Enlightening Symbols: A Short History ofMathematical Notation by Joseph Mazur

Rafe Jones

The Mathematical Intelligencer
ISSN 0343-6993
Math Intelligencer
DOI 10.1007/s00283-015-9589-y

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Author's personal copy

Enlightening
Symbols: A ShortHistory ofMathematical
Notationby Joseph Mazur
PRINCETON: PRINCETON UNIVERSITY PRESS, 2014, 312 PP., US $29.95,
ISBN 978-0-691-15463-3REVIEWED BY RAFE JONES

mathematically minded person can hardly imagine aworld without the symbols that form the connective
tissue of mathematical statements. So woven are
they into the fabric of our mental processes that many ofthem come prewritten on any keyboard: +, -, =,\, and thesymbols 1 through 9, just to name a few. This doesn’t eventouch on familiar algebraic representations such as lettersto denote unknown quantities, superscript numerals todenote powers, radicals to denote roots, and functions
written as f(x). It was not always so. Until as recently as thesixteenth century, not only did most mathematical expressions contain no symbols recognizable to us today, but they
were purely rhetorical. In his entertaining and quirky
Enlightening Symbols: A Short History of Mathematical
Notation, Joseph Mazur drives home how this transformation has shaped our mathematical consciousness byquoting the following translation of a passage from alKhwarizmi’s Algebra (ca. 820 AD):If a person puts such a question to you as: ‘‘I havedivided ten into two parts, and multiplying one of theseby the other the result was twenty-one’’; then you knowthat one of the parts is thing, and the other is ten minus
thing (p. xv).
In modern notation, this is x ð10  x Þ ¼ 21. In addition to its
compactness, the modern expression allows us to focus on
the essential quantities and to ask key questions moreeasily: what if we changed the values 10 and 21? Is there ageneral method for solving all such problems? Mazur, formerly on the mathematics faculty at Marlboro College and aself-described ‘‘mathematics journalist embedded with thelearned troupes’’ (p. 233), is careful to note that the passage
mentioned earlier represents an extreme, and that someauthors used positional Hindu-Arabic numerals and denoted unknown quantities with letters well before thesixteenth century. Nonetheless, I could not help but beshocked at the recentness of our seemingly timeless way ofwriting mathematics.
Through a series of short chapters that read as vignettes,
Mazur recounts the advent of mathematical symbols starting from the very beginning—cave paintings 40,000 years
ago—and ending with the calculus symbols introduced by
Leibniz and Newton. In between, I found a smorgasbord ofA

captivating observations, reflections, anecdotes, historicalnarration, and quotations (the last from a dizzying array of
personages, including historians, mathematicians, philosophers, psychologists, and novelists). For example, I learned
that in his marvelously titled 1557 work, Whetstone of Witte,
Robert Recorde introduced the twin lines of the symbol =,calling them ‘‘Gemini lines,’’ to denote that the quantities
on either side are as identical as possible, and invoking themythological Gemini twins, Castor and Pollux (p. 133). I
also learned that Cardano, famous for his publication of thegeneral solution of the cubic polynomial, somehow foundroom in his mind to accommodate nonreal complex
numbers, but not the notion that the product of two negative numbers must be positive (p. 123).
In addition to the aforementioned smorgasbord, I also
found puzzling loose ends, hard-to-follow chronology, andodd repetitions. This is a great book to pick up when you
have half an hour to spare—at almost every turn there issomething interesting and unexpected—and a more frustrating book to read straight through. Read it to whet yourappetite for mathematical history, to spur ruminations onwhat mathematical notation accomplishes, and to add toyour store of tidbits and anecdotes to tell to students or toother mathematically interested people. Don’t read it toobtain a comprehensive and clear account of the historicaldevelopment of modern numerals and other mathematicalnotation. For that I recommend Karl Menninger’s Number
Words and Number Symbols (or for a more modern andcomparative study of number systems, Stephen Chrisomalis’s Numerical Notation) and Florian Cajori’s A History ofMathematical Notations.
The book is divided into three parts, titled ‘‘Numerals,’’‘‘Algebra,’’ and ‘‘The Power of Symbols.’’ The first tworevolve, respectively, around the historical development of
positional representation of numbers via Hindu-Arabic
numerals, and the development of modern algebraic notation. The last part is a collection of reflections on such topics
as why notation is important, what makes good notation, andwhat physiological processes occur in the brain when weperceive mathematical symbols. This third section contains
some of the most thought-provoking material in the book.
Mazur has broken free from the shackles of historical narrative (which he seems to have been itching to do all along),
and we roam with him as he meditates on deep questions
about our mental processes. For instance, in ‘‘The Good
Symbol,’’ he examines how good notation can suggest newmathematical structures rather than merely represent known
ones. In ‘‘Invisible Gorillas,’’ the most ambitious chapter ofthe book, he surveys contemporary cognitive-psychologicalresearch on how the brain responds to perceptions of words
and symbols. In the chapter titled ‘‘Conclusion,’’ he makes
the provocative move of likening symbols in poetry to thosein mathematics.
In ‘‘Numerals,’’ we find the history of numerical representation, from crude tallying schemes to positional notation
requiring the use of 0 as a number. To illustrate the simultaneous fascination and puzzlement I experienced in reading
this section, and indeed the whole book, consider p. 57. Just
after the interesting observation that the expense of paper
prevented the modern system of numerical representation
Ó 2015 Springer Science+Business Media New York
DOI 10.1007/s00283-015-9589-y

Author's personal copy
from spreading more quickly (older systems allowed forcomputations that could be done on an abacus or sandtable), there is a new subsection, one of only a handful in thebook, given the brevity of the chapters. Already an odd
organizational gambit, the new subsection begins with a very
brief paragraph explaining why Pythagoras appears in acertain illustration. Immediately following, and seeminglyunrelated, we have this statement: ‘‘Caliph stories provide
the backdrops for so many anecdotal yarns that we sometimes forget that they are mostly the Western folk myths of an
exotic bygone civilization.’’ For a reader such as I, who was at
the moment of reading discovering the existence of caliph
stories, this is a puzzling way to introduce a new section. Indeed, only on the following page does one discoverthat the purpose of this subsection, which is less than three
pages long, is to provide some account of how mathematicalideas passed from the Greeks to the Indians to the Arabworld. I also found the chronology of ‘‘Numerals’’ particularly difficult to follow. To be fair, knowledge of such longago happenings is fragmentary, and the history itself is highly
nonlinear. There is also an effort made to address thechronological thicket by including a foldout timeline at theend of ‘‘Numerals.’’ But the appearance, for instance, of two
different illustrations of the morphological development ofmodern numerals (pp. 37 and 78), containing overlapping
dates but completely different content, does little to clarify
what happened when.
In ‘‘Algebra,’’ Mazur takes us from the algebraic notationof Euclid and Diophantus to that of Leibniz and Newton.
Although still subject to the pleasures and frustrations ofMazur’s informal style, I generally found this section moreenlightening and compelling than the previous one. It contains, for instance, a detailed description of the work of oneof history’s earliest known algebraic innovators, Diophantusof Alexandria; an insightful essay on Descartes and hisbreakthrough linking of geometry and algebra; a chapter inpraise of the notational acumen of Leibniz, many of whose
innovations persist to the present; and, immediately following, a chapter on Newton’s use and representation of theinfinitesimal.
Mazur’s writing is best described as conversational, withboth the positive and negative implications of that word in

THE MATHEMATICAL INTELLIGENCER

evidence. Reading the book feels like having a long chatwith a very knowledgeable friend who tells episodic stories, dispensing fascinating factual nuggets from his
impressive store of reading, and peppering them with his
own meditations. But his stories sometimes feel disjointedand meandering, and he is prone to the occasional
overindulgence. There can be no doubt that this conversational quality enhances the book’s readability, andsubstantially broadens its appeal beyond that of a scholarly
text. Mazur’s freewheeling style sometimes provides eruptions of beauty; witness the passage ‘‘We might think that
[Diophantus’s] notation must have hindered clear algebraic
thinking. Perhaps, but routine and familiarity are the tailwinds of conception’’ (p. 108). It also sometimes yields
cringeworthy moments, as when we are reminded that‘‘there are no tweets telling us what went on in the minds ofearly contributors to mathematics’’ (p. 80).
Finally, I would be remiss not to mention the book’s quite
marvelous cover illustration, by Marcella Engel Roberts. It
suggests a great wheel of life, decorated with diminishingconcentric circles of symbols, some familiar but most ancientand mysterious. It provides yet another reason to buy a copyof Mazur’s insightful, delightful, and frustrating book: eachrediscovery of the cover image from under a pile of papers or
tucked away on a shelf will give you the same frisson, thesame sudden sense of the oneness of math with the great
human effort to comprehend the world.
Department of Mathematics and Statistics
Carleton College
Northfield, MN 55057
USAe-mail: rfjones@carleton.edu

REFERENCES

F. Cajori, A History of Mathematical Notations, Open Court Publishing
Company, 1928, 1929.
S. Chrisomalis, Numerical Notation: A Comparative History, Cambridge University Press, 2010.
K. Menninger, Number Words and Number Symbols: A Cultural
History of Numbers, MIT Press, 1969.