Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers by Joseph Mazur (PDF)

Description

An entertaining look at the origins of mathematical symbolsWhile 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.

User’s Reviews

Editorial Reviews: Review “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.” ― 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”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”Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history.” ― Library Journal”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”Symbols like ‘+’ and ‘=’ are so ingrained that it’s hard to conceive of math without them. But a new book, Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Power, offers a surprising reminder: Until the early 16th century, math contained no symbols at all.”—Kevin Hartnett, Boston Globe”Enlightening Symbols retraces the winding road that has led to the way we now teach, study, and conceive mathematics. . . . Thanks to Mazur’s playful approach to the subject, Enlightening Symbols offers an enjoyable read.”—Gaia Donati, Science”If you enjoy reading about history, languages and science, then you’ll enjoy this book. . . . The best part is the writing is compelling enough that you don’t have to be a mathematician to enjoy this informative book.”—Guardian.com’s, GrrlScientist”[I]nformative, highly readable and scholarly.”—Brian Rotman, Literary Review”[T]his insightful account of the historical development of a highly characteristic feature of the mathematical enterprise also represents a valuable contribution to our understanding of the nature of mathematics.”—Eduard Glas, Mathematical Reviews Clippings”Joseph Mazur’s beautiful book Enlightening Symbols tells the story of human civilization through the development of mathematical notation. Surprises abound. . . . The book is visually exquisite, great care having been taken with illustrations and figures. Mazur’s discussion of the emergence of particular symbols affords the reader an overview of the often difficult primary literature.”—Donal O’Shea, Sarasota Herald-Tribune”At whatever depth one chooses to read it, Enlightening Symbols has something for everyone. It is entertaining and eclectic, and Mazur’s personal and easy style helps connect us with those who led the long and winding search for the best ways to quantify and analyze our world. Their success has liberated us from ‘the shackles of our physical impressions of space’–and of the particular and the concrete–‘enabling imagination to wander far beyond the tangible world we live in, and into the marvels of generality.'”—Robyn Arianrhod, Notices of the Notices of the American Mathematical Society”Mazur introduces the reader to major characters, weaves in relevant aspects of wider culture and gives a feel for the breadth of mathematical history. It is a useful book for both student and interested layperson alike.”—Mark McCartney, London Mathematical Society”[T]his is a good book. It is well written by an experienced author and is full of interesting facts about how the symbols used in mathematics have arisen. It would certainly interest anyone who studies the history of mathematics.”—Phil Dyke, Leonardo”Mazur is a master story teller.”—John Stillwell, Bulletin of the American Mathematical Society Review “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 From the Back Cover “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 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). Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐We’ve all been disappointed when we got a Kindle title (or any e reader) and the MathML or LaTex was slaughtered. Since this book has numerous symbols but not a lot of complex formulas, Kindle is not a problem. Where there are “sentences” in math symbols, they are handled well.That said, the hard cover is a treasure. This is a very well produced historical survey that should be in every library, including yours if you’re into history, math or even language. I personally find that I understand the subtleties of (English) language better if I understand the etymology of a word. This is equally true albeit in a slightly different way with the fine treatment of symbolic evolution in this gem of a book.Understanding how a summa symbol came to be won’t tell you how to use it in combinatorics or a power series, or in double form in a logistic regression error term. However, the good news is we don’t have to understand that to really enjoy this book. Flip side, if you ARE into math, you’ll REALLY love this. I like books that don’t just parrot formulas but give an idea of the “intuitive” meaning or reason for the formula, like “putting these numbers in a matrix makes it easier for a computer to change coordinate axes which is one reason we do it.”This text takes that a step further, and not only gives context, but HISTORICAL context in genuinely useful human endeavors– the WHY and how of the underlying value as use evolved. This turns on a LOT of light bulbs if you’re into math, and is still fun and a great “detective” read even if you’re not. Which is my final point: don’t get turned off due to the math side if you love history and detective (deductive) logic– the writing is crisp, well paced and “leads” you quickly forward– IOW NOT DULL or dry. And frankly, as I look at other math books, the symbols also seem a little more friendly and familiar, and less intimidating, knowing about the neighborhood in which they grew up. Would also make a great gift for a detail oriented picky person in your life!

⭐First off, I loved this book and highly recommend it. With respect to readability, this was much easier for me to read than

⭐. We used

⭐for a mathematics history course when I was an undergrad and to me, Mazur’s book was similar to that in readability, but with a focus on symbols of mathematics.As a math educator, I especially loved “Sans Symbols” which introduces the development of symbolic algebra. The start of the chapter has an exceptionally interesting account of the author reading in person the oldest surviving manuscript of Euclid’s Elements.I read many parts of this book aloud to my wife. She listened and appreciated those excerpts. This speaks to the quality of writing, and I cannot say this of very many mathematics books I’ve read.

⭐Somebody should mention that this book is about the symbols used to represent mathematical numbers, i.e. 1,2,3. The mathematical operators are not covered beyond addition. Thus most of the book is about the history of the addition process. While the history is interesting and fascinating, I was hoping for a history of the semiotics used for the operators in math at a higher level than the third grade. That aside, I would recommend the book if for no other reason than it helps examine the role of symbolic representation in expressing elementary logic. The symbols both enable and limit the logic. The efforts used to overcome those limitations are important milestones in technological progression at any level.

⭐The book deals with a subject that lends itself to an orderly presentation. The author clearly knows his subject, but the book suffers from three drawbacks: 1. There are occasional lapses into very colloquial usage, not fit for academic writing; 2. The non-English characters are misrepresented (for example, the Hebrew sequence for the number 9686 in the third chapter is ordered from left to right, although Hebrew, and Hebrew characters used as numbers, are ordered from right to left; the Greek word for ten is Deka and not Peka, and the symbol for that number must thus be delta and not pi); 3. The author often strays from the subject, and especially in the last two chapters wanders in the fields of amateur psychology and his own imagination rather than in the history of the symbols.In short, this is an interesting book, in need of a sympathetic but firm editorial hand.

⭐This is the highest rating, because this book sheds light on one of the toughest philosophical questions. The question is what accounts for the “Unreasonable Effectiveness of Mathematics in the Natural Sciences”? Mazur via his entertaining history that spans many places, times and cultures, explains that the mathematical symbols we commonly use did not come down from the mountain written on a tablet, but evolved with a lot of passion, genius, and serendipity. Our present mathematics have been pushed and cajoled into shape via symbols. This book may not be a complete explanation as to the effectiveness of mathematics, but it is an enlightening peek under the curtain. If you like history and have a mathematical bent (no need to be a mathematician, and mathematicians would like it) you would enjoy this book.

⭐A very interesting and informative book, but I could have done without his rather speculative, semi-mystical interpretations of the effect that the use of symbols might have on our thinking.The print is rather small, and though the hardback is nicely put together, the inner margins are also rather small so that the book has to be opened out more than is comfortable.

⭐I always find Joseph Mazur engaging, and this is no exception, It is fascinating to see how our present way of writing mathematics evolved — and one has to admire the mathematicians who produced important results when the tools they used were still so primitive. There were one or two places where I thought proof-reading could have been better — for example in the discussion of the general quadratic equation we have one expression and two equations, but all are described as equations (at least, that’s what I see in the Kindle version). Well worth reading, though.

⭐Not quite what I was expecting, I thought it would give historical information about the first appearance of modern mathematical symbols such as +, = , sigma etc. Interesting nevertheless.

⭐Arrived on time and as described. Thanks!

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