MATHEMATICS ACROSS CULTURES

SCIENCE ACROSS CULTURES: THE HISTORY OF NON-WESTERN SCIENCE

V O L U M E 2 M A T H E M A T I C S ACROSS C U L T U R E S

Editor:

HELAINE SELIN, Hampshire College, Amherst, Massachusetts, USA

MATHEMATICS ACROSS CULTURES

The History of Non-Western Mathematics

Editor

HELAINE SELIN Hampshire College, Amherst, Massachusetts, USA

Advisory Editor

U B I R A T A N D ' A M B R O S I O

State University of Campinas /I)NICAMP, Säo Paulo, Brazil (Emeritus)

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

A catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-0260-1 ISBN 978-94-011-4301-1 (eBook) DOI 10.1007/978-94-011-4301-1

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Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000

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INTRODUCTION TO THE SERIES: SCIENCE ACROSS CULTURES: THE HISTORY OF

NON-WESTERN SCIENCE

In 1997, Kluwer Academic Publishers published the Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. The encyclopedia, a collection of almost 600 articles by almost 300 contributors, covered a range of topics from Aztec science and Chinese medicine to Tibetan astronomy and Indian ethnobotany. For some cultures, specific individuals could be identified, and their biographies were included. Since the study of non-Western science is not just a study of facts, but a study of culture and philosophy, we included essays on subjects such as Colonialism and Science, Magic and Science, the Transmission of Knowledge from East to West, Technology and Culture, Science as a Western Phenomenon, Values and Science, and Rationality, Objectivity, and Method.

Because the encyclopedia was received with critical acclaim, and because the nature of an encyclopedia is such that articles must be concise and compact, the editors at Kluwer and I felt that there was a need to expand on its success. We thought that the breadth of the encyclopedia could be complemented by a series of books that explored the topics in greater depth. We had an opportunity, without such space limitations, to include more illustrations and much longer bibliographies. We shifted the focus from the general educated audience that the encyclopedia targeted to a more scholarly one, although we have been careful to keep the articles readable and keep jargon to a minimum.

Before we can talk about the field of non-Western science, we have to define both non-Western and science. The term non-Western is not a geographical designation; it is a cultural one. We use it to describe people outside of the Euro-American sphere, including the native cultures of the Americas. The power of European and American colonialism is evident in the fact that the majority of the world's population is defined by what they are not. And in fact, for most of our recorded history the flow of knowledge, art, and power went the other way. In this series, we hope to rectify the lack of scholarly attention paid to most of the world's science.

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VI INTRODUCTION TO THE SERIES

As for defining science, if we wish to study science in non-Western cultures, we need to take several intellectual steps. First, we must accept that every culture has a science, that is, a way of defining, controlling, and predicting events in the natural world. Then we must accept that every science is legitimate in terms of the culture from which it grew. The transformation of the word science as a distinct rationality valued above magic is uniquely European. It is not common to most non-Western societies, where magic and science and religion can easily co-exist. The empirical, scientific realm of understanding and inquiry is not readily separable from a more abstract, religious realm.

The first two books in the series are Astronomy Across Cultures: the History of Non-Western Astronomy, and Mathematics Across Cultures: the History of Non-Western Mathematics. Each includes about 20 chapters. Most deal with the topic as it is perceived by different cultures: Australian Aboriginal Astronomy, Native American Mathematics, etc. Each book also contains a variety of essays on related subjects, such as Astronomy and Prehistory, or East and West. The next four in the series will cover Medicine, Nature and the Environment, Chemistry and Alchemy, and Physics and Optics.

We hope the series will be used to provide bqth factual information about the practices and practitioners of the sciences as well as insights into the world views and philosophies of the cultures that produced them. We hope that readers will achieve a new respect for the accomplishments of ancient civiliza­tions and a deeper understanding of the relationship between science and culture.

Acknowledgments

About the Contributors

Introduction

TABLE OF CONTENTS

Communicating Mathematics across Culture and Time Leigh N. Wood

Anthropological Perspectives on Ethnomathematics Ron Eglash

East and West Edwin J. Van Kley

Rationality and the Disunity of the Sciences David Turnbull

Logics and Mathematics: Challenges Arising in Working across Cultures Helen Verran

A Historiographical Proposal for Non-Western Mathematics Ubiratan D'Ambrosio

The Uses of Mathematics in Ancient Iraq, 6000-600 BC Eleanor Robson

Egyptian Mathematics James Ritter

Islamic Mathematics Jacques Sesiano

The Hebrew Mathematical Tradition Y. Tzvi Langermann and Shai Simonson

vii

ix

Xl

xvii

1

13

23

37

55

79

93

115

137

167

viii TABLE OF CONTENTS

Inca Mathematics Thomas E. Gilsdorf

Mesoamerican Mathematics Michael P. Closs

The Ethnomathematics of the Sioux Tipi and Cone Daniel Clark Orey

Traditional Mathematics in Pacific Cultures Walter S. Sizer

Aboriginal Australian Mathematics: Disparate Mathematics of Land Ownership Helen Verran

On Mathematical Ideas in Cultural Traditions of Central and Southern Africa Paulus Gerdes

Accounting Mathematics in West Africa: Some Stories of Yoruba Number Helen Verran

Chinese Mathematical Astronomy Jean-Claude Martzlojf

The Mathematical Accomplishments of Ancient Indian Mathematicians T. K. Puttaswamy

The Dawn of Wasan (Japanese Mathematics) Jochi Shigeru

Development of Materials for Ethnomathematics in Korea Kim, Soo Hwan

Index

189

205

239

253

289

313

345

373

409

423

455

467

ACKNOWLEDGMENTS

I would like to thank Maja de Keijzer and her staff at Kluwer Academic Publishers. I never believe that Kluwer is a large company, because of all the individual attention the staff gives me. I would also like to thank my colleagues at Hampshire College; I am so fortunate to be able to work with them. Special thanks go to Aaron Berman, Brian Schultz, and Gai Carpenter for recognizing the value of my work and giving me time to do it. And mostly I offer innumer­able, unquantifiable thanks to Bob and Lisa and Tim.

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ABOUT THE CONTRIBUTORS

HELAINE SELIN (Editor) is the editor of the Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Kluwer Academic Publishers, 1997) and Science Librarian and FacuIty Associate at Hampshire College in Amherst, Massachusetts. In addition to editing the new series, Science Across Cultures, she has been teaching a course on the Science and History of Alternative and Complementary Medicine.

UBIRATAN D'AMBROSIO (Advisory Editor; A Historiographical Proposal for Non-Western Mathematics) is Emeritus Professor of Mathematics at the State University of Campinas/UNICAMP in Sao Paulo, Brazil. He is President of the Brazilian Society of History of Mathematics/SBHMat, President of the International Study Group on Ethnomathematics/ISGEm, and President of the Institute for Future Studies/I.E.F. He is also a Visiting Professor and graduate advisor at several universities. He was the Director of the Institute of Mathematics, Statistics and Computer Science of UNICAMP, Brazil from 1972-80, Chief of the Unit of Curriculum of the Organization of American States, Washington DC (1980-82), and Pro-Rector [Vice-President] for University Development, of UNICAMP, Brazil from 1982-90. His most recent publications include Ethnomathematics. The Art or Technique of Explaining and Knowing and History of Mathematics in the Periphery: The Basin Metaphor (Berlin: Max-Planck-Institute fUr Wissenschaftsgeschichte, 1999).

MICHAEL CLOSS (Mesoamerican Mathematics) is a Professor in the Department of Mathematics and Statistics at the University of Ottawa, Ottawa, Canada. He is editor of the volume, Native American Mathematics (University of Texas Press, 1986, 1996). In addition to his work on mathematical develop­ment in different Native American cultures, he has written many research articles on the mathematics, chronology, astronomy and hieroglyphic writing of the ancient Maya.

RON EGLASH (Anthropological Perspectives on Ethnomathematics) holds a B.S. in Cybernetics and a M.S. in Systems Engineering, both from the

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xii ABOUT THE CONTRIBUTORS

University of California at Los Angeles. Following a year as human factors engineer at National Semiconductor, he returned to school for a doctorate in cultural studies of science and technology through the University of California at Santa Cruz History of Consciousness program. He received a Fulbright grant for his work on ethnomathematics, which included sites in Senegal, Gambia, Mali, Burkina-Faso, Cameroon, Benin, and Ghana. The resulting book, African Fractals: Modern Computing and Indigenous Design, has just been published by Rutgers University Press. Dr Eglash's software simulations, which allow students to learn about fractal geometry from African designs, is available from Dynamic Software. His other research areas include history of information technology, community computer networks, and anthropology of communica­tion systems. For more see: http://www.cohums.ohio-state.edu/comp/eglash.htm

PAULUS GERDES (On Mathematical Ideas in Cultural Traditions of Central and Southern Africa) is Professor of Mathematics at the Universidade Pedagogica (Maputo, Mozambique) and Director of Mozambique's Ethnomathematics Research Center - Culture, Mathematics, Education. Since 1986, Dr Gerdes has been the chair of the African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA). Among his books published in English are: Geometry from Africa: Mathematical and Educational Explorations (The Mathematical Association of America, 1999); Culture and the Awakening of Geometrical Thinking (MEP-Press, 1999); Women, Art, and Geometry in Southern Africa (Africa World Press, 1998); Lusona: Geometrical Recreations from Africa (L'Harmattan, 1997); Ethnomathematics and Education in Africa (lIE/University of Stockholm, 1995); and African Pythagoras: a Study in Culture and Mathematics Education (Ethnomathematics Research Project, Maputo, 1994).

THOMAS GILSDORF (Inca Mathematics) is Associate Professor and Chair of the Department of Mathematics at the University of North Dakota. He holds a Ph.D. in mathematics from Washington State University. In 1992-1993, he held the Solomon Lefschetz Postdoctoral Fellowship in mathematics at the Centro de Estudios Avanzados in Mexico City. During his stay there he became interested in the mathematics of indigenous groups of the Americas. That interest increased when he returned and began working with several minority groups, especially Native Americans, at the University of North Dakota. His current research interests include functional analysis and Native American mathematics. Most of his publications have been in the area of locally convex spaces. The article in this book is his second on Native American mathematics; the first, 'Native American number systems', will appear in the proceedings of the Midwest History of Mathematics conference, October, 1998.

JOCHI SHIGERU (The Dawn of Wasan (Japanese Mathematics)) is a histo­rian of science and technology in Eastern Asia. His work involves analyzing traditional mathematics and astronomy and their relationship to Eastern Asian

ABOUT THE CONTRIBUTORS xiii

culture. He was born in Tokyo, and obtained the Ph.D. at the University of London. He is now an Associate Professor at the National Kaohsiung First University of Science and Technology, Taiwan, and a research commissioner of the Institute of Wasan in Tokyo. His publications include The Influence of Chinese Mathematical Arts on Seki Kowa (Takakazu), Ph.D. Thesis, University of London, 1993, and 16 papers and 11 international conference papers. He has also contrbuted to the Encyclopaedia of the History of Science, Technology, and Medicine in Non- Western Cultures (Helaine Selin, ed. Kluwer, 1997) and the Encyclopedia of Folklore (Kano, Masanao, Shunsuke Tsurimi and Nakayama Shigeru, eds. Sanseido, 1997).

KIM, SOO-HWAN (Development of Materials for Ethnomathematics in Korea) is Head of the Department of Mathematics Education at Chongju National University of Education in Korea. He holds a D.Ed. and M.Ed. from the Korea National University of Education. He is also the Financial Director of the Korea Society of Mathematical Education. His current research interests include ethnomathematics, the history and pedagogy of mathematics, and assessment in mathematics education. He has published articles on ethnoma­thematics and mathematics education.

Y. TZVI LANGERMANN (The Hebrew Mathematical Tradition) is Associate Professor of Arabic, Bar Ilan University, Ramat Gan, Israel and Senior Research Associate, Institute of Microfilmed Hebrew Manuscripts, Jerusalem. He holds a Ph.D. in the History of Science from Harvard University. His current research interests center around Science and Philosophy in Jewish and Muslim cultures. His latest books are Yemenite Midrash (HarperCollins, 1997) and The Jews and the Sciences in the Middle Ages (Ashgate-Variorum, 1999).

JEAN-CLAUDE MARTZLOFF (Chinese Mathematical Astronomy) is Directeur de Recherche and belongs to the Chinese Civilization Research Center of the C.N.R.S. (Centre National de la Recherche Scientifique/French National Center for Scientific Research) Paris. He has studied mathematics, astronomy and Chinese at Paris University and classical Chinese at the University of Liaoning Province, Shenyang, China. He became a full time C.N.R.S. researcher in 1980 and has published numerous articles on the history of mathematics, astronomy, and the Jesuit mission in China (17-18th centuries). After many research sojourns in Mainland China (Academia Sinica), Japan (Japanese Society for the Promotion of Science) and Taiwan, he is now writing a book on traditional Chinese calendrical computations and mathematical astronomy.

DANIEL CLARK OREY (The Ethnomathematics of the Sioux Tipi and Cone) earned a Ph.D. in Curriculum and Instruction in Multicultural Education from the University of New Mexico in 1988. He is currently Professor of Mathematics and Multicultural Education at California State University, Sacramento. In 1998, he served as a Fulbright visiting scholar at the Pontificia Universidade

xiv ABOUT THE CONTRIBUTORS

Cat6lica de Campinas in Brazil, and has served as an adjunct faculty member at D-Q University, in Davis, California. He has been a Tinker International Field Research Grant recipient for ethnographic field research using children and computers in Puebla, Mexico; a Mellon Interamerican Field Research Grant recipient for ethnographic field research using Logo programming lan­guage in a highland Maya school in Patzun, Chimaltenango and the Colegio Americano de Bananera, Izabal, Guatemala; and a Title VII Fellow at the University of New Mexico. His primary research interests and objectives stress the use of an ethnomathematical perspective which seeks to facilitate the documentation, development, study, and evaluation of diverse learning environ­ments, with special emphasis on issues of access and equity in mathematics for all learners.

T. K. PUTTASWAMY (The Mathematical Accomplishments of Ancient Indian Mathematicians) is Professor Emeritus of Mathematical Sciences at Ball State University in Indiana. He also taught at the University of Mysore and Madras University in India. He has received M.Sc. (University of Mysore), M.S. (University of Minnesota), and Ph.D. (State University of New York at Buffalo) degrees in mathematics. He has published several papers on the global behavior of solutions of ordinary differential equations in the complex plane. His research papers have earned him regular invitations to speak at various international conferences around the world. Ball State University recognized his earlier research accomplishments in 1976-77, when he received its Researcher of the Year award. He has served as chair for contributed papers at the regional and national meetings of the American Mathematical Society and various interna­tional conferences including the International Congress of Mathematicians held at Berkeley, California during the summer of 1986.

JAMES RITTER (Egyptian Mathematics) is co-director of the program in History and Philosophy of Science in the Department of Mathematics at the Universite de Paris 8 and of the collection Histoires de science, published by Presses Universitaires de Vincennes. His work is concerned with the history of 'rational practices' in the ancient world (Egypt and Mesopotamia) and with the development of general relativity and unified theories in the twentieth century. In the former guise he has published on mathematics, medicine and divination in A History of Scientific Thought (ed. Michel Serres, Blackwell) and co-edited Histoire des fractions, fractions d'histoire (Birkhauser) and Mathematical Europe (Presses de la Maison de l'Homme). In the latter, he has written on unified theories in Albert Einstein: (Euvres choisies (Le Seuil) and on cosmology and literature in Melancholies of Knowledge (SUNY). He has also co-edited The Expanding Worlds of General Relativity (Birkhauser).

ELEANOR ROBSON (The Uses of Mathematics in Ancient Iraq, 3000 BCE-300 CE) is a British Academy Postdoctoral Fellow at the University of Oxford and a Research Fellow of Wolfson College, Oxford. She has a degree in mathematics and a doctorate in Oriental Studies, which was published in 1999

ABOUT THE CONTRIBUTORS xv

as Mesopotamian Mathematics, 2100-1600 Be (Oxford: Clarendon Press). She has taught the archaeology, history, and languages of the ancient Middle East at Oxford since 1995. Her research interests are in the social and intellectual history of mathematics in the pre-Islamic Middle East, focussing especially on the scribal culture of early Mesopotamia (Iraq). She is currently writing a study of an 18th century scribal school, provisionally titled The Tablet House.

JACQUES SESIANO (Islamic Mathematics) studied theoretical physics at the Swiss Federal Institue of Technology in Zurich and has a Ph.D. in History of Mathematics from Brown University. He is a Lecturer in the history of math­ematics at the Swiss Federal Institute of Technology in Lausanne.

SHAI SIMONSON (The Hebrew Mathematical Tradition) is Professor of Mathematics and Computer Science at Stonehill College in North Easton, Massachusetts. He earned his B.A. in mathematics from Columbia University in 1979 and his Ph.D. in computer science from Northwestern University in 1986. He has taught at the University of Illinois, Northwestern University, and Tel Aviv and Hebrew Universities. His research interests are in theoretical computer science, computer science education, and history of mathematics. He has published a number of papers in and received NSF funding in all these areas. His most recent work is about Levi ben Gershon, a medieval Jewish mathematician. 'Gems of Levi ben Gershon' will appear in a special history focus issue of Mathematics Teacher, 'The Missing Problems of Levi ben Gershon, A Critical Edition, Parts I and II' will appear in Historia Mathematica, and 'The Mathematics of Levi ben Gershon' in Bekhol Derakhekha Daehu. His work on medieval Jewish mathematics and its use in the classroom, was sponsored by Grant STS-9872041 from the National Science Foundation.

WALTER SIZER (Traditional Mathematics in Pacific Cultures) is a Professor of Mathematics at Moorhead State University in Minnesota. He has also taught in Malaysia and as a Fulbright lecturer in Ghana. He received a Ph.D. from the University of London, doing work in abstract linear algebra, and has been indulging an interest in the history of mathematics for fifteen years. Publications include papers in mathematics, the teaching of mathematics, and the history of mathematics, particularly the mathematical concepts of Pacific islanders.

DAVID TURNBULL (Rationality and the Disunity of the Sciences) is a senior lecturer in the School of Social Inquiry at Deakin University working in Science Studies. He has published Maps are Territories: Science is an Atlas (University of Chicago Press, 1993), and Masons, Tricksters and Cartographers: Makers of Knowledge and Space (Harwood Academic Press, 1999, in press). He is currently working on encounters between knowledge traditions with special interests in Geographic Information Systems (GIS) and indigenous knowledge; theories of cultural and technical change; and space, place and narrative in the creation and performance of knowledge.

xvi ABOUT THE CONTRIBUTORS

EDWIN J. VAN KLEY (East and West) is currently Professor of History, emeritus, at Calvin College, where he taught from 1961 to 1995 after receiving his Ph.D. from the University of Chicago. Most of his research and writing explore aspects of the impact of Asia on Europe during the seventeenth century, most of it published in articles such as 'Europe: Discovery of China and the Writing of World History', (The American Historical Review 76(2): 358-385, April, 1971). His major publication, co-authored with Donald F. Lach, is Asia in the Making of Europe; A Century of Advance, Books 1-4 (Chicago: University of Chicago Press, 1993). He continues to work on this project.

HELEN VERRAN (Logics and Mathematics; Australian Aboriginal Mathematics; Accounting Mathematics in West Africa) is senior lecturer in the Department of History and Philosophy of Science at the University of Melbourne, Australia. For six years in the 1980s she worked as lecturer/senior lecturer in the Institute of Education, Obafemi Awolowo University, Ile-He, Nigeria. On returning to Australia in the late 1980s she took up a position with Deakin University training members of Australia's traditional Aboriginal communities as primary school teachers. The course was delivered to students in their own communities. This was a research based course in which curriculum development in Aboriginal schools was a focus. It was the starting point for twelve years of research on logic with the Yolngu community; this research continues today. Her most recent work is African Logics: Towards Ontology for Postcolonial Times and Places (forthcoming). She is currently working on Logics and Places: Learning from Aboriginal Australia.

LEIGH WOOD (Communicating Mathematics across Culture and Time) is the Director of the Mathematics Study Centre at the University of Technology, Sydney. The Centre works with students individually and in small groups to help them reach their academic potential. Leigh teaches a range of subjects to mathematics majors, including differential equations and mathematical com­munication. Her interest in mathematical discourse as part of mathematics started in 1985 as a result of working with refugees from Southeast Asia. Leigh has developed many teaching and learning packages including 2 textbooks and 10 videos, all of which have a strong mathematical communication component. Her research areas are language in mathematics and curriculum design.

INTRODUCTION

Every culture has mathematics. That is not to say that every culture has forms of deductive reasoning or even that every culture has counting. In most places, mathematics grew from necessity, and not every culture had reasons to ask 'How many?' or 'How much?' Without trade, and without herding and agricul­ture, societies saw little need to enumerate. But enumeration and calculation are only parts of mathematics; a broader definition that includes 'the study of measurements, forms, patterns, variability and change'l encompasses the math­ematical systems of many non-Western cultures.

I was a Peace Corps volunteer in Central Africa when I first realized that there were other forms of mathematics. I noticed that people counted differently on their fingers. Whereas I would signal one using my index finger, Malawians used their thumbs. To indicate four, I would use all fingers except the thumb; Malawians used those same fingers, but arranged them so that the two adjacent fingers touched. When they got to five, they made a fist instead of extending all their fingers. At the time this seemed like another fascinating cultural distinction, part of the whole process of learning to see myself as a product of my society and to realize that things I thought were universal human activities were in fact American ones. What intrigued me more was that Malawians indicated 100 by shaking their fists twice, as if knocking on a door. This seemed to me an unusual mathematical organization and was my first hint that numbers and how we express them are connected to our culture and upbringing.

Until the past few decades, histories of mathematics have virtually ignored the mathematics of non-European cultures, even after Egyptian mathematical papyri and Babylonian clay tablets illustrating complex mathematical problems were discovered. This neglect grew from the colonial mentality, which ignored or devalued the contributions of the colonized peoples as part of the rationale for subjugation and dominance. The disparagement by European colonial societies is particularly ironic because the early Europeans, the ancient Greeks themselves, acknowledged their intellectual debt to the Egyptians. Herodotus believed that geometry had originated in Egypt, emerging from the practical need for re-surveying after the annual flooding of the Nile. Aristotle thought that the existence of a priestly leisure class in Egypt prompted the pursuit of

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H. Selin (ed.), Mathematics Across Cultures: The History of Non-Western Mathematics. xvii-xx. © 2000 Kluwer Academic Publishers.

XVlll INTRODUCTION

geometry. Modern Euro-American scholars, blinded by their belief in their own concept of rationality, disregarded the origins of their own discipline by ignoring Aristotle and Herodotus.

The views of Herodotus and Aristotle also illustrate two theories concerning the beginnings of mathematics, one arising from practical necessity and the other from priestly leisure and ritual. This dichotomy appears in many of the articles in this collection, along with a third element that is also present. Mathematics is either closely connected to religion and ritual or to the needs of daily life, but occasionally it seems to be a purely intellectual activity, as in the case of magic squares.

Eurocentric views of scholarship have changed radically in recent years, and newer histories of mathematics, such as George Ghevergese Joseph's The Crest of the Peacock: Non-European Roots of Mathematics (London: I. B. Tauris, 1991) and Frank J. Swetz's From Five Fingers to Infinity: A Journey Through the History of Mathematics (Chicago: Open Court, 1994), reflect this trend. They illustrate the debt we owe to the Arabs in bringing together the technique of measurement, evolved from its Egyptian roots to its final form in the hands of the Alexandrians, and our remarkable number system, which originated in India. Although these books and other new scholarship have contributed significantly to the study of the mathematics of non-Western cultures, they focus almost entirely on cultures whose mathematical systems are precursors to our own. The mathematics of other cultures is still largely overlooked.

The Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, published by Kluwer Academic Publishers in 1997, contained many articles on world mathematics. Mathematics Across Cultures is an extension of that encyclopedia. Our aim is to add depth to the articles that, as encyclopedia articles must, covered the subjects as broadly as possible. The collection is divided into two sections. Essays in the first section discuss the connection between mathematics and culture, the field of ethnomathematics, the concept of rationality and how it applies to the study of culture, and the transmission of knowledge from East to West. The chapters in the second part describe individual cultures and their mathematics.

The book begins with Leigh Wood's essay, 'Communicating Mathematics across Culture and Time.' Condensing a great deal of history and philosophy in her discussion, Wood analyzes mathematical communication in societies without written language. She goes on to examine the challenges to successful communication even within cultures that have left written records. Ron Eglash's essay on ethnomathematics compares ways of looking at mathematics in different cultures. In one view, ethnomathematics is the study of mathematical concepts in indigenous cultures; in the other, the word applies to the connec­tions between mathematics and culture regardless of geography or type of societal structure. Edwin Van Kley's paper, distilled from his massive and impressive work, Asia in the Making of Europe, describes the centuries-long movement of science and technology from East to West. Among the scientific achievements that the West adopted was the Indian system of arithmetical notation, trigonometry, and the system of calculating with nine Arabic numbers

INTRODUCTION xix

and a zero. Van Kley sees the dominance of Western culture, especially in the last century, as a threat both to the world's traditional cultures and to tradi­tional Christian Western culture.

David Turnbull's provocative essay in the sociology of scientific knowledge, 'Rationality and the Disunity of the Sciences', proposes rethinking the notion of the unity and universality of science and mathematics. He sees the concept of local knowledge as a means of comparing knowledge production systems across cultures; he goes on to dissect the concepts of rationality, objectivity, and scientific method. In his view, 'rather than rejecting universalizing explana­tions what is needed is a new understanding of the dialectical tension between the local and the global.' In her paper, 'Logics and Mathematics: Challenges Arising in Working across Cultures', Helen Verran also tackles the connection between a universal concept -logic - and its effect on cultural misperceptions. By looking at Australian Aboriginal and Nigerian classrooms, she demonstrates how logic is perceived and carried out in dissimilar societies.

In the final chapter of the first section, 'A Historiographical Proposal for Non-Western Mathematics,' Ubiratan D'Ambrosio shows how colonialism led to a disparaging and belittling of the colonized cultures and their mathematical and scientific achievements. He proposes a new way of looking at mathematics that recognizes other systems of intellectual and social knowledge.

Fifteen culture areas are studied in depth in the second section. Eleanor Robson explores Mesopotamian mathematics in her article on the uses of mathematics in ancient Iraq. James Ritter examines Egyptian mathematics, urging us to study what it meant and how it was used by the Egyptians themselves. Jacques Sesiano describes the extraordinary synthesis of Arab mathematics from its Mesopotamian, Indian, and Greek heritage. He analyzes arithmetic, algebra, geometry, and number theory and ends with a detailed look at magic squares. Y. Tzvi Langermann and Shai Simonson look at two different branches of mathematics in their essay on the Hebrew mathematical tradition. Langermann investigates the arithmetic and numerology of Abraham Ibn Ezra and goes on to describe Hebrew contributions in geometry; Simonson focuses on algebra and its evolution from a geometric to a combinatorial subject.

Thomas Gilsdorf considers several environmental and cultural influences on Inca mathematics, emphasizing the interdependence of mathematics and non­mathematical factors. Michael Closs uses glyphs and available texts to interpret the mathematics of several Mesoamerican people, including the Olmec, Zapotec, Maya, Mixtec, and Aztec. He examines their mathematical astronomy and the connection between the calendar and the state. For North America, Daniel Orey investigates one group of native Americans, the Sioux, and the mathematics used in building and setting up their tipis.

Walter Sizer's essay on traditional mathematics in Pacific cultures examines numeration, geometry, games, and kinship relations in Polynesia, Melanesia, and Micronesia. Helen Verran explores a different part of the Pacific region in her article on land ownership mathematics among Australian Aboriginal

xx INTRODUCTION

people. She believes that questions concerning different forms of logic and mathematics are central to cross-cultural understanding.

In his paper on Central and Southern Africa, Paulus Gerdes presents evidence for early mathematical activity and discusses details of geometrical ideas as expressed in mat weaving, house wall decoration and sand drawings. Helen Verran explores the West African tradition as she relates stories of Yoruba number use. She shows how the properties of a number system reflect the needs of the culture that creates it and stresses the importance of the link between a society and its workings for understanding its number system.

Jean-Claude Martzloff condenses hundreds of years of history in his study of Chinese mathematical astronomy. He discusses the sources themselves and places them in their historical, political, and epistemological context. The second part of his paper explores in depth the mathematics of the Chinese calendar. T. K. Puttaswamy's chapter on India describes the richness and complexity of the mathematics of ancient Indians by focusing on early works such as the Sulbasiitras and great mathematicians such as Aryabhata I and Brahamagupta. Jochi Shigeru's essay, The Dawn of Wasan (Japanese Mathematics),' provides us with an excellent illustration of a way to do math­ematics that is unique to the Japanese culture. He explains its evolution, relates the stories of some important mathematicians, and also studies magic squares in detail. Kim Soo Hwan takes a completely different approach when he shows how to use games and flags to reveal Korean ethnomathematics.

There are many reasons to study other systems of mathematics, including their potential contribution to our own mathematical discourse and to our appreciation of the richness of cultural and scientific diversity. We invite you to explore different ways of doing mathematics with us.

NOTE

Helaine Selin Amherst, Massachusetts

Winter 2000

1 Australian Academy of Sciences, Mathematical Sciences: Adding to Australia (1996: ix), cited in Wood, Leigh, 'Communicating Mathematics across Culture and Time', in this volume.