*Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, *Joseph Mazur, 2014, 285 pp. $29, hardcover. ISBN 978-0-691-15463-3. Princeton University Press, 41 William Street, Princeton, NJ 08540

In the last two decades, a variety of popular books on selected topics in mathematics have appeared on the market. Many of their authors have employed an historical approach to their topic. In his latest such book, Joseph Mazur takes his reader into the intriguing evolution of mathematical notation and examines the power and effect of these symbols on mathematical thinking itself. Mazur is a mathematician and has worked in and taught mathematics throughout his life; therefore, he brings the necessary experience and knowledge to this task. But despite this relevant background, he modestly refers to his efforts as an exercise in mathematical journalism.

A good book, lecture or lesson must be not merely composed but also crafted; that is, each piece of the total effort should be carefully chosen to lead the reader to a level of better understanding of the matter under concern. At first appearances, this book seems to be well crafted. The contents are divided into three sections: “Numerals,” “Algebra” and “The Power of Symbols.” Each section is comprised of several brief chapters, all prefaced with a catchy or appealing heading, such as “The Indian Gift,” “The Timid Symbol” and so on, as well as appropriate introductory quotations. Supporting illustrations, tables and explanatory notes are abundant, and a rich listing of references is supplied. In appearance and organization, *Enlightening Symbols* is an appealing book. The author’s task presents something of a quandary: he must explain symbols using symbols since words, themselves, are symbols – visual markers – for thoughts and concepts. The challenge is met: Mazur’s writing style is conversational and each topic of focus is modified by historical and cultural perspectives, environments I found to be lively and interesting.

Where more historically attuned readers might find some difficulties is with the evolutionary tacks Mazur has taken. It is always good to begin at the beginning. Thus, perhaps the discussion on numeration could have elaborated more on the use of tally symbols and their initial 1:1 correspondence which, as quantities increased, eventually gave rise to code symbols and the concept of groupings or “base” for a numeration system. The first formal numeration system considered is the Babylonian “so-called sexagesimal (base 60)” (p. 10) system, but the reader has not yet been told what “base” means. The cuneiform tablet Plimpton 322 is identified as being considered “a proto-Pythagorean theorem” (p. 13); I am not aware of such a specific association. At least one modern scholar has even rejected the previous interpretation of the tablet as a list of Pythagorean triples; see Eleanor Robson’s discussion of Plimpton 322 in [1]. In beginning his examination of Babylonian numeration, the author should have mentioned the research of Denise Schmandt-Besserat [2], which traces the origins of Babylonian numerals to the use of counting tokens and their eventual imprinting into wet clay. In this scenario the following path of evolution would clearly emerge: quantity of concrete object \(\rightarrow\) semi-abstract object (token) \(\rightarrow\) abstract symbol (clay impression) \(\rightarrow\) cuneiform numeral. Schmandt-Besserat’s ultimate hypothesis is that writing resulted from mathematical notation, a rather startling discovery!

In Chapter 3, “Silk and Royal Roads,” where the Chinese numeration system is examined, the opening quote is erroneously credited to Master Sun in the Chinese mathematical classic, *The Nine* *Chapters.* Actually, the speaker is a Master Sun, Sun Zi, who existed, but his words are from the preface to his own classic, *Sun Zi* *suanjin,* which was compiled at least 400 years after *The Nine Chapters.* This work supplies the first instance of a mathematical problem solved by the “Chinese Remainder Theorem.” Chinese numerals adhere to a base 10 positional ordering. On page 29, the Chinese characters for the number 26,999 are written as a horizontal string; actually, traditional numbers were written as vertical strings of characters. The numerals demonstrated had two symbolic functions: they served as number symbols and number words, a unique dual responsibility that is ignored in this chapter. The Chinese use of computing rods as a device for performing computations is noted, but no mention is made of their configurations serving as a set of numerals nor that they were positioned in arrays where an empty space served as a place-holder, the natural “zero.” In Ancient China, several sets of numerals were employed at the same time, for different purposes: commercial trade, accounting, etc. Rod numerals were used in scientific work. Here again, we have a numeration system that evolved from the concrete to the semi-concrete and finally to the abstract/symbolic, but it is not recognized. The rod numeral system was finally absorbed into the design and functioning of the *suanpan*, the Chinese abacus, an innovation forced by imperial decree to make mathematics more accessible to the common people.

In the section on algebra, no mention is made of Luca Pacioli’s abbreviation of the Italian word *cosa* for “thing” (or “unknown”) in his *Summa* (1494). A page from the *Summa*, one that dealt with “The Rule of Three,” say, would show how the Italian word *cosa* appeared numerous times on one page, so many times that the replacement by the abbreviation/symbol* co* was warranted. The Italians were leaders in the development of algebra for many years and, as a result of this experience, algebra was associated with the Italians and widely performed outside of Italy under the name “Cossic Art.” Another beginning missed!

The craftsmanship in the writing of this book might have been a little sharper had the author approached his task as an historian or teacher rather than a journalist. Despite the discontinuities in this rendering of the tale of mathematical notation, *Enlightening Symbols* remains an attractive, interesting book and offers much information for a perceptive and somewhat knowledgeable reader.

### References

[1] Eleanor Robson, “Words and Pictures: New Light on Plimpton 322,” *The American Mathematical Monthly,* v. 109 (2002), pp. 105-120.

[2] Denise Schmandt-Besserat, *How Writing Came About*, University of Texas Press, 1996.

*Editor's note:* For another discussion of the same book, see the review by Tushar Das in *MAA Reviews.*