Africa and Mathematics - From Colonial Findings Back to the Ishango Rods [1 ed.] 978-3-030-04036-9

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Mathematics, Culture, and the Arts

Dirk Huylebrouck

Africa and Mathematics From Colonial Findings Back to the Ishango Rods

Mathematics, Culture, and the Arts Series editors Jed Z. Buchwald Caltech, Pasadena, CA, USA Marjorie Senechal Smith College, Northampton, MA, USA Gizem Karaali Pomona College, Claremont, CA, USA

The series Mathematics in Culture and the Arts publishes books on all aspects of the relationships between mathematics and the mathematical sciences and their roles in culture, art, architecture, literature, and music. This new book series will be a major resource for researchers, educators, scientifically minded artists, and students alike. More information about this series at http://www.springer.com/series/13129

Dirk Huylebrouck

Africa and Mathematics From Colonial Findings Back to the Ishango Rods

Dirk Huylebrouck University of Leuven Ghent, Belgium

ISSN 2520-8578     ISSN 2520-8586 (electronic) Mathematics, Culture, and the Arts ISBN 978-3-030-04036-9    ISBN 978-3-030-04037-6 (eBook) https://doi.org/10.1007/978-3-030-04037-6 Library of Congress Control Number: 2018964713 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: First image: Africa Museum Tervuren, Belgium (General Register Number photo: 1969.59.636, negative G 4878). Second image: photo by the author Third image: Royal Belgian Institute for Natural Sciences, Belgium. Fourth image: drawing by the author based on an illustration in “Les Peintures Murales Intérieures des Habitations du Migongo, Rwanda”, Africa-Tervuren XVII-2, 1972, by Celis G., and Celis Th., with approval from the authors. Fifth image: Royal Belgian Institute for Natural Sciences, Belgium. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

Ishango. A name that sounds like a river, a continent, an era, or a lost people. A river: close to the river Semliki, at the shores of Lake Edward, not far from two historical routes, one following the Congo River, and one along the sources of the Nile, an important axis for trade and exchange between civilizations. A continent: Africa, the birthplace of human civilization on Earth. An era: the time of the first human hunters and gatherers, several thousand years ago. The epoch of exploration of the Ishango site in the middle of the twentieth century by several Belgian expeditions and, later, by an international archeological group. A lost people: who were they, these early hunters and gatherers? When Professor Jean de Heinzelin did excavations at the Ishango site, he brought many apparently irrelevant objects with him. Later, this Belgian archeologist would earn international fame. Among the findings was a tiny rod, seemingly without major significance, but undoubtedly several thousand years old. Until a closer look was taken at the carved lines on the rod. Suddenly, after being buried for about 200 centuries under many layers of earth, the rod was revived: it was studied, analyzed, recorded in an inventory, discussed, and recognized as the undeniably oldest mathematical tool of humanity. What an honor! Suddenly the history of mathematics shifted from the supposed cradle of civilization around the Mediterranean Sea to the heart of Africa, not far from where, according to the commonly accepted hypothesis, the human race was born. Then, after this short moment of ephemeral glory, the Ishango rod fell again into obscurity, to be forgotten for about 40 years in the depository of the Royal Belgian Institute for Natural Sciences. Until another scientist, Belgian mathematician Dirk Huylebrouck, took steps to allow it to be recognized at its true value. A new era of intense activity surrounding the Ishango rod was born. Recognized by the scientific community and the cultural and political worlds, the Ishango rod resurfaced and enjoyed its deserved glory. The government of the Brussels Region made it the symbol of its program to motivate young people to get an education and in particular to choose a career in science. The Ishango rod has become the subject of new studies and analyses. Innovative v

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hypotheses have been raised about its mathematical interpretation, and many new scientific papers have been devoted to it. Finally, it was displayed in the permanent exposition of a museum, so that the public could admire it, in a separate arrangement, dedicated solely to the study of the Ishango era and Ishango civilization. I happened to meet Dirk Huylebrouck in 1994, in the Central African country of Burundi, where he had invited me for some lectures at the University of Bujumbura. This engaging and polite man, specialized in ethnomathematics and passionate Africa enthusiast, had several projects in mind, including one to draw to the Ishango rod the attention it deserves and its true worth recognized. Back in Europe, because the outbreak of genocide in neighboring Rwanda had ended his contract, he exerted considerable effort to bring to light this artifact of human civilization. The Ishango rod, small in length, about 10 cm, but massive in terms of historical meaning and value, is not a rod in the true sense of the word but rather a small bone, with remarkable carvings, at regular distances and in small groupings, which makes it possible to interpret the rod as humanity’s first calculating tool. Moreover, the Ishango rod provides a space-time relation between, on the one hand, the first civilizations to attempt calculations and develop mathematics and, on the other hand, our modern technological civilization based on science and mathematics. It links the African continent to the Western world. This book sketches the discovery of the extraordinary Ishango rod and the different scientific hypotheses regarding the object. The book can be read as a novel, without many mathematical equations or proofs, and yet it explains in a simple way how our African ancestors came to count in different systems, how the different hypotheses about the rod were retained or rejected, and how the history of mathematics was turned upside down. Bold conjectures, but eventually confirmed when the existence of a second rod was revealed by the discoverer of the first Ishango rod, Jean de Heinzelin, on his death bed. Go and admire these little rods at the Museum of Natural Science. Somebody used it to count some 22,000 years ago, like our children do at school to this day. Vladimir Pletser Former next astronaut for Belgium, Director of Space Training Operations (2018–), Blue Abyss, United Kingdom Chinese Academy of Sciences (2016 – 2017), Beijing, China European Space Agency (1984 – 2016), Noordwijk, The Netherlands

Introduction

About the Contents This first book about ethnomathematics in Central Africa groups several papers that appeared in scientific and popular science journals. The text is also based on a series of lectures for specialists as well as on talks for a larger lay audience. Other parts of the book find their roots in various scientific publications, requiring high-level mathematics, sometimes up to the research level. However, the nonmathematical reader can safely skip the few specialized mathematical passages, which are independent of the other parts. These difficult notions were retained in the hope of winning the esteem or at least a positive reaction for African mathematics. Mathematics released from its utilitarian aspects often seems a more noble kind of mathematics, especially when these applications are executed on muddy stones, silly little shells, or tiny bones. Consequently, the few difficult passages seemed appropriate in order to avoid having some readers look down on ethnomathematics and see it as mere recreational mathematics, lacking any scientific grounding. Thus, some parts of the book will seem too austere to some readers since a rigid style is usual in mathematical writing, while to others many parts of the book will look too much like a popularization. Still, overall, an anecdotal approach was avoided as much as possible.

About the Language Numerous problems arise when translating names and words from certain African languages into English. Examples include r/l-related sounds in some African words. Sometimes, two sources both use an r or an l for the same word, not because the r or l cannot be pronounced but because the actual sound is somewhere in between. Another problem is that many African languages use prefixes that are not recognized as such in English. In this connection, one should not talk about “Bantus” but about “tus” because both  the “ba” and the “s” point to a plural form. Literally, ”Ba-ntu” indeed means “human-s,” and the prefix “ba-” stands for the suffix “-s.” vii

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Geographical Considerations This book first appeared in Belgium, in Dutch and later in French as well, and so there is a special focus on the Democratic Republic of the Congo (henceforth called the Congo), Rwanda, and Burundi, since until 1960 Congo was a Belgian colony, while the latter were territories mandated to Belgium until 1962. The book completes a gap in the ethnomathematical literature, since publications  from France tend to concentrate on Senegal and other West African countries, while English publications often emphasize data about South or East Africa. For those historical reasons, references to other continents are limited, though ethnomathematics covers a large area, from the Andes Mountains to Siberia. This geographical restriction to the Central African region raises another question. Many remember the terrible genocide in Rwanda or the images spread through movies such as Hotel Rwanda, while Congo is often associated with rebellions, diseases, and misery. Thus, readers of this book may blame the author for neglecting the tragic situation and only presenting a different view from a nonpolitical and nonjournalistic point of view. Yet, for most dissertations about the history of the exact sciences in the West, this is the case as well, and only seldom is a remark made that no attention was paid to the Hundred Years’ War or to the rebels in the Balkans, the Basque Country, or Northern Ireland. This book shows it is possible to discuss Central Africa without falling into the customary political morass. A more substantial comment about ethnomathematics points to the danger of imposing a prejudiced view about ethnomathematics on readers. For that reason, the first part of the book presents examples of situations where African mathematics could have been used, so that the reader gets an idea about the typical setting. As the reader progresses, he will thus be able to get into a frame of mind for personally judging the value of the final chapters and the importance of the second part of the book. However, this does not imply an intention to replace a refutable Eurocentrism by an equally disprovable Afrocentrism. Of course, the focus on African realizations is unavoidable here because it is the subject of the book, but while some chapters could create the impression of an overenthusiasm for “Ishango mathematics”, other chapters supply many counterarguments and critical observations. If some authors quoted here criticize Greek mathematical realizations as if Hellenistic achievements were “all stolen from Africa,” the author certainly does not share their point of view.

General Considerations This book tries to deal with the nonmathematical backgrounds of African mathematics in a similar way to how books about Western mathematics do. Accordingly, just as books about European mathematicians contain some stories about the death of Descartes at the court of Queen Christina, these stories, of course, matter very little for the appreciation of Descartes’ mathematical work. Knowing whether

Introduction

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Christina was Swedish or Russian is interesting, but the Cartesian coordinates do not change regardless. It was sometimes difficult to decide whether an additional description of some mathematical concept should be included or, on the contrary, whether it would hinder the narrative. For example, the question of where exactly the Nyali people lived was omitted, since they were mentioned only occasionally, but the Yoruba are described in greater detail, as half a chapter is devoted to them. Yet, some considerations of a more general order were intentionally omitted. For instance, the question of whether this or that civilization was of a lower or higher mathematical level than another is not a priority to most mathematicians (though this is often brought up at talks!). Also, the question of why and how some regions developed while others seemed to stagnate is another topic for historians or sociologists. Finally, the author would like to express his disagreement with the notion that mathematicians should not waste their time searching for the roots of their science. By contrast, linguists and art historians routinely search for the oldest forms of written or spoken words of a language or discuss the oldest art forms. Mathematics began with a quest for logical structures, even in the absence of a counting– apes or parrots count too, but they don’t do mathematics. “Mathematics is as old as humanity,” Belgian Academy member Prof. F. Buekenhout proclaimed: man’s skill for a structured and intentional reflection distinguished himself from animals.

Acknowledgments Words of special thanks go to the following people: Antonio Arnaiz-Villena (Department of Immunology and Molecular Biology, Madrid, Spain), for the e-mail correspondence about DNA research; Tijl Beyl (architect and Africa traveler, Belgium) for some exceptional information about West Africa; friar Christian Blondeel (Gent, Belgium), for the graphic analysis of African sculptural art; Kenneth Brecher (Department of Astronomy at Boston University, USA) for debunking the myth about the round world of the Zulus; M. Caluwé and Vancleynenbreugel (Royal Museum for Central Africa, Tervuren, Belgium) for their help in searching for collection items; Ivan Cnop (VUB, Free University Brussels, Belgium) for the information about algorithms; M.-E. Dehousse (astronomer, Belgium) for the information about monoliths in Rwanda; Paulus Gerdes (Mozambique) for the many encouragements; Jos Gansemans (formerly of the Royal Museum for Central Africa, Tervuren, Belgium) for the help on ethnomusicology; Chris Impens (UGent, Belgium) for some skeptical considerations; Désiré Karangwa (University of Rwanda, Rwanda; University of Bujumbura, Burundi) for information about counting in Rwanda; Jean Meeus (Royal Observatory of Uccle, Belgium) for the elaboration of solar eclipse maps; Frank Michiels (formerly at the Royal Museum for Central Africa, Tervuren, Belgium) for the help in ethnomusicology; Joseph Ndarishikanye and Mathias Nduwingoma (formerly at the University of Bujumbura, Burundi) for the interviews about astronomy with their family members; Pierre Nzohabonayo (University of

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Bujumbura, Burundi) for the help on traditional astronomical considerations; Sofie Ponsaerts (KU Leuven, Belgium), who dedicated her master’s thesis to ethnomathematics as a follow-up to a conference given by the author; Jeffrey Shallit (Waterloo University, Canada) for the assistance in the problem of dating the Ishango rod. Many images were acquired thanks to the good services of Hein Vanhee, in charge of the Collection Management Division of the Royal Museum for Central Africa, Tervuren, Belgium, and of Laurence Cammaert, of the ADIA service of the Royal Belgian Institute for Natural Sciences in Brussels, Belgium. Architect Frederic Delannoy (Brussels, Belgium) adapted numerous illustrations posing miscellaneous problems. Finally, the encouragements of Dr. William Hawkins, University of the District of Columbia, must be highlighted: without his editorial assistance publication of this book would have been impossible.

Contents

Part I Mathematics in the Heart of Africa 1 Rationale and Sources ����������������������������������������������������������������������������    3 Is Mathematics Universal?������������������������������������������������������������������������    3 Teaching Geometry������������������������������������������������������������������������������������    4 Consequences for Textbooks���������������������������������������������������������������������    6 Ethnomathematical Studies������������������������������������������������������������������������    9 Collecting Data������������������������������������������������������������������������������������������   11 Oral History ����������������������������������������������������������������������������������������������   13 Recent Observations����������������������������������������������������������������������������������   16 Archeological Findings������������������������������������������������������������������������������   16 2 Storytelling and Music����������������������������������������������������������������������������   19 Stories��������������������������������������������������������������������������������������������������������   19 The Moon and Lunar Months��������������������������������������������������������������������   20 Conceptions of the Universe in Traditional Rwanda ��������������������������������   21 Names for Heavenly Phenomena in Rwanda��������������������������������������������   23 Science Fiction in Rwanda������������������������������������������������������������������������   26 Music����������������������������������������������������������������������������������������������������������   29 3 Creative Counting������������������������������������������������������������������������������������   35 Number Bases��������������������������������������������������������������������������������������������   35 The Doubling Principle and Mixed Counting Forms��������������������������������   37 Counting Gestures��������������������������������������������������������������������������������������   39 More Creativity������������������������������������������������������������������������������������������   44 More Complicated Number Bases ������������������������������������������������������������   45 Words for Larger Numbers������������������������������������������������������������������������   46 The Question of Base 12����������������������������������������������������������������������������   48 Classical Explanations for Base 12������������������������������������������������������������   49 Duodecimal Hand Counting����������������������������������������������������������������������   51 Base 12 Vocabulary������������������������������������������������������������������������������������   51

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4 Drawings ��������������������������������������������������������������������������������������������������   55 Graphs, Mathematics, and Africa��������������������������������������������������������������   55 Stories About African Graphs��������������������������������������������������������������������   56 Sand Drawings, Memory Aids, and Drawing Patterns������������������������������   57 Reasoning with Sand Drawings ����������������������������������������������������������������   58 More Mathematics in Sand Drawings��������������������������������������������������������   59 Forms, Shapes, and Their Analysis������������������������������������������������������������   63 African Forms in Mathematics Books ������������������������������������������������������   65 Geometric Patterns������������������������������������������������������������������������������������   66 “Pure” Drawings����������������������������������������������������������������������������������������   68 Frieze Patterns and Crystallography����������������������������������������������������������   71 African Fractals������������������������������������������������������������������������������������������   72 5 Reasoning Without Writing��������������������������������������������������������������������   77 Multiplication on the Hands����������������������������������������������������������������������   77 Mental Arithmetic��������������������������������������������������������������������������������������   78 Little Ropes as Means of Notation in Burundi������������������������������������������   79 Counting Strings and Sticks in Africa��������������������������������������������������������   81 The Use of Counting Strings in the West and Asia������������������������������������   84 Games of Strategy��������������������������������������������������������������������������������������   87 Mancala������������������������������������������������������������������������������������������������������   87 Mancala Rules��������������������������������������������������������������������������������������������   90 Example of a Game������������������������������������������������������������������������������������   93 The More Complicated Igisoro Game ������������������������������������������������������   94 Examples of First Moves ��������������������������������������������������������������������������   96 Example of an Igisoro Game ��������������������������������������������������������������������   99 Notes����������������������������������������������������������������������������������������������������������  104 6 Multiplication in the Yoruba and “Ethiopian” Way ����������������������������  107 The Yoruba Region������������������������������������������������������������������������������������  107 Description of 1887������������������������������������������������������������������������������������  108 Yoruba Multiplication��������������������������������������������������������������������������������  111 “Ethiopian” Multiplication������������������������������������������������������������������������  113 Tailor or Hieroglyph Algorithm ����������������������������������������������������������������  115 Variations on Doubling Algorithm������������������������������������������������������������  116 Fanciful Representation of Ethiopian Multiplication��������������������������������  117 Examples of Calculation Game ����������������������������������������������������������������  117 Mathematical Considerations��������������������������������������������������������������������  120 Algebra and Osteopaths ����������������������������������������������������������������������������  122 Part II The Ishango Rod(s) 7 The Ishango Site��������������������������������������������������������������������������������������  127 Choice of Site��������������������������������������������������������������������������������������������  127 The Excavations ����������������������������������������������������������������������������������������  129 The Ishango People������������������������������������������������������������������������������������  133

Contents

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A Particular Finding����������������������������������������������������������������������������������  134 Dating the Rod ������������������������������������������������������������������������������������������  136 8 Mathematical Carvings ��������������������������������������������������������������������������  141 Description of Carvings ����������������������������������������������������������������������������  141 Hypothesis of de Heinzelin������������������������������������������������������������������������  143 A Lunar Calendar��������������������������������������������������������������������������������������  145 Some Objections����������������������������������������������������������������������������������������  147 A Third Look at the Carvings��������������������������������������������������������������������  148 Circumstantial Evidence for Alternate Interpretation��������������������������������  150 9 Missing Link ��������������������������������������������������������������������������������������������  153 Linguistic Missing Link����������������������������������������������������������������������������  153 Archeological Missing Link����������������������������������������������������������������������  154 Mathematical Missing Link ����������������������������������������������������������������������  156 Cultural Missing Link��������������������������������������������������������������������������������  159 Genetic Missing Link��������������������������������������������������������������������������������  161 Ethnographic Missing Link������������������������������������������������������������������������  163 Too Many Missing Links ��������������������������������������������������������������������������  166 10 Not Out of Africa��������������������������������������������������������������������������������������  167 Other Opinions������������������������������������������������������������������������������������������  167 Still Older Mathematical Artifacts ������������������������������������������������������������  168 Antiracist Mathematics������������������������������������������������������������������������������  170 The Skeptical Inquirer��������������������������������������������������������������������������������  171 The Black Athena Debate��������������������������������������������������������������������������  173 A 100-Year-Old Authority on Ishango������������������������������������������������������  175 11 A Second Rod ������������������������������������������������������������������������������������������  177 The Rediscovery of the Ishango Rod��������������������������������������������������������  177 A Second Ishango Rod������������������������������������������������������������������������������  180 A Second Opinion��������������������������������������������������������������������������������������  184 Part III Epilogue 12 Museum Visit, Teaching, Research ��������������������������������������������������������  191 Mathematical Tour Through an Africa Museum����������������������������������������  191 A General-Level Ethnomathematical Quiz������������������������������������������������  197 Answers to Quiz������������������������������������������������������������������������������������  201 A High-School-Level Ethnomathematical Quiz����������������������������������������  204 University Level: Examples of Theses������������������������������������������������������  206 Scientific Research������������������������������������������������������������������������������������  211 Origins of the Illustrations������������������������������������������������������������������������������  217 References ��������������������������������������������������������������������������������������������������������  223 Index������������������������������������������������������������������������������������������������������������������  227 Africa + Mathematics: Summary.������������������������������������������������������������������  229

About the Author

Dirk Huylebrouck  worked at universities in the Congo for about 8 years until a diplomatic incident between Belgium and President Mobutu of the Congo interrupted his stay. Then he went to the University of Aveiro in Portugal and the European Division of Maryland University, until the majority of his American (military) students were sent to Iraq. He returned to Africa, specifically to Burundi, but for only a short time, because of the genocide in neighboring Rwanda. In 1996, he finally agreed to teach in the Faculty of Architecture of the KU Leuven (Belgium). Fortunately, in his mind he can still escape abroad, as he has been editing a column titled “The Mathematical Tourist” in the journal The Mathematical Intelligencer since 1997. However, he may soon have to flee abroad again, literally, because he has become (in)famous in Belgium for his work in popularizing errors in, for example, the Belgian Atomium landmark, the work of Leonardo da Vinci, a runway of the Brussels airport, the interpretation of the forbidden fruit in the “Mystic Lamb” by Jan and Hubert Van Eyck, and, most recently, in Norbert Francis Attard’s Fibonacci artwork.

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Part I

Mathematics in the Heart of Africa

Chapter 1

Rationale and Sources

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Is Mathematics Universal? Ethnomathematics concentrates on the importance of native culture for mathematics. In relation to mathematics, with its absolute or universal character, this may sound surprising since many think mathematics may be valid even “beyond” our universe. On occasion, it is suggested that “All mathematics is the same, wherever in the world it is done.” And, of course, it is. However, a more nuanced answer to this appropriate remark, which partially evades the question, underlines the importance of teaching methods. Indeed, imagine two students learning a foreign language. The first uses an old-­ fashioned scholarly method of parsing sentences using grammatical rules with declinations and conjugations. The second learns the language through a modern approach of short sentences that are repeated over and over, with or without audio help. After a while, both will understand the same language, but the two will seldom have an identical, thorough command of and linguistic feeling for the language. In mathematics, the situation is comparable. It can also be learned in several different ways, for instance, through a so-called classical method, with all kinds of geometrical and trigonometric drawings, or through more so-called modern mathematics with abstract structures and logical analysis or using the even more contemporary approach based on problem solving. In both situations, the average student will come away with a different view on the subject. A good reason to pay attention to the field of ethnomathematics certainly is the basic education it may provide for quantitative fields. Two Americans, J. Gay and M. Cole, carried out one of the first studies in the field, though it did not see much follow-up in mathematical circles (Gay and Cole, 1967). The authors went as Peace

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_1

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Corps volunteers to Liberia and wondered if there were any appreciable differences between the mathematical skills of American and African students. Gay and Cole subjected Liberian Kpelle people and groups of Americans to several sets of mathematical problems. In this way, they obtained statistics showing differences and similarities in “quantitative” skills such as estimating volumes or distances, measuring time, and others. Thus, the Kpelle appeared to be very good at visually estimating quantities, but they had more difficulties with counting large quantities. These skills can be useful for doing elementary arithmetic, but to classify Gay and Cole’s work as a study in the field of mathematics seems an exaggeration. In any case, the illustration shows an example of how Gay and Cole asked about estimating time intervals of 15, 30, 45, 60, 75, 90, 105, and 120 s. The Americans probably used the well-known trick of saying “1 Mississippi, 2 Mississippi, 3 Mississippi”, and so on, until they hit 15, 30, and eventually 120, to count seconds. Whatever the Kpelle’s method may heve been, the test showed no noticeable difference between the two groups for estimating longer periods. This test was chosen here as an example because it contradicts a widespread prejudice – that Africans have a “different notion” of time. Following Gay and Cole, this is not so, at least for larger intervals of time. Note other experiments do indeed reveal relevant differences in skills between the Kpelle and Americans (Fig 1.1). 40

Fig. 1.1  Graph based on data from Gay and Cole’s book Error in percent

30 20 10 0

15

30

45

60

75

90

105

120

–10 –20

Time in seconds Americans

Kpelle

Teaching Geometry Another good reason to attach importance to culture and mathematics is analogous to the previous one but is related to geometry and forms (Huylebrouck, 1995). The observation of some drawn representations and spatial forms do in fact differ. A very common opinion has it that Zulus, for example, do not observe some well-­ known optical illusions the way Westerners do. For instance, a set of straight lines crossed by another straight line seems skewed to many observers. Allegedly, the illusion does not have the same effect for Zulus because they live in a completely

Teaching Geometry

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round world, with only circular forms and without straight lines. However, as Kenneth Brecher (USA) pointed out in a private conversation, this story seems to be a apocryphal based solely on the exaggerated imagination of some overenthusiastic anthropologists (Fig. 1.2).

Fig. 1.2  Some well-known “optical illusions” where Zulus note no oddity, unlike, probably, the reader

Still, Martin Gardner too found it very surprising that some African traditions point to a completely circular view of the world. Gardner was a well-known American writer of popular mathematics books who earned a reputation through his battle with pseudoscience such as numerology or parapsychology. He illustrated his reflection by a few lines of a poem by John Updike: “Zulus Live in a Land without a Square.” When Zulus cannot smile, they frown, To keep an arc above the eye; When asked distances to town, They say: “as flies the butterfly.”

A Western version might read as follows (created by the author): When Europeans cannot smile, they look grumpy, To keep a straight line above the eye; When asked distances to town, They stretch the arm and say: “that far.”

Traditional architecture in Rwanda and Burundi suggests a similar view of the world in the so-called “Region of the Great Lakes”; this is the region around the lakes Tanganyika, Kivu, and Rutanzige. The Zulu culture of South Africa and the Central African people of Burundi did meet, despite the enormous distance between them. It is known, for instance, that a branch of the Zulu army reached the border of Burundi and was defeated there in 1854. The architectural archetype that settled itself through time in Burundi is the so-­ called rugo, a word meaning “many huts.” Surrounded by a kind of corral, it has separate sections for parents and children, as well as other parts for cattle. The outside walls of the huts, the separations between the different parts, and even the internal subdivisions within the huts are all circular (see Acquier, 1986). From a mathematical point of view, there is reason to prefer a curved line to a straight one. Sure, the latter corresponds to a linear and, thus, easier approximation in many problems, but a circle has the pleasing property that it surrounds the largest

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surface area with the smallest circumference. Thus, mathematically, a round corral seems a better solution, since in this way the smallest amount of fence will confine the largest quantity of cattle (Fig. 1.3).

Fig. 1.3  The round universe of traditional rugo houses and corrals from Burundi

Another example is related to descriptive geometry and the explanation of the construction of perspective drawings. The “naïve” painting of Central Africa is well known today thanks to the attention paid to it in art museums, in the media, or in public places. Some scenes do not seem to follow the usual rules of perspective. This disturbs the unconsciously trained Western eye, inasmuch as some think that the representation is erroneous. Yet the African representation is correct: it corresponds to a parallel projection. This difference in the traditional representation of a three-dimensional object can have consequences for teaching methods in geometry or perspective classes (Fig. 1.4).

Consequences for Textbooks Around the world, students struggle with exercises of the kind “a train leaves town A at a speed of 100 km/h, while another train leaves city B at the same time, but at a speed of 50 km/h. If the distance between both cities is 450 km, when will these trains cross?” Yet, in Burundi, for instance, there are no trains. Following some ethnomathematicians, similar misguided problems about purchasing a fancy car or planning a holiday cruise would serve as a bad (negative) model for those students who can barely pay for their own studies. Thus, these socially committed ethnomathematicians suggest mathematical problems of a different kind be posed. A question adapted to Buganda culture would read as follows: Let 1 cow = 2500 shells; 1 male slave = 1 cow;

Consequences for Textbooks

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Fig. 1.4  The Western perspective and a parallel projection

1 female slave = 4 to 5 cows; 5 goats = 1 cow; 1 ivory tooth = 1000 shells. How many ivory teeth does one get for 2 goats?

Note that feminist readers will be amused to see that a woman is worth four to five times as much as a man. Of course, a similar argument would hold with respect to Western children, who might not be very interested in questions that have nothing to do with their lives. Some mathematics teachers propose changing textbooks by adding more socially motivated exercises  (Nelson et  al., 1993; Lerman, 1994). For example, students could be asked to compute the (low) percentage of the total sales price that goes to a banana grower, or they could do compile statistics on social problems. Such exercises have their merits but also many additional ramifications and indirect considerations; they can distract students’ attention away from problems’ mathematical content (Fig. 1.5).

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Banana: 1.01€

0.11 Farmer 0.35 0.15 0.15 0.1 Packing Salesman Transport Importer Wholesaler 0.15

Fig. 1.5  Socially tinted mathematical problems: how much more does a cultivator get, proportionally (left)? Play with statistical diagrams (right)

Some companies that specialize in mathematics educational material sell so-­ called “multicultural posters” designed to appeal to a range of students, from Navajo youngsters to Irish students. A picture of Africa is accompanied by the following caption: “The Ashanti weighed gold dust and yellow brass based on a meticulous measuring system.” These posters illustrate the importance of ethnomathematics for the self-respect and image of students from those cultures. Professor William A.  Hawkins edited a poster about African and African-­ American Pioneers in Mathematics, with pictures of mathematicians, the mathematical Ahmes (or Rhind) papyrus, and the oldest mathematical finding, the Ishango rod. Hawkins was the head of the Strengthening Underrepresented Minority Mathematics Achievements (SUMMA) program of the Mathematical Association of America and so was able to distribute the poster to many universities and colleges (Fig. 1.6). Ethnomathematics can also be important for textbooks in other fields. Anthropologists, for instance, study “acculturation,” the phenomenon of important cultural changes resulting from long, intensive contact between different communities. Some authors use the term acculturation only when an additional element of compulsion is present, such as with the submission of one group to another. Colonization is such a situation, and in Africa this sometimes had unexpected consequences. In Burundi, for instance, rectangular ground plans gradually replaced the already mentioned traditional round ones. In the period before 1920, the rugo dominated, made entirely from local materials and constructed around a hearth in the middle. Around 1940 changes occurred as the ground plan became square and the hearth lost its central place. Finally, around 1950, a more European subdivision was imposed (Fig. 1.7). Yet there might be other reasons for these changes aside from domination by European culture. The population increased, so a need emerged for more rooms in the same space and, thus, for additional subdivisions. Others suggest the importation of rectangular zinc roof plates caused these changes since a permanent roof covering is a sign of prestige. Still, it remains noteworthy that the usual examples of acculturation from anthropology often concern only religion or tradition, but they can be found in less esoteric matters too.

Ethnomathematical Studies

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Fig. 1.6  Hawkins’ poster “African and African-American Pioneers in Mathematics”

Ethnomathematical Studies Cultural forms, religious traditions, and art forms of people in Central Africa in traditional communities as they were known before colonization have been studied extensively, but little has been said about those communities’ scientific achievements. Forging has received some attention, as has traditional medicine or medicinal plants.

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Fig. 1.7  Evolution of ground plans in Burundi; some elements, such as the location of the hearth or the bed, are evident

In the past, mathematics seemed to have completely escaped the attention of anthropologists and ethnographers, to the point where one might wonder whether this is not confirmation of the absence of something like mathematical activity in societies that had no writing system. In the past 20 years, a new field has emerged, known as ethnomathematics, that aims to recover what can be found of this part of cultural heritage. Brazilian historian of mathematics and mathematics teacher Ubiratàn D’Ambrosio claims to have been the first to coin this term, though in a more general sense, “to express the relationship between culture and mathematics.” Some authors prefer the expression multicultural mathematics, to encompass research about mathematical achievements in different cultures.

Collecting Data

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Nevertheless, knowledge about traditional African mathematics still remains very fragmentary and is limited to some very specific situations, for particular reasons. First there are the Yoruba of Nigeria, about whom much has been published. Second are the Kpelle of Liberia, whose mathematical expertise was discovered mainly by American foreign aid volunteers (such as the already mentioned Peace Corps). South Africa and Arab Africa are two other parts of the continent that have attracted attention in the field of ethnomathematics for obvious historical or economic reasons. Mozambique and Angola have attracted wide attention, mainly through the extensive work of Paulus Gerdes, a Dutch mathematician who spent most of his life in Mozambique, where he acquired citizenship (1952–2014; Fig. 1.8).

Fig. 1.8  Front page and logo of one of the first editions of the African Mathematical Union

Regarding Central Africa, until about 30 years ago there was only one ethnomathematical article in an international journal. Of course, the lack of written documents is a problem for the collection of information about traditional societies, but in addition, some people who still know these societies are getting older and are more difficult to track down, often because of unstable regional situations. However, several Africa museums possess numerous written testimonies, which are only rarely consulted from a mathematical point of view.

Collecting Data Information about African ethnomathematics is collected in four different ways, even if some methods yield only fragmentary data. These are classical Greek writers, oral chronicles, recent observations of traditional customs and artifacts, and archeological findings.

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There are a few records by American authors about the vocabulary or the mathematical abilities of newly arrived African slaves (two stories are given in this book), but such items are very rare. Exceptional too are Greek written references to Africa and to elements of (mathematical) knowledge on the continent. Perhaps the Greeks obtained parts of their knowledge from the Egyptians: writings from Herodotus (450 BC) to Proclus (400 BC) bear witness of Egyptian astronomy, mathematics, and land surveying. Other Greek writers mention the priests of Memphis as the true founders of science. Many suppose that Pythagoras (500 BC) and Thales (546 BC) went to study in Egypt, and for Aristotle (350 BC), whose teacher Eudoxos had studied in the land of the pyramids, “Egypt was the cradle of mathematics.” It is not surprising that the Greeks knew the location of the sources of the Nile. Homer, Hekate, Herodotus, Aristotle, and Hipparchus all placed them in Central Africa. Astronomer and mathematician Ptolemy put them south of the equator. Ptolemy grew up in southern Egypt, which perhaps makes him more African than his Greek name suggests. Many maps based on information given by Ptolemy use the reference “Lunae Montes” or “Mountains of the Moon” to designate the region of the sources of the Nile. Roman legends speak about Pygmy people living at the sources of the Nile in these Lunae Montes, and Egyptian texts also refer to “little men of the forest and land of the spirits at the foot of the mountains of the moon.” Arab tales place the biblical Garden of Eden in the region of these mountains and call them Jebel Kamar (Figs. 1.9 and 1.10).

Fig. 1.9  Title page of a text about storytelling in Africa

Oral History

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Fig. 1.10  Map mentioning the Lunae Montes, or Mountains of the Moon, or, in an African language of today, the Unyamwezi region

Today, Unyamwezi, or “region of the moon,” still refers to the area south of Lake Victoria and north of the Tanzanian city of Tabora. The Efé are a Pygmy people who live in a forest to the east of these mountains, which they call Baba Tiba. The African names as well as the Latin and Arab denominations all refer to the “land or mountains of the moon.” It is amusing that, according to some people, the word rugo, the term for the traditional habitat in Burundi as described earlier, is related to urugo or “corona of the moon”; thus, the Central African landscape is indeed that of the “circles of moonlight.”

Oral History The tradition of reciting long stories, from generation to generation, exists from time immemorial in some regions in Africa. These stories constitute important sources of information. The chronicles, songs, and verses are important oral testimonies of history. Yet, their content is sometimes doubtful, because the courtiers who told these stories often preferred to please the sovereign instead of telling the true factual history including possible unpleasant facts. Accordingly, some scholars warn that the stories should be treated as legends rather than as trustworthy historical reports.

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For instance, there is the story of the Dogon and the star Sirius. Supposedly, the Dogon knew it was a double star, long before the arrival of the Europeans and without telescopes. The authenticity of the story is doubtful, and thus the Dogon chronicle was omitted here. Clearly untrue are some additions on maps “following Ptolemy,” such as comments about the existence of a one-eyed African people called the “monoculi” (Fig. 1.11).

Fig. 1.11  Another map “following Ptolemy” with the additions of the so-called monoculi

The correctness of the oral traditions can sometimes be verified quite accurately, when astronomical phenomena like comets or solar eclipses are mentioned. For instance, a story from Rwanda reveals a solar eclipse, or ibwirakabiri, occurred during the first months after the accession to the throne of King Mibambwe III Sentabyo. Modern computer software for maps of solar eclipses that could have occured in Rwanda in that time span shows that a total solar eclipse indeed occurred on 13 June 1741, starting at 6h 29 min and ending at 11h 56 min. There were other eclipses in the region afterwards though, that can be determined with a similar precision – in 1741, 1763, 1774, 1781, and 1792. In combination with the context and the estimations of the ages of certain individuals, some have proposed accepting the earliest possibility of 1741, while others have opted for the year 1792. In any case, this case proves that the middle part of the eighteenth century is a good approximation for dating the king’s accession to the throne.

Oral History

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A more recent eclipse that is easier to compare with known events was that of December 1889, when King Rutarindwa ascended the throne. In addition, the associated stories can be indirectly verified: from a king in a neighboring region, Olimi I, king of Nyori, one knows of a solar eclipse in 1506. Olimi’s son was at war with Rwanda, and this provides another possibility for dating events in Rwandan stories (Figs. 1.12 and 1.13).

Fig. 1.12  Map of 1741 eclipse. The meridians and parallels of latitude are represented each 10°. Dotted lines: isomagnitudes of 20%, 40%, 60%, and 80%, north and south of total eclipse

Fig. 1.13  Maps of eclipses of 1763 and 1792, used for dating an event in Rwanda

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Recent Observations A third source of information stems from societies that still follow traditional forms of living. Some practices used in games and counting methods in markets or features shown in ornaments and designs make it possible to glean information, even if not immediately evident (Fig. 1.14).

Fig. 1.14  A walk through a landscape (left) and an aerial picture of it

For example, a walk through the Burundi landscape would not seem to provide any information, but an aerial photograph allows one to reconstruct a ground plan from which a logical pattern of a complex of huts can be recreated (Fig. 1.15).

Archeological Findings A fourth source of information is archeology: excavated teeth or bones sometimes show structured patterns or abstract drawings. The term protomathematics refers to this domain of research. Yet, as with oral traditions, a critical state of mind is necessary to estimate its importance in this case as well since the interpretation of patterns is sometimes subject to fantasy. For example, there is a 30,000-year-old carved flat stone found in Blanchard in the French Dordogne. Alexander Marshack thinks he recognizes some phases of the moon in this “diagram” (Fig. 1.16), but Marshack’s interpretation is rather subjective and is not universally accepted. So as not to get mired in similar disputed interpretations about which archeologists do not yet agree, only facts are mentioned here about which a broad consensus exists among historians. Another example that will not be discussed here is a Kenyan megalithic site from around 300 BC, with 19 pillars oriented toward some constellations. Or else, there is  the placement of bird statues in the great walled town of Zimbabwe, which would follow the pattern of the Southern Cross, but that interpretation will be left to others.

Archeological Findings

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Fig. 1.15  Derived logical structure

The association with preposterous New Age Stonehenge stories causes the skeptical observer to shudder, though they are intriguing ideas indeed. However, they unnecessarily diminish the complete story, instead of adding something essential to the ethnomathematical prerequisites for the interpretation of the Ishango rods.

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Fig. 1.16  Marshack’s interpretation of the Blanchard stone starts with the two markings in the middle of the stone. They supposedly correspond to the last visible crescent moon and the new (invisible) moon. Next, moving up and to the right and then turning down toward the left, we come to the full moon at the first group of four lines at the second turn. When the line turns right again, the four black spots correspond to the next full moon. At the fourth turn, downward to the left, there is another full moon and five more markings

Chapter 2

Storytelling and Music

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Stories Traditional storytellers in Rwanda were charged with the official task of memorizing the history of the country (Huylebrouck, 1997). The interpretation of their poetic stories is sometimes ambiguous, not only because of differences in translation or because of a lack of knowledge about the historical anecdotes, but also because of their style, which is unusual to Western readers. This chapter describes those stories’ distinctive character, though it leans more to astronomy and science in general than to ethnomathematics  (Coupez and Kamanzi, 1957, 1962, 1970; d'Hertefelt et al., 1954, 1962; Smith, 1970; Vansina, 1962). Some of these aspects will be useful in the subsequent discussion. To set the tone, imagine being part of the court of an African king. An exercise given to youngsters in Rwanda during a kind of ritual for the so-called Lyangombe went as follows: To prove the imperfection of his means against the heavenly court of gods and spirits, and standing in front of the meeting, the Lyangombe gave the candidate impossible riddles to solve, so that the weakness of the human species, limited to its own means, was shown. For example, he asked his listener to grasp some stars from the heavens and bring them to earth. To obey, the candidate went to the hearth in the middle of the room and threw some charcoal to the heavens with all his force. Their pitiful fall to earth proved in a selfevident way that man was not capable of such a task. The carbon stars he threw up were nothing compared to the stars, which flicker in the night.

This may serve as an example of traditional teaching.

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The Moon and Lunar Months The change of day and night dominates in tropical regions, as the sun divides light and darkness in exactly two halves of 12 h of day and 12 h of night. The moon too, with its repeating cycle from crescent to circle, from darkness to light, sparks the imagination. These events demand explanations, through dancing and singing ceremonies, with instrumental music and recitation, expressed through symbols such as femininity, death and renewed birth, and fertility. During the night of the full moon, the Baya of the Central African Republic dance in a circle. On the day of the full moon, the royal “makondere” of the Banyoro in Uganda play in their special “Mukama” musical group. In tropical Rwanda, there are four seasons: the long rainy season, the short dry season, the short rainy season, and the long dry season. The lunar month starts with the month of “nzeri” at the first rain falls in the second half of September or in early October. Because of the weather, the beginning of the year might not correspond to a phase of the moon or to a special solar moment. Since the moon has a cycle of about 29 ½ days, and since the solar year exists for 12 lunar months plus another approximately 11 days, the use of lunar-months has the additional consequence that the solar year is not an exact multiple of the lunar month. These difficulties were solved in Rwandan traditions by supposing that the year has 12 months, and by the introduction of an extra period, between the months of April (mata) and May (kamena). This month in between was called “gicurasi” and, thus, corresponded to the eventual discrepancy in days while the people waited for the first rains. Starting with the new moon in this period, the yearly repeating ritual of the “way of the moon” took place. A period of mourning characterizes this ritual, announced by the following phrases that are several hundred years old (Coupez and d’Hertefelt, 1964): When gicurasi appears The drums are not present. When the moon is invisible, One waits 5 days. Then the king goes home And the drum hammers and shaft are presented to him. He takes his seat on the throne, And the drum of welcome greets. Meanwhile, the drums were put in place; They abundantly beat the Karaza rhythm But they do not beat the Timbo. When they stop beating, The drum hammers are presented to the king. The head of rituals enters the corral And says: “Listen, people! The drums have withdrawn. Nobody can marry,

Conceptions of the Universe in Traditional Rwanda

21

Nobody can realize a wish, Nobody can claim his honor, Nobody can express thanks. The Twa women announce the dusk By the “iyombe” song, With the Impara and the flute players. No drum beats any longer, Except the drum of salute.” Each time this one announces the dusk at court, They go to Cyirima Announcing the dusk there where the Karinga is situated, Until the “kamena” moon appears. The authorities see it, Then, the next day, The big leading drum is put in its place Near the stake at the entrance of the corral. When the king has to be greeted, He returns, and the drum hammers And shaft are presented. The drums of greeting salute. Next, the king brings the leading drum, Puts it upright on a ritual animal, And beats it twice Saying: “The day after tomorrow very early!” He goes to Cyirima. He announces the moon, He goes in all corrals Overlooking the entrance of the main corral And does the same. The next day, one proceeds as before And the king says: “Tomorrow very early!” All those in charge of rituals and skillful In beating drums Have come in the meantime... A lengthy explanation of what was going to happen in the ceremony itself follows this announcement. In the end, it is told how life starts all over at the next appearance of the moon.

Conceptions of the Universe in Traditional Rwanda Abbé Kagame published a complete series of studies about the language and traditions of Rwanda  (Kagame, 1951, 1952, 1955, 1956, 1959a, b, 1960). He was a pioneering Rwandan scholar but had to deal with such a flood of criticism from

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Western Africa specialists that the impression prevails that the personal feelings of some specialists got the upper hand of sound scientific work. To this day, a battle rages about who was the first to have written down the narrative history of Rwanda. While some accuse Kagame of making up information or at least changing some, others charge him of disregarding other data, such as stories about creation. To many indeed, the king and founder of Rwanda, Gihanga, came down from the heavens a long time ago, accompanied by thunder, just like the kings of the Sumerians, the Mesopotamians, or the Cretans. He brought the cow, the seeds of life, the layout of huts, and the fire with him to the mortals. The fire was a holy fire that served to commemorate and conserve the memory of Gihanga, and it burned eternally at the court, day and night. It was extinguished only in 1933, when the king converted to Catholicism. In a long Aristotelian dissertation, Kagame had two men exchange ideas about narratives and legends in Rwanda. He had two imaginary characters, Kama and Gama, say the following about the concepts of the universe: Kama: - Let us underline that cosmology differs from cosmogony. The latter is the science about the origin of the universe. Civilizations that did not know writing were also concerned with the question. In this case, these problems are those, not of philosophy, but of ethnology. Gama: - That is quite correct, my dear! Let us leave cosmology for the cosmologists so as not to make mistakes about the use of inadequate expressions in Rwandan philosophy. Yet it seems to me you can discover some philosophical expressions in traditional cosmogony without negatively impacting the significance of these stories. Kama: - We cannot, to use this expression once more, apply the word cosmogony to the Rwandan situation. Indeed, I know of no mythical story from Rwanda about the creation of the world. We have recourse only to a scientific ordering describing the structure of the universe. Your proposal is self-evident, even more so because the Greek philosophers themselves, by observation and explanation of the universe, finally came to the examination of the soul itself. I do not have to explain the story about the conception of the universe, because this is a subject of cultural ethnology. Thus, I will limit myself to philosophical aspects of the problem: the universe following Rwanda was divided into three levels. The middle level concerned the earth where we live. Below was the world of the depths, the home of the “Bazimu” (the ghosts of death); that is why this lower region was called “i Kizimu” (with the nonliving). Above the earth was a higher level of heaven = “ijuru,” the home of God and of some beings with extraordinary strength, such as thunder. The blue luster of the sky was a kind of rock in the form of a dome; it served as an immovable base and as a roof for the earth. The giant dome stretched from the circular border of the earth. These are the most important facts that we can use and that indeed have a connection with philosophical points of view. Gama: - You are holding back too much, my dear! Could not you tell us more about the heavenly bodies, in particular about the sun and the cosmic powers? Kama: - That would be a useless trespassing into the field of cultural ethnology. We have already considered the philosophical side of the heavenly bodies, in the chapter about the categories, where it was pointed out that they are “ibintu” (“ikintu”), that is, “things without soul.”

Names for Heavenly Phenomena in Rwanda

23

An important comment to make is that we should not forget that the Rwandan people never had an idea about the universe as such. They never intended to form an image about the things around them forming the universe; all together, they do not conceive of it as a single entity. Customarily, they will use expressions like “everything that is” (everything that exists) or “of all things.” Gama: - In the same way one can talk about the earth; by this word one understood the nations, people, countries, trees, but it never meant “planet earth.” This is so, certainly, since the expression “the end of the world” is emphasized as “there where the world is carried by stakes and sticks!” (“iyo litrwa inkingi n’ inkenke”), or else as “there where the land ends” (“iyo gihera”), or even as “there where the things fall” (“iyo bigwa”). No expression translates the term “the earth” as we know it today.

The earth was imagined as a disk covered by a dome of blue rocks. Wonderful legends narrate how some nether people, helped by spiders making fantastic ladders, were capable of seeing the “imana” (god) above the dome, while others had gained the confidence of the moles and reached the region below (Loupias, 1908). During the day, the sun slowly crosses the sky, but at night, the heavenly body returns in the opposite direction, above the head of the people. They never saw it during its passage in the opposite direction, simply “because it was night.” Yet there were rays piercing through the blue rocks, and this gave rise to the stars. They were evidence of the eternal source of light existing in the universe. Another version has it that when the sun set, a mythological people in the West ate it. After they cut the “meat of the sun” into pieces, the “middle bone” was thrown back over the azure wall. Upon first contact with the soil in the East, it changed again into a young sun, and repeated its travels, to the great delight of the humans. One anecdote concludes in an amusing way about how well the previous conceptions of the universe were adopted: when umbrellas were seen for the first time in Rwanda, they were called “ijuru,” or heaven, because of their round shape, just like the heavenly dome (Fig. 2.1).

Names for Heavenly Phenomena in Rwanda What follows is a small lexicon of words in Kinyarwanda, the language of Rwanda. The translation gives but an approximate idea. For instance, the word “kibon-umwe” is translated as “meteorite,” but the actual meaning is “the one seen by a single person.” Still, in English, the combination of words “falling star” does not correspond to the phenomenon of a meteorite, either (Figs. 2.2 and 2.3) (Pauwels, 1969). Q. And the sun, did you consider it a god? A. No. The sun was not considered divine. What was seen as divine were the sheep. Yet, none could participate in the cult of “kiranga” until sundown. Q. What did the sun represent in the life of a person? A. One looked at the sun to know what kind of work has to be done at a certain moment. There were times to put the cattle outside to pasture (at about 7 a.m., “gusohora inyana”), to bring it back (around 11 a.m., “gucura inyana”), to let the cows drink (around 1 p.m., “kunywesha inka”), and to return the cattle at night to the “rugo” (around 5 p.m.,

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Fig. 2.1  A poetic expression of the universe: an illustration found in a collection of Rwandese poetry from the 1950s “amkwaza”). The moment at around 5 p.m. when the birds begin to sing was called “umutwenzi,” the “laughing of the birds.” Q. And the stars, how were they perceived? A. When we woke up early, there was a big star, which was seen as a reference, and called “shingabuce,” that is to say, the “one that announces the beginning of the day.” As for the rest of the stars in the heavens, we did not really have an idea about them.

In confirmation of this, another interview was held, in Gitega, the second most important city of Burundi. At about 10 km from there, on Gishora Hill, lived the family of another student, Ndarishikanye Joseph. The interviewed community was of very advanced age. The participants gave another name to the bright morning light. They called it “ndawishinzi”, and affirmed that a traveler had to look at the heavens to see when the star appeared on the horizon. The name “shingabuce” was less well known. They used the name of “umugore-ukwezi” or “women of the moon” for a special star close to the moon. They confirmed that it began shining before the moon and that it could be seen before 6 p.m. In both cases, the bright light was probably Venus, ignoring the fact it was twice the same heavenly body. They also talked about a group of stars, which they considered “inyenyeri yama iruhande y-ukwezi”, which simply means “a number of stars close to the moon.”

Names for Heavenly Phenomena in Rwanda

25

General ijuru

heavens, firmament

ubukana bw' inkuba

thunderclap

umukororombya

rainbow

inkuba

thunder, lightning

ikirere

firmament

About the sun ikizubazuba

solar disk

ubwirakabiri

solar eclipse

kurasa

to appear, to rise

kurenga

to set

About the moon ukwezi

the moon

indembero

crescent for the new moon

kurembera

to wane, after the last quarter

mu myijima y' ukwezi

the invisible phase of the moon

imboneka ly' ukwezi or ukwezi kuraboneka umutaho

the new appearing crescent

akerera or mu myezi

small crescent before the first quarter first quarter of the moon

agateganayo or inzora y' ukwezi or ukwezi kwazoye ingengero

full moon

mu mfundo

waning moon

umwezezi

shade of the moon

last quarter

Venus imboneranyi

the glittering

nyamuhiribona

the one that gives to the one that can see the servant of the moon

umugaragu w' ukwezi Phenomenon of the heavens kibonumwe

falling star, meteorite

nyakotsi

comet (the one that smokes)

inyenyeli

Star

Fig. 2.2  Words for heavenly phenomena in Kinyarwanda, the language of Rwanda

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Fig. 2.3  In the “rugo” of a student’s grandparents, an inquiry was made into knowledge of the stars in traditional Burundi

In another interview, the name “ukwezi” was given to the moon. “Igihete” corresponded to the new moon and “ingasiro” to the full moon, while no particular name could be given to the first and the quarters. When the size of the moon increased, it was a good sign (there would be no battles, and it would be the right time for harvest). In the opposite case, bad times were coming. The interviewed individuals confirmed that there were exactly 14 days between two phases of the moon. They did not distinguish any constellation of stars, but a certain group of stars was called “inyenyeri nyinshi,” which simply means a “multitude of stars.” They admired the movement of the stars during the night from East to West, but thought that the sun and the stars were much further away than the moon and that the sun was much further away than the stars. The word for a solar eclipse was “ubwirakabiri,” or a “double sunset.” In the case of an “ubwirakabiri,” they believed that the sun and the moon waged war against each other. Professor P. Nzohabonayo, of the Physics Department of the University of Burundi, helped to translate the expressions. He pointed out that “(gi)shingabuce” (“the one that announces the beginning of the day”), “ndawishinzi” (“the one that informs travelers about their hour of departure), and “inyenyeri y’ ubuca” (“the rising star”) are all names to designate Venus.

Science Fiction in Rwanda Although science fiction has little to do with astronomy, it illustrates the general fascination for the world “out there” (Fig. 2.4). Tales involving heavenly adventures existed in traditional cultures, too, as an example from Rwanda shows.

Science Fiction in Rwanda

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Fig. 2.4  This entrance to a school in Rwanda illustrates the varied interest students should have

Indeed, a long time ago, the kings loved to wander under the starlight at night. During these nightly walks, the king had to stop at the first source near his palace. A legend ensured that God himself, who had enough of the spectacle of the humans, used to walk over the hills, secretly, hidden from the view of the people. Other kings, like Yuhi III Mazimpaka, looked at the skies during the day. “He-who-­ sorted-quarrels” was well known as an intelligent king. Get the laughers at his hand. He is the central figure in the following anecdote (Pagès 1933, 1934, 1935). Once upon a time … when the king [Yuhi] sat among his courtiers, he heard a rumbling sound, far away, like thunder: - “Was that not thunder?” the monarch shouted, “did you not hear that too?” - “No,” answered the onlookers. The king turned his head to the sky and saw a kind of pirogue flying through the sky, so far above the ground the human eye could hardly see it. He was the only one to observe it, because he had a clairvoyant soul. -“But do you not see a large ship in the sky?” he yelled at his devotees. - “We don’t see a thing,” the latter answered, “and we hear even less.” Mazimpaka let the “biru” continue, because they were the confirmed magicians at the court. It was useless: the heavens were not revealed to their eyes. It was only given to the king to see the vessel floating above clouds as a bird: - “I can see it clearly, including the people inside; it moves fast and approaches. I can unmistakably hear the sound of oars... I feel the owners of the ship will become masters of Rwanda and bring order and peace...” In surprise, the audience did not dare to talk; they looked at each other, lost in amazement. The same idea went through their heads. They remembered that the king had a weakness for beer and honey; he was often drunk. Maybe he had gone crazy from too much drink. But the clairvoyant could guess the reason for their silence. Their disbelief spurred him on: - “You do not want to believe me. I speak for the descendants; your greatgrandchildren will bring me due. They will confirm the truth in my words, because this ship will bring them to the kingdoms of their time.”

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2  Storytelling and Music This only stirred up the emotions of the courtiers, instead of tempering them: - “Our king has lost his head,” said one of them. - “He drank too much,” said another. “Is he dreaming or enchanted? How can he see things we do not see ourselves? Is his hearing better than ours? He speaks about a ship floating through the sky...though prawns are made to float on waterIn "though prawns are made to float on water," please check the use of "prawn." A prawn is a crustacean (a sea creature). Earlier, "prow" was used but was changed to "ship." Was "prow" meant here? The use of "prawn" is very confusing.. Yuhi has lost his mind, he must be ill.” The king, becoming increasingly sad, went on: - “I am tired of your mockeries and suspicions; your grandchildren will prove what I confirm. They too will not believe it at first. Their amazement will not be less than yours. You all, my subjects, my magicians, my sorcerers, my children, you will know that this ship, when it comes over our country, will not be prevented from landing (through your disbelief and mistrust). I know your skepticism about this; you do not mind what I say. I predict our great-grandchildren will use this ship and that it will allow them to travel (good and well) over a frightening lake. At that time, you will remember me. I count on the descendants.”

Here the original description from 1935 ends, but 40 years later, another author added information (Perugia Del, 1978). Historians doubt its truthfulness, so here is the sequel, with some reservations: The aeronauts could have bad intentions for Yuhi’s kingdom. Thus, he asked for his most powerful bow and for a jar of the purest honey. After a prayer, he shot several arrows, drenched in honey, in the direction of the flying ship. When he shot them all, he admired the heavenly saucer flying to the east. - “They went back to their country,” he said to the astonished audience around him, “but because they now know that we are a friendly people, they will come back to visit us.” And following (Desouter, 1982) the story ends by: Yuhi III Mazimpaka would have been so sad because of his disciples’ disbelief that he withdrew without drinking any beer for four days and four nights.

It is amusing to see how the interpretation of such a story by Western specialists depends on the epoch in which they lived. In the earliest times, they interpreted it as a prediction of the arrival of Europeans in Rwanda. Yet the inhabitants of the country had long been aware of the Europeans’ arrival as they heard stories about passing expeditions from Arab slave traders. Later, missionaries saw in this story an announcement of the new religion. For them, the heavenly ship became the “means of God” and the frightening lake it would cross was “hell.” In more recent times, the age of science and space discovery, the interpretation as a visit from extraterrestrial beings in a flying saucer may be more popular. The inhabitants of Rwanda loved these kinds of stories. However, Yuhi’s visions were not always so farsighted: he died jumping off a rock in a lake where the water, unfortunately, existed only in his imagination. In a personal correspondence with the author, the Belgian astronomer M.-E. Dehousse, who knows Rwanda very well, doubted that Rwandans accepted the flying saucer version. Yet he did notice that two so-called “snakestones” in the southeastern part of the country were venerated by the local people as they thought it were be monoliths from outer space.

Music

29

Fig. 2.5  J.-P. Harroy was of high rank in Rwanda when the country was a Belgian mandate. He wrote that the central African drum music made him “think of the music of Johann Sebastian Bach” (on the photo: Burundi drummers)

Music In 1958, the royal drums of Rwanda came to Europe, in particular Belgium, on the occasion of the World’s Fair in Brussels. The Belgian audience was not prepared: the 24 drummers left an impression of “a horrendous and intense banging,” according to one listener. Others politely admitted that “the racket prevented you from concentrating after a short while” (Fig. 2.5). Yet the true problem could have been a lack of understanding of the underlying structure, just as someone who has not studied musicology will find it hard to appreciate a piece of classical music. Someone who is familiar with a certain art form will be more open to it and discover properties or structures more distant observers do not even want to see. The drum music from Rwanda and Burundi obeys three structural features: the hemiola effect, additive rhythm, and the Gestalt effect  (Kubik, 1968; Merriam, 1953a, 1964; Nketia, 1979). Hemiola refers to proportions used in rhythmic models and how these proportions are arranged in time intervals. Fixed intervals are subdivided into an equal number of subintervals by successive beats. Starting with the same interval of time, the following subdivisions (and only these) of these intervals exist: –– 2, 4, 8, or 16 beats or –– 3, 6, 12 or 24 beats.

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Thus, the proportion of the time periods between beats is in both cases 2:3. Usually, an intermediate rhythm of four or six beats follows, accentuated by hand clapping or knocking on wooden sticks: this is called “simple idiophony.” Sometimes, a slow rhythm of two or three drum beats is used to accentuate the basic rhythm. Faster cadences of 8 or 16 and 12 or 24 beats serve as the basis for a more melodic rhythm of more penetrating percussion. These elements form the basis of the structure. Yet there are often subdivisions that cannot be placed in this two-beat or threebeat principle. One can represent them as proportions of changing two or three beats and, thus, as successive realizations of the 2:3 proportions. This drum structure is called a hemiola: a combination in series of two or three subdivisions, each of the same length with eventual additional subdivisions. Brandel pointed out that a change in a reversed sense also exists, that is, a change of a two-grouping to a ­three-­grouping (Brandel, 1961). He gave as an example a piece of music of “royal” drums in Rwanda (Figs. 2.6 and 2.7):

Fig. 2.6 Hemiola

Fig. 2.7  A piece of drum music from Rwanda with a hemiola

Music

31

Sometimes, the 2/8 groups are organized in 3/4 measure (17 bars in the subdivision), and in that case, the 3/8 groups are organized into 3/8 measures (22 bars in the subdivision). Even then, a true hemiola can be evident when two 3/8 bars are combined. The consulted references confirm that this hemiola rhythm makes African music so different from Western music. Connections with the Middle East, Hindu rhythms, or the Ancient Greek hemiola (using five units for the leading percussion) do not make it unique, but the fast succession of unequal leading beats in a 2:3 proportion is typically African. The music is distinguished by intervening changes in the main rhythm: many changes occur in a short time, usually within one measure of time. Additive rhythms differ from the more Western rhythms, although they are both methods to subdivide intervals of time. In African music, the use of unequal groupings is preferred, with a resolute asymmetry. To describe an additive rhythm, we consider an interval of 12 units. It can be grouped into two parts of 6 + 6, but also into groups of 7 + 5 or 5 + 7. In the same way, a measure of 8 beats can be decomposed into 5 + 3 or 3 + 5, or even as 3 + 2 + 3, 2 + 3 + 3, or 3 + 3 + 2. Within one interval of time, an equal period is lengthened or shortened, but of course, all pieces together yield the same sum (here, 12 or 8). On a staff, this situation can be represented very clearly (Figs. 2.8 and 2.9).

Fig. 2.8  African drum subdivisions of time intervals

Fig. 2.9  Additive rhythm

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The so-called Gestalt effect is the third notable feature of drum music in Rwanda. According to Brandel: “The coincidence of hemiola lines inevitably carries with it some kind of Gestalt effect, almost as if a new rhythmic pattern, resulting from the composite interplay of all the lines, emerged. Very often, the preponderance through timbre, pitch, etc. of one line over the others makes it suitable for single-line listening no matter how complex the entire work”. The features of this Gestalt variation again point to a similarity with Mediterranean and Asian music, notes Günther, and others again see a link with Ancient Greece (Günther, 1964). In the given examples, the deeper toned drums in the ensemble change from a 2-grouping, 3/4, to a 3-grouping, 3/8. The lead drum continues its 2/8 figure but is overshadowed by the basses. Here, Brandel makes the following comment (Figs. 2.10 and 2.11):

Fig. 2.10  A more complex piece of African drum music

Fig. 2.11  A piece of Rwandan drum music Because of its lesser obtrusiveness, the listener does not really hear the total counter-­ rhythm - he merely feels it. The dynamic accent in the leading drum is almost lacking and the 2/8 grouping is achieved by means of very subtle timbre contrast.

These properties are noticeable in a more complicated piece of percussion from the royal drums in Rwanda: Despite the galloping strength of the lowest line, the 3-grouping of the top line somehow makes itself quite apparent, and the eventual result is a complex pull in two directions.

Music

33

Therefore, and in contrast to the remarks made in connection with the 1958 World’s Fair held in Brussels given at the beginning of this section, it is not the lack of structure and logical constructions that make this music difficult to access for Western listeners, but rather its abundance. Frank Michiels worked as a researcher at the Africa Museum of Tervuren and had fun playing these staffs on his computer. With ethnomusicologist Gansemans as promoter, he defended his thesis at the Belgian University of Leuven on Rwandan drum music (Michiels, 1984). Indeed, there is much more to say about what seems but simple drum rumbling. The deeply established respect for drums in Rwanda is illustrated by the official ban on playing on four “royal” drums. At the royal court, the drums served a function comparable to that of a scepter or flag. A sentence was pronounced by placing a drum close to the body, so that it seemed as if the condemnation came from the sacred drum, and not from the convicting mortal. Furthermore, a large quartz crystal was placed inside these drums, representing their “souls” (Fig. 2.12).

Fig. 2.12  The large drums had a quartz crystal inside, their “souls” (on the photo: Burundi drums)

Chapter 3

Creative Counting

|||

Number Bases The Western number system with base 10 clearly comes from counting on the fingers of the hand. Yet it is neither the best, nor the most elegant, nor the easiest system. Several authors have written about the beginning of counting (Menninger, 1969; McGee, 1900). Early on, elementary arithmetic was uncommon, and naming numbers often was the only “mathematical” operation. In many cases, the latter even is an overstatement. Here is a summary of some remarkable or amusing facts from a mathematical point of view. The figures show two maps, one following G. G. Joseph, with base 2, 4, 20, and 10 systems, and one following John D.  Barrow, showing base 20 and 10 systems (Figs. 3.1 and 3.2; Barrow, 1993; Joseph, 1992). The simplest number systems can in fact hardly be called counting methods, as they more closely resemble methods of enumeration. In some regions in Central Africa and South America, counting goes like this: one, two, two-and-one, two-two, many. The Gumulgal of Australia count very logically: 1 = urapon; 2 = ukusar; 3 = ukusar-urapon; 4 = ukusar-ukusar; 5 = ukusar-ukusar-­ urapon; 6 = ukusar-ukusar-ukusar.

This is similar to the South American Bakairi system: 1 = tokale; 2 = ahage; 3 = ahage tokale; 4 = ahage ahage; 5 = ahage ahage tokale; 6 = ahage ahage ahage.

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_3

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3  Creative Counting

36

Base 20

Base 2 Base 20

Base 2

Fig. 3.1  Maps of number systems, one by Joseph (left) and one following Barrow

Fig. 3.2  Map of Barrow’s “neo-2-counters”

37

The Doubling Principle and Mixed Counting Forms

Both of these counting methods seem to stop at 6, but others put three dots at the end; does this mean one would eventually express the number one thousand, 1000, in this way? Another Australian community has a simpler solution for large numbers: 1 = goona; 2 = barkoola; 3 = barkoola-goona; 4 = barkoola-barkoola; 5 = barkoola-­ barkoola-­goona; 6–7 … = many.

Some binary systems are not very evolved. In many cases, they are not real binary mathematical systems, in the sense that 4 is not necessarily 22, and is 8 not 23. Other counting systems do not even reach 10, and even then, there often are no special words for 4 or 8. In fact, only the counting method of the Bushmen goes as far as 2 + 2 + 2 + 2 + 2: 1 = xa; 2 = t’oa; 3 = ’quo; 4 = t’oa-t’oa; 5 = t’oa-t’oa-xa; 6 = t’oa-t’oa-t’oa; 7 = t’oa-­ t’oa-t’oa-xa; 8 = t’oa-t’oa-t’oa-t’oa …

The Doubling Principle and Mixed Counting Forms Some users of base 2 have the following counting notation for 6, 7, and 8: 6:

||| |||

7:

|||| |||

8:

|||| ||||

Reading these arrangements horizontally, they yield the combinations 3  +  3, 4 + 3, and 4 + 4. The expressions are the essence of the neo-two counting system and illustrate a primary evolution, starting from the easiest counting systems, then with slight additional reasoning, using doubling. Similarly, some people in Australia count using the following schema: one, two, three, two-two, two-three, three-three. Sometimes, seven becomes three-three-one and eight turns into two-four. Anthropologists recorded the evolved counting system with base 2 imagined by Barrow during a study of calendars in the Indian region of Madras. There, numbers are represented with shells or stones, though 8 does not completely follow the predicted doubling format (Fig. 3.3).

Fig. 3.3  Madras’ representation of the number 1687 in shells

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3  Creative Counting

Such a system for forming numerals by adding smaller numerals is sometimes called an additive counting system (Fig. 3.4). Some people prefer the addition of two equal numbers, or numbers that differ by at most one unit. Fig. 3.4 Gesture illustrating the doubling principle: 4 = 2 + 2

Perhaps the popularity of the additive method for the numbers 6–9 was due to its use in mental calculations. For example, one can get the double of 7 through the addition of 7 + 7 = (4 + 3) + (4 + 3), as 4 + 3 + 3 = 10, and thus the answer is 10 + 4. In Sub-Saharan Africa, mental calculation was a tradition for centuries. Before colonial times, the (mental) calculating techniques were based on repeated doublings, and sometimes they still are (Chap. 6). The Mbai use such an additive system: 6 = muta muta or: 3 + 3; 8 = soso or: 4 + 4; 9 = sa dio mi or: 4 + 5. For the Sango from northern Congo 7 = na na-thatu or: 4 + 3; while 8 = mnana or: 4 + 4, and 9 = sano na-na or: 5 + 4. The expression of numbers as sums of smaller numbers makes counting in everyday applications easier. A small number base has analogous advantages. If 5 is the base, then 7 plus 8 becomes: “5 + 2” plus “5 + 3” and since 2 + 3 = 5, the operation can easily be changed to 5 + 5 + 5 or 10 + 5. Thus, it is not surprising that the given counting systems, bases, and methods occur in mixed forms, so the advantages of several systems can be combined. In South America, one community counts as follows: 1 = one; 2 = two; 3 = three; 4 = four; 5 = hand; 6 = hand-one; 7 = hand-two; 8 = hand-­ three; 9 = hand-four; 10 = two-hand; 11 = two-hand-one; …; 15 = three-hands; 16 = three-hand-one; …; 20 = four-hands …

Counting Gestures

39

This is almost a base 5 system. Somewhere in Paraguay, the numbers are mixed in an even more ingenious way: 1; 2; 3; 4; 5 = 2 + 3; 6 = 2 × 3; 7 = 1 + (2 × 3); 8 = 2 × 4; 9 = (2 × 4) + 1; 10 = 2 + (2 × 4).

Back to Africa, the Bulanda in West Africa use base 6: 7 is 6 + 1, 8 is 6 + 2. The Makoua from Northern Mozambique mix bases 5 and 10. “Twen-ty” is not “2–10” nor is “thir-ty” “3–10” as in English, but: 6 = thanu na moza or 5 + 1; 7 = thanu na pili or 5 + 2; 20 = miloko mili or 10 × 2; 30 = miloko miraru or 10 × 3.

The Bété from the Ivory Coast are said to use three bases, 5, 10, and 20: 56 = golosso-ya-kogbo-gbeplo, and this is decomposed as 20 times 2 and 10 and 5 and 1. In the tiny country of Guinea-Bissau, different methods are used side by side since the Bijago use a purely decimal system, the Balante mix bases 5 and 20, the Manjaco use a decimal system, with exceptions such as 7 = 6 + 1 and 9 = 8 + 1. The Felup have mixed bases 10 and 20, with exceptions such as 7 = 4 + 3 and 8 = 4 + 4. Counting words can also differ with respect to what has to be counted, such as cows, beans, or people  (Bigirumwami, 1967). This is not so surprising since in many languages, the definite and indefinite articles differ with respect to the word that follows (i.e., whether a word is neutral, masculine, or feminine). For instance, in Burundi, cattle is counted in groups of five, if possible. In this way, 6 can sometimes be “itano n’umwe” or “5 + 1,” but it can also be “itandatu” or 3 + 3 (Fig. 3.5). “Indwi” or 7 can change to “itano n’ iwiri” or 5 + 2.

Fig. 3.5  Six can be expressed as: “5 + 1 cows” but “3 + 3 bananas”

Counting Gestures In colonial times, information was gathered about all kinds of curiosities in Africa (Delafosse, 1928). Long before the word “ethnomathematics” was invented, there was an awareness of the great diversity in the representation of numbers, as shown by one document dating from 1909 and kept at the Belgian Africa Museum of

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Tervuren. Below is the translation of such a (French) document, and the next illustrations show the photos mentioned in it (Fig. 3.6). Answer on 6134 of July REPORT ABOUT NATIVE MONEY, AND THE COUNTING 30 and 6425 SYSTEMS, IN USE IN THE DISTRICT. of August 20 - 1909.

Using pictures, the enclosed table gives the different counting methods, used by the natives belonging to different races living in the district. The Babomas, who inhabit the entire region Sector of the of the Mushié, between the Kasai and the Lake Fimi. The Wadia, who, more the North … Subject: I°) COUNTING.

Fig. 3.6  Announcement in a colonial report of the photos presented in the next illustration

One can try to imagine the surprising scene of someone in topi and lightweight suit with a camera taking pictures of counting hands, in the middle of a roaring crowd surrounded by jungle (Figs. 3.7 and 3.8). The illustrations give a detailed view of the gesture for “one,” as shown in the previous general picture. A contemporary version of some signs makes them perhaps easier to understand (see Figs. 3.9 and 3.10). As Zaslavsky has pointed out, there can be differences between the basis of a counting system for the pronunciation of numbers and the corresponding gesture (Zaslavsky, 1973). The Maasai, who live in the region north of the Tanzanian city of Arusha, seldom utter numbers without simultaneously showing them with the fingers. Sometimes a number is not spoken, and the listener is supposed to confirm indications of the fingers orally, so that both are sure about the agreed number. For example, they bring the top of the forefinger to the thumb and the top of the middle finger to the forefinger to indicate 3. But when a stretched forefinger rests on a stretched middle finger, it means 4. In New Guinea, the Papuans verbalize all counting words for 2, 3, 4, 19, 20, and 21 as “doro,” which means “finger,” and the spectator must watch carefully to determine which number the raised fingers indicate. In Rwanda and western Tanzania, 4 is shown by holding the forefinger of one hand against the ring finger, until it rests, with a snap, on the middle finger. This is not easy; practice makes perfect. At demonstrations of this counting method during lectures, preference is given to volunteers not playing the piano or violin, so that the demonstrator may bend their fingers slightly. Eastern Bantu people say 6 = 3 + 3 and 8 = 4 + 4, while 7 becomes “mufungate,” or “fold three fingers”: 7 = 10–3. The Songora for their part say 7 = 5 + 2, but still

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41

Fig. 3.7  Creativity in counting signs, recorded in a so-called colonial source, held at the Belgian Africa Museum of Tervuren

Fig. 3.8  Detail of photos taken in 1910 about counting on the hands. The title says “different ways of counting, in use around Lake Leopold II”

Fig. 3.9  Contemporary photos of the number 1 following Yao and Shamba

1 = thumb 1 = left little finger 1 = right little finger 1 = right forefinger Fig. 3.10  Map following Barrow with gestures for “1”

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9 is “kenda” or “take away one”: 9 = 10–1. The Soga show 6 by holding the left forefinger near the closed right hand, and 7 by adding the left forefinger and the middle finger. The Chagga say the numbers from 6 to 9 by taking the fingers of the right hand, starting with the little finger, with the entire left hand. In contrast, the Tete place their fingers over their left thumb (Atkins, 1961; Seidenberg, 1959, 1962; Moiso and Ngandi, 1985, Moiso, 1991) (Figs. 3.11 and 3.12).

Fig. 3.11  The sign for 3 (left) and for 2 times 3, or 6 (right), in Rwanda

Fig. 3.12  The sign for 4 in Rwanda: keep the forefinger first on the ring finger (left), and release with a snap (right)

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Lastly, note that the Kinga, Hehe, and Nyaturu indicate the number 5 by a closed fist with the thumb placed between the middle and ring fingers. This is understood as 2 + 1 + 2, but in the Western world, it often has another connotation.

More Creativity Zaslavsky emphasized the great creativity in number words by the different gestures used in representing numbers. These are still seen in markets and other public places where they accompany conversation. Yet the words are not always simple reflections or translations of their physical equivalents. The Shamba language provides some counterexamples (Fig. 3.13).

6 7

Spoken

Meaning

mutandatu = ntatu na ntatu mufungate = funga ntatu

3 and 3

8

munane = ne na ne

9

kenda

Gesture

show three fingers on each hand, or 3+3. take away 3 four fingers on the right fingers, 7 = 10-3 hand and three on the left hand, or 4+3. 4 and 4 four fingers on each hand, or 4+4. take away one, 9 = five on the right hand, and 10-1 four at the left hand or 5+4.

Fig. 3.13  Differences for the Shamba between words and gestures for numbers

To express numbers larger than l0, the Sotho from Lesotho work with several people together. For example, to represent 368, the first person raises three fingers of the left hand (3 × 100), the second person the thumb of the right hand (6 × 10), and a third raises three fingers of the right hand (8 units). This is in fact a positional system: a person shows units, tens, hundreds, or thousands, depending on his place in the group. The Bushongo from East Congo draw lines in the sand with three fingers of one hand, and after three groups of three, one line completes ten: ||| ||| ||| |. The Fulani from Niger and northern Nigeria put little sticks on the ground to show how many goats they possess. Two short sticks in the form of a V means 100 animals, while a cross, X, indicates 50. Horizontal sticks indicate tens, while vertical sticks represent units. Thus, = ||| means 23, while VVVVVVX|| stands for 652. Sometimes the depiction is more expressive: for the Bambara from Mali and Guinea a person represents 20 (toes and fingers), and a mat on which a man and woman sleep is 40, or “débé.” An additional indication is due to Mubumbila Mfika, a Gabon chemist. He thought he could distinguish groupings or counting signs in graphical symbols,

More Complicated Number Bases

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carved on arrows, sticks, or a body. To him, a longer line at the end of a triplet revealed the Bashongo grouping by threes (Mubumbila, 1988, 1992). This longer line would furthermore be a symbol for a counting space. The Bambala would then, again as in the West, count by fives and place a space between groups. In the counting method of the Bangongo, Bohindu, and Sungu, one could distinguish a grouping by four. Mubumbila’s indications fit well in this context (Fig. 3.14). Fig. 3.14  Table of Mubumbila Mfika, with graphical counting symbols and their grouping, by different Bantu people: A Bashongo, B Bamabala, C Bangongo, D Bohindu, E Sungu

More Complicated Number Bases The sections on the Yoruba (Chap. 6) describe a number system with base 20 in detail, although Central Africa provides a similar base 20 example with the language of the Kele. The Yasayama from Congo use a system based on the number 5. Even today, this can be deduced from the numerals in their language (Maes et al., 1934): 1 = omoko; 2 = bafe; 3 = basasu; 4 = bane; 5 = lioke; 6 = lioke lomoko; 7 = lioke lafe; 8 = lioke lasasu; 9 = lioke lane; 10 = bokama; 11 = bokama lomoko; 12 = bokama lafe …

This is no true base 5 system, because 25 = 5 × 5 plays no particular role, comparable to 100 = 10 × 10 in a base 10 system. Perhaps these numerals are a remnant of such a system, and the adaptation to base 10 has more recent origins.

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The Baali system is a Congolese system, too, but here 4 and 6 are the base numbers. It is more remarkable than the preceding system, because 4 × 6 = 24 plays the role of the number 10 in the decimal system. Indeed, when 576 = 242 is reached, a new word is invented, and the counting method starts over again: 1 = imoti; 2 = ibale; 3 = isyau; 4 = zena; 5 = boko; 6 = madia; 7 = madea neka (6 + 1?); 8 = bapibale (6? + 2); 9 = bapibale nemoti (8 + nemoti = 8 + 1); 10 = bapibale nibale (8 + nibale = 8 + 2); 11 = akomoboko na imoti (10? + 1); 12 = komba; 13 = komba nimoti; 14 = komba nibale; 15 = komba nisyau …; 24 = idingo; 25 = idingo nemoti …; 36 = idingo na komba; 37 = idingo na komba nemoti …; 48 = modingo mabale; 49 = modingo mabale nemoti …; 576 = modingo idingo (= 242); 577 = modingo idingo nemoti (= 242 + 1) …

A question mark indicates adapted forms that are difficult to explain. Other modifications are straightforward; for example, the relation between “idingo” and “modingo” is one of singular to plural, thereby illustrating the use in many African languages of prefixes instead of suffixes for indicating a plural, a diminutive, or another derived form. Combinations of number bases also occur. The Nyali from Central Africa employ a mixed system using 4, 6, and 24 = 4 × 6, and even for larger numbers: 1 = ingane; 2 = iwili; 3 = iletu; 4 = gena; 5 = boko; 6 = madea; 7 = mayeneka; 8 = bagena (= plural of four) …; 24 = bwa …; 576 = mabwabwa (= 242), …

Inhabitants of the same region, the Ndaaka, have 10 and 32 as the base numbers. Thus, 10 is “bokuboku,” 12 is “bokuboku no bepi,” and for 32 there is a particular word, “edi.” Then 64 becomes “edibepi” (= 32 × 2), while 1024 is “edidi” (or 322). A number such as 1025 is therefore expressed as “edidi negana” or 322 + 1.

Words for Larger Numbers Rwandan mathematician Prof. Dr. Désiré Karangwa shared the interest of the author for ethnomathematics when they were colleagues at the University of Burundi. During a conversation about counting methods, Prof. Karangwa pointed to the existence of traditional words for large numbers in the language of Rwanda, Kinyarwanda (Hurel, 1951; Meeussen 1959). Numerals are formed as follows: 101 = ijana na umwe, or: one hundred one. 2110 = ibihumbi bibili na ijana na icumi, or: two thousand one hundred ten. 10,000 = inzovu, or: an elephant. 20,000 = inzovu ebyilli, or: two elephants.

Thus, with some additional plural forms, 98,780 = inzovu cyenda na ibihumbi munani na magana arindwi na mirongo inani. According to the linguists Rodegem and Coupez, employed until recently at the Africa Museum of Tervuren, Kinyarwanda did not have a way to express larger numerals (Coupez, 1960; Rodegem, 1961a, b, 1967). However, a Belgian clergyman, Pauwels, who left many notes about traditional Rwanda, reported numbers up

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to 100,000, while the Rwandan Abbé Kagame, who was highly esteemed by African university professors, mentioned even larger numerals (Pauwels, 1955). In any case, here is the continuation, with reservations, following Pauwels and Abbé Kagame: 100,000 = akayovu, or: a small elephant 1000,000 = agahumbi, or: a small thousand 10,000,000 = agahumbagiza, or: a small swarming thousand 100,000,000 = impyisi, or: a hyena 1,000,000,000 = urukwavu, or: a hare 1.999.999.999  urukwavu na impyisi cyenda na uduhumbagiza cyenda na uduhumbi cyenda na utuyovu cyenda na inzovu cyenda na ibihumbi cyenda na magana urwenda na mirongo urwenda na icyenda, or: a hare and nine hyenas and nine small swarming thousands and nine small thousands and nine small elephants and nine elephants and nine thousand and nine hundred and ninety and nine.

The English translation for the word for 10 million is probably not correct. It seems to mean something like “there are so many things to count that their number is incomprehensible.” A multiple of the word for one billion, “a hare,” did not seem to exist, so that two billion minus one is the greatest number that could be named in traditional Kinyarwanda. These numerals might have been in use at the royal court in Rwanda, where all kinds of things from different provinces were received as taxes, although it is ­difficult to imagine someone counting as far as, say, 1000, for practical purposes. The author’s experience with graduating mathematicians in Burundi was that young Africans were very surprised to learn about these large numbers. What in heaven’s name could their great-grandparents have counted with these numbers? The intellectual feat of inventing these numbers sometimes filled them with disbelief. Why should their ancestors have wanted to understand “a swarming thousand”? The language of Burundi is related to that of their northern neighbor Rwanda. Besides some grammatical differences, there are but slight nuances: 101 = inane na rumwe (or na imwe) 10,000 = ibihumbi cumi 100,000 = ibihumbi ijana 9999 than becomes “ibihumbi icenda na amajana icenda na mirongo icenda na icenda”

In the Buganda kingdom, north of Rwanda, Luganda is the language spoken. There 10,000  =  “mutwalo,” and this was some kind of unit when counting with shells. Greater numerals existed, too, such as 10,000,000 or an uncountable quantity. Remarkably, there existed a plural, so that, for example, the uncountable quantity times two stands for 20,000,000. The Bangongo language, spoken in Congo, doesn’t go as far: 100  =  kama; 1000 = lobombo; 10,000 = njuku; 100,000 = losenene. The Tanzanian Ziba language has a clear Swahili influence: 100 = tsikumi; 1000 = lukumi; 10,000 = kukumi. People in Nigeria who count in base 20 (Chap. 6) had a word for 204 = 160,000, namely “nnu khuru nnu,” which means “400 meets 400.” This is a large number, but they did not count to infinity. For them, “pughu” was “an uncountable large number.”

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The approximate translations for large numerals are less strange when the etymology of large numerals in English is considered. The word “thousand,” for instance, would go back, in its German form, to the Old Norse “pushundrad,” where “pus” refers to the Indo-European root meaning “to swell,” “to rise,” or “to grow.” Thus, “thousand” roughly means “a swollen hundred” or “a strong hundred.”

The Question of Base 12 The origin of base 12 (duodecimal) and base 60 (sexagesimal) systems is a frequently raised issue  (Ifrah, 1985; Huylebrouck, 1997). Readers of mathematics journals regularly confront redactions with related queries, and it may even be one of the rare mathematical questions discussed in newspapers or on the radio (Fig. 3.15). The commonly held opinion is that the occidental duodecimal and sexagesimal scales are due to the Babylonians. Over 5000 years ago, they divided one complete revolution, or the full angle that intercepts the entire circumference, into 360 degrees (360°). The degree symbol of an angle in such a subdivision, the little circle as an exponent, “°”, refers to the astronomical knowledge of the Middle East, since it is a small depiction of the sun.

Fig. 3.15  Even today there is a Dozenal Society, preferring base 12 over base 10

The Babylonian answer sets the question about the origin of these number systems back in time about 5000  years, but the question now is why this scale of Babylonia was based on the number 60. Some point to the size (60°) of the angles from the middle of a hexagon to the six sides and to the frequent use of this six-sided polygon as an approximation for the circumscribed circle because its “radius” equals its side. However, this does not enlighten us about the relation between the 360 subdivisions and the number system with base 12. The use of a decimal base requires little justification. On the contrary, teachers spend a lot of time teaching their students not to count on their fingers. The relation 60 = 5 × 12 suggests a connection between counting the fingers of one hand and the duodecimal base, but most books about the history of mathematics conclude that there is no “palpable” explanation for the Babylonian choice of the duodecimal base. Nevertheless, the question continues to arouse curiosity because base 12 is used frequently even today: –– in the word dozen and gross (1 gross is 144 or 12 dozen), the numerals preferred for counting eggs or oysters, for example;

Classical Explanations for Base 12

49

–– in the number of hours in a (half) day and the number of months in a year; –– in the (old) English lengths and monetary units, where 12 inches correspond to 1 foot, and, up to few decades ago, 12 pence equaled 1 shilling. Furthermore, a counting system with base 10 is not the most practical, because 10 has only a limited number of dividers. Thus, the collection and grouping of data or the implementation of statistics in tables becomes more tedious. Fortunately, the base 10 system has only 10 digits to be recalled by our limited human memory. In the case of base 20 (or 30), many digits (individual numerals) must be memorized from 1 to 20 (or 30). By contrast, a base 2 or base 3 system would require many repetitions of a few digits, of which in turn the number of repetitions would have to be memorized. Base 12 has the same advantage as given earlier for base 10, while 12 has more divisors, and more importantly, 12 corresponds better to our calendar system. Parenthetically, Charles XII of Sweden (1682–1718) wanted to make this duodecimal system mandatory in Scandinavia, but the clock struck 12 for the king before he could have his way. On the European continent, the metric system and battlefields had to deal with Napoleon. The little general thought more decimally about these matters. Nevertheless, even in 1955, only a few decades ago, a French civil servant named Essig tried to introduce the duodecimal system for weights and measures. The number 12 was important in ancient times: why don’t we say “one-ten” and “two-ten” in English? To some, the words “eleven” and “twelve” stem from “ain-­ lif” and “twa-lif,” meaning respectively “one-lif” and “two-lif.” Here, “lif” is related to “left.” This instinctively refers to the previously discussed ways Africans form number-words, and some language experts suspect that people on the European mainland were, in the distant past, in contact with migrants counting on the hand in different ways. Others claim that 11 and 12 were formed through the well-known addition procedure, just like 13 (three-ten) or 14 (four-ten), but that the so-called “constituents” became unrecognizable for all kinds of different reasons.

Classical Explanations for Base 12 Back to the Babylonians and their duodecimal (or sexagesimal) base. The question about the origins of this duodecimal preference remains. In general, it is explained in two very different ways, although they provide little satisfaction when looked at more closely because they are a posteriori explanations. If one knows the decimal as well as the duodecimal (or sexagesimal) system, then, say these two explanations, the latter presents an advantage. The two widely spread justifications are known as the arithmetic and the astronomical explanation. The arithmetic explanation for the origin of the sexagesimal scale puts an emphasis on the simplicity of calculations. Fractions with a numerator of 1, called principal fractions, play an important role. Their notation using a decimal point is the

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well-known “decimal point notation,” but here the latter word can lead to confusion. In base 10, the principal fractions with a numerator of 2–9 are 1 , 1 , …, 1 , and of 2 3 9 1 1 1 1 these , , , and have a nonterminating but (of course!) repeating so-called 3 6 9 7 decimal representation: 0.333…, 0.666…, 0.999…, and 0.142857142…. In the sexagesimal system, on the other hand, the base 60 has divisors 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, and consequently many principal fractions can be represented by a finite number of digits after the decimal point. This is probably the reason why some businesses prefer to calculate with 360 days per year instead of 365. Theoreticians find nonreducible fractions, on the other hand, to be an advantage, and these occur more often in larger numbers if the base is a prime number, such as 7 or 11. Because a prime number is by definition only divisible by 1 and itself, there are less theoretically boring simplifications (such as 4 = 2 ). 10 5 Nonetheless, these considerations remain a posteriori. The advantage of one base above another becomes clear only when one already has some knowledge of arithmetical properties and some understanding of fractions. At a time when arithmetic still had to be invented, this was rarely the case. The astronomical explanation rests on the relation between the duodecimal and sexagesimal bases: the length of a lunar month is about 30 days, so there are approximately 12 months in a year. In addition, the Babylonians identified 12 signs in the zodiac, and this again yields 30 days per astronomical constellation. The Sumerians, the Assyrians, and the Babylonians subdivided the cycle of a day and a night into 12 equal parts, which they called danna; thus, a danna was equivalent to two of our hours. The Babylonians had a special name for 1/12 of an eclipse and, more generally, of a circle, namely a beru. To this day, the Berber people express the number 45 as “one month and three hands.” In the English word “month” the relation to the “moon” cycle is also obvious. The astronomical justification underestimates the observational possibilities of the earliest people. Furthermore, one must be able to count before enumerating star cycles. The zodiac hypothesis is easily eliminated, by observation, since the sun does not only go through 12 constellations called Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, and Sagittarius, but through a 13th group of stars as well, the less well-known constellation of Ophiuchus (or even more, since constellations are simply a product of the human imagination). As for the commonly distinguished 12 months, we probably need to express more gratitude to Julius Caesar and Emperor Augustus for the two months that bear their names than to a hidden duodecimal heritage. Furthermore, it was known for a long time that there are less than 30 full days in a lunar month and, thus, that the solar year corresponds to more than 12 months. Of course, many civilizations tried to fit the lunar cycle into the solar year. Therefore, because they knew what the exact intervals of time were, they eventually added a thirteenth month. In Rwanda, for example, such a thirteenth month was called “gicurasi” (cf. Chap. 2).

Duodecimal Hand Counting

51

These reasons show that the arithmetic and astronomical explanations may be too well thought out. Yet acquaintance with counting customs provides a third account of the base 12 issue, based on a simple physiological interpretation and related to finger counting.

Duodecimal Hand Counting The creativity reflected in the diversity of the various African counting methods makes an alternative explanation acceptable. Indeed, a technique using phalanges as numbers from 1 to 12 remains in use in Egypt, the Middle East (Syria, Turkey, Iraq and Iran), Afghanistan, Pakistan, and India (some authors add Southeast Asia as well). The thumb of the same hand counts the phalanges of one hand, and four fingers have an obvious total of 12 phalanges. The fingers of the left hand record the number of dozens counted on the right hand, including the left thumb, since it is not counting. Because 5 × 12 = 60, this provides an additional indication why the numbers 12 and 60 so often occur together even up to the present for counting days, hours, and minutes. A possible explanation for the asymmetric use of left- and right-hand counting lies in the frequent necessity of counting on one hand when the other hand is holding something. For instance, when a soldier wants to count, he is carrying his weapon under his left arm, keeping his right hand free to catch the weapon in case of emergency. This would explain the initial success of base 5, because the right hand can count the fingers of the left hand and stop at 5. At the same time, it would clarify why so many right-handed people use the left hand to count, and vice versa. This explanation is not a posteriori, like the arithmetic or astronomical ones. In the earliest civilizations, the duodecimal base is often encountered in many very elementary examples. For matriarchal communities, the number 1 was associated with woman, the number 3 with man, and 4 with the union of man and woman. Alternatively, in a later evolution, 3 stood for man, 4 for woman, and 7 for both together. The number 4 seemed the most widespread of all mythical numbers. It was associated with colors, social organization, and various uses for many people. The use of 6 as a mythical or sacred symbol was less common than the 4 cult, but sometimes a mythology of a 4 cult changed into a 6 cult. For example, the four cardinal directions (north, south, east, and west) were completed by two other points, the zenith and nadir, when convenient. The associations using 3 and 4 are not a posteriori, unlike the counting of lunar cycles or the notations with a decimal point, for which some basic mathematical ability is needed. The examples are very simple and merely a motivation to use 3 and 4 when counting on the phalanxes of the hand (Fig. 3.16).

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Fig. 3.16  A duodecimal counting method

Base 12 Vocabulary The physiological explanation for the duodecimal base is only a hypothesis, but number words provide additional support. N. W. Thomas reported on number words used by West African people in the region of Nigeria. Between the Benue and Gurara Rivers, which flow from the west to the Niger River, live the Yasgua, the Koro, and the Ham. It is important to emphasize that they live in an isolated region enclosed by rivers, because that may be why they retained their particular vocabulary. The Yasgua counted as follows: 1 = unyi; 2 = mva; 3 = ntad; 4 = nna; 5 = nto; 6 = ndshi; 7 = tomva; 8 = tondad; 9 = tola; 10 = nko; 11 = umvi; 12 = nsog; 13 = nsoi (=12 + 1); 14 = nsoava (=12 + 2); 15 = nsoatad; 16 = nsoana…; 17 = nsoata; 18 = nsodso; 19 = nsotomva; 20 = nsotondad …

The Koro did not repeat their word for 12 but used another prefix, “pl”: 1 = alo; 2 = abe; 3 = adse; 4 = anar; 5 = azu; 6 = avizi; 7 = avitar; 8 = anu; 9 = ozakie; 10 = ozab; 11 = zoelo; 12 = agowizoe; 13 = plalo (=12 + 1); 14 = plabe (=12 + 2); 15 = pladsie; 16 = planar; 17 = planu; 18 = plaviz; 19 = plavita; 20 = plarnu …

Base 12 Vocabulary

53

The Ham people counted in a similar way, again using a different choice for the number words. A comparable description comes from the Birom region in central Nigeria. L. Bouquiaux gave the following list of numerals (Bouquiaux, 1962): 1 = gwinì; 2 = bà; 3 = tàt; … 9 = aatàt (12–3); 10 = aabà (12–2); 11 = aagwinì (12–1); 12 = kúrú; 13 = kúrú na gw gwinì (12 + 1); 14 = kúrú na v bà (12 + 2); 15 = kúrú na v tàt (12 + 3); … 20 = kúrú na v rwiit (12 + 8); … 24 = bákúrú bibá (12 × 2); … 36 = bákúrú bitát (12 × 3); … 108 = bákúrú aabitàt (12 × 9); … 132 = bákúrú aagwinì (12 × 11); … 144 = nàga …

H. F. Mathews reported on still another community in Nigeria using a base 12 vocabulary. He added that the system was gradually replaced and that the specific number names for 11 and 12 were dropped, while larger numbers were formed through multiples of 10. He wrote this in 1917, so that it could be interesting to substantiate almost 100  years later how much of the duodecimal heritage still remains.

Chapter 4

Drawings

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Graphs, Mathematics, and Africa Graph theory is a standard part of mathematics. A classic example is finding the shortest path a salesperson has to follow for visiting, just once, all capitals of the 50 US states. This so-called traveling salesman problem remains an open one, though several “good” solutions exist. The “path” between two cities can be represented by a line, where an arrow is added if the orientation is important as well. The form of the line can be straight or a roughly drawn curve, but this is less important. The applications of graph theory are sometimes a part of an operations research course and are numerous, from economics to computer science. The introductory example in most operations research textbooks is a reference to Euler’s problem about how to walk over the seven bridges of the former German city of Königsberg. Today it is the Russian city of Kaliningrad, and the seven bridges no longer exist, but back when they did exist, Euler showed that it was impossible to walk over the seven bridges if the same path cannot be used twice. An important mathematical feature is that calculations can be made with these graphs, allowing mathematicians to obtain surprising results an intuitive glance cannot detect. The mathematical study of Euler’s graph, for example, is carried out by numbering the three left points, from above to below, as 1, 2, 3, and to add the number 4 for the point on the right. In a list, called a matrix, a 1 is used to denote a path from the point mentioned in front of the row to the point mentioned above the column. When there is no path, a 0 is used. In the given example, there are zeroes on the main diagonal because there are no loops in the points themselves (Fig. 4.1). The mathematical power of graph theory does not stem from the amusing anecdote of Eulerian bridge walking or other fun sketches but from the mathematical technique using these “matrices,” allowing conclusions to be drawn that would © Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_4

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bridge

1 bridge bridge

bridge

2 bridge

4

bridge

3

bridge

Fig. 4.1  Euler’s problem in a sand version (left) and its mathematical version (right)

o­ therwise have been unpredictable in the tangle of lines. The strength of this mathematical version is illustrated by its many applications, such as the four-color theorem (four colors are sufficient to color any map such that no two adjacent countries are the same color). African examples can also serve as an introduction to graph theory. Different cultures around the world make drawings in sand, using a finger or a twig, with or without lifting it, and lines full of twists and turns reinforce their mythical character. Sometimes symbols are added to clarify the purpose of the drawing. This shows that not only dance, music, or dramatic expression was a source of inspiration for rituals, but mathematical schemes were as well.

Stories About African Graphs The Tsjokwe, from the border area of the Congo and Angola, call their sand drawings sona. The diagrams illustrate stories, recited during gatherings in the center of the village. During a 6–8 months of initiation ceremony, each generation learned these traditional representations and the accompanying proverbs, recitations, or riddles. Another event where these drawings were sometimes used were mourning ceremonies. The story goes that a village chief died and that there were three candidates for his succession. A geometrical drawing represented this situation: a large white dot in the middle for the dead chief, and three small black dots, here numbered 1, 2, and 3. A closed curve surrounded the three pretenders to the throne and the dead chief. Two candidates cannot reach the dead chief without crossing the line, but the middle one can, and thus he becomes the new chief. In another graph, a single line links a newborn to the dead; the sun and the moon, which play a role in the life of each mortal, were added (Fig. 4.2).

Sand Drawings, Memory Aids, and Drawing Patterns

57

Fig. 4.2  A graph to explain who becomes the new chief (left) and the “graph of life” (right)

Sand Drawings, Memory Aids, and Drawing Patterns An oft-used technique for executing the drawings starts by placing a just few points and making a movement as if a curve is woven around these points. The inner edges of the rectangular “frame” are considered to be mirrors, where the curve bounces, as if it were a ray of light. Next, some additional mirrors are placed in the diagram to generate the desired curve (Fig. 4.3).

Fig. 4.3  Five rows of six dots and two “mirrors” (left), crossed by an imaginary ray of light, while the squares are successively colored black, white, black, white, and so forth (middle) to generate the final pattern, reminiscent of African fabrics

A rectangular network is constructed around the dots, so that these dots are situated on the crossings of the network lines. If, in our illustration, a ray of light is inserted at a 45° angle from the lower left, starting from the first subdivision point on the lower border of the rectangular frame, the ray will describe an imaginary route, bouncing in the above right corner back to the right over a very small path, to return against the right border to the left corner below, and so on. While the imaginary ray of light crosses the network, a square is colored black, then white, then black again, and so on, until the entire network is colored. Finally, a pattern is obtained in which the original curve can hardly be recognized. It brings to mind African fabrics.

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It is possible to arrange the imaginary mirrors differently, or to use more or fewer points, and the result will be different. The procedure for obtaining the square patterns (here using black and white boxes) is sometimes reversed in graph theory. This is done, for instance, in mathematical studies of chess movements on a checkerboard. African drawing experts drew the figures expertly and without hesitation. The coordinate system here is a mnemonic, reducing the memorizing task to just a few numbers and to one geometric algorithm (Fig. 4.4).

Fig. 4.4  Other curves from other initial conditions

Reasoning with Sand Drawings One of the first to draw attention to these sand drawings and their relation to graph theory was Marcia Ascher (Ascher, 1988, 1994). She made lists of similar drawings from different continents, classifying the drawings with respect to region or people. She also noticed properties of a mathematical nature, such as doubling a graph (Fig. 4.5).

More Mathematics in Sand Drawings

59

Fig. 4.5  Doubling a graph

The Myubo people use dots to represent ancestors and other dots for the huts of a village. The surrounding line illustrates that the spirits of ancestors influences the life of a clan. When the number of inhabitants increases, the drawing has to be adapted, and this is called “doubling” the graph. In other examples, a graph can even receive a threefold representation. Another mathematically inspired property explored by Ascher was the presence of symmetry in drawings, such as bilateral, double bilateral, or rotational symmetry. Many examples show that it can be interesting to group the drawings differently, for example following the number of points used. Yet the structure is not always so logical and straightforward. African creativity again shows surprising alternatives (Figs. 4.6 and 4.7).

More Mathematics in Sand Drawings When the philosopher Ludwig Wittgenstein tried to provide a definition of the essence of mathematics, he pointed to graph theory, of which he asserted that “everyone would immediately recognize its mathematical character.” Paulus Gerdes, who began working in the field of the ethnomathematics almost at the same time as Ascher, tried to formulate some truly mathematical theorems about these African curves (Gerdes, 2002). Gerdes followed a different approach with matrices than in the first example of Euler’s bridges. Inspired by the color patterns deduced earlier, a 0 now corresponds to a given color of a square, a 1 to another color, a 2 if a third color is desired, and so on (Figs. 4.8 and 4.9). Two such matrices can be added, so that a new pattern emerges (Fig. 4.10). Gerdes studied a special type of Angola Lunda drawings; he called them “Liki designs” after his daughter. Such a Liki design with its associated matrix is presented in the illustrations (Fig. 4.11).

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Fig. 4.6  Drawings arranged following the number of dots. Omitting or adding a dot makes it possible to obtain analogous cases, given in parentheses

Fig. 4.7  A completely different approach to symmetric drawings

More Mathematics in Sand Drawings

se es

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Fig. 4.10  Matrix addition and color patterns

0 2 1 1

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Fig. 4.9  Matrix deduced from African sand curve pattern

0 1 0 1

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philosop

Fig. 4.8  Foreword written by author for one of Gerdes’ books (Gerdes, 2001)

1 0 1 0

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61

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4 Drawings 0 1 1 0 1 0

1 0 0 0 1 1

1 0 1 1 0 0

0 0 1 1 0 1

1 1 0 0 0 1

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Fig. 4.11  Liki matrix deduced from African sand curve pattern

The following properties characterize Gerdes’ Liki matrices: 1. Along borders every “major” grid point (bold black dots in illustration) always has two different adjacent colored squares. 2. Of the four unit squares between two arbitrary neighboring “major” grid points (in vertical or horizontal direction), two adjacent squares are always of one color, and the other two are of another color (Fig. 4.12).

Fig. 4.12  Illustration of Liki property

Following Gerdes, the two properties would imply that a square Liki design is composed of cycles. This is illustrated here for a Liki design with 3 × 3 grid points (Fig. 4.13).

0 1 0 1 0 1 0 1

1 1 0 first cycle

Fig. 4.13  Liki cycles

1

0

1 0

0 1 0 1

0 1 1 0 second cycle

0 1 0 1

0

1 1 0 third cycle

1 0

Liki matrix with all cycles of first order

Forms, Shapes, and Their Analysis

63

The usual multiplication of matrices transforms these matrices of zeroes and ones into matrices that no longer contain zeroes and ones, yet Gerdes saw some remarkable properties in these square Liki matrices (Fig. 4.14). Fig. 4.14  Structure of second order

Theorem Liki matrices A and B commute: AB=BA. Theorem AB has a second-order cyclic structure (see illustration). Gerdes extended these theorems to matrices with more raster points and formulated other variants on these properties. This could be an initial application of a more common mathematical analysis on these sand drawings.

Forms, Shapes, and Their Analysis Some architects and mathematicians love to draw all kinds of rectangles, triangles, and circles over pictures of classical buildings and paintings. Greek temples, medieval cathedrals, or da Vinci’s works are favorite subjects of these so-called ‘studies’. An important role in this ‘analysis’ is played by the so-called golden number ϕ (also called the golden section, golden ratio, or divine proportion). It turns up when a line segment with length x (>1) is divided into two parts of lengths 1 and x − 1 and when this subdivision is such that the proportion (x − 1)/1 equals the proportion 1/x. It implies the quadratic equation x2 – x − 1 = 0, where (1 + √5)/2 = 1.6180... = ϕ is the positive solution. The smaller part, x  −  1, with length 0.618…, is called the minor, the longer one, with length 1, the major. A rectangle with width 1 and length ϕ is known as a golden rectangle. Discovering golden rectangles, even in situations where it cannot be present, is a favorite occupation of some pseudoscientists. Statues whose head was broken off ages ago are interpreted using the golden section, as if the sculptor knew clairvoyantly that the statue would lose its head. The disciples of the Romanian diplomat Matila Ghyka, one of the instigators of the golden section myth, succeeded in producing the golden section in four different ways for the same temple. In a first drawing, some golden section adepts start with the lower step, in another one others begin with the second step, or else with the third step, and finally some take the foot of the temple as lowest point for their drawings (Figs. 4.15 and 4.16).

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Fig. 4.15  Following the golden section myth, the rectangle on the left would be “too flat,” the rectangle on the right “too square,” and that in the middle would be of an “ideal shape”

Fig. 4.16  Four different ways to “discover” the golden section in one and the same building. It illustrates the lack of a rigorous approach in these pseudoscientific investigations

Various books repeat these drawings, over and over, without concerning themselves with the fact that some drawings contradict each other. Students confronted by exercises to “draw a golden rectangle over a temple” will pass the test only if they are on exactly the same wavelength as the examiner. The lack of critical thinking possibly stems from the acceptability these claims seem to have. They seem to underline the importance of classical Greek civilization, the Renaissance, and (parts of) actual Western culture. By discovering proofs of well-reasoned thought reflected in artistic expressions, the golden section helps support the illusion of European superiority. In a book reissued in 1984 with a luxurious color cover, French author D. Neroman considered Greek statues to be ideally beautiful, but African and Jewish

African Forms in Mathematics Books

65

people as “not yet mature” because the proportion of the height of the navel to the distance between the navel and the top of the head is less than the golden ratio (Fig. 4.17).

Fig. 4.17  Neroman’s study of the height of the navel of Greek, African, and Jewish women, and a Senufo statue with major–minor proportions

Unfortunately, others react to this myth by analogously “discovering” the most fantastic golden sections in African art. With the same inventive creativity, they detect golden rectangles and major–minor line segments in African objects. This, of course, is not very difficult since a similar fantasy can be applied to African art as to Occidental art. On the other hand, this can also be appreciated, because it is rather exceptional for artistic minds to take pleasure in mathematically inspired subjects. However, it has nothing to do with science or mathematics. In addition, for the emerging field of ethnomathematics, which itself is still subject to criticism, it is most imprudent to go astray on such treacherous terrain.

African Forms in Mathematics Books The study of form in African art is not very popular because it is widely thought that African art is merely emotional (whatever that might mean). At the base of this notion is perhaps an African exclamation from the early 1960s: “Si l’Europe nous a donné la raison, l’Afrique nous a donné l’émotion.” It sounds strong in English too: “Europe gave us reason, Africa emotion.” If this statement were correct, it would support the point of view that African art expressions are almost purely emotional manifestations, while European art would be based more on rational thought. This prejudice still dominates and has probably prevented many from taking a technical

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drafting approach to African sculpture. We will not discuss this statement here, since the following examples clearly illustrate that the opposite is quite possible. The Mathematics Teacher is a magazine published by the National Council of Teachers of Mathematics in America and once featured some African illustrations (Zaslavsky, 1970). The relation between creativity and mathematics teaching was emphasized, and the forms were compared to specific mathematical curves (Figs. 4.18 and 4.19). Professor Njock, from Yaoundé, is an African scientist who holds strong views about the importance of mathematics and art for his continent (Njock, 1976, 1979): African art opens fundamental perspectives to the world about the social and cultural history of Black Africa: it gives shape to over two thousand years. […] Pure mathematics is an art of creation and imagination. Consequently, black art is essentially mathematical. Indeed, it stimulates the entire personality, the moral and cognitive possibilities, the imagination and creative attitude. In analyzing the contribution of black art, one cannot neglect the individual knowledge, or the numerous feelings of emotion and imagination. […] The accepted pedagogical theories note the great benefit of imagination and appreciation here. One can ultimately think of the importance allowed by educators of free artistic expression, of the necessity to express oneself through spontaneous activities and emphasize the originality of each individual, of the creative evolution justifying man’s thoughts, not only as a base of artistic activity, but also as a dimension of human life. Black art is one of the most important parts of the magnificent structure of universal civilization.

Geometric Patterns Some woven fabrics seem to have a mathematical inspiration. The Ashanti of Ghana use geometric patterns very intensively in their woven textiles, while many similar weavings appear in the Ivory Coast and Burkina Faso (Fig. 4.20). Here, too, there exist stories describing fabrics and their particular names. Thus, a zigzag margin means that the traveler will always return home. Still, the patterns do not always have a geometrical inspiration and often result from simple practicalities. For instance, they were often made on small looms, and this could have given rise to the flounces of Ashanti weavings. Afterwards, they are simply sewn together to produce larger fabrics. An interesting observation is that modern African textiles still imitate these strip patterns, although today they are made in one piece, so no technical requirements related to working with small strips come into play (Figs. 4.21, 4.22, 4.23, and 4.24) (Pauwels, 1952). Internationally, the Ashanti drawings are the most famous, even though they rival South African drawings in their reputation. In Central Africa, many drawings and patterns can also be found on enclosures around huts and on baskets, covers, milk jars, and drum decorations.

Geometric Patterns

67

Fig. 4.18  Examples of a well-reasoned form analysis used at the art school of Nyundo in Rwanda

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Fig. 4.19  Two illustrations from the magazine The Mathematics Teacher

Fig. 4.20  Weaver and sewed finely woven fabrics from West Africa

“Pure” Drawings During his stay in Rwanda, Belgian mathematician Celis wondered why geometric, nonfigurative drawings enjoyed such a strong preference over images of people and animals or storytelling illustrations (Celis, 1972). In particular, he observed this in an isolated part in the southeastern region of the country. It used to be rather inaccessible, and thus it is believed that most drawings were original traditional concepts and not the result of exchange with other cultures or the consequence of ongoing acculturation.

Fig. 4.21  A modern piece of Ashanti cloths imitating the strip motif

Fig. 4.22  Interior of a Rwandan hut with abstract decorations

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Fig. 4.23  Pauwels’ list of Rwandan drawings; the number indicates the associated description

The phenomenon of decorating huts by the so-called imigongo seems to go back about 300  years. Oral narratives recount how the legendary notable Kakira ka Kimenyi came to introduce the tradition of decorating hut walls (Fig. 4.25). Many deeds of his life show that Kakira ka Kimenyi was obsessed by the idea of pureness; his cattle were held in huts and slaughtered there too, so that no fly could touch them [...] He hated mud and sat on a stone in the pouring rain. His neatness was so legendary that it became a proverbial expression to say “isuku ni ya Kakira” or “as neat as Kakira” Rich in initiatives as he was, Kakira would have made these paintings for no other purpose than his own enjoyment and a concern for neatness; first he made them for his father, [...] and next for his own hut. [...] Having made these paintings of his own, he encouraged young girls of the aristocracy to copy his idea. In this way, the paintings spread.

G. and T. Celis noticed that all of the patterns they came across were combinations of just a few basic constructions. The vertical and horizontal directions, together with three skew lines and their symmetric directions with respect to a vertical axis, alone suffice to form all motifs. With these eight directions, the imigongo can be reduced to a few cases, and only parallel lines, rhombi, isosceles, and equilateral triangles play a role. A coincidence was that these geometric observations led to the rejection of certain drawings as unoriginal because they did not follow the prescribed rules (see Chap. 12). Others used the Kakira rules to program a computer so that traditionally correct patterns were created. It is ironic that through his preference for abstraction, Kakira ka Kimenyi, who was legendary for his purity, made a contribution to mathematics, the “pure” science.

Frieze Patterns and Crystallography

71

Fig. 4.24 Traditional imigongo from Rwanda

Frieze Patterns and Crystallography J. Williams defended the use of Chinese, Arab, and African drawings to teach students 6–16 years of age: Crystallography studies the classification of patterns following their symmetry groups but multicultural examples of patterns and designs can illustrate this as well. Zaslavsky reproduced an image of embroidered cloths of the Bakuba from Congo (now in the British Museum) that give a complete collection of seven different one-dimensional frieze patterns (that is, linear patterns in strips on Greek temples). These patterns are associated with one-dimensional transformations, such as 180° rotation and horizontal and vertical reflections. Group theory can be used to prove that only seven such patterns can exist.

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Fig. 4.25  Patterns from previous figure

The use of so-called group theory is a more mathematical way of appreciating African geometrical figures than drawing up a simple descriptive inventory of all kinds of drawings explaining their shapes, purposes, or interpretations. Using group theory mathematicians can prove, for instance, that only 7 frieze patterns are possible regarding the symmetries in the patterns or, if a color is allowed in the patterns, that the number of possibilities increases to 24. As for (noncolored) two-­dimensional patterns, that is, in the plane, 17 mathematical patterns are possible. D. Crowe studied repetitive patterns in the art of the Bakuba of Congo and in Benin (Crowe, 1975). For frieze patterns, he found examples of all 7 possibilities, but for plane patterns only 12 out of 17 (Fig. 4.26).

African Fractals In popular mathematics, fractals are well known, probably because they can be represented graphically by nice drawings, reminding the shapes of clouds, ferns, cauliflowers or psychedelic MTV-videos. The notion is over a 100 years old, but it is thanks to the Frenchman B. Mandelbrot (who emigrated to the US) that this mathematical subject could escape the mathematical ivory tower and get sufficient media attention (Figs. 4.27 and 4.28).

African Fractals

73

1° pattern: …LLLL… generating isometry: 1 translation. 2° pattern: …LΓLΓ… generating isometry: 1 slide reflection. 3° pattern: …VVVV… generating isometries: 2 reflections. 4° pattern: …NNNN… generating isometries: 2 half turns. 5° pattern: …VΛVΛ… generating isometries: 1 reflection, 1 half turn. 6° pattern: …DDD… generating isometries: 1 translation, 1 reflection.. 7° pattern: …HHHH… generating isometries: 3 reflections. Fig. 4.26  In Benin all seven mathematically possible frieze patterns can be found

Fig. 4.27  This fractal was named in honor of Benoit Mandelbrot, who popularized the topic in the media

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Fig. 4.28  Illustrations characterizing the book African Fractals: the interpretation of a Ba-ila settlement as fractal

Like fern leaves, a fractal has the property that zooming in reveals a new image that looks exactly like the original. This zooming goes on to infinity, because for a fractal the definition of a drawing on a smaller scale follows from the drawing on the larger scale. This is why fractals seem to have smaller and smaller ‘tentacles’. When they more or less cover the plane or fill space, they lead to a so-called fractal dimension. For a fractal curve on a surface, this dimension lies between one (for a straight line) and two (the dimensions of a surface) and for a spatial fractal between two and three (total dimensions of space). For instance, a thread, which is of course one-dimensional, knitted in a very warm sweater creates an interlaced surface of a dimension of almost two. Fractals appear to be very useful in describing natural phenomena using a very simple mathematical formulation. The prominent African American scientist Benjamin Banneker used a so-called quincunx fractal. In Senegal, it is not unusual to encounter this fractal pattern as a decoration on small leather bags that are worn around the neck. Ethiopian crosses represent other examples where the mathematical fractal structure has been rediscovered (Fig. 4.29). More imagination is necessary, though, to see fractals in African hairstyles or statues. As long as these are arithmetically and geometrically correct (which was not the case in the golden section mysteries mentioned earlier), there is, of course, nothing wrong with such an exercise. Ron Eglash wrote a whole book about African fractals. It may be correct to say that many African shapes have been inspired by nature and living scenery and thus that those shapes would incorporate the fractal structure of nature (Fig. 4.30).

African Fractals

75

Fig. 4.29  Banneker’s fractal on a Senegalese leather bag (above) and on Ethiopian crosses

Fig. 4.30  An African statue in a fractal interpretation

On the other hand, scientists like David Avnir (The Hebrew University of Jerusalem, Israel) have shown that the fractal property of nature is debatable: very often there are only two or three scale levels, while the possibility of an infinite continuation is a part of the definition of a fractal. In Eglash’s so-called African fractals, too, there often are at most three levels. Avnir went as far as writing

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Mandelbrot in person objecting to his fractal interpretation of nature. Their discussion did not reach a definite conclusion. Patrick Fowler (University of Sheffield, UK) made a witty remark in private correspondence: “Why isn’t there a book about, for instance, ‘Catholic fractals’, since in some religious art, a saint holds a statue of Mary, holding the child Jesus? Is that fractal art too?” Is such a statue proof that Catholic artists invented fractals 600 years ago? Similarly, perhaps there are Hindu or Viking fractals (Fig. 4.31).

Fig. 4.31  Catholic, Hindu, and Viking ethnomathematical fractals?

And so, for what it is worth, here is an example where sand curves, drawn patterns, and fractals elegantly converge. With some mathematical imagination, Gerdes obtained a fractal starting from an African sand curve, which he first converted into a drawing pattern (Fig. 4.32).

Fig. 4.32  Gerdes’ fractal obtained from a pattern of simple sand curves

Chapter 5

Reasoning Without Writing

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Multiplication on the Hands Writing and mathematical reasoning seem inseparable. Yet great chess masters can memorize enormous logical sequences without writing anything, for example when playing blind simultaneous games. Some mathematical geniuses carry out involved operations without writing a single number on paper. We do not wish to defend the hypothesis that mathematics is possible without writing, but some examples do illustrate that reasoning without writing is possible. In Chap. 3, hand counting, for instance, clearly showed different creative ways to represent numbers. Even some elementary arithmetic can be done on the hands, as shown in a method for multiplying small numbers, common to some regions, in the West as well. For instance, suppose you need to multiply 7 × 8. Because 7 is two units more than 5, two fingers on one hand are folded down, and because 8 is three units more than 5, three fingers on the other hand are folded down. In total, five fingers are folded down, and they are said to represent the number 50. The nonfolded fingers, three on one hand and two on the other, are multiplied: 3  ×  2  =  6. The total is 50 + 6 = 56 = 7 × 8 (Fig. 5.1). The method works for all numbers between 5 and 10, and one need only memorize the multiplication tables of numbers from 1 to 5. Another example: 7 × 9. The first number, 7, leads, as previously, to two folded fingers; now 9 is four units more than 5, and that produces four folded fingers on the other hand. In total there are six folded fingers: 60. The nonfolded fingers are again multiplied: 3 × 1 = 3. The total is 60 + 3 = 63 = 7 × 9. Mathematically the method is easy to explain. Imagine x × y must be executed, and both x and y lie between 5 and 10. The number of folded fingers is then x − 5 and y − 5; the number of nonfolded fingers is 5 − (x − 5) and 5 − (y − 5). The sum © Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_5

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Fig. 5.1  7 × 8 and 6 × 6 on the hands

of the folded fingers is multiplied by 10, which will be represented by 10  ×  [(x  −  5)  +  (y  −  5)]. The product of the nonfolded fingers is, symbolically, [5 − (x − 5)] × [5 − (y − 5)]. The sum of both is indeed x × y as some elementary algebra easily confirms: 10 × ( x − 5 ) + ( y − 5 ) ]+[ 5 − ( x − 5 )  × 5 − ( y − 5 )  = 10 ×  x + y − 10 ]+[ 10 − x  × [10 − y ]



= 10 x + 10 y − 100 + 100 − 10 x − 10 y + x × y = x × y.



The procedure was traditionally used to multiply numbers between 10 and 15, or 15 and 20, and even numbers between 20 and 25. The extension to these larger numbers needs some more algebra, but it is not much harder. The fast execution of this method transforms it into a magic trick, by which a mathematician can easily twist an audience round his finger.

Mental Arithmetic The description of the doubling principle in counting (Chap. 3) already referred to a kind of tradition for mental arithmetic that existed in Sub-Saharan Africa. Well known are the legends of people who following the arrival of an explorer or a missionary suddenly started to recite multiplication tables. These stories are hard to verify, but there is one written source, about an African named Thomas Fuller, who was brought to America as a slave in 1724. The following notes were taken in Philadelphia in 1788 (Fauvel and Gerdes, 1990): Account of a wonderful talent for arithmetical calculation, in an African slave, living in Virginia.

Little Ropes as Means of Notation in Burundi

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There is now living, about four miles from Alexandria, in the state of Virginia, a negro slave of seventy years old, of the name of Thomas Fuller, the property of Mrs. Elizabeth Coxe. The man possesses a talent for arithmetical calculation; the history of which, I conceive, merits a place in the records of the human mind. He is a native of Africa, and can neither read nor write. Two gentlemen, natives of Pennsylvania, William Hartshorne and Samuel Coates, men of probity and respectable characters, having heard, in traveling through the neighborhood, in which this slave lived, of his extraordinary powers in arithmetic, sent for him, and had their curiosity sufficiently gratified by the answers which he gave to the following questions. First. Upon being asked, how many seconds there are in a year and a half, he answered in about two minutes, 47,304,000. Second. On being asked how many seconds a man has lived, who is seventy years, seventeen days and twelve hours old, he answered, in a minute and a half, 2,210,500,800. One of the gentlemen, who employed himself with his pen in making these calculations, told him he was wrong, and that the sum was not so great as he had said – upon which the old man hastily replied: “top, massa, you forget de leap year.” On adding the seconds of the leap years to the others, the amount of the whole in both their sums agreed exactly. Third. The following question was then proposed to him: suppose a farmer has six sows, and each sow six female pigs, the first year, and they all increase in the same proportion, by the end of eight years, how many sows will the farmer then have? After ten minutes he answered: 34,588,806. The difference of time between his answering this, and the two former questions, was occasioned by a trifling mistake he made from a misapprehension of the question. In the presence of Thomas Wistar and Benjamin W. Morris, two respectable citizens of Philadelphia, he gave the amount of nine figures, multiplied by nine. […]

An anonymous source affirmed that Fuller was brought to America when he was 14 years old, whence Fauvel and Gerdes deduced that he must have developed his abilities in mental arithmetic already at that time. During my stay in Burundi, I was informed that a young girl living in the capital Bujumbura had exceptional calculating talents too, but due to the tumultuous situation there, this information was hard to check. Still, it does provide a nice transition to the following section.

Little Ropes as Means of Notation in Burundi At the time, the course “Special Methodology for Mathematics Teachers” at the University of Bujumbura (Burundi) had a section on the study of the non-European roots of mathematics. One of the subjects was Inca knot tying as done in the Inca quipus of Peru. These coding methods were important for the Inca state: they were memory aids, used in bookkeeping, official transactions, and to record history and laws. The quipus are short or long series of ropes, tied together in different ways, on which beads are stringed. They use a place notation with base 10 (Fig. 5.2). No real computations were done with these quipus, but still they would have been more than a set of knots to aid memory. After all, they have a logical and spatial ordering to record information. A quipu with an upper string to which strings are attached, along with other strings, looks like an upside-down tree structure.

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Fig. 5.2  Standard illustration of quipus

In mathematics, Gustav Kirchhoff formulated this notion for the first time and applied it in 1847 to the study of electric networks, while Arthur Cayley used it in 1857 for chemical isomers. Some students at the University of Bujumbura pointed out that their ancestors also used short strings as counting devices or calendars (Fig. 5.3). Historians at the university confirmed it, and during a conversation in the village of a student named Nduwingoma Mathias (cf. earlier discussion), the subject was discussed. Some answers seemed strange, such as the consideration about the pregnancy of a woman, immediately followed by the calving of cows, but this merely illustrates the importance of cattle in this culture. Not everything that was said is scientifically correct, but what follows is the account of the interview:

Fig. 5.3  Congolese counting strings with 10 knots, very similar to Burundi strings

Counting Strings and Sticks in Africa

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Q. As far as you can remember, was there always someone who used pieces of strings to count ? A. Yes, there was. The use of pieces of strings was particularly very common for counting the months in which a woman was pregnant and for counting the number of months when a cow would calve. Q. How did one proceed? A. The woman took a rope made from the tree from which the elderly made cloth. These trees are called “umuhororo” and “umumanda.” Then, in the first month when a woman knew she would be pregnant, she tied a knot, corresponding to the first month. The second month she tied a next knot at the appearance of the moon. Thus she went on counting until the ninth lunar month was reached. It went in the same way for a cow. While he noticed the first signs of fertility, the owner of the cow started to tie the first knot, and he went on each time the moon appeared, until the total was 10. Q. What happened with the string later on? A. After giving birth, the woman burned the string and ate the ashes. Q. How did she store the string? Where? A. She buried it in order not to lose it. Q. Did someone keep this string? A. No, after birth, the woman buried the string and ate the ashes!

The burning of the short ropes after giving birth explains why they were seen so seldom in Burundi.

Counting Strings and Sticks in Africa A search for more information about the Burundi strings revealed that the use of counting strings was widespread in Africa. Lagercrantz dedicated survey papers to African counting strings, counting sticks, and even cuts or tattoos on the body  (Lagercrantz, 1970, 1973). He wondered why these objects were so rarely seen in museum collections around the world. He explained it as being a function of their temporary character, since they were discarded as soon as they had fulfilled their use. Moreover, the material from which they were made, such as dried banana bush leaves, was not permanent. Lagercrantz made an explicit exception, though, for the Africa museum of Tervuren, but even there, these objects are not on display in the permanent collection. It was only through a special inquiry that collaborator Marleen Caluwé could pick up the trail of these small items in the very extensive collection, with the help of her colleague Van Cleynenbreugel, a man who had stored the museum’s complete collection in his head (Figs. 5.4 and 5.5). In many cultures, knotted strings and carved sticks were used as elementary calendars, for recording information about the passing of the days, weeks, months, or years, or for recording payments, loans, distributions of water, goods to transport, sales of ivory, and so forth. They were discovered during an expedition by Grant and Speke, where carriers used them long before missionaries could have imported them.

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Fig. 5.4  Various counting strings

Fig. 5.5  Lagercrantz’s map on use of counting strings and an example of such a rope, after an illustration by Lagercrantz

Counting Strings and Sticks in Africa

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In trade, knots were tied on both ends of a string, at one end for the sale, at the other for purchase. A debtor gave a string to his creditor as a promise, and if the creditor lost the string, he lost his right to receive repayment (Figs. 5.6, 5.7, and 5.8).

Fig. 5.6  A string with 152 knots and one with 10 × 30 knots, used to determine when a cow would calve

Fig. 5.7  Different kinds of counting sticks, with clear carvings

A Makonde man, who left for a voyage of 11 days, tied 11 knots in a rope and said to his wife: This knot (the first one) is today, the day that I leave. Tomorrow (and he touched the second knot) I will still be on my journey and have two more days to travel. But here (the fifth knot) I arrive at the end of my journey. There I will stay the sixth day, and the seventh I set off on the trip home. Do not forget to untie a knot every day from the string. The tenth day you will have to cook for me, since on the eleventh day I will be back home.

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Fig. 5.8  Sticks to count to 10, 20 (base 10?), and up to 24 and 84 (base 12?)

In the kingdom of Monomotopa the court historian was bound to tie one knot upon accession to the throne of a new king. In 1929 this rope had 35 knots, and all kings could be identified going back to the middle of the fifteenth century. Strings were very important to keep track of counting, but other objects served this purpose too, such as carved sticks or specially designed wooden objects. During interactions regarding debts they were presented as a kind of written proof that a certain amount was counted.

The Use of Counting Strings in the West and Asia In our era of writing and computers, it may be surprising that strings, sticks, stones, and even bones were used to keep a record of numbers. Yet this custom was widespread in various cultures. Section 1333 of the Napoleonic Code refers to “la taille” or “tally” of a carved stick on which collected taxes were recorded. Sometimes a carving stick was cut lengthwise and both parties kept half; in case of argument, they were brought together again.

The Use of Counting Strings in the West and Asia

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The modern Arab subha is a counting device using strings, to recount the 99 names or blessings of Allah (pronouncing each name is, of course, a blessing). There are smaller models, too, with 33 small beads or pearls. Often there are some pearls in between, to facilitate counting, and a larger bead at the end, so that a subha to count 33 blessings will in reality have 35 or 36 pearls (Fig. 5.9).

Fig. 5.9  Subhas, small and large

Arab stores all over the world sell them in all colors and shapes, and it is amusing as a mathematical tourist to walk around markets and shops counting pearls. On the other hand, Jewish prayers are counted on a “taliet” or “prayer shawl” with knots on the sides. In a Japanese Shinto temple, the author found 30 little pearls on a string (Fig. 5.10). To many Catholics, rosaries or paternosters are well-known counting ornaments. On a paternoster, there are 5 × 10 smaller beads, which represents 10 times the “Hail Mary,” separated by 4 larger beads, also called mysteries, which stand for the Lord’s Prayer (Pater Noster is Latin for “Our Father”). At the beginning and the end are 2 larger beads as well, which yields a total of 56 beads. Together with the 3 beads attached to a smaller rope holding a crucifix, this makes 59 prayer indications. The smaller string with the crucifix and the circle with Mary’s prayers are connected by another little object, sometimes a bead, or a heart, which gives a total of 60 beads (Fig. 5.11).

86 Fig. 5.10 Japanese counting string

Fig. 5.11 Catholic paternoster, from an abbey near Brussels (Belgium)

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Mancala

87

For such a typically Western Catholic counting device, the total of 60 is notable. Because there were 12 Apostles, there is another remarkable relation with the Babylonian or African sexagesimal and duodecimal bases. Perhaps the African missionaries actually imported with their paternosters a counting system to the people from whom it in fact had originated.

Games of Strategy John von Neumann was one of the most important mathematicians of the last century, if not of the modern era. Besides numerous other writings from the most abstract algebra to quantum physics, he left a work about game theory, written together with the German-born economist Oskar Morgenstern. Although another mathematician, German Carl Friedrich Gauss (1777–1855), had already considered some chess-related problems, it was only after von Neumann’s study in particular that mathematicians started to appreciate the analysis of games. Thus, exotic games, where the rules are hardly known or described, are seen as new challenges. Indeed, when a computer program beats a chess grandmaster for the umpteenth time, it is sometimes remarked that this is rather a result of the greater calculating power of the machine and of the many preprogrammed games that exist than by the intelligence of the computer program. This progress is sometimes seen as a victory of technique and hardware more than of a smarter software solution. For newer exotic games, these allegations are not valid, and mathematicians were, for instance, very interested in the non-European game Go (Ball, 1989). The game of the Dogon known as Sey is more a game of skill where objects must be caught as fast as possible out of the sand, but the West African Yoté is an elementary game of strategy. It is a kind of simple version of checkers played on a 5 × 6, with each player having 12 pawns. The pieces are moved orthogonally to an empty adjacent cell, not diagonally as in checkers, and pieces are captured in that direction too. Another difference with checkers is that one can place his pieces where he chooses in the beginning of the game. A Guinean brainteaser is more or less related to a different mathematical field of study, topology, and in particular to the theory of knots. This Guinean puzzle was used by people who lived on the coast of that West African country and can easily be made using a piece of wood, a ring, and a (preferably rather thick) rope. It is ­better to try it out first a few times to get a feeling for the degree of difficulty, instead of going right away to the solution given below (Figs. 5.12 and 5.13).

Mancala The African strategy games of the mancala family test one’s counting skills; they are among the oldest strategy games in the world. R.C. Bell, a specialist in board games, classified it in the top 10 of the world’s games (Bell, 1988). Researchers at

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Fig. 5.12  A brainteaser from Guinea: move the ring to the right

Fig. 5.13  Solution to brainteaser from Guinea. First, the ring is pulled from below left through the middle loop. Next, this middle loop is pulled from the back through the opening so it is in front of the player. Now the ring can move under the different loops of the rope, which is later again pulled to the back. The ring is now on the right and can finally be brought under the middle loop to the right to reach the required final position

the prestigious Massachusetts Institute of Technology were the first to program the board game on a computer. Today, several computer variants are available on the Internet (enter “mancala” in any search engine). Since Nokia preprogrammed the related “bantumi” game on its very popular cellphone model 3310, no young person can doubt the prestige of the African game.

Mancala

89

The word “mancala” comes from the Arab word for “move” and refers to moving little pawns from one hole to another. The game has other names such as “boa” in Tanzania, “omweso” in Uganda, “ayo” in West Africa, “(igi-)soro” in Rwanda and parts of East Africa (Merriam, 1953b), and so forth (Hall, 1953; Pauwels, 1955b; Nsimbi, 1968). These pawns can be any kind of small object, from dried beans over crown caps to little shells. Here we prefer to use little cowry shells, which are common counting instruments in many parts of Africa (Chaps. 6 and 12) (Fig. 5.14).

Fig. 5.14  An old version of mancala, carved in stone

The game is also played in four rows (see subsequent section on igisoro), or with more or fewer holes, or with or without an additional hole to collect captured shells. North of the equator, a board with two rows is more common, while three rows are widespread in Ethiopia. There can be 6–50 holes in a row. The rules of the game vary as well, and in this way it is possible to study cultures, not with respect to their language or dialect, nor to their traditions or religion, nor to their music of artistic expressions, but with respect to their way of thinking, that is, to the way in which they play this board game (Figs. 5.15 and 5.16). Furthermore, the playing of a strategy game such as mancala creates a form of communication without having to know the language of one’s teammates. In this way, logical thinking shares a property with graphic arts or music. Avid chess players know, for example, that a player who starts by moving a pawn from the edge of the board is actually wasting a move. This indicates scorn for the skills of one’s opponent. The author of the present book experienced something similar when playing the Rwandan version of the game, igisoro. After losing a few games, he noticed that his opponent always started in the same way. It was only after asking some questions that the explanation came: it was a way to show to spectators that the author was a very weak player. As a consolation prize he received an igisoro board to practice on.

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Zande Ngombe

Down Congo

Teke

Logo

Ba(a)li

Hunde Shi Rwanda Fulero Rundi

Kuba

Maniema region

Tsjokwe

Luba

Fig. 5.15  Map of Congo with circled regions where a strategy game is played in similar ways

Mancala board

Kalaha

Kalaha

Fig. 5.16  A mancala game board with four prepared pawns

Mancala Rules The game is played crosswise between two opponents. To put it in geometric terms, the longest axis of the game is perpendicular to the line connecting the players. Each player controls six holes, those directly in front of him, which make up his camp, and one so-called “kalaha,” situated to the right of his own camp. The object of the game is to capture as many pieces as possible from one’s opponent. At the beginning of the game there are four pieces in each hole (normally fewer for beginners), except in the kalahas at the edge. In the beginning each player thus has 24 pieces at his disposal. The pieces in one’s own camp do not need to be of a different color from those of one’s opponent because they can change camps during the game.

Mancala Rules

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The first player starts by picking up all his pieces in one of the holes. He himself decides where to start within his own camp. He distributes these pieces, one by one, in separate holes around the board, moving counterclockwise. A piece can also be dropped in its own kalaha and in the holes of one’s opponent if there are enough pieces. If a player’s last piece lands in his own kalaha, he gets another turn. These are the elementary and almost only rules for moving pieces. Each player moves on his turn. Capturing pieces in the camp of one’s opponent happens when the last piece of a player lands in an empty hole of his own camp. The player takes all pieces from the opposite hole of the opponent’s camp and puts it in his own kalaha together with the piece with which he started. Then it is the opponent’s turn. When all 12 holes in the middle of the board are empty, the game ends, and the winner is the player with the most pieces in his own kalaha. When the game is almost over, one of the players might not have any more pieces. In that case, he can skip his turn and wait to see if another piece from his opponent’s camp lands in his own camp, but most experienced players decide very quickly how to end the game. It is possible to start playing with this (overly) simplified version of the rules. Note that the last piece can occasionally reach the opponent’s kalaha, but for this to happen, at least eight pieces are needed in the rightmost hole of one’s own camp. In that case, though, no extra move is allowed in most variants of the game. Many variations of the game do not even allow one player to drop a piece in the kalaha of the other player; he must simply skip it when he gets to it. “Making a bridge” is an additional move that is often allowed. It means that if the last piece falls in a hole where there are still other pieces, the player can take these pieces and continue the move with these extra pieces. This can also happen with pieces in the opponent’s camp. Of course, similar variations on the base rules must be clearly agreed upon before the game starts. To learn the game faster, and to get a better feel for a suitable strategy, one can start with two pieces per hole. During demonstrations, using little cowry shells as pieces has the additional advantage that exactly 20 shells are needed when playing a 5-hole game. Thus, this prepares one for the calculations of the Yoruba (Chap. 6); thus, here a modified setup is used, as given in the illustrations (Figs. 5.17, 5.18, 5.19, 5.20, and 5.21). a2

b2

c2

d2

e2

a1

b1

c1

d1

e1

Fig. 5.17  A simplified mancala setup with only two pawns in each hole; this makes it easier to learn “ethnomathematics”

a2

b2

c2

d2

e2

a1

b1

c1

d1

e1

Fig. 5.18  A simple mancala move: the little shells of c1 were distributed to d1 and e1

a2

b2

c2

d2

e2

a1

b1

c1

d1

e1

Fig. 5.19  A subsequent move: the opponent moves the shells of d2 to c2 and b2. This is not a good move because the first player can now capture shells by moving his shells from a1 to b1 and c1

a2

b2

c2

d2

e2

a1

b1

c1

d1

e1

Fig. 5.20  The last little shell fell in the empty hole c1 and this allows the player to capture three shells in c2

a2

b2

c2

d2

e2

a1

b1

c1

d1

e1

Fig. 5.21  The three shells of c2 together with the last shell that fell in c1 are collected in the kalaha of the first player

Example of a Game

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For the sake of clarity: a player can only start with a move with shells from his own camp. He can only take away shells from a hole of the opponent opposite an empty hole where his last played shell fell. After taking pieces, the opponent plays in any case. The figures illustrate a possibility that is not necessarily the best but serves here as an instructive example.

Example of a Game To demonstrate the game quickly and in a simple way, we indicate the number of shells in a player’s camp with five digits, followed by the number of shells in his kalaha. Thus, the initial situation is (2,2,2,2,2; 0) for both players. Next, it is advisable to have a game board on hand or, lying on a beach, to fill holes with little shells or, in a bar, to use cups with crown caps or, in a classroom, to put pieces of paper on a large drawing of the game board (Fig. 5.22).

Fig. 5.22  A traditional scene of igisoro players

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Thus, suppose the first player moves two shells from the fourth hole: (2,2,2,0,3; 1), so that he gets an additional turn because the last shell was dropped in his own kalaha. He now decides to empty the second hole and thus obtains (2,0,3,1,3; 1). The last shell fell in an empty hole of his own camp, so that two shells from his opponent’s camp can be obtained. Together with his own shell, they are dropped in his own kalaha: (2,0,3,0,3; 4), while the opponent still has (2,0,2,2,2; 0); the score is 4-0. Now it is the opponent’ turn; he decides, for example, on (0,1,3,2,2;0) by emptying his first hole. Then the first player makes (2,0,3,0,0; 5) by emptying the fifth hole, but two of his shells go over to the opponent’s camp: (1,2,3,2,2;0). The score is now 5 for the first player and still 0 for the second. The situation does not look too good for the second player (Fig. 5.23). 1 a

3

4

5

6

7

nteba

c

nteba ugutwi

8 ugutwi

ugutwi

b

d

2

North

South

nteba

nteba ugutwi

Fig. 5.23  Igisoro board with names of some particular locations on the board

But the second player can score starting at the fourth hole: (1,2,3,0,3; 1), after which he can immediately form (1,0,4,1,3; 1) by moving the shells from his second hole. Now it is 5-1. The first player now acts in a smart way by starting at hole 3: (2,0,0,1,1; 6), and he can take four shells from the second player: (0,1,0,1,1; 11). Thus, he will certainly be the winner as there are only 20 shells to share out. Moreover, he can improve his score by emptying hole 6: (0,1,0,1,0; 12), and by moving the fourth, and taking a shell from his opponent: (0,1,0,0,0; 14). His opponent now forms (0,0,0,1,0; 2), while the first player changes camp, which went from (1,2,0,0,0; 14) to (1,0,1,1,0; 14). A small consolation for the second player is to take one shell from his opponent to form (0,0,0,0,0; 4), but now the second player has no more shells. The first player can move his shells by himself or collect them immediately: the final score is 16-4.

The More Complicated Igisoro Game A more extensive version of mancala is “igisoro.” Coupez and Benda drew up a description of the game while editing a dictionary of the Rwandan language (Coupez and Benda, 1963). They made numerous inquiries and wrote down in particular the vocabulary of various aspects of the game.

The More Complicated Igisoro Game

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The game involves two opponents playing on a large wooden board, about 60 × 30  cm. The board has 32 round holes arranged in four rows and eight columns, divided into two parts. The players move 64 pawns with the aim of collecting enough of them so that there are none left for the opponent to take. Traditionally, pawns must be small, round, and smooth balls and are called, even outside of the context of the game, “ubusoro,” somewhat like the name of the game itself (the stem “-soro” suggests “little ball”). Yet in the game they are called “inka” (derived from “-ka,” meaning “cow”). Here we shall call them simply pawns. In igisoro, players move their pawns only on their own part of the board. In mancala, players move their pawns on the whole board. They crisscross in the direction indicated in the illustration, again counterclockwise. For the sake of simplicity, we shall specify the two camps by North and South. Occasionally, the specific names of some holes will be used because they will play a particular role in what follows. The holes do have exact names in Kinyarwanda, the language of Rwanda, but still we introduce a kind of Western and nontraditional system of coordinates to make it easier for the reader. A move can start at any hole containing at least two pawns, again contrary to the rules of mancala, where a single pawn is allowed to move. The choice of a hole is a matter of tactics. A player will collect all pawns in a hole and drop them one at a time in each hole, starting with the one after which the pawns were collected. If the last pawn falls in an empty hole, the move ends. If not, the player continues with the pawns that were in the hole where the last pawn was dropped (“making a bridge” in mancala). Depending on whether one makes a bridge or not, the move is called a simple or a multiple move. Pawns of the opponent can be captured when a move ends in a hole of the upper row (of that player’s camp, of course), containing at least one pawn, and this as long as the two opposite holes of the opponent (in the same column) both contain at least one pawn (Fig.  5.24). Capturing is mandatory. When a player continues a move instead of capturing when possible, voluntarily or not, he gives his opponent a choice: either forcing him to capture anyway or letting him continue. In the latter case, he can leave the pawns that he could have taken or he can collect them in one of the two holes, thereby preventing them temporarily from being captured. Anyway, during the move immediately following, he does not have the right to start the move with that hole. If a player indeed collected some of his opponent’s pawns, he goes on with the collected pawns. Note that he then starts in the hole just after the one where he made his last simple move, before ending in that last hole (that is, eventual bridges not included). This is very confusing for the beginner because one must remember where one started the move which led to the capture, then start again one hole later counterclockwise from that spot. There he drops the pawns he captured, one by one. This movement may cause a new capture, followed by a new movement, and so on, until a move regularly ends, because the last pawn fell in an empty hole. To make it even more complicated, there is an additional rule. When someone starts a move, or starts a bridge, at a hole called “nteba” (b2, b7, c2, c7) or “ugutwi” (a1, a8, d1, d8), then the player can reverse the sense of the move if, by doing so, he

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1

2

3

4

5

6

7

8

a b c

North

South

d

Fig. 5.24  Capturing in igisoro. Only when player South succeeds in ending a move by dropping the last pawn in c2 can he collect his opponent’s pawns. Here this is possible by starting in d5, making a bridge at c6 to get to c2. There, the player continues with the pawns taken in a2 and b2, and starting all over in c5

can achieve a situation where the opponent’s pawns can be taken – without making a bridge! The reverse procedure can start again with the pawns captured from the opponent. Advanced players start by making opening moves simultaneously, by bringing their pawns to the desired position through a series of movements following the given rules. When one of the players has ended the second move of his opening, then he has the right to take his opponent’s pawns, at which point the actual game starts. The initiative to attack can be taken by any of the players when it is the first match and the timing is a matter of tactics. If one has already played other matches, then whoever won the last game takes the initiative. Thus, the winner gets priority, not the loser. The idea is that this enables the winner to confirm his victory. Each player makes a move on his turn (simple or multiple), ultimately continued or not by a capture, until one of them no longer has enough pawns to capture his opponent’s pawns. It is rather exceptional if the loser is stripped of all his pawns. Igisoro must be played fast. Penalties are imposed on players who hesitate. Making a mistake against the rules is called “gukanga” or “to cheat.”

Examples of First Moves As in chess, there exist various openings. They are not called Sicilian or Indian as in chess, but “madondi,” “kabakigi,” “rucugiita,” or “ihene.” Here, too, every opening has its particularities. Sometimes the two players choose the same opening as a kind of challenge. A player who wants to show off his skills can choose a more difficult opening. The madondi opening refers to the recurring dry and repeated hits, and this refers to the sound of the pawns. Beginners use this opening since it puts the pawns out of reach of the opponent for a long time (Figs. 5.25 and 5.26).

Examples of First Moves

1

97

2

3

4

5

6

7

8

6

7

8

c d Fig. 5.25  First madondi move: start in c3, arrive at d2

1

2

3

4

5

c d Fig. 5.26  Second madondi move: leave c4, bridge in d1, arrive at d3

The kabakigi is untranslatable, but it goes as follows. The first kabakigi move is like the previous opening (Figs. 5.27 and 5.28). 1

2

3

4

5

6

7

8

7

8

c d Fig. 5.27  Second kabakigi move: start from c1, arrive at d5

1

2

3

4

5

6

c d Fig. 5.28  Third kabakigi move: leave d2, bridge in d4, arrive at d6

The rucugiita mugabo, or just rucugiita, opening makes opponents nervous. Its name comes from the fact that it offers many possibilities for an attack, but since it demands the distribution of many pawns, it must be executed fast so as not to be interrupted by an attack by the opponent (Figs. 5.29, 5.30, 5.31, and 5.32).

98

5  Reasoning Without Writing

1

2

3

4

5

6

7

8

7

8

7

8

c d Fig. 5.29  First move: start from c6, bridge at c2, and arrive at d4

1

2

3

4

5

6

c d

Fig. 5.30  Second rucugiita move: start from c1, arrive at d5

1

2

3

4

5

6

c d Fig. 5.31  Third rucugiita move: start from d1, bridge at d3, and arrive at d6

1

2

3

4

5

6

7

8

c d Fig. 5.32  Fourth rucugiita move: start from d4, arrive at d7

Finally, there is the “ihene” or “goat,” an opening the author faced when playing the game in Rwanda. Like a goat stands up and shows what is beneath it to viewers, the player exposes himself by emptying hole c8. It is a dangerous opening that illustrates the player’s strength (Figs. 5.33, 5.34, and 5.35).

Example of an Igisoro Game

1

99

2

3

4

5

6

7

8

c d Fig. 5.33  First ihene move: start from c8, bridge at c4, and arrive at d2

1

2

3

4

5

6

7

8

6

7

8

c d Fig. 5.34  Second ihene move: start from c1, arrive at d5

1

2

3

4

5

c d Fig. 5.35  Third ihene move: start from d2, bridge at d4, and arrive at d6

Example of an Igisoro Game The following notations are used: North  a 4 b4 c 4 d 4 e4  a3 b3 c3 d 3 e3   a 2 b 2 c 2 d 2 e2   a1 b1 c1 d1 e1

f 4 g 4 h4  f 3 g3 h3  . f 2 g 2 h2   f 1 g1 h1 

South





The initial situation is



0 4  4  0

0 4 4 0

0 4 4 0

0 4 4 0

0 4 4 0

0 4 4 0

0 4 4 0

0 4  . 4  0

100

5  Reasoning Without Writing

Here we let player South start. He spreads the pawns out in e2, with the objective of getting many pawns in the lowest row:



0 4  0  1

0 4 5 1

0 4 5 1

0 4 5 1

0 4 0 1

0 4 4 0

0 4 4 0

0 4  . 4  0

Player North decides to make an identical move, symmetrically, and opens at d3.



0 4  0  1

0 4 5 1

0 4 5 1

1 0 5 1

1 5 0 1

1 5 4 0

1 5 4 0

1 0  . 4  0

0 4  1  2

0 4 0 2

0 4 6 2

1 0 6 2

1 5 1 0

1 5 0 1

1 5 4 1

1 0  . 4  0

South now spreads f2:





North starts at a3, with the aim of leaving empty holes in his lowest row (the upper row for South, seen from the standard position as shown in the illustrations):



0 0  1  2

1 5 0 2

1 5 6 2

0 1 6 2

2 0 1 0

0 6 0 1

2 6 4 1

2 1  . 4  0

South then spreads out b1, taking pawns from h3 and h4, starts again at h1, and captures again, at g3 and g4. This yields



0 0  1  2

1 5 1 0

1 5 7 3

0 1 7 0

2 0 2 1

0 6 1 2

0 0 6 0

0 0  . 7  3



Example of an Igisoro Game

101

Now North continues at b3:



0 0  1  2

1 0 1 0

1 6 7 3

0 2 7 0

2 1 2 1

0 6 1 2

0 1 6 0

0 0  , 7  3



Yet South plays e2 at his next move, leading to the capture of the pawns at c3 and c4:



0 0  2  3

1 0 2 1

0 0 9 0

0 2 9 1

2 1 0 2

0 6 1 3

0 1 6 1

0 0  . 7  3

0 2 6 1

0 1  . 7  3

North starts with d3, to finally capture f1 and f2:



0 0  2  3

1 0 2 1

0 0 9 0

0 0 9 1

2 3 0 2

0 9 0 0



South does e1:



0 0  3  0

1 0 0 2 0 0 0 3 3 10 10 1 2 1 2 1

0 9 1 1

0 2 7 0

0 1  . 0  4

North executes an additional move: spreading in the opposite sense produces pawns when starting at g3. Next, dropping pawns in f3 and e3 follows, so that e1 and e2 can be captured. Next another start is taken at h3, in the usual sense:



0 0  3  0

1 0 0 2 0 0 1 0 0 0 4 10 0 2  . 3 10 10 0 1 7 0   2 1 2 0 1 0 4

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5  Reasoning Without Writing

South now decides to start at c2, with the goal of getting a pawn in as many holes as possible, so that an eventual loss can be limited on a future move.



0 0  5  2

1 0 5 4

0 0 1 3

0 0 0 4

2 0 0 1 4 10 0 2  . 1 2 8 1  2 3 2 1



North starts at e3 and arrives at h4. By a reverse move, North can capture in g1 and g2. The start is made in g4:



1 1  5  2

2 1 5 4

1 1 1 3

1 0 0 4

3 1 1 0 0 11 2 4  . 1 2 0 1  2 3 0 1

South begins at f1, lands in h2, where a bridge is made to f2, so that f3 and f4 are captured. Next, a3 and a4 can be incorporated:



0 0  0  4

2 1 8 6

1 1 4 5

1 0 0 6

3 0 4 1

0 0 0 2

1 2 3 3

0 4 . 1  1



South seems to have the winning hand, but North launches a counterattack through e4, and first b1 and b2 and then also c1 and c2 are captured:



0 4  0  4

1 5 0 0

3 4 0 0

1 2 0 6

2 2 4 1

0 2 0 2

3 4 3 3

0 6  . 1  1

The odds seem to have shifted, but South has g1 in store (takes g3 and g4):



0 4  0  4

1 3 1 2 0 0 0 5 4 2 2 2 0 6  . 0 1 1 5 1 5 3  0 0 6 1 2 0 3



Example of an Igisoro Game

103

North begins at a3, taking e1 and e2.



0 0  0  4

1 7 0 0

3 6 1 0

1 4 1 6

2 4 0 0

0 3 1 2

0 1 5 0

0 6  . 3  3



South does a1 in a reverse sense, taking d3 and d4 and, later, e3 and e4.



0 0  1  2

1 7 3 1

3 6 4 2

0 0 1 0

0 0 3 3

0 3 4 2

0 1 0 3

0 6  . 2  6

North now starts at b3, with a defensive purpose, to protect himself against capture by spreading out his pawns. Later on, this will turn out to be a bad move.



0 0  1  2

1 0 3 1

3 7 4 2

0 1 1 0

0 1 3 3

0 4 4 2

0 2 0 3

1 7  . 2  6



South starts stabbing his opponent in the heart, h1, and taking c3 and c4.



0 0  2  3

1 0 0 0 0 0 0 1 1 4 4 6 3 5 6 0 3 1 3 2

0 2 2 3

1 7  . 4  0

North tries g3, with a reverse capture of e1 and e2.



1 1  2  3

0 0 4 0

1 0 6 3

1 1 3 1

1 2 0 0

1 5 6 2

1 0 2 3

1 8  . 4  0



C1 is the final blow: taking h3 and h4, later a3 and a4, and finally f3 and f4.



0 0  4  4

0 0 6 1

1 0 0 1

1 1 5 3

1 2 2 2

0 0 9 1

1 0 5 6

0 0  . 8  0

104

5  Reasoning Without Writing

The sole pawn North can still capture is e3, and then there is just one pawn in North’s camp, so that North must taste defeat.



0 0  4  4

0 0 6 1

1 0 0 1

1 1 5 3

1 0 2 2

0 1 9 1

1 1 5 6

0 0  . 8  0



South is the winner (Fig. 5.36).

Fig. 5.36  The game board is set down between the players

Notes In some cultures, the husband and his wife-to-be must play a mancala-type game as a kind of test. If they play the game without fighting, it is seen as insurance for a quiet future. Despite the long playing time and the large number of possibilities, igisoro is a game of strategy, where chance does not play a role. This is why the Abbé Kagame, a clergyman, already mentioned earlier, wrote in a book of poetry over 50 years ago, “Let the almighty Imana or God play igisoro with his divine hand.” Kagame’s literary work was written in Kinyarwanda, the language of Rwanda, but the added illustration speaks for itself. Albert Einstein made a similar

Notes

105

comparison in his protest against the use of probability in quantum physics. Einstein’s assertion that “God does not play dice with the universe” is well known, but Einstein would certainly have agreed, had he known, that in the Rwandan version, the Imana (God) plays igisoro (Fig. 5.37).

Fig. 5.37  A 50-year-old poetry collection shows a divine hand and the igisoro game

Chapter 6

Multiplication in the Yoruba and “Ethiopian” Way

|||||

The Yoruba Region The origins of the Yoruba people from southwestern Nigeria are shrouded in the mystery of time. Oral traditions suggest that they came from the East, and a number of similarities between the customs and traditions of the ancient Egyptians and the Yoruba confirm this. Their recent history starts with the establishment of the Oyo state around the first century of the second millennium. Commercial and cultural contacts with the North encouraged scientific and artistic activities in the region. In the next centuries the kingdom of Benin expanded, independently of the Oyo kingdom. British colonialists crushed both at the end of the nineteenth century, although the name Benin remains in use to this day (Fig. 6.1). In the earliest mention about the discoveries of other number systems, the first reports were sometimes confused about the interpretation of the information they contained. In 1886, Dr. O. Lenz saw some people count in the region of Timbuktu, and he recording the following information. In the region of the Segu, of Massina and more to the north to Timbuktu, one makes a kind of unity of the number 16 times 5 cowry shells. It is not evaluated as 80 but as 100. Thus, one gets 16 × 5 = 100 […]. In this way one has instead of in reality 100,000 only 80,000 (oginaje temedere) 10,000 only 8000 (oginaje sapo) 1000 only 800 (gine oginaje) 100 only 80 (debe).

Other authors also tried to interpret these first reports of number systems. F.-J. Nicolas pointed to the influence of other cultures and changes over time (Nicolas, 1979). European prehistoric discoveries suggest that cowry shell counting could be © Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_6

107

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6  Multiplication in the Yoruba and “Ethiopian” Way

Fig. 6.1  Map of Yoruba region

Algeria Mali Region of the Yoruba Nigeria

Benin Lagos

Egypt Sudan Ethiopia Congo

Tanzania

Fig. 6.2  Cowries, little shells of the porcelain snail, were used in West Africa as money, counting devices, and ornaments. Today, specialized shops sell them as hangers

2000 years old. In some parts of Africa it only began in the fourteenth century, while in other places cowry shells were noticed only in the past century (Fig. 6.2). Following Nicolas these shells were often taken in groups of five in the region of Timbuktu, while in earlier times groups of four shells were more common. Even piles of three would have been in use in an intermediate period. This is striking because in the latter case the number 60 shows up again, and this is a frequently occurring total since it is formed as 3 × 20. Eighty was nevertheless the most important quantity, expressed as 4 × 20, and later by forming 16 groups of 5 shells.

Description of 1887 A description of the customs of the Yoruba by Mann dates from 1887 and provides more details. This author describes a magician-calculator who presented himself with a large pile of shells. He took them by fives, and grouped them by fours. Five groups of 20 little shells thus gave 100 little shells in total. He expressed numbers through differences, using 5 and 20.

Description of 1887

109

This is expressed also in the names of numerals  (Delano, 1963). So we have 1 = okan, …, 10 = ewa, and starting from 11 = okanla to 14 = erinla, numbers are formed through additions. This appears in the words: 11, okanla, is 1 (okan) and 10 (ewa); here, ewa is contracted to la. From 15 on, differences play a role: 15 = arundinlogun, where arun = 5, dinl = minus, ogun = 20. Finally, 19 = okandinlogun, and this is analyzed as 20 minus 1. Next, from 21 to 25 additions are used again, but from 26 to 29 subtractions return. At 30  =  ogbon, additions are used until 35 = aarundinlogoji, where the word dinl again points to a subtraction (arun = 5, dinl = minus, ogoji = 40 = ogo-ji = 20 × 2), and so forth. Thus, ogerin = ogo-erin = 20 × 4, which recalls the French quatre-vingt. Danes should find it even easier to understand the Yoruba, because in Denmark 60 is tres or 3 × 20, and 80 is firs or 4 × 20, while in addition 50 (halvtres = 2.5 × 20), 70 (halvfjers = 3.5 × 20), and 90 (halvfems = 4.5 × 20) refer to the original use of base 20. In fact, the English word “score” points to the ancient use of a counting stick, where there are always 20 carvings. Thus, a similar base is not that unusual in Western experience (Figs. 6.3 and 6.4).

600 - 80 +5 Fig. 6.3  Formation of 525

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6  Multiplication in the Yoruba and “Ethiopian” Way

kan meji meta merin maruun mefa meje mejo mesan mewa mokonlaa mejilaa metalaa merinlaa meèedogun merindinlogun metadinlogun mejidinlogun mokondinlogun ogun mokonlelogun mejilelogun metalelogun merinlelogun meéedogbon merindinlogbon metadinlogbon mejidinlogbon mokondinlogbon ogbon

1 2 3 4 5 6 7 8 9 10 +1+10 +2+10 +3+10 +4+10 -5+20 -4+20 -3+20 -2+20 -1+20 20 +1+20 +2+20 +3+20 +4+20 -5+30 -4+30 -3+30 -2+30 -1+30 30

mokonlelogbon mejilelogbon metalelogbon merinlelogbon maruundinlogoji merindinlogoji metadinlogoji mejidinlogoji mokondinlogoji ogoji mokonlogoji mejilogoji metalogoji merinlogoji maruundinlaàadota merindinlaàadota metadinlaàadota mejidinlaàadota mokondinlaàadota àadota mokonlelaàadota mejilaàadota metalelaàadota merinlelaàadota maruundinlogota

+1+30 +2+30 +3+30 +4+30 -5+20×2 -4+20×2 -3+20×2 -2+20×2 -1+20×2 20×2 +1+20×2 +1+20×2 +1+20×2 +1+20×2 -5-10+20×3 -4-10+20×3 -3-10+20×3 -2-10+20×3 -1-10+20×3 -10+20×3 +1-10+20×3 +2-10+20×3 +3-10+20×3 +4-10+20×3 -5+20×3

Fig. 6.4  Yoruba numerals, in another dialectal variant than the one given in the text, but identical from a mathematical point of view in terms of structure

Here are some more Yoruba examples of number construction: 45 = (20 × 3) – 10 – 5

50 = (20 × 3) – 10

108 = (20 × 6) – 10 – 2

300 = 20 × (20–5)

318 = 400 – (20 × 4) – 2

525 = (200 × 3) – (20 × 4) + 5

Yoruba Multiplication

111

The last number, 525, is formed by first taking 3 igba groups of 200 shells each. Of these, 4 ogun heaps of 20 are removed, and finally one arun group of 5 is added. Although this way of forming numbers seems rather complicated to Western eyes, it is simply a matter of familiarity. The number 397, for instance, or “three times one hundred and nine times ten and seven,” is expressed in the Yoruba ­language as “three less than four hundred,” which is much shorter. This comparison becomes even more evident if we check it in a language that is not necessarily the mother tongue (in the native tongue the comparison is so familiar that the difficulties in forming numbers disappear). Thus, in French, 397 becomes trois-cents quatre-­vingt dix-sept, and this corresponds to the rather complicated expression of “three times one hundred and four times twenty and ten and seven.”

Yoruba Multiplication The execution of a multiplication such as 17 × 19 in the traditional Yoruba way is done as follows. First, 20 heaps of 20 cowry shells are put in front of the calculator (Fig. 6.5).

Fig. 6.5  Initial context for the execution of 17 × 19

112

6  Multiplication in the Yoruba and “Ethiopian” Way

Next, from each heap one shell is taken, and these are set apart to form a new heap. Then 3 heaps of 20 are set aside. From one of them a shell is added to one of the three heaps, and next from the same heap another one is added to the remaining heap. Thus the two completed heaps are appended to the new heap (Fig. 6.6).

...

... Fig. 6.6  Execution of 17 × 19

Now the Yoruba magician-calculator reads the result: 17 × 19 = 323 (Fig. 6.7). The Yoruba approach often encounters disbelief during lectures because it does not clearly show the reasoning while handling the shells. First, one wonders why the result of the multiplication can indeed be read rightaway from the given arrangement of the shells. The Yoruba calculator starts with 400 little shells, and from each heap he put one aside. Of the 20 heaps with 19 little shells he singles out 3, so he knows that 17 × 19 will correspond to 400 minus what he has put aside. This is 1 heap of 20 (resulting from 1 shell from each heap), and 3 heaps of 19 little shells. To count the latter more easily, he completes them as much as possible to multiples of 20 by taking from 1 of them 2 shells and adding 1 to each of the 2 others. Thus, 3 heaps of 20 shells were set aside, and 1 of 17 shells. The Yoruba calculator now sees the result before him: 17 × 19 is 17 less than 400 minus 3 times 20 or, literally, spoken using the rules of forming numerals in Yoruba: 17 from 60 from 400. This indeed is our “three times one hundred and two times ten and three, or 323.”

113

“Ethiopian” Multiplication

17×19

17

3×20

Fig. 6.7  Final result which the Yoruba read, in their number vocabulary, as 17 × 19 = 323

Another reaction points out the contrast with the Western practice. Yet how does a Western student learn to execute an arithmetic operation? Addition is learned in some schools using blocks, while others prefer colored rods, but in any case students learns to make “bridges” using 10 or 100. For example, suppose 7 and 8 have to be added. Of course this is 15, but how is this taught? Well, from that 8, 3 units are taken, so that 7 is increased to 10. Of the 8, 5 units remains, and so the result is pronounced as 10-and-5, or, in correct English, 15. When 80 and 50 must be added, a bridge will be made using 100, and because, following the addition of 20 to 80, another 30 units of the initial 50 remain, the addition results in “100-and-30.” The procedure is comparable to the Yoruba method, where the bridge is made around 20 and 400. The only true difference between the Western and the Yoruba schemes is the use of subtractions in expressing numerals.

“Ethiopian” Multiplication A still different multiplication method was at the base of Egyptian arithmetic. It is well known because of two papyri and from secondary Greek and Roman sources. A certain Ahmes wrote the mathematical “Ahmes papyrus” that is now held at the British Museum thanks to a donation by a collector named Rhind (hence the

114

6  Multiplication in the Yoruba and “Ethiopian” Way

alternative name “Rhind papyrus”, after the sponsor). Dated around the year 1650 BC, it was nevertheless based on findings about 300 years older. Another document, the “Moscow papyrus”, dates from 1850 BC and, together with the Ahmes papyrus, consists of 112 mathematical problems. They range from the calculation of the area of a circle, the volume of a truncated pyramid, and a cylinder (or a hemisphere), to the most classical solutions of equations, with the difference that the problems are expressed in hieroglyphs using crocodiles, miniature suns, and snake symbols. Their multiplication method, of which the first written evidence can thus  be dated to around 4000 years ago, was widespread. In Greece as well as in Russia, the Middle East, and Ethiopia, it was a popular method in rural regions. After all, no multiplication tables are needed. A description used in many books identifies the technique with a more European-­ sounding name, the so-called Russian peasant method. The classical explanation asserts that the calculation method spread from Egypt through Greece to Russia and returned to the African continent via the Middle East to end up in Ethiopia. Still, an influence of Egypt on Ethiopia or of Ethiopia on Egypt seems more evident than a detour via Russia. Because the influence of Ethiopia on Egypt will be discussed later, we will use here a similar freedom of interpretation as European authors use when designating the Egyptian method the Russian peasant method and call it the Ethiopian method (Fig. 6.8). Others have stated that one could see tailors in West Africa using this doubling method, but these sources could not be substantiated.

Russia

Fig. 6.8 Classical evolution of multiplication method by doubling

Greece Middle-East Egypt

Ethiopia

115

Tailor or Hieroglyph Algorithm

Tailor or Hieroglyph Algorithm As an example, here is how they would have obtained the result of 17 × 13. First, 17 is doubled several times and simultaneously the corresponding powers of 2 are recalled: 17, 34, 68,136,…and



2, 4, 8, (16 ) , …



When the powers of 2 exceed the second number, 13, the number from the row with doubling is recalled: 136. Next, the difference is considered 13  – 8  =  5, and the procedure is repeated until 5 is exceeded: 17, 34, 68,… and 2, 4, ( 8 ) ,…





Now the number 68 is recalled, and the difference 5 – 4 = 1 is obtained. The procedure is repeated a last time: 17,…and ( 2 ) , …





The retained numbers are now added: 136 + 68 + 17 = 221 = 17 × 13.



In Egypt this method is denoted in hieroglyph writing. The symbol for 1 was |, for 10 it was ∩, and for 100 the sign. The scribe would double the first number, 17, and so forth, and stop when the second number was exceeded. The scribe then wrote down the following (the corresponding conversion is given to the right): |||| ||| ∩ || ∩ || ∩∩ |||| ∩∩∩ |||| ∩∩∩ ||||∩∩ || ∩ |∩∩

|

17

1*

||

34

2

|| || |||| |||| |||∩

68

4*

136

8*

17 + 68 + 136 = 221

1 + 4 + 8 = 13

The hieroglyph resembles a papyrus roll and means “the result is the following.” The star asterisk is, of course, not a hieroglyphic symbol but was used here to indicate the halving of an odd number. The corresponding number in the doubling must then be recalled (Fig. 6.9).

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6  Multiplication in the Yoruba and “Ethiopian” Way

Fig. 6.9  During a Nile cruise, the author was fortunate to come across some mathematical calculations at the port of Kom Ombo

Variations on Doubling Algorithm The more Russian variant on this theme goes as follows. The numbers are written in two columns, and the first is doubled several times, while the second is halved. When that is impossible, because the number is odd, an arrow is inserted beside the odd number, and the half is rounded down (the halves are simply dropped):  17

13

 34

 6

*

 68

 3

*

136

 1

*

Now add all the indicated numbers: 17 + 68 + 136 = 221. A reverse application of the method permits division. In the Ahmes papyrus, the variables x and y are introduced by the words “reckon with y such that x is obtained.” To divide 221 by 17, an Egyptian scribe would consequently think: starting with 17, how many times should I add it to itself to get 221? To do so, 17 is again doubled a number of times, until some of these doublings yield 221. Here it is 17 + 68 + 136 = 221, and the reverse scheme becomes *

 17

1

 34

2

*

 68

4

*

136

8

Examples of Calculation Game

117

Now all indicated numbers are added: 1 + 4 + 8 = 13, and thus 221/17 = 13. If the division does not work out, the Egyptians had a procedure using principal fractions. They knew only those fractions with numerator one, with one exception, the fraction 2 , which they used as well. Thus, their division method was more involved due to 3 the lack of knowledge of fractions with a numerator different from 1.

Fanciful Representation of Ethiopian Multiplication As explained earlier (Chap. 4), some applications of ethnomathematics suffer from an excess of fantasy. An example was the golden section myth, emphasized extensively in many school textbooks, with the encouragement of educators and mathematicians. They seem to agree with the widely held view that art from ancient times and from the Renaissance followed some mathematical pattern, like nature. In African considerations every bit of fantasy in the interpretation of ethnomathematical facts is attacked or, in the best case, not mentioned at all in textbooks. Of course, neither is advisable. Because a justified fantasy probably is the best medicine against an aversion for mathematics, we here use, as a mental exercise, the Ethiopian multiplication method on an African game board. This is not the way in which this multiplication was carried out, but it provides an idea of how one can do so without writing and without being confronted by numerical notation (Huylebrouck, 1996). The igisoro game board can be used in a fairly useful way to represent numbers. The upper half is called North and the lower South, as in Chap. 5. In each part the holes correspond to powers of 2: 1, 2, 4, 8, 16, 32, 64, 128 in the lower row and 256, 512, 1024, 2048, 4096, 8192, 16,384, and finally 32,768 in the upper. It is not coincidental that these last numbers correspond to units known from computer science: 1024 = one megabyte, 2048 = two megabytes, 4096 = four megabytes, and so forth. In this way, one can use North and South together to represent numbers up to 1 + 2 + … + 32,768 = 65,535.

Examples of Calculation Game We start with a simple example, 3 × 6, which of course equals 18. Two (white) balls, one in the hole for 2 and one in the hole for 4, represent the number 6 in the North. Next, the number 3 is shown in the South by two balls, one in the hole for 1, and one in the hole for 2. The method now says that the first number should always be halved; when this yields an odd number, the second number is first recalled and then doubled. The procedure stops when the halving of the first number results in 1. Thus, 6 is divided into 3, what is easily carried on the game board by simply moving the two balls in

6  Multiplication in the Yoruba and “Ethiopian” Way

118

the North 6 over one to the closest border (indeed, 4 becomes 2, 2 becomes 1). The doubling of 3  in the South is as simple by moving the balls to the other side: 1 becomes 2, 2 becomes 4. This is repeated, but when 3 is halved, 6 must be recalled first, because half of 3 is of course not 1. This is easily noted in the execution because in moving the balls to the North, one ball drops off the game board. We thus put the dropped ball to the side, in exchange for two other balls (in the illustration it is for simplicity represented by black balls, to avoid confusion) that we put in the South together with the white balls. Next, the white balls in the South are doubled: 2 becomes 4, and 4 becomes 8. The procedure stops because 1 is reached in the North. The recalled numbers represented in the South must be added now. There are two balls, one black and one white, in the hole for 4, and these are replaced (4 + 4 = 8) by one ball in the hole for 8. The two balls in the 8 hole are replaced by one in the 16 hole. There still remains one ball in the hole for 2: the sum is 16 + 2 = 18, which indeed is the product of 3 and 6 (Fig. 6.10). Summary: 6 × 3:

6 → 3* → 1* 3 → 6* → 12* 6 + 12 = 18 = 6 × 3

An advanced example is 13  ×  5. 13 is represented in the North by 3 balls: 13 = 1 + 4 + 8. In the South 2 balls are sufficient for 5 = 1 + 4. 13 is halved, by moving the balls one hole. Incidentally, the description of the method in words is more long-winded than the actual execution. One does not even have to check about whether a number is even or odd because one simply sees a ball falling from the board when moving the balls around. When that happens, one uses black balls to record this halving of an odd number. Upon the consecutive doublings of 5 two balls are moved all the time, and one makes sure to take only the white balls (Fig 6.11). Summary: 13 × 5:

13*

→6

→ 3*

→ 1*

5*

→ 10

→ 20*

→ 40*

5 + 20 + 40 = 65 = 13 × 5.

The addition of the recorded numbers, 5, 20, and 40, happens on the game board indeed as if it were a game: two balls are replaced by a single ball in the hole right next to it. A lot of time is lost when one must always add the translation that 2 balls in the hole for 4 are replaced by one in the hole for 8, and so forth. After all, this replacement method recalls the way of playing the game itself.

Examples of Calculation Game

119

←6=2+4 ←3=2+1

6:2=3→ 3×2=6→

←1=3:2 ← memorize 6 ← 12 = 6 × 2

Stop: remainder 1 → Add →

← 2 + 16 = 18 = 3 × 2 Fig. 6.10  3 × 6 on an African game board

On an expert level, here is another assignment: 241 × 17. The summary should speak for itself (Fig. 6.12): 241 × 17:241* →120 →60 →30 →15* →7*     17* →34 →68 →136 →272* →544*     17 + 272 + 544 + 1088 + 2176 = 4097 = 241 × 17.

→3* →1088*

→1* →2176*

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6  Multiplication in the Yoruba and “Ethiopian” Way

← 13 = 1 + 4 + 8 ← 5=1+4

13 : 2 → memorize 5 → 5×2→

←6:2 ←5×2

3 : 2 = 1 and stop → first, memorize 5 × 2 next × 2, and add everything →

← 1 + 64 = 65 Fig. 6.11  5 × 13 on an African game board

Mathematical Considerations Using modern notation, the Ethiopian multiplication method can be explained as follows. First one recalls the property that every number can be expressed as a sum of powers of 2. Thus:

13 = 1 + 0 × 2 + 1 × 4 + 1 × 8.

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121

← 241 = 1 + 16 + 32 + 64 + 128 ← 17 = 1 + 16

241 : 2 → memorize 17 → 17 × 2 →

← 15 : 2 ← memorize 272 ← 272 × 2

Stop: remainder 1 → Add →

← 1 + 4096 = 4097 Fig. 6.12  17 × 241 on an African game board

The product of 13 and 17 is then executed term by term: 13 × 17 = (1 + 0 × 2 + 1 × 4 + 1 × 8 ) × 17 = 1 × 17 + 0 × 2 × 17 + 1 × 4 × 17 + 1 × 8 × 17 = 17 + 4 × 17 + 8 × 17 = 17 + ( the double of the double of 17 )

+ ( the double of the double off the double of 17 )



= 17 + 68 + 136 = 221.



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6  Multiplication in the Yoruba and “Ethiopian” Way

The doublings correspond to the terms in the expression of 13 that had a coefficient of 1. They match up to where half of an odd number was taken. One may wonder whether this Ethiopian multiplication is much more difficult than Western multiplication. The study of the “amount” of work a particular multiplication procedure requires is a part of the mathematical field that studies algorithms. Multiplication can indeed involve more or less time, even when executed by persons or by identical computers, depending on the method employed. For example, the usual multiplication for 13 × 11 runs as follows:    13   ×11    13   + 13. = 143 A faster method to execute a multiplication by the number 11 consists in simply adding the ciphers of the second number and placing them between both. Indeed, 1 + 3 = 4, and thus 13 × 11 = 1 4 3. This requires much less time than writing down the general method (similarly, 27 × 11 = 2 9 7, or 62 × 11 = 6 8 2, and so forth). In a similar way, the described Ethiopian and Western methods can be compared with respect to their performance by standing by, for instance, with a stopwatch beside two identical computers executing a large multiplication operation simultaneously, following different preprogrammed methods (of course, the computer can also measure the time needed itself, but doing it with a stopwatch is more fun). There exists another interesting link to computer information science. Niklaus Wirth, the creator of the computer language Pascal, was a prominent specialist in algorithmic mathematics and wrote popular books about the fundamentals of programming. One of them carried the remarkable title Algorithms + Data Structures = Programs (Wirth, 1976). One exercise in this book consisted in writing a program to multiply two numbers only by adding, halving, or doubling them, just as in the Ethiopian method. Of course, the comparison of a computer relay with a hole in an African game board is evident. A ball in a hole can represent a closed relay, while an empty hole can stand for an open relay. Doubling a number is a simple shifting of balls, an instruction also well known from certain computer programming languages.

Algebra and Osteopaths In the example of the Western multiplication of 13 × 11, the numbers were as usual arranged from the right, starting with the last cipher, and not following the first one. A dot after the number 13  in the next-to-last line recalls this. This notation for

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123

numbers, from right to left, is an Arab legacy, as are the names “algebra” and “algorithm” themselves. Muhammad ibn Musa al-Khowarizmi, or Mohammed the son of Moses of Khorezm, was a member of the Houses of Wisdom of Caliph al-Ma’mum of Baghdad (AD 813–833), where he made observations. In 825 al-Khowarizmi wrote a dissertation about equations entitled “Hisãb-al-jabr w’al-muqãbala,” or the “Calculation of reduction and restoration.” This concerns the rearrangement of terms on one side of an equation (restoration) and the addition of equal terms (reduction). Robert of Chestar in AD 1140 translated the “al-jabr” into Latin. His Latin translation, Algebrae Liber et Almucabala, popularized the name “algebra” for the theory of equations. It is interesting to note that jabr means in Arabic also “the placement of a bone”, because it is related to the word “reduction.” During the Middle Ages the Moors brought the word to Spain, where algebrista became the name for an osteopath or bone manipulator. This provides a fluid transition to the next part of this book.

Part II

The Ishango Rod(s)

Chapter 7

The Ishango Site

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Choice of Site The sources of the Nile and knowledge of their location (cf. Chap. 1) play a major role in this part, since at the farthest sources high in the mountains on the border of the Congo and Uganda is where Ishango is located. At Lake Albert, there are two important branchings, of which the western comes from Lake Kioga, and further upstream from Lake Victoria, even further south the water flows from the Ruvuba River. It is the southernmost source of the Nile and reaches all the way to Burundi. The eastern branching is formed upstream by what was formerly called Lake Edward and later Lake Idi Amin; today it is more appropriate to use the African name Lake Rutanzige. Even further south lie Lake Kivu and the elongated Lake Tanganyika, but these flow to the Congo and, thus, to the Atlantic Ocean. Lake Rutanzige is a small lake by African standards. The water that flows out of it forms the Semliki River and flows, as mentioned, to Lake Albert. In earlier times the flow of the river was in reverse, because at the exact location where the river rises from the lake there is a delta, like an estuary. At some distance from this spot there is the little village of Ishango, almost the sole center of some importance in this scarcely populated area (Figs. 7.1 and 7.2). About 70 years ago, European explorers discovered a number of bone harpoon heads, apparently from prehistory, at the foot of the steep cliffs all around this delta. In the hills of the surrounding heights, on the right shore, biologist H. Damas made a first archeological sounding in 1935. He found different fossils and two fragments of human lower jaws, which were examined at the University of Brussels. The jaws that were found indicated what people had lived there, but it appeared that they could not have produced the fossils that had been found.

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_7

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7  The Ishango Site 29o

Algeria

Egypt

Mali

1o

Soudan Nigeria

South Africa

Mo

Angola

zam biq ue

Ouganda Kenya Congo

Congo

Lake Albert

Sem lik i

2o

30o

0

-1o -2o -3o -4o

N ile

128

LakeRutanzige (Edward)

32o

33o

34o

Uganda Lake Kioga

Mount Ruwenzori Kilembe mines Katanda Katwe Ishango

Rwanda

Burundi Most southern source of the Nile

Kenya

Lake Victoria

Tanzania

Unyamwezi-region: mountains of the Moon.

Fig. 7.1  Location of Ishango

Fig. 7.2  Views on the Semliki Valley, in the year 1980. At that time, many politicians came to visit Ishango, but they never went to the site of the oldest mathematical finding

Until 1950 no attention had been paid to this particular location, but then a happy coincidence took place. Victor van Straelen, head of the Institute of National Parks of the Belgian Congo, was at that time also the head of the Royal Belgian Institute for Natural Sciences of Belgium (RBINS). He wanted to organize an expedition because Ishango lay in what was then the Belgian Congo, and so it came to be that van Straelen charged RBINS-geologist Jean de Heinzelin de Braucourt with a mission (see Arnold, 1999). It is to the credit of de Heinzelin that he quickly realized that the Ishango people were Homo sapiens and that the site and objects were more important than generally recognized based on knowledge current at that time. The geologist would seize the

The Excavations

129

opportunity and make a detailed archeological examination. His report, a thick book, is conscientious and illustrated with photographic plates; it appeared in 1957 (Fig. 7.3).

Fig. 7.3  The site of Ishango, 50 years ago. Is that de Heinzelin on the terrace?

The Excavations At Ishango, de Heinzelin unearthed remains of various settlements, which accumulated to a depth of 4 m. After all, in the Nile Valley, in West and East Africa, various civilizations of hunters and fishermen had developed during the last Stone Age, one after the other, or even in the same period. But the Ishango settlements had but a short lifetime of some hundred years. A volcanic eruption would bury them and so change the region in a kind of African Stone Age Pompeii. The excavations of de Heinzelin took place between April 23, 1950, and July 22, 1950. His team would excavate about 500 m3 of earth. It was the colonial period, but still the scientist was certainly not disrespectful to the people with whom he worked. He described their collaboration explicitly as follows (Fig. 7.4):

Fig. 7.4  Excavation works of de Heinzelin

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7  The Ishango Site

Fig. 7.5  The archeologist in traditional clothing of the time; he is presenting trench N43GE and layer N.F.Pr

The supervising personnel (guards, clerks) belonged to the organization of the “Parc National Albert” (“The National Albert Park”). The work force was a group of 20 young Bashu and Banisanza. Despite their lack of experience, they followed the given instructions very easily and accomplished their work very carefully. Their sharp sight turned out to be very useful for the sorting tasks. […]

De Heinzelin dug two large trenches (Figs. 7.5 and 7.6). The field of excavation was described by the directions N43GE and N143GE. They formed a right angle, and the mathematical reader will immediately note that an angular scale of a hundred degrees was used. This is why the symbol G instead of the Babylonian symbol, a little sun as a superscript °, was used. The scale of hundred degrees is not unusual for geometers, in particular for those with a French preference or nostalgia for the times of Napoleon. The right angle and the directions can be seen in the illustrations. Letters are harder to distinguish, but they simply refer to particular findings of less importance to the mathematical Ishango story (thus t stands for tibia, shinbone, and so forth). The richest layer was the niveau fossilifère principal or “principal level of fossils,” where many tools, remains of fauna, and human bones were found. It was designated by “N.  F. Pr.” and other layers received similar abbreviations in the detailed descriptions. For instance, the layer “S. X.” stands for sable graveleux or “granular sand.” To get an idea about the meticulousness of de Heinzelin, here is a short extract of a detailed description. In this short passage out of a much longer similar list the reader can try to determine the exact spot of a very remarkable mathematical finding. Thus one can form an idea of the archeological inquery and the confrontation with the numerous trivial and less prosaic objects, where yet suddenly a remarkable object seems to be uncovered (Figs. 7.7, 7.8 and 7.9): S. X. = pebbly sand in crossed stratified layers, inconsistent, which collapses when dry. Slight local hardenings in contact with the bones and other organic remains, sometimes in the form of roots. Violet coloring in layers, more or less general, abundance of heavy minerals.

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131

Fig. 7.6  The excavation field was situated on the drawing above, in the middle, where a little black cross, +, stands, under the symbol for the camp. The drawings have scales 1:1500 and 1:250 - inventory: industry with stones on quartz and quartzite; worked and knocked off boulders; fragments of iron oxide; industry with bones present in large numbers; harpoon heads with one or two rows of teeth; rod with parallel lines; remains of bones. N. F. Pr. = “Niveau fossilifère principal” (principal fossil level), formed by the accumulation of remains of bones and shells. The rest of the bones are in general splinters, caused by hard beats on the anvil, and almost all fragments, even the smallest ones, show traces of artificial fractures to extract the marrow. There are monsters of stone and bone industry mixed in it, but many were already out of use. Some bones are round or rubbed off, but most have sharp edges. It is clear that this artificial accumulation results from the dumping of cooking residues and remnants of handcraft, in the waters of the lake

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7  The Ishango Site

Fig. 7.7  Cross section of layers of excavation

Fig. 7.8  A number of discovered objects; left: plate with vaguely geometric drawing; right: some bones, perhaps used for mashing or at the shore of a small beach, not far from a fishing and hunting village. Their fragmentary character implies that the bones are not easy to identify. They provide but a representation of the fauna hunted by men at the time of the removal of the place, and no natural amalgamate. The shells, mainly “melanoides tuberculata” and “viviparus unicolor,” are in very good shape of conservation, and sometimes with mother-of-pearl or skin, and some even seem to be alive. None has been smashed to pieces, and the mollusks probably have not been eaten.…

The Ishango People

133

Fig. 7.9  In this trench, approximately in the middle, the Ishango rod was found

Jean de Heinzelin brought the excavated material back to Belgium, where a whole row of drawers in the RBINS of Brussels stored the many findings from Ishango on its 19th floor.

The Ishango People The Ishango people earned their living by fishing or by gathering food and some occasional farming, depending on the season. They were not nomads but belonged to a sedentary or semisedentary group. Many grindstones and polishing stones point to a developed culture: humans began to use them when they learned to crush pigments for decorative purposes and pound seeds for food. There were no traces of pottery, and the hundred discovered quartz crystal objects of different sizes and shapes were not delicately wrought, in contrast to other African deposits. The weapons used in hunting and fishing pointed to an advanced technique. Furthermore, a comparison with weapons found in other parts of Africa suggested that the Ishango site was the source from which a certain production technique spread over a large part of the continent. The form of the harpoon head was observed closely: the number of teeth, in double rows or a single row, the presence or not of a cavity for fixation, or the position of the teeth. All together this gave a unique image, following de Heinzelin, because nowhere else in such a framework of time was such a combination of properties evident.

134

7  The Ishango Site

The Ishango human was a true Homo sapiens, part of a local population of an older Paleolithic race. This interpretation corresponds well to the present-day view of human evolution. Because of the aforementioned volcanic eruption, this population virtually disappeared, and the site’s modern inhabitants have no immediate ties to it. On the contrary, the distant ancestors of the people who live there today probably are coresponsible for the almost complete disappearance of the Ishango people: In any case, it is suitable to reserve the denomination “Civilization of Ishango” or “the Ishango era” for the harpoon industry stopping with the eruption of the volcanic explosions of Katwe. The group staying at that time in Ishango must have spread out, or emigrated, or died there. The invasion of the Hamites and the Bantus, for whom the succession is not very clear but who regardless cannot go back in time for very long, has pushed back little by little the ancient truly autochthonous groups of the population to the South. There exist some remote groups of them as in the Kalahari, which is their last hiding place, but they are now threatened by extinction; other groups have actually disappeared. These impoverished groups of population, who were chased everywhere by black and European pressures that aimed for identical levels of destruction, have been preserved but very few aspects of their life and their ancestral forms....

These words of de Heinzelin about European destruction date from of 1957, 3 years before the independence of the Congo. He added the following footnote: I can only indicate that the inhabitants of the region of the Semliki attribute mostly those places and old villages, characterized by an abundance of worked quartz crystals, spontaneously to the “Bambuti.”

The Ituri Forest, bordering the Semliki Valley of Ishango, was frequently in the news in May–July 2003 because of a genocidal campaign against entire populations. The supposed heirs of Ishango, the Bambuti (which simply means “pygmy” in Swahili), live there to this day, or at least until a few years ago.

A Particular Finding As described in the inventory of layer S. X., de Heinzelin found a rod with parallel stitches (see Fig. 7.10). It is a small carved rod, only as large as a pencil (10.2 mm.), and it even looks like a writing device. There is a small piece of quartz enclosed at one end that could not be removed without damaging the bone. It sticks out of the bone only 2 mm (Fig. 7.11). This part in the quartz could have been used to make carvings or incisions. At the time, it was thought there were no other such carved objects (Chap. 11, “Outcome” will contradict this), but perhaps hides were carved, or even the human skin, since there existed a long ancestral tradition of tattooing. The object is petrified, and it may have been subject to some chemical changes through the action of water and other elements. Yet the carvings are clearly visible on the dark brown object. Even following de Heinzelin, the groups of carvings are undeniably not decorative. Except for the pointed quartz crystal and the nondecorative

A Particular Finding

135

Fig. 7.10  Six views of Ishango rod, following official picture of the Royal Institute for Natural Sciences

Fig. 7.11  Two views on quartz crystal

136

7  The Ishango Site

lines, the Ishango rod is exceptional for a third reason: it is in the form of a utensil one can grasp. Objects with a shaft to take hold of were rare in similar prehistoric sites. De Heinzelin described how he found it: I discovered this object personally in the sand in the crossed layers S. X. in the neighborhood of the mark of 12 m of section N 43G E. It was very encrusted, covered by quartz crystals attached to each other, in particular at the side of the object that pointed upwards. The other side, which was more to the open, showed the beginning of some markings. The same kind of crust was also attached to the bones and to the harpoon heads of the principal fossil level. This is due to the penetration of rainwater from the above lumped layers of peat. Due to the strange nature of this piece, I wish to validate its authenticity, so that there is no doubt about it.

Again he added a footnote, as if wanting to confirm that there was a witness present when the carvings were revealed: It [the rod] was revealed from its casing by “préparateur” technician J. de Kleermaeker who, with the admirable dexterity proper to him, placed the carvings in evidence without damaging one of them.

To this day it is not known what animal this bone comes from. In some publications it is said that it is the fibula of a baboon, but this is a result of some confusion (see Chap. 10, where another engraved bone, from a baboon in this case, is mentioned). Some important researchers who specialize in the identification of bones from the Ishango era were working in the Africa museum of Tervuren (a few miles from Brussels) when de Heinzelin brought the bone to the KBIN of Elsene (a “district” in the Brussels region). In the hope that this small distance might some day be bridged, the bone still awaits a thorough examination of its origin (Figs. 7.12 and 7.13).

Dating the Rod For a long time the dating of the Ishango rod presented a problem. It was such a difficult challenge that de Heinzelin stopped devoting much attention to Ishango when it turned out that his first estimates were wrong. However, no scientific error could be ascribed to him if one takes into account the knowledge available at that time. Indeed, not far from Lake Rutanzige is a chain of mountains whose tops were active volcanoes thousands of years ago. Their eruptions increased the amount of nonradioactive carbon-12, whence the common proportion of carbon isotopes in the lake changed. Old shells thus seem inappropriate for accurate dating using the carbon-­14 method. Charcoal was at the time another alternative to the C-14 dating, but it was not found in the Ishango village. De Heinzelin finally dated the Ishango site to between 6500 and 9000 BC, on the basis of the best archeological and geological evidence. As a geologist, de Heinzelin compared, for example, the age of the various layers of earth. According to his

Dating the Rod

137

Fig. 7.12  The oldest mathematical finding

analysis, Ishango was a stable settlement for some hundred years before a volcanic eruption buried the whole region. The work of Yellen, Brooks, Mehlman, Stewart, and Cornelissen − the latter is attached to the Belgian Africa Museum of Tervuren − shed more light on these dating problems (Brooks, 1995). In the 1980s they organized new excavations in Ishango. A publication from 1995 dates some objects found in the neighborhood where the bone was discovered to 75,000–90,000 years old. As early as 1987, one publication set the age for the Ishango findings at 18,000– 23,000 BC.

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7  The Ishango Site

Fig. 7.13  By copying this model to a size of 10.2  cm (4.01 in.) one gets a true model of the Ishango bone

In any case, the Ishango settlement seemed to have peaked about 20,000 years ago, even if the bone is not always dated to the same period. How then can the differing opinion of a prominent scientist like de Heinzelin be explained? Following Anne Hauzeur, an archeologist of the RBINS who knew de Heinzelin very well, he based his calculations on an examination of the tools found at the site and, in particular, on their morphology. Indeed, this study provides valuable information about the age of the carved rod.

Dating the Rod

139

In addition, there are stone objects made from a local material, a kind of white river quartz of average quality. It occurs especially as small scrapers and scratchers, of a quite disappointing aspect and not very conspicuous. At the time of the discovery, de Heinzelin had placed the collection at the end of the period of hunters and gatherers (Late Stone Age) because of the small size of the objects. Since then, many other sites, older as well as more recent ones, have furnished evidence of the same kind of technique, so these features were not very distinctive. The processing of bones was evident in particular among a remarkable collection of harpoon points and toothed heads, which turned out to be more accurate characteristics of the time periods than the quartz objects. The site of Ishango, which was occupied repeatedly, enhances our understanding of the evolution of these harpoons. First there were two opposite rows of teeth, and then a single row. The carved bone of Ishango is associated with the harpoons with two rows. The site of Ishango had a very old stratum of harpoon production, a model of which was distributed starting from the region of the Great Lakes to the West as well as to the North (Sudan and Egypt). Remarkably enough, there is just one exceptional site to the South, in Botswana, with the same kind of harpoons. The appearance of the harpoons seems related to a change in climate and an increase in rainfall. The life style changed to more intensive fishing, and this continued after sedentism and after the introduction of cattle breeding. Ishango must have been a juncture on the way to civilization, corresponding to a much earlier time than could have been thought before (Fig. 7.14).

Fig. 7.14  Harpoon points from Ishango, Khartoum, Taferjit, Shaheinab, West Africa, and Nagada (in that order, from left to right), with the evolution between the first and the difference from the last (the locations of the sites are given in the next chapter)

140

7  The Ishango Site

Finally, the author’s personal contact with J. Shallit, of the University of Waterloo in Canada, and telephone inquiry to Allison Brooks suggest that 20,000 BC was quite a safe estimate for the mathematical Ishango rod. In a recent lecture at the Royal Belgian Institute for Natural Sciences, Brooks concluded that the small object was certainly older than 22,000 years. An age of 20,000 years is thus a very safe estimate of the age of the Ishango rod, because there is even a margin of 2000 years. This cautious estimate represents the consensus of most archeologists. As for mathematicians, it does not matter that much. Even with an age of 8500 years as calculated by de Heinzelin, it is older than any similar mathematical finding. Historically, however, an age of 22,000  years means that about another 15,000 years of evolution remained in the era of the first pharaohs.

Chapter 8

Mathematical Carvings

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Description of Carvings The small bone contains three columns of carvings, distributed lengthwise. One of the columns lies along the most curved side, and it can play the role of a central column. De Heinzelin called them M, G, and D, after the French words milieu (middle), gauche (left), and droite (right). Within each column the carvings are put in groups labeled a, b, c, and d, and for column M there are the additional groups e, f, g, and h. The Ishango rod thus carries three columns of carvings.: Column G: a:11, b:13, c:17, d:19. Column M: a:3, b:6, c:4, d:8, e:10, f:5, g:5, h:7. Column D: a:11, b:21, c:19, d:9. The carvings are almost parallel in each group, but sometimes they are of different lengths and a distinct orientation. They are clearly arranged in groups. For example, in the case of column M, the groups Ma and Mb are close to each other, and they are followed by some spacing, after which come Mc and Md, also closely grouped. Then there is again some space, and then follow Me, two 5s in Mf and Mg, and then a final 7 in Mh. There is some uncertainty about the number of carvings in the Me and Mf groups of 10 and 5 carvings. A part of the bone was seriously damaged by infiltrating water, as mentioned in the archeological report of de Heinzelin. This is why in the Me and Mf columns there are also Me’ – Me” and Mf’ – Mf”, as noted in the explanatory figure (Figs. 8.1 and 8.2).

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_8

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8  Mathematical Carvings

142

11 11

3 6

19

21 13

17

4 8

10

9 19

5 5

7

Fig. 8.1  A traced-out version of the carvings

Fig. 8.2  The names used by de Heinzelin, and the corresponding numbers, arranged following the original table of de Heinzelin

Numerous observations immediately suggest themselves: –– The numbers in column G are prime numbers, which are only divisible by 1 and themselves, between 10 and 20; –– The successive subgroups of column M are related by doubling: 3-6, 4-8, or by halving: 10-5, except for the two last groups, 5 and 7;

Hypothesis of de Heinzelin

143

–– The markings of column D, row 2, obey the rule 10 + 1, 20 + 1, 20 – 1, 10 – 1; –– The total number of carvings of columns G and D is 60 = 5 × 12, and for column M it is 48 = 4 × 12. The interpretation of these logical relations is a subjective matter, but it can be strengthened by circumstantial evidence. The D-column could point to a number system with base 10, while other observations indicate that an arithmetic operation was being carried out. The latter facts could suggest some kind of lunar calendar, because the sums 60 and 48 could indicate two lunar months and one and a half lunar months. These suppositions are explained in more detail in what follows.

Hypothesis of de Heinzelin De Heinzelin was very intrigued by the markings on the bone. He consulted many mathematicians and acquaintances: Mr. and Ms. Defrise, Capiaux, Casteels, Libois, Pelseneer, and Lancelot Hogben. The latter is well known: he is the author of a then well-known popular work, Mathematics for the Million. In this way, de Heinzelin proposed a list of arithmetic operations, which he represented in his very first report (Fig. 8.3). De Heinzelin confirmed having contacted a “critical mind,” his friend the skeptical philosopher François Deknop: Is this arithmetic or simply a chaotic ordering? The table with the series of relations suggests, by insertion of the accepted symbols, very noticeable operations and correlations. Well one recognizes no absolute general rule to acknowledge the complete attribution of an arithmetic hypothesis, analogous to, for example, the table of Pythagoras. Yet, the doublings from 3 to 6, from 4 to 8, the disposition of the prime numbers, the numbers greater and smaller than 10, and the addition of even numbers hardly allow one to escape from the idea that one here has to deal with the testimony of an intention. In the opinion of the mathematicians I have consulted, no logical method can demonstrate whether these numbers must be assigned to, or not, the kind of coincidence that can, for example, occur when counting hunting trophies or earnings. On the other hand, everyone will dare to pretend, I think, when explaining the table with numbers, that the feeling for human habits does tend to a conjecture of arithmetic.

In his considerations based on the very limited knowledge of ethnomathematics in the 1950s, the estimation of African calculating abilities is very intuitive. On one hand there is an underestimation, when one’s thoughts are directed only to the possibilities of bases 2 and 10. These are, of course, the first thoughts that come up to a Western mind, because 10 is the usual base and 2 refers to computer calculations. On the other hand, there is an overestimation, because there is not a single proof of the knowledge of the abstract notion of prime numbers any time before the Greeks. As de Heinzelin concludes, it is important to seek additional circumstantial evidence, because these were nonexistent in his time: If there was some arithmetic present on the bone, it is surely based on bases 2 and 10; the primitive use of it is not surprising, because they are the most natural for men.

144

Fig. 8.3  De Heinzelin’s table with various arithmetic properties

8  Mathematical Carvings

A Lunar Calendar

145

Base 2 appears on the rod through the principle of doubling and the table of 4; base 10, by its central position, the repetition of the operations leading to 10 and the multiples of it, the separation of M ≤ 10, G and D ≤ 9, and the numbers 10 ± 1 and 20 ± 1. The choice of the prime numbers also lets us suppose that the knowledge of the higher operations of multiplication and division would be known. In short, it would be a “magic table” revealing the advanced arithmetic knowledge:   (a) addition, subtraction, doubling, and multiplication (with reservation for the division),   (b) base 2 and 10, where 10 is seen in the form “9+1”,   (c) prime numbers,   (d) the highest number mentioned is 21,   (e) the highest number obtained through operations is 60. It speaks for itself that these interpretations remain open for discussion. My goal is to put it forward for consideration rather than to seek approval. A possible proof would be the discovery of similar objects in related or contemporary civilizations.

The idea that the Ishango carvings furnish evidence of an arithmetic game or of an experimental curiosity was, following de Heinzelin, supported by the already mentioned Prof. P. Libois. He imagined that the properties of numbers were observed during the manipulation of small heaps of pebbles, bones, grains, or pits. As the first part of the present book showed, de Heinzelin’s imagination was not, as of 1957, far off the reality anthropologists would record during the postcolonial days.

A Lunar Calendar In 1972 the American journalist A. Marshack defended the use of the Ishango rod as a lunar calendar. His alternate interpretation should not immediately be discarded, even if it is rather surprising in light of the illustrations. Indeed, there is convincing circumstantial evidence, such as the ropes and other objects used by African civilizations as calendars (Chap. 5). The role of the moon in many customs, traditions, and stories has been discussed (Chap. 2, “Story Telling and Music”). A quick examination of the Ishango rod already pointed to an association with a lunar phenomenon: the sum of rows G and D is 2 months, and the total of the M carvings is one and a half months. Furthermore, not only the number of carvings but also their difference in size and shape, as well as the order of the subgroups, could have significance. Marshack carried out a detailed microscopic examination of the bone and compared the markings in different forms and sizes against the phases of the moon (Figs. 8.4 and 8.5). He adds a little carving to the 11-13-17-19 row, so that it became an 11-13-18-19 row (the upper row in the illustration). He doesn’t change anything to row M, but lengthens the carvings representing 7 to get 7 lines of equal length. He considers the row 11-21-19-9 not only in reverse order (that is, 9-19-21-11) but also adds a short little carving, so that this row becomes 11-21-20-9. In row G, 11-13-18-19, Marshack saw something special in the notation of the 19 lines as 14 carvings separated from 5 others. He interpreted the five lines as the

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8  Mathematical Carvings

Fig. 8.4  Marshack’s second drawing of Ishango carvings

Fig. 8.5  Marshack’s interpretation of Ishango rod as a lunar calendar

days of a short lunar period, such as the days during the full or the new moon. He felt supported in this by the fact that, for example, the Bushmen celebrate the full moon for 3 days, and not 1 day. Thus he thinks that if there are in fact 16 days from an invisible moon to a full moon, there are never as many days counted effectively, but instead 13, 14, or 15 days. Why there are longer series of lines than others, and in what direction the three rows must be read, he explains by suddenly proposing the following drawing, where the sequence can be seen between the rows. The smaller and thicker lines would then yield the distinct phases of the moon. Some kind of rhythm or pattern would be evident. In his voluminous Roots of Civilization Marshack concludes: It represents a notational and counting system, serving to accumulate groups of marks made by different points and apparently engraved at different times. Analysis of the microscopic data shows no indication of counting by fives and tens but that the groups of

Some Objections

147

marks vary irregularly in amount. That this is an early system of notational counting is clear; however, this does not necessarily imply a modern arithmetic numerical system. I have tracked the origins of such early notational systems back to the Upper Paleolithic cultures of about 30,000 BC. It is hardly likely that this artifact was unique in the culture of the Ishango area. The newly evolved techniques of microscopic analysis may reveal other such examples. It is also possible that tallies of this sort were made in a more perishable form – on wood, on animal skins, or by an accumulation of pebbles or seeds; if this was so, the evidence would have been lost in the course of time. When did man first have occasion to keep mathematical records? Was it to note the passage of time, to predict the season for planting seeds, the flooding of rivers, the coming of the rains? The first calendars, notches on a bone, were probably lunar, following the phases of the moon.

In Ishango, a good lunar calendar was likely very important. At the beginning of the dry season, people came to the lake; in the rainy season they lived on the hills and in valleys. Those who lived permanently along the lake looked in the dry season to animals and birds coming to the lake in search of water. It thus was important to count the months in between both seasons. Some even see a relation with a quotation from Homer’s Iliad, where he talks about the “Oi Pygmaioi” who lived near the shores of the Nile and “engaged at the seasonal appearance of the cranes, in a bloody battle to defend the crops they lived from.”

Some Objections The lack of circumstantial evidence was a shortcoming in de Heinzelin’s interpretation of the Ishango rod as an arithmetic game. Naturally, in the time when he found the object, little was known about arithmetic games of strategy in Africa, but even the comparison with known ethnomathematical activities does not argue for this point of view. As stated, there is no evidence of the notion of prime numbers before the time of the Greeks. Naturally, 11, 13, 17, and 19 are precisely those numbers between 10 and 20 that are not divisible by any number besides 1 and itself, but one must admit that the Greek notion of prime numbers reaches much further. It is seen as a more abstract concept. Furthermore, the other arithmetic facts on the bone do not correspond to a notion such as prime numbers. Indeed, the other columns show only additions or differences with 1, doublings and halvings. The prime numbers 3, 5, and 7 are missing in the row, and the only even prime number, 2, is missing. Other simple concepts such as squares or other powers do not figure on the bone, although these are of course not really necessary for a good understanding of prime numbers. Yet it would be a strange jump to go immediately from elementary counting concepts to prime numbers. Another criticism found in the literature is that it is at least very surprising to attribute the development of advanced arithmetic notions to a small group of Neolithic people living in relative isolation on the shores of a lake apparently cut off

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from other traceable settlements of any size and importance. Yet it is not correct to say that Ishango communities lived completely isolated from others. This is manifestly contradicted in Chap. 9. Still, the claim of very important scientific progress seems impossible, starting from quasi-mathematical ideas all the way up to the idea of prime numbers in one single stroke of the pen, or better, in a few strokes of a quartz crystal. On the other hand, the interpretation as a lunar calendar fits better in the context of the Ishango settlement, but the explanation itself is extremely complex. Thinking about his similar explanation on the Stone of Blanchard from the French Dordogne (Chap. 1), the impression of a far-fetched interpretation is hard to discard, since in this way in any succession of lines or holes a calendar can be discerned. Furthermore, the archeologists who were contacted at the RBINS of Brussels were very reserved about this theory, and they did not wish to endorse it. This does not mean that the work of Marshack must be discarded. He started as an interested journalist on an archeological adventure and later received prizes in the establishment scientific world. He had a position at the prestigious Harvard University. A differentiated interpretation can nevertheless unify the strong points of both the arithmetic reading of de Heinzelin and the astronomical analysis of Marshack through a combination of very elementary mathematical facts with solid circumstantial evidence.

A Third Look at the Carvings Dr. Ir. V. Pletser, a former physicist at the European Space Agency now working for its Chinese counterpart (see the preface), took another look at the carvings after he had seen the Ishango rod in a rather accidental way and became fascinated by it (Pletser and Huylebrouck, 1999a, b, c). He suspected that they not only represented numbers but that the way in which they were engraved on the bone could also provide information. The middle M column clearly occupies a central position on the bone. The approximate length of each carving, the vertical distance between the eight groups, and their orientation yielded an interesting outcome (Figs.  8.6 and 8.7). From top to bottom, the first four groups suggest a doubling, but the process looks more like a rearrangement in a new group, in a particular way. Mb looks like a set of three carvings of the same length, such as in Ma, together with two longer ones, at each side, and a still longer one to make six. In the same way, Md looks like it has a subgroup of two carvings in the middle, with a smaller group of three carvings that are longer than the first two on each side but of the same length. In this way, Pletser undertook a detailed study of the carvings, where he also took into account the angles at which they were placed. If s, m, and L stand for small, medium, and Long, then his observations can be summarized as follows: The group Me shows a singularized last carving, which involuntarily prompts us think of the significance of numbers often occurring in a previous chapter, that is,

149

A Third Look at the Carvings Fig. 8.6  The part of Pletser’s study about column M 3

Ma: 3s;

6

Mb: 1m + 3s + 1m + 1L;

10∞

4

Mc: 4L; Md: 3L + 2s + 3m;

8 10∞

10

G 11 13

←

17 19

← ←

60

M

Sum

3+6+4 4+8+9 8+9 9+5+5 1 7 48

10∞

5

Mf: 3m (?) + 2L;

5=6-1

Mg: 1m + 4L (?);

7=6+1

Mh: 3m + 4L.

D 11 →

21

→

19 9

Sum

Me: 8m+1L+1s;

10∞

G 2×6-1 2×6+1 3×6-1 3×6+1

60

Fig. 8.7  Two alternative interpretations with simple patterns

M 3, 2×3 4, 2×4 2×5, 5 6-1 6+1

D 10-1 2×10+1 2×10-1 10-1

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“kenda,” or “take away one.” Mf and Mg show two different ways to express 5, but they appear in a very damaged place. Probably the first three lines of Mf are of the same length, whence the indication 3 m. In Mh the number 7 is broken down into 3 plus 4. From these probably unconscious subdivisions, in could be concluded that the numbers 3 and 4 play a particular role. If Me points to a decomposition of 10 into 9 + 1, because the tenth carving is somewhat separate, then there is a kind of relation such as that found on old slide rules used by engineers in precalculator times. The logic in the D column, with the numbers 10 ± 1 and 20 ± 1, is straightforward. In light of the suggested use of bases 3 and 4, it thus seems to make more sense to read the G column as 12 ± 1 and 18 ± 1. If the M column is now seen as three doublings, 3-6, 4-8, and 5-10, then the remaining 5 and 7 of this column can be seen as the beginning of this G series: 6 ± 1.

Circumstantial Evidence for Alternate Interpretation Following the previous interpretation, the Ishango rod can therefore be seen as simple evidence of a counting procedure in a community where bases 10 and 6 (or 3 and 4) were still mixed up and both used. This explains why the sums of the columns are 60 and 48. If, for instance, the bases had been 10 and 8, then probably the numbers 80 or 40 would have appeared. Such a counting process even appears familiar to Western eyes. When, for example, the result of an election must be tallied and the outcome is counted on paper or on a board, the numbers are often grouped in clusters of five and in clusters of two sets of five. In the West, the base 10 system is indeed customary: |||| ||||; |||| ||||; |||| |||| … Quantities of little objects, such as eurocents or chips, are, on the other hand, often counted by twos, as follows: 2, 4, 6, 8, 10, … These methods do not relate to a certain “superior” mathematical ability but a convenient practice when 10 is the counting base. Examples of the mixed use of bases 6 (or 12) and 10 were given in Chap. 3. There were, for instance, the Nyali, who still actively use numerals with bases 4 and 6. Thomas, quoted in the discussion of duodecimal bases, observed that base 6 for numbers from 7 to 9 was used in Guinea. Still following him, the Bulanda people would use combinations of 4-6 such as 10  =  6  +  4 for the denomination of this numeral, and the Bola would have expressed 12 as 6 × 2 and 24 as 6 × 4. The relation between graphical representations and these counting techniques was then discussed in Chap. 4, so that it looks very plausible that these carvings were drawn as to emphasize the counting. Chapter 5 showed in addition how some people still use elementary notational techniques (see Chap. 9 as well). Furthermore, the only clear arithmetic operation on the Ishango rod, doubling, is indeed an essential operation that appears, for instance, in a name for the number 8, “ne-na-ne” or “four-andfour,” or in the Ethiopian multiplication method of Chap. 6. The indications of Dr. Mubumbila Mfika about counting in distinct groupings of tally signs (Chap. 3) are very apropos in this context. Counting sticks, rods engraved

Circumstantial Evidence for Alternate Interpretation

151

with markings, and even tattoo tallies have been well documented throughout this book. Most of the cited authors rarely refer to Ishango, so it can be confirmed that their conclusions were made independently of the Ishango story. Admittedly, there sometimes are immense distances between the indicated locations and even enormous differences in time. However, this is also the case when it comes to, for instance, “Greek” mathematics, which spread over distant separate poleis (or cities) and over about 10 centuries, or arithmetic in China or in the West. A balanced interpretation makes the Ishango rod seem less spectacular because it does not involve prime numbers or astronomy. In other words, it does not involve mathematical overkill, and perhaps this is better. Yet the argument based on circumstantial evidence, given in previous chapters, is at least as credible as the interpretation of the bone as a lunar calendar. At the time of de Heinzelin, and even in Marshack’s time, these facts were unknown.

Chapter 9

Missing Link

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Linguistic Missing Link In his paper from 1920 about West African people, Thomas confirmed that the use of base 12 stems from ancient times. It was referred to in the written document “Travels of Ibu Batuta” dating from before the fourteenth century. Yet Thomas supposed that in view of European prehistoric discoveries, it should go back 2000 years at least. He knew about Babylon as “the best known reference to such a base” but wondered whether there was a link between them. He suspected such a relation with the Middle East, and in particular with Egypt, because “as regards burial customs the foreign element is conspicuous.” He thought that from Egypt there would have been an influence reaching to Nigeria. In these early colonial times, 1920, it was “evident” that Thomas only thought in the direction of an influence from North to South, and not vice versa. Thomas concluded (Fig. 9.1): It has frequently been assumed that the duodecimal system, which is in Europe crossed with the decimal system, is a product of Babylonia; how far this view is still accepted I do not know. But it is clear that, even though Egyptian influence in West Africa may be well established, we can hardly accept such a far-reaching theory as Babylonian influence on numbers below 20, which would surely imply both early and close contact, in the absence of other evidence of Asiatic influence in this area. It remains to add that if we find no duodecimal system among any people likely to have been in contact with Nigerian tribes, we must assume an independent origin for the system. If it had been transmitted from Babylonia via Egypt, it must surely have left some traces on its road. For those who believe the duodecimal notation can have been invented once only, it is an interesting problem to bring the Nigerian duodecimal area into relation with Babylonia.

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_9

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Fig. 9.1  Simplified map of triangle Ishango-­ Nigeria-­Egypt, with the directions of influence as thought in Thomas’ times

Middle-East: base 12, dozen, 360°... Koro, Ham, Yasgua: 1=unyi; 2=mva; ...; 12=nsog; 13=nsoi (=12+1); 14=nsoava (=12+2); ...

? ?

Huku-Walega: base 12

The Ishango rod could be the missing link Thomas was looking for over 80 years ago; not from Egypt to Nigeria across the Sahara, nor vice versa, but from Ishango to Nigeria as well as Egypt, there could have been an influence. This is a much more obvious way: from Ishango to West Africa there is a corridor South of the desert, along rivers, lakes, and oases, while from Ishango to Egypt the valley of the Nile shows the route. Thomas had such a suspicion himself after all, because at the end of his publication he adds that he believed there was another community in Africa engaged in practices identical to those had studied in West Africa. He mentioned the Huku-­ Walegga, for whom 7 is expressed as 6 + 1, 8 as 2 × 4, and 16 as (2 × 4) × 2, while the next three numbers are again verbalized as sums, and then 20 was again 10 × 2. The Huku-Walegga live, still following Thomas, in a region northwest of the Semliki River. Exactly at the shores of this river the Ishango rod was found, 30 years later. But Thomas could not foresee that in 1920.

Archeological Missing Link In his excavation report of 1957, the discoverer of the Ishango rod, Jean de Heinzelin, already compared the harpoon heads found together with the Ishango rod with findings in other regions in Africa. He made maps of Africa about the distribution of harpoon heads made in bone and about the distribution of harpoon points in iron, following data from other archeological excavation sites (Fig. 9.2). In 1962 de Heinzelin summarized his further findings in the popular science magainze Scientific American: From Central Africa the method of Ishango seemed to have spread north. In Khartoum, near the Upper Nile, there exists a settlement that was inhabited much later than that of

Archeological Missing Link

155

Fig. 9.2  Map showing distribution of harpoons in bone (left) and in iron (right)

Ishango. The harpoon heads that were found there show a diversity of styles. Some have carvings that were first thought of in Ishango. Near Khartoum, in Es-Shanheinab, there is site where the harpoon heads illustrate the influence of Ishango. The Ishango technique spread from Es-Shanheinab westwards and northwards. This second branch along the Nile valley reached the Egyptian Nagada.

In this way the suggestions of de Heinzelin strengthened the hypothesis of a corridor south and east of the Sahara. The itinerary must start somewhere, of course, and this missing link seemed to lie according to the archeologist in Ishango (Fig. 9.3).

Fig. 9.3  De Heinzelin’s map showing influence of Ishango based upon archeological data (left), and a simplified version (right)

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Mathematical Missing Link In the traditional Eurocentric vision, the origin of mathematics lies in Greece, where it suddenly exploded onto the scene like a nuclear bomb and radiated from about 600 BC to AD 400. When the Greek mathematical nuclear cloud cooled off, the sciences languished until the Renaissance of the sixteenth century, when a true nuclear power plant was established that would generate scientific energy to this day. Eventually, that account would add that we owe the Arabs a bit for transporting mathematics from Greece to Europe (Fig. 9.4).

(Western-) Europa Greece Arab world

(Western) Europe Arab world

China

Greece Persia Mesopotamia Egypt

India

Fig. 9.4  Classical historical view of science’s path (left) and alternative itinerary by G. G. Joseph (right)

The mathematician Morris Kline, author of Mathematics in Western Culture, formulated his opinion about this part of the history of mathematics rather expressively (Kline, 1953): Mathematics finally secured a new grip on life in the highly congenial soil of Greece and waxed strongly for a short period… with the decline of Greek civilization the plant remained dormant for a thousand years … when the plant was transported to Europe proper and once more imbedded in fertile soil.

The expression “Europe proper” might spur protest, even in Eurocentric circles. G. G. Joseph, the author of The Crest of the Peacock cited earlier, pointed nevertheless to a greater lapse in the statement, namely overlooking the contribution of non-­ European cultures. Thus there are the Indian Sulbasutras (800–500 BC), the Egyptian papyri, and the clay tablets of the Middle East. Joseph proposed an

Mathematical Missing Link Fig. 9.5  Some more alternative paths for the history of mathematics

157

(Western) Europe Arab world West Africa

China

Greece Mesopotamia Egypt

India Region Sudan

Central Africa

alternative route showing how mathematics reached Europe. Joseph stressed that the Arab contribution was more than moving it from one place to another, because they summarized results from ancient Egypt (up to the Alexandrines), India (with the introduction of the zero), and China (Fig. 9.5). Joseph also paid attention to the mathematics of the Mayas, but this falls way out of the scope considered here. The influence of Egypt on Greece was already noticed in the description of the sources of ethnomathematics, when Herodotus, Proclus, Thales, and Aristotle were mentioned (see Chap. 1). Note that Plato recommended the Egyptian writings to learn mathematics in his Laws. In Phaedrus, Plato has Socrates say the following: He [Theuth] it was who invented numbers and arithmetic and geometry and astronomy, also draughts (checkers) and dice, and, most important of all, letters. Thamus reigned as a king of Egypt; he lived in the great city of the upper region which the Greeks call the Egyptian Thebes, as they call the god himself Ammon.

Yet the Egyptian origin of Greek mathematics is a matter of debate. There is a consensus that numerical values for the Pythagorean theorem can be found in clay tablets from Babylon (and some even refer to earlier sources in India). Though it is said that Pythagoras stayed in Egypt for a long time, some doubt he was a real figure at all, that perhaps he never even existed. Moreover, Eurocentrists defend Greek mathematics by observing that the Pythagorean theorem is simply the 47th theorem in the Elements of Euclid. In other words, it is just a small fragment of the Euclidean (Greek) oeuvre. Chapter 10 takes up this polemic. For the sake of clarity and despite all ethnomathematical considerations mentioned here, let there be no doubt that Euclid features on the hit parade of the greatest mathematicians of all times. Nobel Prize

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winner Bertrand Russell, philosopher and mathematician, even named Euclid his favorite mathematician. Others place Euclid a few places lower on the list of greatest mathematicians, claiming that Euclid had just summarized what was known in his time, like an encyclopedia: but Euclid’s work is to mathematics what the Oxford dictionary is to English literature. Anyway, there is no doubt that Euclid remains one of the foremost mathematicians of all time. In Chap. 6, an example was given of the multiplication method in Egypt. Its style looked more like the related African mathematical methods than that of the Greeks. Yet many Egyptian problems are closely related to the classical style. Here, for example, is an Egyptian problem about the volume of a truncated pyramid with a square base of side 4, height 6, and square upper face with side 2. In the Egyptian method this is solved as follows: 1 . Square this 4. Result = 16. 2. Square 2. Result = 4. 3. Double 4. Result = 8. 4. Add together this 16, 8, and 4. Result = 28. 5. Take 1/3 of 6. Result = 2. 6. Double 28. Result = 56. 7. Behold it is 56! This procedure looks more like a recipe from a cookbook than a mathematical formula, but the final prepared dish tastes good. An Egyptian priest had to hide his knowledge to protect his privileges, and he would not have been very keen to publicly reveal proofs or methods, if he had had those. The Egyptian treatment is clearly different from the Archimedean method that reasoned the formula for the volume of a pyramid by deducing it from the volume of a beam. The question of the truncated pyramid clearly brings Greek mathematics to mind, while the multiplication method creates more of an African impression. It seems logical that this hieroglyphic mathematics forms a crucial transition between both. Archeologist de Heinzelin formulated it as follows: The first example of a well-worked-out mathematical table dates from the dynastic period in Egypt. There are some clues, however, that suggest the existence of cruder systems in predynastic times. Because the Egyptian number system was a basis and a prerequisite for the scientific achievements of classical Greece, and thus for many of the developments in science that followed, it is even possible that the modern world owes one of its greatest debts to the people who lived at Ishango. Whether or not this is the case, it is remarkable that the oldest clue to the use of a number system by man dates back to the central Africa of the Mesolithic period. No excavations in Europe have turned up such a hint.

The earlier ethnomathematical examples can therefore complete Joseph’s alternative routes for the history of the sciences with some references to Central and West Africa (Fig. 9.6).

Cultural Missing Link

159

Egypt, MiddleEast: base 12-60.

1

Bulanda: base 6 7 is 6+1; 8 is 6+2 2 Ishango: evidence base 12 Creative counting Mathematics, 1998

EastAfrica to India: base 12 on phalanges

12

Fig. 9.6  Summary of some indications based on African counting methods

Cultural Missing Link Courses about the history of central Africa generally start with the colonial era and the unavoidable Stanley and Livingstone. The traditional view of Africa is even in the most positive cases associated with a large collection of isolated groups of huts, without any evolution since prehistory. It creates a fictitious Africa that is a mixture of exoticism and mysticism, sometimes very popular in African American circles. Not only is this a romanticized image a myth that is quickly weakened by the realistic reports of equally unavoidable travelers like Burton and Speke, but later on there were indeed some exchanges between different kingdoms and cultures (Fig. 9.7). In an analogous way, classical Greece was no homogeneous, unchanging unit; it was composed of many different city-states, a classical period (600–300 BC), and a period after Alexander (300 BC–AD 400). Yet, the image of a unified Greece fits better into the Eurocentric mythology of a people who from nothing established the most impressive civilization and invented democracy and science, while other peoples slept all that time. Different writers formulated hypotheses about the influence of sub-Saharan Africa on the Egypt of the pharaohs. In 1976, Noguera gave his book How African Was Egypt? the subtitle A Comparative Study of the Cultures of Ancient Egypt and Those of Black Africa.” One had to wait until 1987 before this opinion found a staunch defender in Martin Bernal. In Black Athena, Bernal supported the idea that the African aura reached Greece (Figs. 9.8 and 9.9).

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189-199: Pope Victor I, an African, unifies the Catholic Church Fenicia: destruction Carthago in -146

-712: Piankhi, an Ethiopian, becomes pharaoh f usa o sa M ge to n a a :M m Askia Muhammad founds 1324 n pelgri o i a l c a c university of Sankore in Meroe kingdom in Nubia M Me 1490 in Timbuktu and Kush: - 1100 tot 400 1230-1255: Sundiata Songhay kingdom destroyed Axum: high point under king Ezana assumes power in Mali by Morocco in 1590 (320-350) 700-1200: Ghana Oya kingdom kingdom. Powerful reaches peak in king: Tenkhamenin 1630 1100 (1400?) - 1900: Hima kingdoms in Central Africa

-500 to 200: the Nok people Alfonso of Congo lets terra-cotta statues ascends trone in 1506 Bantus from West Afrika reach Congo in -500 and southern Africa in 400.

Zanzibar; trade route to India and China Zimbabwe kingdom expands in 1100 under Monomotapa; centre of this Shona empire is a walled town

Fig. 9.7  Some precolonial facts about the history of Africa

Fig. 9.8  Illustrations by Noguera: a Peul woman, and an image from Tassili in the Sahara – not from Egypt, such as one might have erroneously imagined

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Fig. 9.9  In Egypt, tourists can behold Queen Ti and warriors on temples. As the author confirmed personally, Egyptian guides consider black skin a sign of fertility and not as genuine proof that, for example, Ti was black

There is no consensus about the truthfulness of the arguments, which are sometimes difficult to judge and often subject to interpretation. Bernal did win many supporters (Bernal, 1991) and I. Van Sertima published a collection of essays on the same topic in Black Athena Revisited, edited by Mary R. Lefkowitz and Guy MacLean Rogers (Van Sertima, 1984, 1995). Anthropologists such as Brace may very well reject the theories of the Senegalese Sheikh Anta Diop on the African origins of the first Egyptians, but others agree with the notion that an invasion of a new dynastic race took place, and that this explains the sudden new form of economy in the Nile Valley. Some indications of the latter hypothesis is that certain plants, animals, customs, architectural realizations, and funeral rites were foreign to Egypt and were suddenly imported. Today, he AfricanAsiatic character of the Egyptian language represents a field in which there is agreement. The illustration shows some clear indications of an exchange between Africa, ancient Egypt, and Classical Greece (Fig. 9.10). Statues of a woman holding snakes in both hands seem to come from Crete as well as from Nigeria, while traditional dress in Uganda and Egypt seems identical. How far the relations as drawn in the figure point to the Central Africa of Ishango will of course never be settled with complete certainty through cultural indications (see Chap. 10, as well).

Genetic Missing Link DNA research should be able to provide more and clearly tangible evidence. A  group associated with the researcher A.  Arnaiz published a number of papers where relations were shown to exist between Northeast Africa, West Africa, and the

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Fig. 9.10  Cultural relations between sub-Saharan Africa, Egypt, and Greece

Athenians (see Arnaiz-Villena et al., 2002). Here, so-called HLA genomics were studied (whatever that means). To show that the Greeks shared an important part of their genetic material with sub-Saharan Africans, the so-called Chr 7 markers were traced (again such a mysterious term from biology!). Some e-mail correspondence with the very obliging author, Arnaiz provided permission to translate large parts of his research and to adapt them (Fig. 9.11):

Dimension 2

0.4 Spanish-Basques Berbers French Portuguese Moroccans Spaniards Algerians

Orama Amhara

Fulani Moroccan-Jews Greeks Mossi Italians (Attica) Cretans Egyptians Rimaibe Iranians Macedonians Libanese San (Bushmen) Turks -0.2 -0.6 0.6 Dimension 1

Fig. 9.11  Data analysis diagram grouping related people

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Our analyses place the Greeks as an outgroup among other Mediterraneans, including Macedonians and Cretans. Quasi-specific high-frequency DRBl Greek alleles were sought throughout Asia and Africa in order to explain this discrepancy. Sub-Saharan and Sahel African populations share these alleles with Greeks, i.e. Mossi, Fulani, Rimaibe (from West Africa and sampled in Burkina-Fasso) and Nuba (Sudan), Oromo and Ahmara (Ethiopia, east Africa). Neighbor joining and correspondence analyses put Greeks together with the above-mentioned sub-Saharan groups. The following explanations of how Negroid populations could have reached Greece (and not Crete) may be put forward: It is possible that the densely populated Hamitic Sahara (before 6000 BC) may have contained an admixture of Negroid and Caucasoid populations and some of the Negroid populations may have migrated towards present-day Greece. This could have occurred when arid Saharan conditions became established and large-scale migrations occurred in all directions from the desert. In this case, the more ancient Greek Pelasgian substratum would come from a Negroid stock. A more likely explanation is that some time during Egyptian pharaonic times a Black dynasty with their followers were expelled and went towards Greece. Indeed, ancient Greeks believed that their religion and culture came from Egypt. Also, Herodotus states that the daughters of Danaus (who were black) came from Egypt in great numbers to establish a presence in Greece. Otherwise, the Hyksos pharaohs and their people were expelled from Egypt and may have reached Greece by 1540 BC. However, the Hyksos are believed to come from modern Israel and Syria. Other gene input from Ethiopians (meaning “Blacks” in ancient Greek) may have come from King Memnon from Ethiopia and his troops, who went to help the Greeks against Troy according to Homer’s Iliad. The fact that Crete does not show a Black African input may be due to the fact that the Ethiopian emigration may have occurred in Minoan times, when Crete had a strong sea empire and did not allow the invaders into Crete. Black Fulani or Peul and the associated Mossi and Rimaibe came from East Africa and now live in regions in Mauritania, Mali, Burkina-Fasso and Niger. Genetic studies show that they are related to people from the actual Sudan and Ethiopia. Nor [neither?] the Eastern nor the Western black Sahel people are related to South African or indigenous Senegalese blacks. The Fulani are one of the few people in the region who use cow’s milk and the byproducts for themselves and to do business. Some authors pretend that they come from the Egypt of the pharaohs. The Nubians are now widely spread over Sudan, and descendants of the old Nubians who reigned over Egypt between the 8th and 7th century BC. Later they established their kingdom in Meru North of Khartoum.

The map based on genetic studies shows a link between Greece and Ethiopia along the Nile Valley south to around Khartoum (Fig. 9.12). Some questioning to Arnaiz for studies that also compare the genetic material to data from the Ishango region has not yielded any immediate results. Moreover, in recent times, DNA research on the African roots of some Greek civilizations has been critisized as well. Other specialists in this field could not provide an answer either, and thus the missing genetic link, if there even is one, will remain a missing link.

Ethnographic Missing Link For the genetic missing link, a southern itinerary that starts somewhere in the Sudan and reaches Ishango is lacking, not because it could not exist but because no materials are available for study. On the other hand, the ethnographic material is abundant

164 Fig. 9.12  Summary of genetic influences following Arnaiz

9  Missing Link

Greece, Athens, Aegean region

Fulani Mossi Rimaibe

Nubians Oromo Ahmara

Genetics, 2002

but confusing, at least for the average mathematics reader. Peppered with all kinds of anecdotes and names of the most diverse people and tribes, it is difficult to see the coherence in the studies of Lagercrantz, cited in the paragraphs about counting strings (Chap. 5). Yet we can deduce some elements for a conclusion from his papers: Counting by means of stones, nuts, sticks, blades of grass and similar objects is an old practice that is in particular remarkable for South-Sudan. The counting method can be compared to counting strings. The latter are spread more to the North, but they are part of the same black culture. The ancient cultural elements also include counting by lines drawn on the ground or painted on doors and walls. This in particular has a Western orientation all the way to the Tuareg.

The whole of the available information is still too faint to see a distribution here from the Congo to northern Sudan and more to the west to the Tuareg. As for a particular ritual, namely that of counting through a form of tattoo, Lagercrantz is clear “cut” (Fig. 9.13): The practice to enumerate the number of killed enemies by carvings on the warrior’s own skin has a specific and well-known distribution, but is seen as archaic. On one side there is the region around Lake Albert (Rutanzige around Lake Albert (Rutanzige), while), while on the other side there is the central part of South Africa. This reflects the welldocumented ethnographic relations between Northeast and South Africa.

Still, Lagercrantz does believe that the Hottentots from the Kalahari acquired their knowledge about weekly and monthly calendars from the Dutch, who landed long ago in nearby South Africa. How these Dutchmen influenced the people around Lake Rutanzige is not clear, unless this influence only counted for calendars, and the counting with tattoos had to be seen separately from it. In any case we see, on the added map, that counting with tattoos is indicated with ▼ symbols, northeast of Lake Rutanzige and in southern Africa.

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Fig. 9.13  Lagercrantz’s map on the use of counting sticks and carvings on the body (left); a circle indicates the use of carving sticks and a triangle the use of body carvings (right)

Now de Heinzelin had already noticed that the penetration of Hamites and Bantus in Central Africa had pushed back the ancient autochthones of Ishango back to the south, even up to the Kalahari, where some remote groups remain. On a map he indicated a similar relation between people east of the Ishango region and southern Africa. The relations between groups all over Africa are evident following the burden of proof given by de Heinzelin. The ethnographic material on similarities in counting methods does not contradict this for sure but involves only a few groups, and that material is very modest in general (Fig. 9.14). Fig. 9.14 Relations between groups of populations following ethnographic data

Soedan Tueareg

Ishango

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Too Many Missing Links In 1920, the linguist Thomas did his research, in which he wondered if there was a relation between the expressions with base 12 found in West Africa, Egypt, and the region of Ishango. In 1950, 30  years later, the Ishango rod was found, and de Heinzelin made his map about the influence of Ishango on Egypt and West Africa. He formulated his well-documented conclusions in 1962. In 1999, Pletser interpreted the Ishango rod arithmetically as a function of base 12. Due to these significant differences in time and in field of study, the linguist Thomas, the archeologist de Heinzelin, and the space scientist Pletser were not aware of each other’s work. The latter did know about the Ishango finding as such, but not about the map with the distribution of the findings of harpoon heads. The cultural considerations from the 1980s complement this well. Added to it are the partial missing links on ethnographic field data from the 1970s and genetic facts from this century, which also stem from very remote fields of study. All in all, too many pieces fit the puzzle for them to be interpreted as mere lucky coincidences. However, as for mathematics, such a missing link does not need to exist: mathematics is not a science that needs to develop linearly from one axiom to the next theorem. It happened more than once in history, and probably it is even the case today, that mathematical notions were and are forgotten and in the best case invented or rediscovered. Perhaps Ishango, Egypt, and West Africa came to the same base 12 conclusions independently, without any mutual influence. The last word may again belong to de Heinzelin: One need not even think about granting some kind of mark of respect, under the form of some universal posthumous certificate, to the man of Ishango. On the contrary, I personally believe that most civilizations that came afterwards and were more similar to modern civilizations, starting with the late Paleolithic, the Mesolithic and finally those of the Neolithic phases, of agricultural villages and cities, had to invent at different times the same thing, at about the same moment of development, in different places, yet at different moments in time. This important idea imposes itself, for the invention of ceramics, writing, and arithmetic as well as for the domestication of animals.

Chapter 10

Not Out of Africa

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Other Opinions This chapter gives an overview of the critique of ethnomathematics and of some conclusions related to the Ishango rod. For some, the proposed interpretations do not go far enough because 1 . There are other interpretations for the carvings (fair enough!); 2. There are other older mathematical objects (even better!). To others, ethnomathematical theories reach too far because: 1 . Mathematics and culture do not go together (shame!); 2. Pseudoscientific nonsense must be combatted (indeed!); 3. It’s all part of a political agenda (gosh!). The reader will be able to estimate the value of the arguments, even if we present only short summaries here, because some ideas and developments go far beyond the Ishango story (or is it vice versa?). Internet sites and more extensive comments can be found in the references at the end (Fig. 10.1). Let us recall the three interpretations of the carvings on the Ishango rod: 1 . Arithmetical game, by J. de Heinzelin; 2. Lunar calendar, by A. Marshack; 3. A series of carvings, resembling the pronunciation of numerals, by V. Pletser. Others have interpreted the carvings as a calculation of a woman’s menstrual cycle and deduced from it that the first mathematician must have been a woman.

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6_10

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Fig. 10.1  A game: Hinud-Arab ciphers with as many Ishango type strokes as they indicate; seven has a forged solution (the stroke across in non-Anglo-Saxon notation), and for nine even more creativity is necessary

Still others have seen in the Ishango rod a new missing link, this time to poetry. Following Roy Harris, the origin of writing is not in drawing figures or in copying speech, but in recording numbers (Harris, 1986). He claims the oldest writing is on the Ishango rod: Counting is in its very essence magical, if any human practice at all is. For numbers are things no one has ever seen or heard or touched. Yet somehow they exist, and their existence can be confirmed in quite everyday terms by all kinds of humdrum procedures, which allow mere mortals to agree beyond any shadow of doubt as to “how many” eggs there are in a basket or “how many” loaves of bread on the table. As every serious student of versification has always understood, versification is about counting language. The reason to write verse is less to score the voice than to imbue words with the magical quality of counting. That is why meter, or measure, is at the heart of debates over all verse forms, including free verse. Number is one of the creative grounds of poetry, and the idea that writing grew out of counting is the missing link in studies of the graphic in versification. It is almost uncanny that lines of verse look exactly like the most primitive ways of counting—parallel scorings that can be numbered. Verses are countable in exactly the way that token-iterative digits are countable, from either end of the sequence. Each one indicates only its singularity, not a number. Every poem in lines effaces, or predates, the distinction between writing and drawing in the same way as the lines on the Ishango Bone; lines of verse combine functions of writing, drawing, and counting.

Still Older Mathematical Artifacts Traces of counting on bones have been found all over the world. In 1937, Karl Absolon discovered in Vestonice (in the former Czechoslovakia) a 17-cm-long wolf bone on which 57 thin lines were carved about 30,000 years ago. The first 25 are perhaps in little groups of five and all have the same length. They are marked at the end by a line twice as long, after which again a long line announces a group of 30 thin lines. This Paleolithic object possibly shows a counting method referring to the five fingers of the hand. Beyond this, little is known about the people that carved this bone except that they also left an ivory statue. The bone is now in the Moravian Museum of Brno in the Czech Republic (Fig. 10.2).

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Fig. 10.2 Another 30,000-year-old bone

Fig. 10.3  Old carved bone from South Africa; many carvings, no pattern

In the early 1970s, a remarkable baboon fibula was found during excavations in the Border caves in the Lebombo Mountains, on the border between South Africa and Swaziland. The 7.7-cm-long bone appeared to be 35,000 years old and showed 29 clearly delimited carvings. The archeologist who discovered it, Peter Beaumont, saw a similarity with the calendar sticks used to the present by the Bushmen of Namibia. Yet they are always simply records of tallies, as an aid to remember a certain amount of a counted quantity, without any logical structure or relation between the carvings (Fig. 10.3). A tripling of the 20,000-year age of mathematics according to the Ishango rod is due to Francesco d’Errico  (Errico, 2002). In a cave called Blombos, a 50,00070,000-year-old pebble was found with about seven parallel lines, and d’Errico considers it proof of geometric thinking. To some there is no reason why the capacities of early humans to reason or to develop concepts would be different from those of their modern descendants. The kinds of facts and relations with which humans are confronted have undoubtedly changed over time. This is why Francis Buekenhout, professor at the French speaking Free University of Brussels and member of the Royal Academy of Belgium, states that mathematics should probably be even much older (Buekenhout, 2000). After all, the first humans distinguished themselves from their animal ancestors by their ability to think, so mathematics must be as old as that. In the paper “Les polyèdres de Lucy à Jacques Tits” or “Polyhedrals: from Lucy to Jacques Tits,” Buekenhout gives a summary of the history of polyhedra from the era of the first human, Lucy, until the modern era of one of the most prominent researchers in the field, the Belgian Jacques Tits. Buekenhout summarizes his thoughts as follows: The notion of polyhedra plays remarkably enough a key role in prehistory and history. Its mathematical definition is not always used in the same way, allowing a certain flexibility to adapt it to different situations. The presence of Lucy in the title is purely symbolic. It refers to the earliest fragments of worked stones that go back over 2.3 billion years. The Paleolithic is characterized in a remarkable way and very early by man-made stones. […] The man-made stone can be interpreted as a polyhedra if one allows that the edges do not necessarily have to be straight and that the face does not need to be flat. The polyhedra thus conceived have after all neither a less precise nor a more interesting structure.

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Comparable hypotheses only strengthen the Ishango story. Naturally humans made some carvings here and there before thinking about including a mathematical pattern in these carvings. If they, as Errico claimed, even tried to draw beautiful parallel lines, even better: practice makes perfect. Exact thinking certainly evolved with humans and probably, as Buekenhout expressed it lyrically, since Lucy.

Antiracist Mathematics To the present day, discussion about the African contribution to many fields occasionally generates controversy and in particular in mathematics. In the Anglo-Saxon countries where ethnomathematics is included in some of the official teaching programs (Chap. 12, “Museum Visit, Teaching, Research”), these controversies sometimes take a political turn. In 1987, the prime minister of the United Kingdom, Margaret Thatcher, addressed the annual congress of her Conservative Party. She expressed her disdain for the attention mathematics from other cultures had started to receive with the following words: Children who need to be able to count and multiply are learning anti-racist mathematics, whatever that is.

Someone once placed an online announcement about the book Ethnomathematics: Challenging Eurocentrism in Mathematics Education, in which Arthur B. Powell and Marilyn Frankenstein collected a number of papers. The electronic posting ­carried the subtitle Math for Dummies and launched an online discussion. It is written in the typically erroneous but original and universal cyber English: • This is a totally serious posting, though outrageous enough to be laughable. I find it fascinating that one of the author’s surnames is “Frankenstein” – and is appropos given her dubious expertise in academia. “The Field of Ethnomathematics” sounds like something everyone would write. • Victoms rejoice, for now you may be a victom of math. I wonder how much these people were paid for this “study” and who paid them. • We are training idiot teachers to teach a nation of dumbed down children. What in the hell is going on? We are going to compete in the global economy with this kind of nonsense? Not surprisingly, about half of the children of teachers attend private school. DUUUUUH! • We surly don’t want to teach our kid math. Some day they will grow up and figure the percentage the Government takes from their income and ask us: “What did you do Dad to stop the growth of the Government”. • We must keep our kids ignorant and dependant on Government. We certanly don’t what them to be mean consertives and throw us onto the streets. • In retrospect, one day after having posted the above “ethnomathematics” book review, my feelings are – while I am not surprised that there are loonies out there who believe this nonsense, and I am also not surprised that they can find a

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publisher, I AM shocked that (1) they have university appointments; (2) they are not denied tenure for publishing such garbage; and (3) there are reviewers who purport to take their “scholarship” seriously. Similar press excerpts appeared in newspapers and magazines, but the preceding quotes give a good sense of the general tone.

The Skeptical Inquirer A more serious debate that was no less lively appeared in The Skeptical Inquirer magazine in 1995 and 1996. The discussion between Walter F. Rowe and Beatrice Lumpkin dealt with a series of dissertations written by various authors, meant as source material for teachers. Some papers on science and mathematics were grouped under the heading “African-American Baseline Essays” (Rowe, 1995; Jones, 1954). Rowe rightfully attacked the pseudoscientific statements about pyramids and the alleged theft of Arab discoveries by European scholars, as if Newton and Galileo had no merits. For this nonsense not Lumpkin but another author was responsible, carrying the lyrical name Hunter Havelin Adams III. Rowe razed the man’s theories to the ground and the man himself: “Adams was in fact a hygiene technician with only a high school diploma.” In “School Daze: A Critical Review of the ‘African-­ American Baseline Essays,’” Rowe concluded as follows (Fig. 10.4):

Fig. 10.4  Cartoon about African roots of Egyptian pharaohs

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The science and mathematics essays in the “African-American Baseline Essays” are riddled with pseudoscience and pseudohistory. As tools for the training of public school teachers they are not merely worthless, but are likely to prove pernicious. Their fallacious modes of reasoning may dull the critical faculties of readers. The “scholarly” research displayed in both essays is too shoddy to serve as a model for any teacher or student. The essays will contribute to the growing tribalization of American culture. A purported goal of the “African-American Baseline Essays” is to “eliminate personal and national ethnocentrism so that one understands that a specific culture is not intrinsically superior or inferior to another.” This statement is nothing but cant. Throughout the science and mathematics essays the genuine achievements of Greek, Arab, Persian, and European scientists and mathematicians are ruthlessly pillaged, and credit for them assigned to black African cultures on the flimsiest of grounds.

Adams did not react to the accusation, but Lumpkin did, since her paper appeared in the same collection. Some paragraphs out of the lengthy discussion dealt with the Ishango rod. Here is how the discussion evolved: Beatrice Lumpkin begins her mathematics essay with a discussion of prehistoric African systems of numeration. The discussion centers on the Ishango bone, an artifact excavated in Zaire that has been dated to 6500 B.C. (Marshack 1972). The Ishango bone is engraved with a series of parallel scratches having varying lengths and grouped according to some system. A variety of explanations of the marks have been advanced: They may represent a multiplication table, a game tally, or a calendar. The reader of the mathematics essay is clearly intended to infer that systems of numeration originated in Africa. However, the Ishango bone is a rather recent example of a type of inscribed artifact produced by Paleolithic cultures stretching from the Iberian Peninsula to the Russian steppes. Most of these artifacts have been found in Europe. These facts are easily gleaned from Alexander Marshack’s The Roots of Civilization (1972), a source Lumpkin cites in the mathematics baseline essay and in other writings.

The response of Lumpkin on these allegations contained the following paragraphs: Evidently Rowe was not aware of new work on this subject. Recent scholarship, which Marshack includes in his 1991 revised edition, gives a much older date of 18,000 B.C. to 23,000 B.C. for the Ishango bone. It is based on work by Brooks and SmithReference “Brooks and Smith (1987), Yellen et al. (1995), Nelson (1993), George Gheverghese Joseph (1991) were cited in text but not provided in reference list. Please check. (1987). The Ishango bone grouped numerical values recorded as tally marks and was probably preceded by simpler tally records. A simpler tally record on a fossil baboon bone has been found in Border Cave, between Namibia and South Africa. The bone was inscribed with 29 equally spaced rallies, perhaps a record of a lunar period. Dated about 35,000 B.C., it is the oldest numerical record known to date (Bogoshi et al. 1987). It is possible that modern humans possessed a sophisticated tool kit by the time the species spread from Africa to other continents. Ages of 75,000 to 90,000 years are given for modern-looking toothed harpoon bones found by Yellen et al. (1995) on the Semliki River in Zaire, near the Ishango site on Lake Rutanzige. That discovery may require a correction of current textbooks, which say that such tools were first invented in Europe 40,000 years ago (Yellen et al. 1995).

That was telling him! Yet, Rowe got the final word: Citing more recent research on the Ishango bone does not mitigate Lumpkin’s tendentious use of source materials in her original African-American Baseline Essay. The point of

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Lumpkin’s discussion of the lshango bone in the African-American Baseline Essays is to lend plausibility to a non sequitur: “Since Africa is widely believed to be the birthplace of the human race, it follows that Africa was the birth- place of mathematics and science.” The limited nature of the archaeological record does not at this time permit any conclusion about where mathematics originated. It is instructive to compare Lumpkin’s discussion of the lshango bone in her mathematics essay with the discussions of such artifacts in David Nelson’s essay “Teaching Mathematics from a Multicultural Standpoint” (Nelson 1993) and in George Gheverghese Joseph’s book “The Crest of the Peacock” (1991). Nelson and Joseph (both prominent leaders in the international multicultural mathematics movement) acknowledge the provenance of similar notched bones in both Africa and Europe. In further contrast to Lumpkin, neither of these authors tries to draw from these artifacts any sweeping conclusions as to where mathematics first developed. Joseph ends his discussion of the Ishango bone in The Crest of the Peacock with the following warning: Finally, in the absence of records, conjectures about the mathematical pursuits of early man have to be examined in the light of their plausibility, the existence of convincing alternative explanations, and the quality of evidence available. A single bone may well collapse under the heavy weight of conjectures piled upon it.

In other words, in the eyes of the most critical minds, the Ishango rod is some kind of isolated artifact, a coincidence (they turn out to be wrong – see Chap. 11, “A Second Rod”). About its mathematical nature there is no doubt, but to them, one cannot draw far-fetched conclusions about the bone being the beginning of the mathematics. In any case, there is no doubt about its right to being called the oldest mathematical artifact!

The Black Athena Debate In the previous chapter, Martin Bernal’s statement about the African influence on Classical Greek and Western culture was mentioned. There, mathematics was an addendum and the Ishango rod an incidental event. The larger cultural debate about the African role sometimes takes true Jerry-Springeresque proportions in the United States and is not restricted to influence on the West! On the Internet, statements like the ones below are not uncommon. Ancient Egypt was a black African civilization. And historians wrote that such great lawgivers as Lycurgus studied in Egypt and brought back the legal and political basis for the West’s politics. The early Greek-Roman gods & goddesses such as Apollo, Zeus, Hercules, Athena, Venus, were all Black, being renditions of the Black Egyptian gods. The historian Herodotus himself wrote that “the names of nearly all the gods came to Greece from Egypt.” The Aeneid, like the Iliad, Odyssey and all the other great epics of the world, is a poetic story dealing with Black people! Aeneas, the Trojan hero of Virgil’s Aeneid, was in direct descent from Dardanus, the African founder of Troy. Whites blew off the Africoid nose of the Sphinx! Africans gave us Math, Algebra, Geometry, and Trigonometry! -including the Arabic Numbers! Africans also originated the world’s first known universities. Melanin gives Black people superior physical, mental & spiritual ability.

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10  Not Out of Africa Blacks were first in Asia and pioneered Asia’s early civilization! China’s first historical dynasty and first emperor were Black! The martial arts originated in AFRICA, not Asia! The first people of India were Africans! The word India itself means Black. Yoga & other so-called Eastern teachings originated in Africa! Barbaric whites invaded India, bringing destruction. The Africoid Olmecs was the parent culture of Ancient America. Africans ruled widely in Europe for over 1400 years! Europe’s royal families descended from Black/mulatto rulers! Yes Beethoven was Black. So were five U.S. presidents! Did Jesus have dreadlocks? The word Semite is from semi which means half. Half what? Half BLACK! Is God Black? The Original Man was BLACK, “made in the IMAGE of God” his Parent, according to scared books. Children look like their parents.

The counterattacks came from classics scholars, among others, who felt their honor had been offended. Mary Lefkowitz wrote a book with the revealing title: Not Out of Africa: How Afrocentrism Became an Excuse to Teach Myth as History (Lefkowitz, 1996, Lefkowitz and MacLean, 1996). She showed that education in history “with a good feeling” is dangerous, because it prevents one from studying the real ancient Egypt and the real traditional Africa. Lefkowitz thought that most Afrocentrist statements were historical concoctions, having their origin in old European misconceptions about Egypt (Fig. 10.5).

Fig. 10.5  Ancient Egyptians were called Kam or Kam-Au or blacks; the great black man and woman, Isis and Osiris, had the above hieroglyph

Were the Africans the true inventors of democracy, philosophy and science, instead of the Greeks? Were Socrates and Cleopatra really of African origin? Was the Greek philosophy copied from Egypt? So Afrocentrist writers from Malcolm X to Leonard Jeffries have pretended. Still recently, Martin Bernal suggested, in his book “Black Athena,” that European scholars would refuse to recognize the full appreciation of the debt of the Greek to Egyptian civilization. Clarence E. Walker expresses it along the same lines:Afrocentrism is a mythology that is racist, reactionary, and essentially therapeutic. It suggests that nothing important has happened in black history since the time of the pharaohs and thus trivializes the history of black Americans. Afrocentrism places an emphasis on Egypt that is, to put it bluntly, absurd.

The debate may have reached a deadlock, but anthropologist Wim van Binsbergen tries to revive it using the slogan “Towards the third millennium with Black Athena?” In Europe, the discussion never came alive because two parties are needed for a debate. And at least in Belgium there were mostly opponents (Fig. 10.6).

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Fig. 10.6  Satirical image made by students of the author: maybe the Ishango rod is as important to the history of science as an engraved carrot?

A 100-Year-Old Authority on Ishango Rowe’s critique and the other considerations of the preceding paragraph do not overturn the Ishango story. Indeed, even considering his skepticism, Rowe agrees that the Ishango rod is the oldest mathematical artifact. Rowe opposes in the first place the mishmash of pyramidiological exaggerations that contend, in one

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stroke of the pen, that the Ishango rod is at the origin of all mathematics. As previously stated, this criticism is easily circumvented by not suggesting that the Ishango rod is “the beginning of mathematics.” Rowe also opposes the exceptional symbolic value given to the object, but this, of course, is a matter of appreciation. In any case, it may have a claim on the title of oldest mathematical artifact, even from Rowe’s critical mind. To him, mathematics is not “Out of Africa,” but the oldest artifact, however insignificant it might be, undoubtedly comes from Africa. And that is more than enough reason to pay attention to it. To eliminate any remaining doubts about the value of the Ishango rod, another method of persuasion is now applied, that is, recalling an undisputed scientific authority. In the field of the history of mathematics, Dirk J. Struik of the mathematics department at the Massachusetts Institute of Technology is such an authority. His book, A Concise History of Mathematics, was in its fourth printing in 1987 and had been translated into several languages. In recognition of his many achievements, he received a lifetime membership from the American Mathematical Society, though this may not have been such an enormous gift from the mathematical community since Struik was already 100 years old at the time. His book does not mention the Ishango rod, but it was not very well known at the time he wrote the first edition of this standard volume, many years earlier. In a more recent review of Joseph’s The Crest of the Peacock, Struik remarks: Professor Joseph has also turned his attention to Africa, where the oldest artifact of mathematical interest has been found, the Ishango Bone. This bone tool, dating back to about 35,000 B.C. (and now in Brussels), has two sides with clearly visible notches, arrayed in groups, which may represent an arithmetical game of some sort, or some ritual, or hunting, or perhaps an astronomical record. Professor Joseph does not devote much further space to the records of African mathematics, but this has been done by Claudia Zaslavsky, Paulus Gerdes, and others in their study of present-day mathematical notions of peoples of Nigeria, Zambia, Mozambique, etc. Here pottery, tiles, fishnets, knots, games, decorations, sand drawings (graphs we call them), and even kinship relations present mathematical patterns. Or shall we use the term “proto-mathematics”? But Molière’s M. Jourdain did not know either that he had been talking prose.

Chapter 11

A Second Rod

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The Rediscovery of the Ishango Rod Jean de Heinzelin and his team undertook several excavations in Ishango between 1950 and 1959. In 1957, he wrote a voluminous report about the excavations done until then. A summary appeared in Scientific American in 1962. He would not return to the subject, perhaps because of his many other projects, or perhaps because he was embarrassed by his dating of 11,000 years since, as it turned out, it should have been 22,000  years (Chap. 7). American mathematicians like Claudia Zaslavsky (Africa Counts), Marcia Ascher (Ethnomathematics: A Multicultural View of Mathematical Ideas), and William A. Hawkins (Ishango poster, see Chap. 1) kept the fire burning, though, and so it happened that the Ishango rod became better known in North America than in Africa or Europe). In the middle of the 1990s, the editor-in-chief of the journal The Mathematical Intelligencer, Chandler Davis (University of Toronto, Canada), suggested writing a piece about this Ishango rod for his journal, which is widely read among professional mathematicians. However, at the Royal Belgian Institute for Natural Sciences of Brussels the mysterious African rod was not even on display. It was kept on the 19th floor in one of the countless drawers and could only be seen upon special request. In any case, the 1996 paper in The Mathematical Intelligencer received no negative comments following its publication, though the readers of this journal do not shy away from polemic. CNN reported about it in a 1998 broadcast and more TV networks followed its example (Figs. 11.1, 11.2 and 11.3). Other mathematicians agreed to write joint papers about the Ishango rod, such as Freddy Dumortier, at the time a driving force of the European Mathematical Association. On the occasion of the International Mathematical Year in 2000, it was

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Fig. 11.1  Since its discovery, the Ishango rod was kept in a drawer on the 19th floor of the RBINS among a multitude of other objects found at Ishango

Fig. 11.2  Opening image of CNN report on Ishango rod and official stamp with Ishango marking

agreed that some of the Ishango markings would figure on a specially edited stamp. They had an amusing side effect: some customers interpreted them as a kind of bar code for scanning the price of the stamp. So counting started with little lines on a bone and finally became bar codes in post offices and supermarkets (Huylebrouck and Dumortier, 1996a, 1998). This positive impact did not go unnoticed, so when Belgium held the presidency of the European Union in 2001, Ishango became the centerpiece of its scientific

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Fig. 11.3  Ishango display at Royal Belgian Institute for Natural Sciences in Brussels

research efforts. It had taken about half a century, but now the oldest mathematical artifact was finally put on public display in a prominent place of the Royal Belgian Institute for Natural Sciences (Huylebrouck, 1996a, 1998, 2000a, b, 2003). There is an interactive computer exhibition with information. Today, a sign has been placed on the excavation site in Ishango in the DRC Congo, through the initiative of Congolese journalist and librarian Magloire Paluku. It says: “At this precise spot a 10cm long bone was discovered in 1950 that inspired the calculator.” On September 10, 2015, a glass plate with laser carvings and light effects was inaugurated in front of the African Academy of Sciences in Nairobi, Kenya. In Hong Kong, for many years, a stylized copy of the Ishango rod has been seen in exhibitions (Figs. 11.4 and 11.5). Today, research continues on the Ishango rod. At the Royal Belgian Institute for Natural Sciences, Patrick Semal studies 3D scans of the Ishango rod made with a Skyscan 1076 medical scanner from the University of Antwerp. The 35 μm digitized image was reconstructed in three dimensions with software called Artecore. It showed the inside without breaking the object, and thus a “glue” became visible with which the quartz crystal was firmly fixed on the bone: the crystal was set on the bone intentionally, not accidentally (Fig. 11.6).

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Fig. 11.4  Sign at excavation site in Congo

A Second Ishango Rod Ishango, however, had not yet revealed all of its secrets. During the last of the Ishango excavations held between 1950 and 1959, Marcel Spinglaer, a research technician on de Heinzelin’s team, discovered another notched bone. This explains why the detailed report of 1957 does not mention its existence, but not why the publication in 1962 in Scientific American is silent about it. The information available on the second bone is from de Heinzelin’s notes, written in the autumn of 1998, almost literally on his death bed. On February 28, 2007, on the occasion of the 50th anniversary of de Heinzelin’s report on the Ishango excavations, an Ishango meeting was organized at the Flemish Royal Academy of Belgium, revealing for the first time the information about the second rod. After it had taken about 50 years for the first Ishango rod to become part of the permanent exhibition, it would take another 10 years to get to know what de Heinzelin had actually written about the second rod on his death bed (Figs. 11.7 and 11.8). The second rod was found in a layer close to the first, and there is no difference regarding the dating of the object. The excavated layers were composed of domestic waste, not by foundations of homes, so it remains uncertain whether the carvers of the first and second rods knew each other. However, it is very likely that they lived

Fig. 11.5  A (temporary) Ishango rod statue in Brussels (above left), a reference at the Smithsonian Institution in Washington, DC (above right), a plate in Nairobi (below left), and a stylized copy in Hong Kong (below right)

Fig. 11.6  3D scans of Ishango rod

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Fig. 11.7  Three views of second Ishango rod

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A Second Ishango Rod

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Fig. 11.8  De Heinzelin’s detailed drawings of second Ishango rod

in a similar environment and shared similar cultural milieus. The second bone is slightly longer and has a hollow section in the middle. It is a shaft of a long bone whose anatomical determination is as doubtful as that of the first Ishango rod. The bone appears to have been straightened from scraping and polishing and is well preserved, though it is truncated at one end. There might have been a quartz crystal mounted inside it, as was the case for the first bone, thereby confirming that the Ishango culture used objects for notching, for “notational purposes,” or even for educational objectives.

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The second rod has six rows, which Heinzelin identified by the letters C, D, E, F, G, and H: C: 14 long; 6 short. D: 6 long. E: 17 long; 1 short. F: 6 long. G: 20 long. H: 6 long; 2 short.

De Heinzelin added that the small notch on the E column stands “on the 10th place,” and he wondered about whether this implied a “transition from base 10 to 12.” Here is his more detailed description of the rather “problematic” G column: Short 4’; short 6’; two short twin notchings between 7 and 8; a short notch 9’ after 9; a mat portion between 10 and 12 (two number bases?); two short notches 18’ and 18’ between 18 and 19 (or thirds?); four short notchings between 19 and 20 (or fifths?); short 20’.

Row C has a total of 20 carvings, row E 18, and row G 20, while 6 appears clearly in rows C, D, F, and H.  Thus, bases 6 and 10–20 seem to appear again. Moreover, there are two spatial connections between the rows, at E10 = F1 = G10 and E12 = F2 = G12. De Heinzelin also thought he saw “fractions,” namely thirds and fifths, but this is very doubtful. However, de Heinzelin himself put a question mark next to them, and the area is difficult to see, even on the clearest pictures. That part could very well have been damaged. In addition, the use of fractions has never been recorded in sub-Saharan Africa. This second rod does not confirm de Heinzelin’s arithmetic game interpretation, as there are no “prime” numbers, or any other computational operations. Marshack’s interpretation as lunar calendar is not confirmed either. The second rod looks less spectacular than the first, but there is also some logic to it. This demands an explanation, but perhaps one that need not be more far-fetched than Pletser’s interpretation that it was simply evidence of a counting method in a community that mixed bases 6 and 10. Also, the links between the rows Pletser suggested seem to be confirmed by the two spatial connections mentioned earlier. Thus, both rods indeed recall counting processes with lines, as still used today and described in Chap. 8: sometimes numbers are counted by groups of 5: |||| ||||; |||| ||||; |||| ||||…; in other cases they are counted in pairs: 2, 4, 6, 8, 10,…, while on a slat indents of different lengths are often used: ||||||||||||||| … And so perhaps the Ishango rods were nothing more than that: evidence of a people that counted in a mixed-base 5 (or 10) and base 6 (or 12) system. What they counted, if it was numbers used in games or days on a calendar, could not be confirmed by that second rod, but this does not diminish the Ishango rods’ claim to being the oldest mathematical objects.

A Second Opinion Similar to what Jean de Heinzelin did more than 50  years ago, when he wrote Lancelot Hogben for a second opinion about the notches on the first Ishango rod, we decided to ask for a “second opinion” from a contemporary mathematician. Serbian

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mathematician Slavik Jablan (1952–2015) seemed well placed to play this role of mathematical outsider since he had written a book on prehistoric mathematics. The question posed to him was this: “What is the prehistoric message in the ‘numbers’ on both Ishango rods?” Slavik answered that there is first of all the question about how “in a mathematical way” one can find out what may or may not have been random in groups of dashes or rows of numbers, whether originating from old bones or from a modern calculation or from a result of a lottery. Of course, if asked to continue the sequence 1, 2, 3, 4, 5, …, everyone will give the answer 6. If the row 1, 1, 2, 3, 5, 8, 13, 21 is given, those who recognize the Fibonacci sequence will rapidly give the answer 34, while others may also come to this answer after some more reflection. Once the construction rule is discovered, both rows can be continued in a simple manner, and even to infinity, so to speak. However, if someone gives the row 16, 7, 14, 6, 11, 8, 20, 5, then these are also eight numbers between 1 and 21, but now a decision about what the next number will be will not follow immediately or even after some time. The row is called, in mathematical terminology, random or arbitrary. It would not help, for example, to place the numbers in ascending order. The inverse problem is also of interest to mathematicians, that is, how to obtain a row of random numbers. How can, for example, algorithms be designed to mimic the results of a lottery? Most mathematical software programs include an instruction to generate rows of random numbers. Out of mathematical caution they are generally called pseudorandom, and the word prefix “pseudo-” indicates that the rows of numbers are, after all, not produced by a lottery but by some reasoning. There is also freeware available on the Internet allowing the generation of random numbers. If we ask the computer, for example, to generate 10 random numbers from 1 to 20, it may produce one time “05 13 08 12 01 20 11 11 04 12” and another time “19 10 18 09 19 05 16 05 10 07.” Although these sequences were generated for the purpose of illustrating randomness, in the first row the number 11 occurs twice (and consecutively, no less), while in the second row 19 appears twice. A patient mouse clicker wishing to generate random numbers over and over, should, at some point, get a row of 10 times the same numbers, and that would be purely by coincidence. A more old-fashioned technique uses lists of numbers printed on paper that actually reflect the result of consecutive lottery results. One chooses a starting point, “somewhere,” and uses the subsequent numbers. Such a list of random numbers will probably include the decimals of pi: π = 3. 14159 26535 89793 23846 26433 83279 50288 41971 6939 … And yet, even in this infinite continuous grab bag of decimals of pi, six consecutive nines suddenly show up at decimal 762. This does not contradict randomness, but it would come as a surprise to see these decimals in a row of random numbers. If a lottery ticket with identical numbers were the winning ticket, it would certainly create a media sensation – would anyone have dared to buy it? Note that the decimals of pi are random following a yet unproven conjecture (whence the previously used “probably”) that was tested statistically, and yet pi is the result of a very simple formula. Pi is simply the quotient of the circumference of a circle divided by its diameter.

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Since there are few numbers on the Ishango rod, Jablan thought of another nonstatistical way to comment on those numbers. After all, by entering a row in an “encyclopedia” of series of numbers, one readily finds out whether the sequence corresponds to a known sequence or not. Neil Sloane’s On-Line Encyclopedia of Integer Sequences contains more than 260,000 series (and their number increases by the day). Now on the Ishango rod figure the following numbers in ascending order: 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 19, 21, with some remarkable subsequences: a first subsequence (3, 4, 5, 6, 7, 8, 9, 10); a second (11, 13, 17, 19, 21); a third subsequence of even numbers (4, 6, 8, 10); a fourth subsequence of odd numbers (3, 5, 7, 9, 11, 13, 17, 19, 21). The first and third series of numbers are simple orderings, while the second (11, 13, 17, 19, 21) represents primes. The fourth is simply a row with odd numbers, but without the number 15. When the fourth row is input in Sloane’s On-Line Encyclopedia, the rows A084820 and A065520 are recognized. The first is too complicated to explain here in a few lines, and the second, A065520, is not easy either: “Numbers n with the property that if m is formed from n by dropping any number (possibly zero) of initial or final digits, then there is a prime ending with m.” Thus 4, 6, 8, …, 15 are not in the sequence, but 9 is because there is a prime number ending in 9, namely 19. However, we have already mentioned that it is very unlikely primes were known 22,000 years ago. It is even more unlikely that the construction of the series A065520 would have been used in Africa. The row 3, 6, 4, 8, 10, 5 is literally in Sloane’s encyclopedia: it carries the number A100000. But it does not make us much wiser, because its description reads as follows:“Middle column of marks found on the oldest object with logical carvings, the 22000-year-old Ishango bone from the Congo.”

And so Jablan tried to analyze the Ishango numbers himself. He noticed that the middle column has an “axis of antisymmetry,” here indicated by a vertical line: 3 × 2 = 6, 4 × 2 = 8 | 10 → 5 + 5. This is also true of row G: 11 13 | 17 19 and even row D: 11 21 | 19 9 by grouping the numbers as follows: 11 + 21 = 32 = 30 + 2, 19 + 9 = 30 – 2 = 28, and so 32 | 28. There is even antisymmetry for pairs of numbers: 11 | 9 = 10 + 1 | 10 – 1, 21 | 19 = 20 + 1 | 20 – 1. Jablan noticed an analogy with Roman numbers that use a similar subtraction and addition: IX | XI and IXX | XXI. On the second rod, the only rows that are not multiples of 6 are 14, 17, and 20, that is, 17 – 3, 17, and 17 + 3. Jablan thought this antisymmetry on the Ishango rods fitted well to the prehistoric era. According to him, one of the important principles in prehistoric times is this duality and symmetry of opposites: “black and white,” “odd and even,” “positive and negative,” … The bivalent “yes and no” logic resulted in a widespread decorative method in the Paleolithic, and especially in Neolithic art. The analysis of works of prehistoric humans would bear witness to the mind of prehistoric humans, which would have been different from the standard image of the “wild mammoth hunters” (Figs. 11.9, 11.10 and 11.11). Paleolithic and Neolithic ornamental and symbolic artifacts are very common, probably because they were made on durable materials, such as bones, stones, or ceramic objects. The Mezin (Ukraine) artifacts are examples of masterpieces of

Fig. 11.9  Left to right: symmetry, no symmetry, shape symmetry, antisymmetry

Fig. 11.10  Artifacts from Mezin, Ukraine

Fig. 11.11  “Perfect” Neolithic monochrome patterns (above) and a reconstruction (below)

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Paleolithic art. They date from the same era as the Ishango rod (20,000 BC). The illustrations show similar processes of design sketches and experiments a contemporary artist could make, as well as some final results: birds and bracelets. More frequent in Neolithic art was the use of “black and white” patterns. They were usually “perfect” in the sense that the black figure was identical to the white background. According to Jablan, this antisymmetry on the Ishango rods is not surprising. Perhaps the notches were meant to recall the number of lines that had to be made on a decoration, a sort of “designer’s reminder.” Neolithic humans indeed copied designs of baskets or textiles on strong carriers such as bones. This allowed the “best” patterns to be stored or recounted. Thus, Jablan’s mathematical hypothesis about antisymmetry not only makes sense arithmetically, but it is also supported by circumstantial evidence. Even without knowledge of numbers, one might think that it is just a pattern, but that too is interesting from a mathematical point of view. Whether the “independent opinion” ultimately turns out to be the right one will, of course, never be proven with absolute certainty. The intention of this text was also to prove that creativity exists in mathematical hypotheses as well. In addition, there are many more interpretations of the Ishango rods apart from de Heinzelin’s arithmetic game, Marshack’s lunar calendar, or Pletser’s counting rods in base 6–10: a rhyme scheme (poetry began in Ishango!), a piece of music (music began in Ishango!), a menstruation tally (the first mathematician was a woman!), and so on. Dr. Reinoud M. de Jonge from the Dutch town of Epse interpreted the rod as evidence of a hunting strategy. And, of course there are those who think it is a remnant of Atlantis or a gift from aliens. These are all the Ishango rods’ “price of fame.” The various hypotheses need not contradict each other: perhaps the Ishango rods were counting devices in base 6–10, used to count Marshack’s lunar phases or Jablan’s antisymmetric ornamental patterns. Thus, Pletser’s hypothesis is by far the most prudent. The many bolder interpretations mainly confirm that the bones reflect some type of logic. The reader might even have a third opinion.

Part III

Epilogue

Chapter 12

Museum Visit, Teaching, Research

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Mathematical Tour Through an Africa Museum What can be noticed at present about the topics covered in this book? And what is their importance for current education or even research? Today, many major towns have some kind of museum dedicated to Africa, preserving colonial treasures or donations from Africa travelers and anthropologists. The illustrations given here originate from the Royal Museum for Central Africa in Tervuren, Belgium, but similar objects can be seen in most ethnological museums. There are, for instance, maps such as those mentioned in Chap. 1, on which the sources of the Nile are indicated near the Mountains of the Moon, following the writings of Ptolemy. This is the region where the Ishango rods were found (Figs. 12.1 and 12.2). Most Africa museums have at least one large drum (see Chap. 2). If possible, the visitor should pick it up and shake it to see if there is indeed a quartz crystal hidden inside it (Fig. 12.3). Chapter 4 is illustrated abundantly in most Africa museums in all kinds of woven patterns (Figs. 12.4, 12.5, and 12.6). The skeptical warning about the golden section (Chap. 4) can be illustrated in wooden milk jars (Pauwels, 1955a). The one given here has a diameter of 10 cm (shown by the hand below) and a useful height of 16  cm (indicated by the hand against the object). For fans of the golden ratio, this yields the desired proportion of 1.6(18)… = ϕ (Fig. 12.7). The previous illustrations bring us seamlessly to Chap. 5, but ropes in museums often have cowry shells since younger visitors often find them pretty neat. They recognize them from African hair styles, but here they were actually used for counting. They are the shells referred to in Chap. 6 (Fig. 12.8).

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Fig. 12.1  An old map, “after Ptolemy,” showing the region of Ishango at the so-called Mountains of the Moon near a large lake

Fig. 12.2  A similar old map “after Ptolemy”

Other ropes have a number of objects threaded on them, representing a story: “… we bought eggs (basket) and fish (roll), there was music (little drum)…”  (Hurel, 1922). Others summarize sayings: “Dancer, never dance on the top of a spear”; or: “A child is like a boat: when you carve it, it will let you cross (the river)” (Fig. 12.9). Sometimes the visitor can literally see how these memory-aid ropes transformed into a kind of prehieroglyphic writing. Sub-Saharan African communities did not

Mathematical Tour Through an Africa Museum

Fig. 12.3  A 1.5-m-high Rwandan drum

Fig. 12.4  Patterns on different supports

Fig. 12.5  Patterns on carpets

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Fig. 12.6  Fabric patterns

Fig. 12.7  Tools in which creative mystics can discover the golden section in Africa

know any form of letter or sign writing, although some older civilizations, such as those of Nubia and Kush (and, in more recent times, the Vai in Liberia as well) developed proper notational systems. And so the tasks for a person were sometimes “summarized” on a cushion that was given to him, or else planks with more or less stylized drawings were used, representing what had to be memorized. If they had been found in Egypt, they would perhaps have been considered (pre-)hieroglyphic forms of scripture (Fig. 12.10).

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Fig. 12.8  Cowry shells were used in ornaments (left) and as a counting and trade material (above right), but other materials were used as money as well (below right)

Fig. 12.9  To the right, the top rope says, “We bought eggs (basket) and fish (roll) at the market and there was music (little drum)…” (above); a similar rope from the Walegga, south of Ishango (below)

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Fig. 12.10  Story board with tasks to be carried out, and prehieroglyphic versions of these memory aids

In Chap. 5 but also in Chap. 6, the games of strategy of the mancala, boa, or igisoro type were mentioned. These and other similar games are sometimes on display (Figs. 12.11 and 12.12).

Fig. 12.11  A strategy game with, remarkably enough, seven holes and one kalaha

The Ishango rod from Part 2 can of course only be viewed in the Royal Museum for Natural Sciences in Brussels, but, as mentioned in Chap. 11, some museums display copies of it. The Museum for Central Africa in Tervuren, Belgium, acquired one recently (Fig. 12.13). Chapter 9, about exchanges between different civilizations in Africa, is illustrated abundantly too in Africa museums. For instance, there are many little stools and head rests in all sorts of forms from various regions of Africa – north, south, east, and west. Sometimes they show striking similarities and sometimes, in contrast, striking differences. Some say these objects were used to sit on, while others

A General-Level Ethnomathematical Quiz

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Fig. 12.12  Another strategy game is a kind of “heads or tails” played with small rods

assert that they were meant for resting the head during the night and protecting one’s hairdo or ears against ants (Fig. 12.14). Exiting a museum brings the visitor unmistakably to the usual souvenir shop, and here too the visitor can find some indications about the exchanges between different civilizations in Africa. Some posters sold in these shops show, for example, obelisks from Axum, a city in Ethiopia. The obelisks from sub-­Saharan Africa demonstrate the intense exchanges between the Egypt of the pharaohs and sub-Saharan Africa (Fig. 12.15).

A General-Level Ethnomathematical Quiz Mathematical education doesn’t seem to be taken seriously without a threatening kind of test, and so here is one, including the traditional official presentation with name and date. The reader can check his African ethnomathematical knowledge as the answers are given below.

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Fig. 12.13  Display cabinet with Ishango rod (above left), with harpoon heads (above right); one of the small objects is the Ishango rod (in the exact middle on the picture below)

Fig. 12.14  Seats and head rests from various regions of Africa

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Fig. 12.15  Africa museum souvenir shops often sell posters of Axum, showing an obelisk, not from Egypt, but from Ethiopia

Test – Quiz – Examination Name: … … … … … … … … Date: … … … … … … What are the European and African solutions to the following story? A man has an outstanding debt with Death, who finally came to him. –– “The debt has to be paid immediately!: said Death, “I demand that you pay me with cattle that is neither a bull nor a cow, otherwise you will die.” –– “You charge me with an impossible task!” begged the man, “cattle are always a bull or a cow!” –– But Death made an appointment for the following day to learn the man’s answer to his question. –– The man asked for advice from his son, who gave him the following sage counsel (complete in a few words which advice he would give): –– (European answer): … … … … …… … … … …… … … … …… … … … –– (African answer): … … … … …… … … … …… … … … …… … … …

A man is standing before a river and has to carry a lion, a lamb, and a cabbage to the other side, but the canoe only has enough space for one of them together with himself. How can he bring everything safely to the other side? (Fig. 12.16) Which drawing cannot be a traditional Rwandan wall decoration? (Fig. 12.17)

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Fig. 12.16  Illustration for the riddle of the lion, the lamb, and the cabbage

Fig. 12.17  All but one of these drawings are traditional imigongos

Consider the numbers 33 = 1 + 32 and 3 = 1 + 2, represented on a mancala game board. Calculate the product of 33 × 3 (Fig. 12.18). Draw this sona in a single stroke. Indicate the start and end points using the letters S and E (Fig. 12.19):

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Fig. 12.18  Mancala game board representation for calculating 33 × 3

Fig. 12.19 Sona

Answers to Quiz Question 1 The Greek oracle of Delphi is famous because of riddles and even more mysterious judgments, but in old European popular tales there were from time to time some puzzling stories too. One was about a king who invited a woman on the condition that she would not come on horseback or on foot, and what’s more not during the day, nor at night; the story goes that the woman would finally come seated on a donkey, at twilight. Analogously, the European answer to the proposed question is therefore: “bring an ox.” In traditional Rwanda there was a mythical and strange individual, too, called Ngoma, and he was often confronted with puzzling oracle sayings  (Boucharlat, 1975; Dion, 1971; Nkongori and Kamanzi, 1957). His answer to the posed paradoxical question was as follows: It suffices to place Death in the impossible situation of requiring his nonexistent cattle. When he shows up on the agreed upon day, answer him as follows: “I have finally found what I owed you. Only, to be able to take possession of it, you cannot come at daylight nor at night. During the day one does not see the stars, and at night they are visible. Thus, come in between both moments, and you will get your cattle.”

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The method, which serves to surprise the listener by an almost mathematical paradox instead of through a figure of speech or an amazing fact, thus is not a privilege of Europe. During a long narrative, an apt saying or a baffling aphorism drew the attention of the hearer. Even in civilizations that did not feel any immediate influence of the Greek art of reasoning, recitations with a surprising logical outcome are encountered. The word “aphorism” is thus only Greek in terms of its etymology. Question 2 The problem is well known in many forms (Gardiner, 1992). The wolf-goat-­cabbage is the most popular in Western Europe, while the fox-goose-grain version is heard in England. It existed in Ethiopia, too, in the Cape Verde Islands, in Algeria, and in Swahili versions. In Liberia cheetah, poultry, and rice were more popular. In Zambia an unsolvable version exists with four items: a leopard, a goat, a rat, and a basketful of corn. As for the proposed lion-lamb-cabbage, the minimal solution is as follows: Step 1: Take the lamb to the other side. Step 2: Return empty. Step 3: Take the lion to the other side. Step 4: Bring back the lamb to the original shore. Step 5: Take the cabbage to the other side. Step 6: Return empty. Step 7: Carry the lamb to the other side. There are two possible solutions, because steps 3 and 5 can be exchanged, but still 7 steps are the minimum. The problem can be generalized: suppose someone has to carry five items, n1, n2, n3, n4, and n5, to the other side of a river and that he can carry only two at a time, where n1 cannot be left alone with n2, and n2 cannot be left alone with n3, …, and n4 cannot be alone with n5. How does one bring all of them to the other side? Again seven steps seem sufficient (Gannon and Martelli, 1993). More generally, when someone must carry 2n + 1 items to the other side of a river and he can only carry n simultaneously, where n1 cannot be left alone with n2, and n2 cannot be alone with n3, …, and ni cannot be alone with ni + 1, …, and finally n2n cannot be alone with n2n + 1, then seven steps will still be required. Question 3 Answer C is the nontraditional drawing, though it was found in some huts, so it must have been of a later date that the genuine images. The story of King Kakira may be of help here to guess the right answer (Chap. 4). Question 4 The answer is given graphically (Figs. 12.20, 12.21, 12.22, 12.23, and 12.24). Question 5 This question should normally be solved in the sand. It makes it a lot more difficult when one is used to drawing with one’s fingers and not by moving one’s arms, as is necessary when drawing longer lines. The origin of the line could be between the

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Fig. 12.20  Halving 33 gives 16 but since one half was omitted, the initial 3 = 1 + 2 is “registered” first (white circles) before it is doubled into 6 = 2 + 4 (black circles)

Fig. 12.21  Halving 16 gives 8, and doubling 6 gives 12 = 4 + 8 (black circles)

Fig. 12.22  Halving 8 gives 4, doubling 12 produces 24 = 8 + 16 (black circles)

Fig. 12.23  Halving 4 gives 2, doubling 24 yields 48 = 16 + 32 (black circles)

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Fig. 12.24  Halving 4 yields 1, doubling 48 produces 96 = 32 + 64 (black circles); as the halving is done, the result is read as follows: 1 + 2 + 32 + 64 = 99

second horizontal row point and the first two dots (between a smaller dot and a larger one) where three lines meet each other. The end then lies at the symmetric dot in the second row between the third and fourth points. The choice of this “sona” was partially inspired by a story related to it, about “The Rooster and the Jackal”: The rooster Kanga and the jackal Mukuza wanted to marry the same woman. Both contacted her father with a proposal of marriage. He asked for payment in advance and they immediately agreed. Suddenly, there ran a rumor that the promised woman had died. Kanga started to cry inconsolably, whereas Mukuza only regretted having lost his advanced payment. Then the father, who intentionally had spread the rumor to see who would be worthy of his daughter, gave her to the rooster who had demonstrated his sincerity.

The two larger points in the drawing represent the rooster and the jackal while the larger point below symbolizes the woman. The story must be told while the line is being drawn.

A High-School-Level Ethnomathematical Quiz Some American universities and colleges offer courses in ethnomathematics. What follows is an American quiz or test from a Math 120 course. All data, including the mandatory key words and sources, were reproduced with the permission of the instructor, Delene Perley, Millikin University. Interested readers can have their test graded: [email protected]. Math 120   Name: __________________ Key words: Time, Calendars, Symmetry, Strip Patterns (Outside readings and Chap. 6 – Ascher) – 40 points 1. (4 points) Give two examples of how traditional African cultures view time differently than we do. 2. (3 points) In calculating a day in the week we use mod 7 arithmetic. Explain what this means and why it is useful.

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3. (2 points) Find 234 mod 5. 4. (3 points) Show how to calculate how many days 8 seconds per year is equal to in 5000 years time. 5. (3 points) Explain why Pope Gregory eliminated October 5 through 14 in the year 1582. 6. (4 points) Our calendar is based on two cycles. What are they and why do they complicate things for us? 7. (4 points) Identify all line symmetries and rotational symmetries for the Adinkra stamps of the Ashanti in Ghana (Fig. 12.25). Fig. 12.25  Ashanti stamps

8. (4 points) When evaluating strip patterns, codes such as VVVV... are used. What does this code mean? 9. (7 points) State the code for each of the seven strip patterns identified by a culture. Each code is represented once (Fig. 12.26).

Fig. 12.26  Strip patterns

10. (6 points) Four strips are pictured without indications of what countries are associated with them. Two are Maori and two Inca. Identify from which culture each comes. Explain where one finds strips like these in each of the cultures. How do the decorative strips reflect the cultural values of the peoples who made them? Explain (Fig. 12.27).

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Fig. 12.27  Maori and Inca frieze patterns

University Level: Examples of Theses No ethnomathematical courses are taught in Belgium, but some master’s dissertations have been written about the subject. The first one described here did not come from a faculty of sciences, and there are likely many other studies in various departments that emphasize indirectly perception or pedagogy in Africa. The mathematical aspects are sometimes rather minimal, except maybe in our first example, but the second example again is every inch mathematical. Linguistics Foreign language student Sofie Ponsaerts (KU Leuven, Belgium) devoted her master’s dissertation to the subject of counting words in Africa, with Prof. Dr. Dirk Geeraerts as her advisor (Ponsaerts, 2002). Didier Goyvaerts, professor at the VUB and at the National University of Congo in Bukavu, offered additional advice. For Ponsaert, numbers are language signs, independent of the form in which they appear: as mathematical symbols or letters or sounds, written or spoken. They have a special place in the manipulation of human language since they call on more cognitive properties, some of which have more to do with mathematics than with language. In her dissertation she decided to discuss only the natural numbers 1, 2, 3, …, since they first originated spontaneously. Today, it is considered obvious that humans have abstract notions of numbers, but Ponsaert believes that our capacity to have such notions was preceded by a relatively long development in the history of humanity. In the first part of her dissertation, she discussed, through psychological glasses, how this evolution happened. Next, she argued how the new skill led to the dawn of new counting systems, initially with aids such as knots and lines and then through notation using numbers with symbols. Linguistically she researched what kind of archetypes were at the base of our Indo-European numerals and what they originally meant. Here she objected, for example, that the French words trois (three) and très (very much) were related: the ancestors of the French did not simply count “one, two, very much.” To conclude, Ponsaert proposed parts of Goyvaerts’ study of Logo, a language spoken in northeastern Congo. The approach followed the example of the British language expert James Hurford and focused on the numerals of this language.

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The approach itself is rather abstract, mathematical in nature. Here, mathematics again illustrates its universal strength, because it allows the formulation of linguistic rules that are valid for all languages, despite the enormous diversity among them, as illustrated in this book. Here are the basic number words in Logo (Fig. 12.28): Nummer Frase

Nummer

Nummer

M

Frase

Frase Nummer

Nummer M

Nummer

M

Nummer

Nummer

5 20 drya 5 drya 3 [((1 × 5) × 20) + ((1 × 5) + (1+1+1))] → nyaba nzi drya nzi drya na Nummer Frase Nummer

Nummer

M

Nummer

Frase

Frase

Frase

Nummer M

Nummer

Nummer M

Frase Nummer M

5 20

M

Nummer Nummer Nummer

5 500

5 20 3 300 drya [((5 × 20) × 5) + ((5 × 20) × 3)] → nyaba nzi nzi drya nyaba nzi na

Fig. 12.28  Diagrams from Ponsaert’s dissertation illustrating the formation of 108 and 800

1 = alo; 2 = iri; 3 = na; 4 = su; 5 = nzi; 6 = kazya; 7 = nzi drya iri (= 5 + 2); 8 = nzi drya na (= 5 + 3); 9 = nzi drya su (= 5 + 4); 10 = mudri; 11 = mudri drya alo; 12 = mudri drya iri; 16 = mudri drya kazya (= 10 + 6); 17 = mudri drya nzi drya iri (= 10 + 5 + 2); 18 = mudri drya nzi drya na (= 10 + 5 + 3); 19 = mudri drya nzi drya su (= 10 + 5 + 4); 20 = nyaba alo (= 20 × 1); 30 = nyaba alo drya mudri (= 20 × 1 + 10); 40 = nyaba iri (= 20 × 2); 50 = nyaba iri drya mudri (= 20 × 2 + 10); 60 = nyaba alo (= 20 × 3); 70 = nyaba na drya mudri (= 20 × 3 + 10); …; 100 = nyaba nzi (= 20 × 5);

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200 = nyaba nzi iri (= (20 × 5) × 2); 300 = nyaba nzi na (= (20 × 5) × 3); … 600 = nyaba nzi kazya (= (20 × 5) × 6); 700 = nyaba nzi nzi drya nayba nzi iri (= (20 × 5) × 5 + (20 × 5) × 2); 800 = nyaba nzi nzi drya nayba nzi na (= (20 × 5) × 5 + (20 × 5) × 3); 700 = nyaba nzi nzi drya nayba nzi su (= (20 × 5) × 5 + (20 × 5) × 4); 1000 = nyaba nzi mudri (= (20 × 5) × 10).

The rules of linguistics allow for the formation of other numbers (see illustration). Ponsaert explains the basic symbols used as follows: The symbol / is the semantic representation for the word “one” or a “unity.” For instance, the group of symbols ////////// corresponds in the same way to the word ten, but also to the related morphemes “-ty” and “-teen.” […] M can, in the case of the English language for example, be “-ty”, “hundred”, “thousand” and so forth.

These above signs for slashes, /, undeniably recall the markings the object found at the sources of the Nile. Mathematics An example of a dissertation written in the mathematical field itself is Alain Gottcheiner’s thesis for earning the degree Diplôme d’Études Approfondies (Master of Advanced Studies, a postgraduate degree) in the mathematics section of the faculty of sciences at the Université Libre de Bruxelles (2001). The dissertation was titled “Une classification des systèmes de parenté prescriptifs” or “A classification of prescribed kinship systems.” Professor Gisèle de Meur, who specializes in mathematical kinship models, directed the research, together with Prof. F. Buekenhout. Gottcheiner’s research involves a mathematical model for family relations and the prediction, using this model, of what rules and relations in society are thus possible. We follow here a simple general approach, which we could have included in the main part of the book, but not all anthropologists agree with this mathematical kinship theory. They often prefer their own jargon above the mathematical language, and so we kept this theory as a remark for the end of the book. In small communities, as on isolated islands, sometimes very strict rules were formulated to avoid marriage between close relatives. People belonged to clans (or subgroups or, in specific cases, to “bwelems”) with, for example, a rule that only two people from the same clan may marry each other, but that the sons will belong to another clan and the daughter to a third clan, following certain prescriptions (Fig. 12.29).

Fig. 12.29  Example of kinship ties

clan 1

clan 2

clan 3

clan 2

clan 3 clan 1

clan 3

clan 1

clan 2

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The matrix vector can represent the first row  x1  P ( arents ) =  x2  ,  x3 





while the numbering for the sons and daughters can be represented by respectively  x2   x3   x  and  x  .  3  1  x1   x2 

The relation between these is

 x2  0 1 0   x1  Sons →  x3  = 0 0 1   x2  = S.P  x1  1 0 0   x3 

and

 x3  0 0 1   x1  Daughters →  x1  = 1 0 0   x2  = D.P  x2  0 1 0   x3  An example of a theorem from these kinship relation studies then is as follows. Theorem A man may always marry the daughter of the brother of his mother.

Proof The man is, of course, the grandson of his grandparents on his father’s and his mother’s side. Thus, we can represent the man as the son of the daughter of her parents: SDP. He is in fact the grandson of the grandparents on the side of his own mother. The daughter of the son of his parents, then, is DSP. In other words, she is the granddaughter of the grandparents on her father’s side. One easily confirms that



0 1 0  0 0 1  1 0 0  SD =  0 0 1   1 0 0  = 0 1 0  = I 3 = DS,  1 0 0   0 1 0  0 0 1 

so that the corresponding elements of SDP and DSP are always identical. They belong to the same clan and can always marry.

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O.K.1 In this way, mathematicians formulate further theorems in this field of study. There does not have to be a limitation to three clans, and the rules for marriage can differ from region to region. Marriages can, for example, be monogamous or not, or marriages can be allowed with someone of another clan (marriage of a man from clan 1 to a woman from clan 2, and children in clan 3; or a man from clan 2 with a woman from clan 3, and children from clan 1; and so forth). This is why others follow a still more general approach without matrices, appealing to the mathematical notion of “group,” a name summarizing the calculating rules that some completely abstract elements must satisfy. In the given example, one says that I3 (the identity matrix given previously), S and D formed the group {I3, S, D},·. In another example from Oceania (African examples such as the Samo from Burkina Faso or the southern Tsongabantu are less frequent in the basic literature), the group for kinship ties is generated by the unit element I (this is the clan of the individual), M (the clan of the mother), M2 (the clan of the grandmother), and M3 = I (the clan of the great-grandmother is again that of the individual). For clan F of the father, the property F2 = I is satisfied, while the relation between M and F is given by MF·MF = I. Mathematicians speak of a dihedral group {I, M, M2, F, MF, M2F},· of order 6. Because it is mathematically widely known which finite mathematical groups are possible and which are not, Gottcheiner could classify different kinship systems through their associated mathematical groups. The enormous cultural differences make it difficult to see coherence in the anthropological jumble. Thus one involuntary thinks of crystallography, where group theory also elegantly determines which forms of crystals are possible or not. Thus, for instance, crystals in the form of a prism with a pentagonal base surface are impossible, and when they were nevertheless encountered in more recent times, they were appropriately called quasi-crystals (Fig. 12.30).

 The “Oxford Dictionary” mentions and criticizes the well-known Scottish, Greek, Choctaw Indian, and French explanation of the origins of the word “OK.” The dictionary doubts the abbreviations “Obediah Kelly,” “Old Kinderhook,” and “orl korrekt” and ends with this: “Another theory with some degree of likelihood is that the term originates from the black slaves of West Africa, and represented a word of the sort of “very well, yes indeed” in various West African languages.” In fact, in Wolof, a regional language in West-Africa, “waw kay” means “yes indeed”, and the relation with “OK” is defended, among others, by Prof. J. Weisenfeld of Columbia University. In “The Times,” Dr. Davis Dalby gives an example of a sentence, recorded centuries ago by a planter, from the mouth of a slave: “Oh ki, massa, doctor no need be fright, we no want to hurt him.” Other African words that led to now commonly used expressions are “jev” (“jive talking”), “banana,” “hipi” (“eyes open”), “boogy-wooky” (“happy sweat”). 1

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Scientific Research Fig. 12.30 Traditional representation with arrows and lines of a marriage system in three clans by the Malekula from Oceania. Letters were added to show the Western interpretation

Scientific Research What follows are some examples of ethnomathematical research that may be of less interest to the nonmathematical reader. A motivation for the research is the application of the proposed game theory in artificial intelligence or  in management-theoretical games and for  the testing of computer limits. Incidentally, a beautiful formula turns up, as in the first example below. The next and last example shows how the application of algebraic structures in social sciences to the study of kinship ties in different cultures finally leads to a philosophical result. Pi (π) and Peta Bytes Duane M. Broline and Daniel E. Loeb published an article about the combinatorics of two games in the mancala family, the “ayo” and the “tchoukaillon”  (Broline, 1995). Broline encountered the game when he taught at the University of Ibadan in Nigeria. Broline and Loeb adapted a theorem first established by the Hungarian mathematician Paul Erdös, known from the book The Man Who Loved Only Numbers by Paul Hoffman. Tchoukaillon is a Russian game of Paleo-Siberian or Eskimo origin that disappeared several decades ago. There is a one-to-one relation, called a bijection, between positions in the African ayo and some positions in tchoukaillon. Broline and Loeb could prove this in a theorem, but the reader will intuitively understand this too when considering the rules of the game. Indeed, tchoukaillon is played in different holes dug out in the sand and with one additional hole, the cala (or rouma, or roumba). The object of the game is to get all pawns in the cala (Fig. 12.31).

1

Fig. 12.31  Tchoukaillon board

2

3

4

5

6

7

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The game can be played by one person, where the pawns are first placed in any hole and placed one by one into successive holes in the direction of the cala. If necessary, the player can go in the opposite direction, starting from the cala. There are three possibilities: –– If the last pawn falls in the cala, then the player gets an additional turn. –– If the last pawn falls in a hole (not the cala) where a pawn already lies, then the player deposits these pawns all together one hole further on (making a bridge, as in mancala or igisoro). –– If the last pawn falls in an empty hole (not the cala), then the game ends. If the holes are numbered starting from the cala, then hole i gives a positive result when it contains exactly i pawns. A position from which it is possible to let all pawns land in the cala is called a winning position. Broline and Loeb then deduced from a number of intermediate theorems that if s(n) is the smallest number of pawns necessary for the nth hole to have a winnable position, then the asymptotic value is given by ∞



s( n) ∼ ∑

M =1

Γ ( M + 1 / 2 ) n2 2

4π ⋅ M !( M − 1)!

=

1 / 2 , 1 / 2  n2 ⋅2 F1  ;1  . 4 2  



In the preceding expression, Γ and F are respectively the gamma function and the hypergeometric function, well known to professional mathematicians. Now F is readily evaluated in this particular case using a summation formula due to Gauss: 2F1(1/2,1/2; 2; 1) = 4/π. Consequently, in the case of n holes, the number of pawns leading to a winning position tends asymptotically to n2/π: Theorem: For large values of n, the smallest number of stones necessary in the mancala-­ igisoro game for a winnable end position is n2/π.

This result is remarkable from a mathematical point of view. In the context of an African ethnomathematical game, the preeminently universal mathematical constant π = pi = 3.14159265… shows up. This is totally unexpected, in view of the number of possibilities for this type of game; when playing with only 12 holes, there are about 1024 combinations, or 1 followed by 24 zeroes. If the topic of study had not been an African game but games more acceptable to the Western mind such as bridge (which originated in Russia) or the Indian game of chess, then the n2/π formula would probably have brought international fame to both mathematicians. Related to this ayo game is awari, one of the simplest variants of the mancala-­ igisoro family, but one for which deep strategic insight is necessary. By programming some computers with African game boards, Dutch researchers John W. Romein and Henri E. Bal could announce in September 2002 that they had obtained the complete solution. Indeed, these scientists of the Faculty of Sciences of the Free University of Amsterdam had checked out all possible combinations of the game. They concluded that when both players play “perfectly” (that is, following the optimal strategy at every step), the game ends in a draw. This was an interesting

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outcome because it meant that this game is a perfect game of strategy, where there is no advantage or disadvantage to how one opens the game. The analysis was conducted on a parallel computer with 144 Pentium III processors with 1.0 GHz RAM. 1.0 petabyte of information was exchanged (a word that will become as popular as “megabyte” or “gigabyte” in the future and corresponds to 1015 bit) and a disk space of 1.4 terabyte was used (again a new fantastically large number: 1012 bit), so that their algorithm needed “only” 51 h. The kalah, still another version of mancala, was solved in various instances. For smaller variants, again a Dutchman, Jeroen Donkers, obtained the complete solution, while in larger cases it was well thought out by G.  Irving of the Computer Science Department at the California Institute of Technology (Irving et al., 2000). As students in Pasadena, California, Donkers’ colleague Jos Uiterwijk and Irving wrote a summary paper about the subject. In lectures, these mathematicians are not afraid to formulate bold statements: Donkers expects the solution to Go within 12 years, and has high hopes for Bridge, even if he thinks that it will be one of the last solvable games. Abstract Algebra In the earlier examples from kinship structure, the notion of a mathematical group has already been encountered. By a group, here symbolized by G,*, one understands a set G (which may or may not contain numbers only, because the elements of G may also be arbitrary objects or even persons), together with a binary operation *: G × G → G, (a, b) → a*b. The latter means that to two elements of G, given in that order, one single element from G is associated (or, eventually, that no element is associated, but then one simply says that the operation is not defined there). This associated element is then indicated by the notation a*b. In order for this operation to be called a group operation, the following properties are required: –– Associativity: for all a, b, and c in G: (a*b)*c = a*(b*c). –– Existence of a unit element: there is an element e in G such that for all a in G: e*a = a = a*e. –– Existence of an inverse element for each element: for all a in G, there is an element b in G such that a*b = e = b*a, where e is the unit element from the previous condition. Often, an additional condition is added, though this one is silently contained in the definition of a binary operation: –– Closure: for all a and b in G, a*b again belongs to G. In the earlier examples in the studies of kinship ties, a set of three elements, {I3, Z, D}, and one of six elements, {I, M, M2, F, MF, M2F}, were considered that both form groups using the usual matrix multiplication as binary operation. Another example can be formed in very general terms. Consider an “arbitrary” community C, with n subdivisions C1, C2, … Cn, which could represent clans, subgroups, or bwelems. An individual can belong to C1 but his mother to C2 or C3 or whatever clan. The table summarizing the relation between an individual and the

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mother is denoted by m. To summarize the paternal relationship, the letter f is used, and for relations through marriage, the letter h. These tables thus correspond to the books of records or, in modern versions, to the spreadsheets local government officials keep in registries of births, deaths, and marriages. There is a mathematical property, though: f = h°m, where ° is the composition of maps. This notion of composition of maps is well known to mathematicians, but in this particular context the expression f = h°m simply means that the father f is the marriage partner h of the mother of the same individual. More generally, mathematicians call this a commutative diagram. Together with the operation °, the relations h, m, and f in fact generate a group, the so-called model of Courrège-Lorrain. For the previous examples from Oceania and Burkina Faso and a single instance from southern Africa, this group structure is the evident algebraic translation, but in the larger African context an even more general description for kinship relations is necessary. Here the notion of semigroup arises. Such a semigroup, here denoted by S,* is again a set S with a binary operation *, where now the only imposed condition is one of associativity. When the property of closure is included, one sees that only two of the four group properties are required, whence the name semigroup (in French, it is literally demi-group or half-group). If in addition a unit element exists, the semigroup (sometimes) receives the special name monoid. Now an element a of a semigroup S is von Neumann regular if there is an element x in S such that a*x*a = a. The semigroup is called von Neumann regular itself if all of its elements are von Neumann regular. Finally, a particular subclass is that of inverse semigroups, where for each element x an element x~ exists such that x*x~*x = x and x~*x*x~ = x~. In this case, x and x~ are called generalized inverses of each other. A model of kinship structure in Africa was proposed by Boyd, Haehl, and Sailer. They start from a set of symbols {0, ♂, ♀, −, +, *}, which are combined as letters in an alphabet to form words following certain grammar rules: 1. 2. 3. 4.

u0 = 0u = 00 = 0; xy = 0; aa = a and ♀♂ = ♂♀ = 0; + a− = *;

where u is in {0, ♂, ♀, −, +, *}, x, y in {−, +}, and a in {♂, ♀}. The empty word Ø, a word without letters, plays the role of unit element: Øx = xØ = x for all words x. The combining of base symbols into words is supposed to be “associative,” that is, that x(yz) and (xy)z are treated as identical, xyz. Thus a semigroup is obtained, and the Boyd team in addition noted that it must be an inverse semigroup to be able to contain the desired kinship structures. Well-defined words are thus interpreted as families, and the resulting mathematical structure is called the free inverse semigroup for kinship, or FISK. Suppose, for instance, as is the case in many African families, that different brothers do not play comparable roles (the oldest or youngest son can be obligated to follow another marriage rule). For instance, for the Baoulés two brothers cannot marry two sisters or a boy and his sister cannot marry a girl and her brother.

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For a larger brother-sister group, as proposed in  Fig.  12.32, the following relationships are defined:

Fig. 12.32  Two diagrams, used when brothers and sisters follow an identical (left) or a different rule of marriage (right)

(i) b = {(1, 2), (2, 1)}, the relation of a man to his brother; (ii) z = {(3, 4), (4, 3)}, the relation of a woman to her sister; (iii) x = {(1, 3), (1, 4), (2, 3), (2, 4)}, the relation of a man to his sisters; (iv) x~ = {(3, 1), (4, 1), (3, 2), (4, 2)}, the relation of a woman to her brothers. With these conditions, the following ethnomathematical theorem can be formulated:Theorem: The relations b, z, and a generate an inverse semigroup, with nine different elements: b, z, x, x~, b2, z2, xx~, x~x, Ø.

In his PhD thesis, F. E. Tjon Sie Fat showed how a FISK is used in an African context (Tjon Sie Fat, 1990). He motivated his study by a reference to Belgian philosopher Leo Apostel. Indeed, as Tjon Sie Fat stated: It is the aim to extend the analysis of Lévi-Strauss, indicated by T, so that different hybrid systems can be included. Here, the letter T refers to the ordered foursome R(S, P, M, T), in which Apostel summarized the important variables of a modeling process: a subject S takes, with its goal P, the entity M as model for the prototype T. Thus the purpose of the mathematical study of the family ties structure can be considered as a restructuring system or a system extension (Apostel 1961).

Origins of the Illustrations

Drawings made by the author are not mentioned separately. Number 3 6

7 9 12–13 14–15 16 17 18 19 21 28 35–36

Description The round universe … Hawkins’ poster “African and African-American Pioneers in Mathematics.” Evolution of ground plans in Burundi Title page of a text about oral history in Rwanda. Map of eclipses. A walk through a landscape … Interpretation by Marshack … A poetic representation … In the “rugo” of a student’s … This entrance to a school in J.-P. Harroy had a high function … The large drums had Announcement in a colonial...

Origin Reproduced with approval of Acquier Jean-Louis. Reproduction with approval of author Bill Hawkins: [email protected]. Reproduced with approval of Acquier Jean-Louis. From old journals held at Africa Museum Tervuren (Belgium). Map of Jean Meeus. Reproduced with approval of Acquier Jean-Louis. Drawings by Frederic Delannoy. From Kagame. Reproduced with permission from Acquier Jean-Louis. Photo art school Nyundo, photo C. Blondeel. Photo Wikimedia Commons, author Shabanmasengesho. Phtoto Wikimedia Commons. References Africa Museum Tervuren; Answer to 6134 of July 30 and to 6425 of August 20, 1909, and entitled “Report about native money, and the counting systems, in use in the district around ‘Lake Leopold II’”. Ethnographic File 126. Collection Royal Museum for Central Africa ©. (continued)

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Origins of the Illustrations

218 Number 42 43

Description Table of… Until today a Dozen Society exists… 47, 52–58 Patterns and matrices. 61 Neroman’s study of... 62

63 64 65 66 67 68–69 70 71 72–74 75 76 78 79

80

81 82

Origin See Mubumbila, 1988. Internet publication of de Dozenal Society, http://www. dozenalsociety.org.uk/ accessed on March 25, 2016. P. Gerdes. Left: illustration Neroman; right: drawing Frederic Delannoy. Examples of reasoned … Drawing by Frederic Delannoy, and work of C. Blondeel’s group, Zwarte Zusterstraat Gent (Belgium). Two illustrations from a Drawing by Frederic Delannoy, after illustrations from mathematical magazine. The Mathematics Teacher. Weaver… Drawings by Frederic Delannoy. A modern piece… Photo Tijl Beyl. Interior of a Rwanda hut… Drawing by Frederic Delannoy. Pauwels’ list … From Pauwels. Traditional imigongo from From Celis, reproduced with permission of authors. Rwanda. In Benin one finds all 7 … Drawing by Frederic Delannoy, after Crowe. This fractal was named.... Universal mathematical fractal drawing. Illustrations African Drawing by Frederic Delannoy after work of Ron Fractals. Eglash, with permission from author. Catholic, Hindu, … Author’s picture (left) and Wikimedia Commons (middle, right). Gerdes’ fractal… Author drawing, after Gerdes. Standard illustration about Drawing by Frederic Delannoy. quipu’s. Counting strings. References Africa Museum Tervuren: Left, cordelette à nœuds, 1960 5/5 R.IV.D.a.6, Doss. ethno. 126, dossier report; 74, May 3, 1920. right: cordelette à nœuds; 1960 4/5; R.IV.D.a.5; Doss. ethno. 126; dossier report; 74; May 3, 1920. All kinds of counting References Africa Museum Tervuren (in order): strings corde à nœuds, bicolore, 10,563; corde à nœuds, 1536, R.VII.D.a.4; corde à nœuds, 1960 1/5, R.IV.D.a.2; Doss. ethno. 126; dossier rapport 74, May 3, 1920; corde à compter (grande), 2025, R.VII.D.a.5; corde à nœuds, 1816, R.V.D.a.5. Lagercrantz’ map… Right: drawing by Frederic Delannoy. A string with… References Africa Museum Tervuren: Left: corde à nœuds (152 nœuds ou mosolo); 884; R. IV.D.a.1; Doss. ethno. 77; dossier report 2; région de Basoko; February 7, 1910; Right: corde à nœuds 1816; R.V.D.a.5. (continued)

Origins of the Illustrations Number 83

84

90 98 113 127 128–133 134 135–136 137 138–142 143–144 145 148–149 154–155 156 157 159

164 170

Description Counting sticks of…

219

Origin References Africa Museum Tervuren (in order): bâton à compter (petit avec biseau), 1812, étiquette “n° 2 Uoira” ou “Noira”; bâton encoché, 2144; bâtonnet à compter (grand), 1538, R.VII.D.a.1, Long; 600 mm, région d’Ubangala, ex-Zaire; au nord de Kisangani (territoire de Banalia) et au sud-est de Buta; province du Haut-Zaire/Congo, Ababoa ou Boa, 23/03/1910. Sticks to count to… References Africa Museum Tervuren (in order): Tally stick, 2111 3/2, R.II.D.a.2, cfr 67.63.5695, 2/2, Rég.; Cabinda, administrative subdivision: Tchimbuande, Kakongo; tali stick (roseau formant une baguette et maintenu par des fruits secs ou épices), 65.59.96, “Ami du Musée”, 440 × 19 mm, Rhodésie du Nord, Bangweolo, October 12, 1965; tally stick, 67.63.5695 2/1, ET.38.20.847 ½, 449 × 45 mm, Rég.; Cabinda, subdivision administrative: Tchimbuande, Kakongo May 1938. An old mancala version … Drawing by Frederic Delannoy. A traditional scene Drawing by Frederic Delannoy. A 50-year-old poetry From Kagame, over 50 years old. collection… Views of Semliki Valley Photos © ADIA RBINS Brussels. Photos about the Ishango Photos from de Heinzelin’s report of 1957. excavations, and the site. Six views of Ishango rod Ishango rod, 6 views: photos and infography: Patrick Semal, 15/06/2001; photos © ADIA RBINS Brussels. Photos of Ishango rod and Photos from de Heinzelin’s report of 1957. the site. Official photo of KBIN. Ishango rod: photo: Marcel Springaer; infography: Ivan Jadin, 23/12/1999; © ADIA RBINS Brussels. By cutting out… Drawings from de Heinzelin’s report of 1957. Marshack’s drawings. Author’s drawings, after Marshack. A part of Pletser’s study of Drawing by V. Pletser. the carvings. Map with de distribution of From de Heinzelin’s report of 1957. harpoons … Illustrations of Noguera: a Drawing by Frederic Delannoy aftr Noguera. Peul woman, …. Cultural relations … Author’s drawing, with illustrations from Noguera. Data analysis diagram … Reproduced with approval of Arnaiz. Lagercrantz’s map… Wikimedia Commons, “Lissala man with tattooes.” Reference Africa Museum Tervuren: bambou encoché, 34,992. Cartoons … Drawing by Frederic Delannoy. Sign at … Photo by Magloire Paluku, reproduced with permission. (continued)

Origins of the Illustrations

220 Number 171

Description Ishango rod in Kenya…

172

3D scans of…

173–174 175–177

Three views of… Neolithic patterns

178–180 181

Museum for Central Africa, Tervuren, Belgium Patterns on…

182

Patterns on carpets

183

Fabric patterns.

186 187

At right, the above rope says Story board…

188

Strategy game…

189

Another strategy game…

193 194 202–204

Illustration… All but one… Ashanti stamps and patterns Diagrams…

205

Origin Left: photo by Patricia Mergen, Liaison Officer, Legal Entity Appointed Representative (LEAR) at European Commission, Royal Museum for Central Africa, Tervuren, Belgium. Photos © ADIA RBINS Brussels; TNT project, NESPOS Society. Photos © ADIA RBINS Brussels. Drawings and photos by Slavik Jablan, Belgrade, Serbia; reproduced with permission. Photos taken by author. References Africa Museum Tervuren (in order): coupe/ schaal, 0.0.33710; coiffe/hoofdtooi, 1949.21.1; paravent, 0.0.41670. References Africa Museum Tervuren (in order): tapis/ tapijt, 1968.39.3; tapis/tapijt, 1980.31.2. References Africa Museum Tervuren (in order): 0.0.28094; 0.0.25651; 0.0.27757–37; 0.0.28089; 1959.21.672; 0.0.16482 3; 1950.24.135; 0.0.45084; 0.0.45107; 1950.24.150. Reference Africa Museum Tervuren: 1975.48.1. References Africa Museum Tervuren: 0.0.42871; 0.0.27312. Reference Africa Museum Tervuren: 1969.59.636 (négatif G 4878). References Africa Museum Tervuren: 1974.54.37 1–9; 1974.54.37 1. Drawing by Frederic Delannoy. From Celis, reproduced with permission of authors. Drawings by Frederic Delannoy, after Delene Perley, of Millikin University. Drawings Sophie Ponsaert; reproduced with permission.

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Index

A Africa museum, 33, 39, 41, 46 Algebra, 78, 87, 123, 213 Arithmetic, 35, 49–51, 78–79, 113, 143, 147, 148, 166, 204 Ascher, M., 58, 59, 204 Ashanti, 8, 66, 69, 205 Associativity, 213 B Baali, 46 Bach, J.S., 29 Bal, H.E., 212 Beaumont, P., 169 Bell, R.C., 87 Brandel, R., 30, 32 Broline, D.M., 211, 212 Brooks, A.S., 137, 140, 172 Buekenhout, F., 169, 170, 208 Burundi, vi, viii, 5, 6, 10, 16, 24, 26, 29, 39, 46, 47, 79–81, 127 C Celis, G., 68, 70 Circle, 203 Cole, M., 3, 4 Congo, v, viii, 38, 44, 45, 47, 56, 71, 72, 90, 127, 128, 134, 164, 206 D de Heinzelin, J., v, 128, 134, 141, 154, 167 Demaine, 179

Dogon, 14, 87 Doubling, 37–39, 58, 78, 114–118, 122, 142, 143, 145, 147, 148, 150 Dozen, 48 Duodecimal, 48–53, 87, 150, 153 E Egypt, 12, 51, 114, 115, 139, 153, 154, 157–163, 166, 173, 174, 197 Errico, F., 169, 170 F Fractals, 72–76 Fulani, 44, 163 G Gamma function, 212 Gauss, C.F., 87, 212 Gay, J., 3, 4 Gerdes, P., 59, 62, 63, 76, 79, 176 Golden section, 63, 64, 74, 117, 191, 194 Greece, 31, 32, 114, 156–159, 161–163 Group theory, 71 Günther, R., 32 H Harroy, J.-P., 29 Harvard, 148 Hauzeur, 138 Hemiola, 29, 30 Hindu, 31

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Index

Hogben, L., 143 Hypergeometric function, 212

Ptolemy, 12, 14, 191, 192 Pythagoras, 12, 143, 157

I Igisoro, 89, 93–96, 99–105, 117, 212 Ishango rod, v, 8, 135, 140, 141, 167, 173, 175, 176, 191, 198 Ishango site, v, 133, 136, 172 Isometry, 73 Ivory Coast, 39, 66

R Romein, J.W., 212 Royal Belgian Institute for Natural Sciences, v, 128 Rutanzige, 5, 127, 136, 164, 172 Rwanda, viii, 5, 14, 19, 22, 23, 26–28, 30, 32, 33, 40, 43, 46, 47, 50, 67–71, 89, 94, 95, 98, 104, 105, 193, 199, 201

J Joseph, G.G., 24, 35, 156, 158, 173, 176 K Kagame, A., 21, 47, 104 Kakira ka Kimenyi, 70 Kpelle, 4, 11 L Loeb, D.E., 211, 212 M Mancala, 87–92, 94, 95, 104, 200, 211–213 Marshack, A., 16, 18, 145, 146, 148, 151, 167, 172 Mathematics teacher, 66 Michiels, F., 33 Middle East, 31, 48, 114, 153, 156 Moon, 13, 16, 18, 20–21, 24, 26, 50, 51, 56, 81, 145, 147, 151 Multiplication, 63, 77, 78, 111–114, 117, 120, 122, 145, 150, 158, 172, 213 N Namibia, 169, 172 Navajo, 8 Njock, G.E., 66 Nyali, ix, 46, 150 Nzohabonayo, P., 26 P Pattern, 16, 31, 57–59, 61, 66, 70–73, 76, 149, 176, 205 Pi, 211 Pletser, V., 148, 149, 166, 167 Polyhedral, 169 Prime number, 50, 142, 143, 145, 147, 148, 151

S Semi-group, 214, 215 Semliki, v, 127, 128, 134, 154, 172 Shallit, J., 140 Square, 158 Star, 19, 23, 24, 26, 201 Struik, D.J., 176 Sun, 20, 22, 23, 26, 48, 56, 130, 199 Symmetry group, 204 T Tanzania, 40, 89 Timbuktu, 107, 108 Triangle, 154 U Uganda, 20, 89, 127, 161 V van Straelen, V., 128 Venus, 24, 26, 173 von Neumann regular, 214 W Walegga, 154, 195 Wirth, N., 122 Y Yasayama, 45 Yuhi, 27, 28 Z Zaslavsky, C., 71, 176 Zulu, 5

Africa + Mathematics: Summary

Are the sources of mathematics to be found at the sources of the Nile? In 1950, Belgian archaeologist Jean de Heinzelin found a 22,000-year-old bone with intriguing carvings in Ishango, a village in the volcanic region on the border of the Congo and Uganda, not far from Rwanda. The author defends the hypothesis of Belgium’s space scientist Vladimir Pletser that the bone was neither an arithmetical game with prime numbers nor an astronomical calendar, but a simple tally stick of a people counting in a different, non-Western way. The mathematics of the rod is an example of ethnomathematics, an emerging subfield in mathematics. Oddly enough, examples from the Congo, Rwanda, and Burundi are scarce, though Belgium has historical ties with this region. The author spent about 12 years teaching in the region, and through this experience and examining valuable collections in Belgian Africa museums, he hopes to show the mathematics from Central Africa represent a particular challenge, even for contemporary mathematicians. Still, critical considerations are not neglected and some subsequently turn out to be correct, as de Heinzelin revealed a second Ishango rod on his death bed. The Ishango rods waited half a century in a dusty drawer before their rediscovery – on one hand at the mercy of Moliere’s colonial bourgeois-­gentilhommes, who did not want to admit that Africans did mathematics, and on the other hand the Afrocentrists who saw in the rods the beginning of all mathematics. Perhaps the third scientific dog will finally run away with the mathematical bones.

© Springer Nature Switzerland AG 2019 D. Huylebrouck, Africa and Mathematics, Mathematics, Culture, and the Arts, https://doi.org/10.1007/978-3-030-04037-6

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