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# Native American Mathematics

### By Michael P. Closs

Spanning time from the prehistoric to the present, the thirteen essays in this volume attest to the variety of mathematical development present in the Americas.

There is no question that native cultures in the New World exhibit many forms of mathematical development. This Native American mathematics can best be described by considering the nature of the concepts found in a variety of individual New World cultures. Unlike modern mathematics in which numbers and concepts are expressed in a universal mathematical notation, the numbers and concepts found in native cultures occur and are expressed in many distinctive ways. Native American Mathematics, edited by Michael P. Closs, is the first book to focus on mathematical development indigenous to the New World.

Spanning time from the prehistoric to the present, the thirteen essays in this volume attest to the variety of mathematical development present in the Americas. The data are drawn from cultures as diverse as the Ojibway, the Inuit (Eskimo), and the Nootka in the north; the Chumash of Southern California; the Aztec and the Maya in Mesoamerica; and the Inca and Jibaro of South America. Among the strengths of this collection are this diversity and the multidisciplinary approaches employed to extract different kinds of information. The distinguished contributors include mathematicians, linguists, psychologists, anthropologists, and archaeologists.

- Preface
- 1. Native American Number Systems (Michael P. Closs)
- 2. Numerical Representations in North American Rock Art (William Breen Murray)
- 3. Some Notes on Quantification and Numerals in an Amazon Indian Language (Maurizio Covaz Gnerre)
- 4. The Calendrical and Numerical Systems of the Nootka (William J. Folan)
- 5. Chumash Numerals (Madison S. Beeler)
- 6. Cultural Ecology of Mathematics: Ojibway and Inuit Hunters (J. Peter Denny)
- 7. Tallies and the Ritual Use of Number in Ojibway Pictography (Michael P. Closs)
- 8. A Survey of Aztec Numbers and Their Uses (Stanley E. Payne and Michael P. Closs)
- 9. Decipherment and Some Implications of Aztec Numerical Glyphs (Herbert R. Harvey and Barbara J. Williams)
- 10. Mathematical Ideas of the Incas (Marcia Ascher)
- 11. The Mathematical Notation of the Ancient Maya (Michael P. Closs)
- 12. The Zero in the Mayan Numerical Notation (A. Seidenberg)
- 13. In Search of Mesoamerican Geometry (Francine Vinette)
- References

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Modern mathematics is international in character. Its concepts are transmitted, studied and developed in numerous national languages in all parts of the world. The symbolic description of these concepts is presented in a universal mathematical notation independent of language. For example, as a part of this notation, numbers are expressed in a decimal system using Hindu-Arabic numerals.

The international nature of modern mathematics is a relatively recent phenomenon and represents a continuation of mathematical developments which occurred in Europe during the centuries from 1600 to 1900. The flowering of European mathematics was first nourished and stimulated through contacts with the Arabic world which had experienced an intellectual awakening during the great expansion of Islam. Arab savants had accumulated a repository of mathematical knowledge which drew on sources to be found in India, Persia and the Mediterranean world. These sources were themselves fed by the earlier mathematics of ancient Greece, Egypt and Babylonia. Thus, modern mathematics results from the cumulative effort of diverse peoples over thousands of years.

Historians of mathematics have concentrated on the great main stream leading to modern mathematics and have paid only scant attention, if any at all, to mathematics in cultures not directly contributing to it. There are exceptions to this tendency and some studies of Chinese, Japanese and African mathematics have appeared. In addition, some work has been done on the primitive origins of counting, arithmetic and geometry. The present volume is also exceptional in that it focuses on the mathematical development indigenous to the New World. This is an area about which there is a dearth of information in the mathematical literature. It is my hope that this work will help to remedy this state of affairs and will lay a foundation for future studies in this area.

In my opinion, native American mathematics can best be described as a composite of separate developments in many individual cultures. The contributions to the volume are concerned with several aspects of this development among various native American groups. The papers, considered as a whole, give a good representation of the variety of mathematical experience found in the New World. The papers also give some idea as to the form which the history of mathematics must take if it is to incorporate material outside of its traditional boundaries. It is a form in which an almost total reliance on the historical approach is supplemented or replaced by drawing on the resources and methodologies of other disciplines such as anthropology, archaeology and linguistics.

#### By Michael P. Closs

Michael P. Closs is Professor of Mathematics at the University of Ottawa.

"Of interest to a wide audience, not just students of mathematics and its history, and is highly recommended for personal reading and general library acquisition."

—Historia Mathematica