Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers

by Joseph Mazur

Overview

An entertaining look at the origins of mathematical symbols

While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In Enlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.

Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.

From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

Product Details

ISBN-13:	9780691154633
Publisher:	Princeton University Press
Publication date:	03/23/2014
Pages:	312
Product dimensions:	6.40(w) x 9.20(h) x 1.10(d)

About the Author

Joseph Mazur is the author of Euclid in the Rainforest (Plume), which was a finalist for the PEN/Martha Albrand Award, Zeno's Paradox (Plume), What's Luck Got to Do with It? (Princeton), and Fluke (Basic).

Table of Contents

Introduction ix

Definitions xxi

Note on the Illustrations xxiii

Part 1 Numerals 1

1. Curious Beginnings 3

2. Certain Ancient Number Systems 10

3. Silk and Royal Roads 26

4. The Indian Gift 35

5. Arrival in Europe 51

6. The Arab Gift 60

7. Liber Abbaci 64

8. Refuting Origins 73

Part 2 Algebra 81

9. Sans Symbols 85

10. Diophantus's Arithmetica 93

11. The Great Art 109

12. Symbol Infancy 116

13. The Timid Symbol 127

14. Hierarchies of Dignity 133

15. Vowels and Consonants 141

16. The Explosion 150

17. A Catalogue of Symbols 160

18. The Symbol Master 165

19. The Last of the Magicians 169

Part 3 The Power of Symbols 177

20. Rendezvous in the Mind 179

21. The Good Symbol 189

22. Invisible Gorillas 192

23. Mental Pictures 210

24. Conclusion 216

Appendix A Leibniz’s Notation 221

Appendix B Newton’s Fluxion of xⁿ 223

Appendix C Experiment 224

Appendix D Visualizing Complex Numbers 228

Appendix E Quaternions 230

Acknowledgments 233

Notes 235

Index 269

What People are Saying About This

From the Publisher

"This book provides an insightful synthesis of the historical and mental revolutions that created humanity's most useful symbols—mathematical expressions."—Stanislas Dehaene, author of Reading in the Brain

"Mathematical symbols are much more than squiggles on paper—they serve as potent sources of insight into a wonderfully complex language. In this book, Joseph Mazur takes us on a fascinating journey into the origins of these symbols. You cannot ask for a better guide."—Mario Livio, astrophysicist and author of Brilliant Blunders

"Joseph Mazur teaches us that the history of mathematical notation is the history of human civilization."—Kenneth A. Ribet, University of California, Berkeley

"In this thoughtful, entertaining, and carefully researched book, Mazur lays out the development of the mathematical notations and symbols that we take for granted. He offers a fascinating look at the history of the conventions of writing mathematical ideas, and shows how important our choice of conventions has been for the growth of mathematical knowledge."—Emily R. Grosholz, author of Representation and Productive Ambiguity in Mathematics and the Sciences

"Worldwide, mathematicians speak a common symbolic language. Mazur traces how symbols developed and gradually took hold, uniting arithmetic, algebra, and geometry to give us metaphors as powerful as those of poems. A fascinating history!"—Philip Holmes, coauthor of Celestial Encounters

"A curiosity cabinet of mathematical gems, curated by Joseph Mazur's cosmopolitan good taste."—Siobhan Roberts, author of Wind Wizard

"Enlightening Symbols explains the origins of symbols used in mathematics. But this fun book is so much more than this. Filled with interesting stories, captivating material, and delightful observations, it offers readers a better understanding of the nature of mathematics and a wonderful overview of mathematical history."—Dominic Klyve, Central Washington University

"An enjoyable and informative tour of mathematics history, Enlightening Symbols describes how our modern system of notation led to the abstraction we work with today. This is an important and interesting story."—Anna Pierrehumbert, Trinity School

Editorial Reviews

02/10/2014
Mazur (Euclid in the Rainforest) gives readers the fascinating history behind the mathematical symbols we use, and completely take for granted, every day. Mathematical notation turns numbers into sentences—or, to the uninitiated, a mysterious and impenetrable code. Mazur says the story of math symbols begins some 3,700 years ago, in ancient Babylon, where merchants incised tallies of goods on cuneiform tablets, along with the first place holder—a blank space. Many early cultures used letters for both numbers and an alphabet, but convenient objects like rods, fingers, and abacus beads, also proved popular. Mazur shows how our “modern” system began in India, picking up the numeral “zero” on its way to Europe, where it came into common use in the 16th century, thanks to travelers and merchants as well as mathematicians like Fibonacci. Signs for addition, subtraction, roots, and equivalence followed, but only became standardized through the influence of scientists and mathematicians like René Descartes and Gottfried Leibniz. Mazur’s lively and accessible writing makes what could otherwise be a dry, arcane history as entertaining as it is informative. (Apr.)

Publishers Weekly

"[A] fascinating narrative. . . . This is a nuanced, intelligently framed chronicle packed with nuggets—such as the fact that Hindus, not Arabs, introduced Arabic numerals. In a word: enlightening."—-George Szpiro, Nature

From the Publisher

"Mazur treats only a subset of F. Cajori's monumental A History of Mathematical Notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books, but Mazur adds new findings and insights and it is so much more entertaining . . . and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history."

European Mathematical Society

"Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."

Washington Post

04/01/2014
The arithmetical notation we see today developed over many centuries. Although roman numerals still pop up now and then (look at how we number the Super Bowls), the system of decimal place-value notation and so-called Arabic numerals are used almost universally. In this book, Guggenheim Fellow Mazur (emeritus, mathematics, Marlboro Coll.; Euclid in the Rainforest) traces the evolution of these symbols from the earliest archaeological evidence to the present. The author surveys the work of earlier investigators and, where there is disagreement, gives fair weight to the different competing conjectures. For algebraic symbolism, Mazur nicely summarizes the historic record, which is much shorter and therefore less open to controversy. Today, this notation seems so natural that it is hard to imagine doing mathematical work without it. Mazur emphasizes the strength of the system, describing how, once expressed in its algebraic form, problems seem to solve themselves and even to suggest areas for further research. The concluding chapters discuss how the latest developments in cognitive science shed light on how using good notation helps to produce clear thinking. VERDICT Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history. Recommended.—Harold D. Shane, emeritus, Baruch Coll. Lib., CUNY

Library Journal

"Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."

Washington Post