 Reasoning with Sand Drawings 70 More Mathematics in Sand Drawings 71 Forms, Shapes, and Their Analysis 75 African Forms in Mathematics Books 77 Geometric Patterns 78 “Pure” Drawings 80 Frieze Patterns and Crystallography 83 African Fractals 84 Chapter 5: Reasoning Without Writing 89 Multiplication on the Hands 89 Mental Arithmetic 90 Little Ropes as Means of Notation in Burundi 91 Counting Strings and Sticks in Africa 93 The Use of Counting Strings in the West and Asia 96 Games of Strategy 99 Mancala 99 Mancala Rules 102 Example of a Game 105 The More Complicated Igisoro Game 106 Examples of First Moves 108 Example of an Igisoro Game 111 Notes 116 Chapter 6: Multiplication in the Yoruba and “Ethiopian” Way 118 The Yoruba Region 118 Description of 1887 119 Yoruba Multiplication 122 “Ethiopian” Multiplication 124 Tailor or Hieroglyph Algorithm 126 Variations on Doubling Algorithm 127 Fanciful Representation of Ethiopian Multiplication 128 Examples of Calculation Game 128 Mathematical Considerations 131 Algebra and Osteopaths 133 Part II: The Ishango Rod(s) 135 Chapter 7: The Ishango Site 136 Choice of Site 136 The Excavations 138 The Ishango People 142 A Particular Finding 143 Dating the Rod 145 Chapter 8: Mathematical Carvings 150 Description of Carvings 150 Hypothesis of de Heinzelin 152 A Lunar Calendar 154 Some Objections 156 A Third Look at the Carvings 157 Circumstantial Evidence for Alternate Interpretation 159 Chapter 9: Missing Link 161 Linguistic Missing Link 161 Archeological Missing Link 162 Mathematical Missing Link 164 Cultural Missing Link 167 Genetic Missing Link 169 Ethnographic Missing Link 171 Too Many Missing Links 174 Chapter 10: Not Out of Africa 175 Other Opinions 175 Still Older Mathematical Artifacts 176 Antiracist Mathematics 178 The Skeptical Inquirer 179 The Black Athena Debate 181 A 100-Year-Old Authority on Ishango 183 Chapter 11: A Second Rod 185 The Rediscovery of the Ishango Rod 185 A Second Ishango Rod 188 A Second Opinion 192 Part III: Epilogue 197 Chapter 12: Museum Visit, Teaching, Research 198 Mathematical Tour Through an Africa Museum 198 A General-Level Ethnomathematical Quiz 204 Answers to Quiz 208 A High-School-Level Ethnomathematical Quiz 211 University Level: Examples of Theses 213 Scientific Research 218 Origins of the Illustrations 223 References 227 Index 232 Africa + Mathematics: Summary 234