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

by Joseph Mazur

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.

Genres: Mathematics, History, Nonfiction, Science

312 pages, Hardcover

First published January 1, 2014

About the author

Joseph C. Mazur is Professor Emeritus of Mathematics at Marlboro College in Vermont. He earned his Ph.D. in algebraic geometry from MIT and has held visiting positions at MIT and the University of Warwick. A recipient of Guggenheim, Bellagio, and Bogliasco Fellowships, he has written widely on the history and philosophy of mathematics, with books translated into over a dozen languages.

Reviews

[Eva Elasigue · Rating 5 out of 5]

This volume revolutionized my understanding of mathematical thought. Following the development of symbols walked me through our primal, unstructured mental evolution of grasping intangible concepts that nevertheless apply to everyday reality, through the human context which fostered their understanding. Connected me to the great abstractions of communication. An incredible way to encounter mathematics, revealing a deep level of global research and reference. For instance, I now credit Virahanka for the golden ratio number addition sequence! And I also credit Joseph Mazur for the insightful education presented here.

[William · Rating 3 out of 5]

This is literally a history of mathematical symbols and numbers.

[TG Lin · Rating 4 out of 5]

無論中西都一樣，在中世之前，「數學」都是以「文字敘述」的方式來表達的。比如像中國有《九章》、《大衍求一》等，我們雖然可以自詡領先歐洲，但那些都沒有成為我們後來所謂的「一般化」，都是用口訣與正常文句的描述。

讀了本書才知道，歐洲在十六世紀才開始為數學進行「符號化」的規範，加減乘除代數乘冪開方，逐漸成為我們今日所熟悉的形態。在這一連串的發展之後，「符號」成為數理方面的主流，雖然增加了學問的門檻，但這種「一般化」卻創造出了更多人類智慧的發明與累積。四百年前知識菁英為著某個數學問題所寫下滿滿數頁的篇幅，今天只需要中學生用幾行符號便能解開這種特化的問題（而且還能舉一反三回答更多同類問題）。書後幾章談論到符號對「認知心理學」與「思考」的關連，非常有趣。

[Lloyd Downey · Rating 5 out of 5]

Another book that turned out to be fairy heavy going. I thought it would be a relatively simple story of where our current mathematical symbols originated and how they came to be adopted and used. Well, yes. we do get a fairly exhaustive analysis of this (and where the use of zero came to be used for the first time......seems it was actually in China). But there is a fascinating section 3 in the book which investigates the role that the symbols themselves have come to play in our thinking and how the symbols enabled new ways of looking at the world. Or in many cases, things that did not appear to be OF this world.
I really liked the book and learned a lot. (Most of which I've immediately forgotten.....hence my attempt to recapture it in this review). And I've extracted a number of quotes direct from the book , below, to help me with this. As I said. There is a lot there. It is information dense.
"The ancient alphabets were not just collections of concrete linguistic elements with individual identities, but building blocks ripe for multiple meanings. Fifth century BC Greeks believed that everything in the world could be connected to whole numbers.
The number 2 (the letter “β”) meant opinion, 3 (“ γ”) harmony, and 4 (“ δ”) justice. Odd numbers were male; even numbers female. The number 5 (“ ε”) symbolized marriage.....So we begin to see that all sorts of metaphorical states of mind are emboldened by these ancient number systems.
This Mayan arithmetic system is pre-Columbian, and yet the notion of adding and carrying characters is similar on two continents that had no human contact for over 50,000 years. Similar to the Babylonian system, it made use of the placeholder utility of zero in a system of dots, bars, and columns.
From the beginning of the early Han dynasty in China, numerical characters were established looking very much like the numbers used today. We have to celebrate just how clever this is. From left to right, [the number 26,999] reads: 2 ten thousands, 6 thousands, 9 hundreds, 9 tens, and 9. It is, by all means, a decimal system. Why so clever? The zero is never needed, at least not as placeholder.......We really must admire the Chinese for this.
Directions for multiplication and division calculations in the Sun Zi Suan Jing were the same as those for Hindu-Arabic numbers in al-Khwārizmī’s book on arithmetic. The near identical descriptions for computations in the two systems have led some experts to believe that our Hindu-Arabic system may have been transmitted from China to India.
There is no clearly established, smoothly defined lineage from early scripts to modern ones......... All we really know is that somehow, in some time and place, the clever place-value idea was transmitted from the Indians to the Arabs and later to the Europeans.
For Babylonian, Greek, and Roman counting boards, number representation followed the laws of place-value. There were no symbols for zero; none were needed, for an empty column would indicate that a place rank was held with no numerical value. This idea led to the next: beads threaded on stretched wires—the more modern abacus....Between the tenth and twelfth centuries, the abacus board, with its vertical columns ranking the powers of ten was the main method of studying practical arithmetic in Western Europe.
Although it is likely that Fibonacci’s book (1202) had introduced Arabic numerals to some parts of European society, it is also likely that travellers and merchants in Italy already knew of it.
Our zero, as a number as well as a placeholder, probably appeared for the first time in book form sometime close to the year 628 AD
Khwārizmī, who was the greatest Arab mathematician of his day, learned of the new Indian numbers from the Arabic translation of Brahmagupta’s Brahmasphutasiddhanta, and wrote a textbook on arithmetic using the new Indian numbers.....around 820 AD, Its title in Arabic is Hisab aljabr w’almuqabala, from which we get the word “algebra.”.....Al-Khwārizmī’s On the Calculation with Hindu Numerals, written sometime near 825 AD, may have been responsible for spreading the Indian system of numeration throughout the Arab world.... A more likely story, however, is one about the Indian astronomer Kanka, who visited the House of Wisdom in Baghdad in 770 AD, and brought with him many manuscripts from India, including the Brahmasphutasiddhanta.
By the next century, however, many of the scripts had converged to a standard very close to the one we use today. ..It seems as if the news of the new numbers migrated all over Europe for two or three centuries before Fibonacci wrote his “Liber abbaci”. News, but not practice.
It is quite likely that sometime in the fifth century, Indian numerals had come to Alexandria via a trade route through Syria. [Much more likely that they were transmitted via the huge grain fleets of around 200 ships which travelled between Egypt and India each year for around 200 years in the early Roman Empire, trading slaves and entertainers as well as wheat]. From Alexandria, the numerals moved westward....The abacus had been giving merchants, astronomers, and mathematicians in the East an easy tool for making hard calculations......In one form or another, for almost five thousand years, it had been used as a reasonably efficient calculating tool. It spread westward in the tenth century.
Part 2 Algebra:.
We say, “Euclid proved that…,” meaning only that such-and-such a theorem can be found in the thirteen books of the Elements. More than likely, Euclid learned many of the theorems from others connected with Plato’s Academy, folks such as Eudoxus and Theaetetus...The Elements gave mathematics its fundamental nature, its first model of proof.
We do know that Euclid was active in Alexandria shortly after Alexander the Great founded the city in 331 BC.. By Diophantus’s lifetime in the second century, the city was still a marvel.
Symbolic modern mathematics, at its most rudimentary stage, can be traced back to Diophantus’s Arithmetica...maybe in the first century AD.....Diophantus did not have any symbol for “plus.”.....Diophantus clearly tells us..... Definition IX: Multiplying less than by less than produces more than. Multiplying more than by less than produces less than. And the minus is denoted by the letter cut short and turned upside down. Here we have the first evidence of the symbol for minus......It seems that the upside down ψ is the only Diophantus mark that may be a true symbol with no direct association with the written word. All other markings in Arithmetica seem to be abbreviations.
Close to a thousand years had passed between the time Diophantus wrote his Arithmetica and the time Matritensis 48 was written. ...We might say that Diophantus’s notation is terribly awkward and acutely difficult to process compared with what we have today, and be amazed that he could do any kind of mathematics. The Brahmins of northern India had some idea of algebra long before the Arabians learned it, contributed to it and brought that art to Spain in the late eleventh century.
For many centuries those mathematicians who worked their algebra rhetorically [that is, with words rather than symbols] were not seeing what we now see....The modern student knows that all that has to be done is to translate the problem into symbolic notation, and let the rules of symbolic manipulation take it from there. The initial incentive was the need to abbreviate, but once the equal symbol was in place, something else took over. The concise character of the symbol came with an unintended benefit: it enabled an unadorned picture in the brain that could facilitate comprehension.
Early historians had credited the Arab algebraist al-Qalasādi as being the first Arab to use letters of the Arabic alphabet to denote arithmetic operations. He was born in Bastah, a Moorish city [now known as Baza] in what is now northeast Spain. Al-Qalasādi wrote several books on arithmetic and one on algebra that had mathematical notation made from shortened Arabic words and letters. We should also be aware that they [the symbols] were already used by North African Muslim mathematicians for at least a century: he was not the originator.
By the end of the fifteenth century, algebra went no further than quadratic equations with just one unknown, a level close to the syllabus of any present-day high school course. Back then, the art was still being performed in words; there were still no signs or symbols for the unknown.
Ideas for substitutions are now so elemental to the modern algebra choreography of reducing one problem to a simpler one that we must marvel at its formidable brilliance and wonder how such a work of genius could possibly be done without the use of symbols.
By 1545, the Italians—in particular, Gerolamo Cardano, his student Lodovico Ferrari, and his rival Niccolò Fontana Tartaglia—had solved general cases of cubic and quartic equations.
Cardano’s Ars Magna, was published in that year. It contained everything that was known about cubic and quartic equations up to that time. The Ars Magna was a real breakthrough for algebra. Though it didn’t have the symbols that were soon to be invented. Geometry in those days tended to be a subject of visual logic; one had to see a drawing, at least in the mind’s eye, in order to approve of the inherent logic.......It had been that way ever since the time of the Pythagoreans, Euclid, Apollonius, and Archimedes......So the Ars Magna was a struggle to write and a struggle to comprehend.
The mistakes of writing √-2√-3 = √6 and √-1x√-4 = 2 that Euler would make two hundred years later would have been avoided with notation that would mark the first product as (i√2)(i√3) to get -√6, and the second product as i(i√4) to get − 2.
Harriot had the ingenious idea of first setting the polynomial equal to zero, thereby setting up an equation, a polynomial equation, and asking for the numbers that satisfy the equation.....This ingenious idea was a game-changer—the problem of finding the roots of polynomials quickly became the problem of factoring polynomials......Even more significant: the idea gave mathematicians a sense of proper algebraic form. By putting all terms on the left of an equation and leaving an isolated zero on the right, the form comes to the forefront.
Algebra was not just about equations, but also about forms.....And finally, in 1637 René Descartes had the idea of using numerical superscripts to mark positive integral exponents of a polynomial in his La Géométrie.
Viète used vowels to represent unknowns, and consonants to represent knowns. That convention does two things: it avoids confusion between representations of knowns and unknowns and—more importantly—permits us to have multiple unknowns.
Our more sophisticated concept of number now accepts the square root of a negative number into the family. That enrichment gave us the fundamental theorem of algebra, which tells us that any polynomial of any degree n ≥ 1 always has n roots that may or may not be distinct.
René Descartes’s Geometria had a new idea for notation, a rule: beginning letters of the alphabet were to be reserved for fixed known quantities and latter letters (past p) were to represent variables or unknowns that could take on a succession of values.....To this day, this division of the alphabet at p remains the loose standard rule.....The intimate link between geometry and algebra had been suspected since Plato’s time...In the third century BC, Apollonius of Perga investigated curves that could be produced by cutting a cone by a plane......Geometry had its origins in the interest of working with lines, figures, and solids that could be imagined in the mind. Algebra had its origins in problems involving number.
Descartes (and Fermat too) taught us that we have optional modes for conceptualizing problems.....Descartes gave us a way to switch between modes of conceptualizing, to translate geometric problems to an algebraic coordinate system..The unity of geometry and algebra was one of the greatest discoveries. ....Space and time were linked, not only through indefinite, unreliable geometric pictures caught by the spirit of intuition, but also through algebra.
Different people were responsible for introducing different symbols; Oughtred, Harriot, Descartes, Leibniz. The function concept would go through many revisions before 1837, when it settled for Gustave-Peter Lejeune Dirichlet’s brilliant definition that we now use everywhere in mathematics: “y is a function of x, if for every value of x there corresponds a unique value of y.”
Part 3 The Power of Symbols
Take the equation x2 + y2 = xy + 4. Hmm… if only that term xy were not there, we would have a circle of radius 2, which is governed by the simple equation x2 + y2 = 4. But the term xy is there, so how does it change the circle? It entwines the two variables x and y in such a way that they cannot be separated without some transformation to simplify the equation. However, the symmetry of x and y in the original equation gives a clue to the geometry of the curve. If you swap x and y, you get exactly the same equation. Aha! That can only mean that the curve is symmetric with respect to the line y = x. Indeed, by rotating the axes clockwise through a 45-degree angle, and labelling the new axes s and t, the equation magically becomes 3s2 + t2 = 8. This new form has no st term; s and t are not entwined together by multiplication. Graph this nice equation in s and t coordinates, and the picture is an ellipse centered at (0, 0) and symmetric with respect to the s and t axes. Just as the symmetric form of the equation x2 + y2 = r2 cries out CIRCLE! CIRCLE!, so too does the term xy, a multiplicative adhesion of x to y, immediately scream to the left hemisphere of the cerebral cortex, rotation! rotation! That 45-degree rotation disentangles the variables x and y by making the xy term disappear.
There is something in the cumulative consciousness of civilization that begs for a mental picture of numbers, even if that picture is fuzzy. But to visualize complex numbers requires something more inventive......We now have i, a number that is an action: the act of a rotation of 90 degrees.
The “quaternions,” as the nineteenth-century Irish mathematician William Rowand Hamilton called them, belong to a new number system in four dimensions that contain the complex numbers and a multiplication system that obey all the laws of algebra, except the commutative law. (That the order of numbers doesn’t matter for addition and multiplication).
It is not the job of mathematics to stick with earthly relevance. Yet the world seems to eventually pick up on mathematics abstractions and generalizations and apply them to something relevant to Earth’s existence......From the symbol i that once stood for that one-time peculiar abhorrence √-1, there emerged a new notion: that magnitude, direction, rotation may be embodied in the symbol itself. It is as if symbols have some intelligence of their own.
Images are primal. Written words and mathematical symbols are invented.....However, the moment we read an equation—simple or complicated—images form in the mind along with verbal reflections that suggest multiple metaphorical connections and associations with what had been seen before.
As Poincaré would have put it. He wrote: Among our students …some prefer to treat their problems “by analysis,” others “by geometry.” The first are incapable of “seeing in space,” the others are quickly tired of long calculations and become perplexed.
Allison and McCarthy in 1994 pinpointed two neighbouring regions of the visual processing area of the brain: one reacted to words exclusively, the second only to Indian numerals, and a third only to faces.
The question of difference splits into two competing questions raised by Dehaene: Does our comprehension of mathematical expressions come from our capacity for processing language structures? Or is it language-independent, relying on some visual system for parsing strings of mathematical symbols?
Like any great poem, Maxwell’s equations tell us far more than what appears in the language. They form the basis for all electrodynamics and optics, and even lead to creative thinking about relativity and quantum mechanics.
Though mathematics tends to use a symbolic language that bundles complexities of verbiage to simplify communication, it also draws on an astonishingly quick mental process that unpacks the essentials for making sense. And, like poetry, it uses a linguistic structure that enables readers to know the hidden meanings and the verbally unimaginable.
So what's my overall take on the book. I really liked it. Learned a lot and it's certainly made me think about the actual role that the symbols themselves have come to play in our thinking. My overall take on the origin of the symbols is that most of what we use currently came to Europe via the arabs who derived it from India and there is some evidence that the Indians adopted a lot from the Chinese. An easy five stars from me.

[Chik67 · Rating 1 out of 5]

Un'occasione un po' persa per strada. Questo libro è anzitutto sgraziato. La sua mancanza di grazia è principalmente conseguenza delle sbagliate proporzioni. La lunga, esageratamente lunga, prima parte, sulla storia dei numeri, della notazione posizionale and the like è dettagliata ma non profonda, e soprattutto non originale. Forse, scrivendo, si è accorto che di libri che filosofeggiano (ahimé, abitualmente in maniera superificiale) sulla decisiva introduzione dello zero come cifra ne esistono molti altri e dunque il testo, da questo punto di vista, aggiungeva poco e andava fatto virare in altra direzione. (Sia detto per inciso: dice almeno trenta volte che l'introduzione dello zero tra le cifre sia stata cosa buona e giusta. Ma quanto a convincere... mmmhhh....).
Poi scarrella lungo la matematica dal medioevo all'Ottocento per dirci chi ha introdotto che cosa in un elenco superficiale benché erudito. Son contento di sapere chi per primo ha introdotto il simbolo di potenza, per carità. Ma c'è ben di più da raccontare in questa storia, qui doveva essere il succo del libro, e quando vai a spremerlo, purtroppo, resti come con un arancia fuori stagione. Secca, vizza, asciutta.
Poi, dal nulla, le ultime pagine sono dense di riflessioni sul rapporto tra simbolo e sua importanza nei processi mentali, esperimenti di psicologia cognitiva. E qui il libro si fa semplicemente poco meditato. Una spruzzata di idee e nozioni senza nessuna tesi chiara. E' il regno in -ebbe. Potrebbe, sarebbe, farebbe. Ma se non c'è conclusione non c'è racconto.
Se a questo si aggiunge che la scrittura non sia proprio da catalogare tra quelle scorrevoli si capisce che, alla fine, in mano resta poco.

[James Morrison · Rating 3 out of 5]

I don't have an equation for star ratings. I loved parts of this book and other parts not so much.

In the seventh grade, about 68 years ago, Mrs. Evans introduced us to algebra by saying that in arithmetic numbers are represented by figures while in algebra we use letters as well as numbers to represent numbers, these letters representing any number whatsoever. She used the simplest of equations as examples and I thought I understood algebraic expressions from that time through all my math classes even in college. I accepted everything Mrs. Evans without question. After reading Mazur's book I now understand some of the lengthily and convoluted history that lead to our representing mathematical ideas in symbolic form.
I don't recommend this book for anyone who doesn't already have an interest in the history of mathematics. The text seems well researched and he goes into details I certainly have not read about elsewhere. Some of it very interesting to me. Perhaps he was more interested in being through and accurate than entertaining and that is fine, but personally it could have moved a bit faster in places.

[Stella · Rating 4 out of 5]

There was a discussion between me and my sister on “why she hates arithmetic”. For her, it is ‘forced’ system that imprinted into human brain to accept that 1 + 1 = 2. “Why is it?” “Why plus is +? Why not £ or ! Or # or anything else?”. I tried to explain her with little source I’ve read, that back in the past, a plus isn’t a +. Even quadratic function was used to be written as XX + 2X + 1 instead of using uppercase we know it now. But I was lack of evidences to support this, that my sister still wasn’t impressed.

Enlightening Symbols is what I need to explain my sister. It is not only explaining about the origin of mathematics symbols, but show how civilization contributes to these symbols. From counting on quantity, which contributes on numbering system. (Why 26 read as 20 (twenty) + 6 (six)) to introducing South Asian numerical to Europe and finally how this symbol works in our brain.

This book is very easy to follow and very structural. Really highly recommended…for my sister.

[Joseph Carrabis · Rating 1 out of 5]

I'm not sure what I expected from this book and am sure I didn't get it. Was this book a history of mathematics?
Not quite.
A history of how mathematical thinking changed over time?
Not really.
How about a history of how mathematical symbology and problem solving co-evolved?
Well...closer.
Okay, how about a history of how symbols make problem solving easier?
Umm...
How the use of symbols frees minds from thinking of physical space?
Okay, now you're really reaching (no pun intended).
Evidently Mazur also writes fiction. Haven't read any of it. Not likely to, based on this book.

[Maşuk · Rating 4 out of 5]

A very good book which does justice do its name. It could, however, have been a bit shorter. Still, if you are looking for detailed information about the history of mathematical symbols, this is a good book for that job.

[Ann · Rating 3 out of 5]

Reasonably interesting book about the origin of various symbols used in mathematics from developmnet of numbers to variables, calculus notation, etc. Much of it is pretty dry. The final third of the book is appendices, etc.

[Brett T · Rating 4 out of 5]

Given two objects, add another two identical objects to them, and the result will be four objects. That's not much harder to say or to grasp than "2+2=4," but imagine trying to express a quadratic equation without being able to use mathematical symbols. My junior high report card shows I had enough trouble trying to do it with the symbols; keeping the different unknowns raised to different powers straight without them has just caused a serious panic attack amongst my neurons.

But math didn't come pre-equipped with symbols. They had to be developed. For that matter, so did numbers. Our remotest ancestors who wrote things down probably just made marks in the amount of whatever number they wanted to represent. We had to develop the idea of writing "5" or some other symbol to represent what we had been expressing with something like |||||.

Joseph Mazur's Enlightening Symbols is a fun romp through the development of these symbols and ideas through history as we gradually collapse complicated ideas into simple symbols. He begins with the development of numerals, including the idea of the zero, and continues with how we came to possess plus signs, equals, minuses, square roots, exponents and so on. Obviously the earlier stages are fuzzier, as they happened much longer ago and are represented in few still existing records.

But as we enter the Renaissance, we see different mathematicians develop individual pieces of the puzzle -- sometimes two versions of the same piece, and Mazur quickly sketches how the eventual winner came to dominate. In some cases, new symbols are probably still appearing as math addresses more and more complex areas and requires new ways to talk about them.

Mazur mostly leaves out the truly head-bonking stuff as he takes his quick trip through math history and writes about his subject with a light and fun tone. He includes enough examples of math statements made without symbols to get his point across and sometimes feels a bit repetitive in doing so. But overall Enlightening Symbols is an excellent look at how essential the development of what we call math was to the advancement of society and technology, even of areas that seem to have little to do with math directly.

Original available here.

[Charles · Rating 5 out of 5]

I have a standard opening speech that I give in all of the math classes that I teach and one component of that monologue is the role of notation. I point out that one of the things that make mathematics difficult to understand is the fact that the symbols are compact representations of operations. I also point out that this compact representation is preferable over having every problem expressed as a “story problem”, the problem form that is most widely feared by students.
What I found most enlightening about this book is the clear conclusion that the mathematical education of the masses would not be possible if the compact and efficient notation was not being used. Trying to decipher the rhetorical forms of the problems was really hard, even after reading the explanations. Furthermore, some of the examples where a long paragraph of text is translated to a simple equality are one of the most dramatic alterations you can find in mathematics. Even if you allow for the greater understanding of the rhetorical form based on extensive practice, it is hard to see how large numbers of students could solve them. Finally, it is also hard to see how mathematics could advance as fast as it has using only rhetorical forms or non-standard notation.
This book is written in a popular style, there are a few equations, but nothing that a reader with a background in basic algebra can’t handle. It also has the interesting characteristic that the sections that the non-professional and professional mathematicians will have difficulty understanding coincide. In some ways modern mathematical notation has a short history and we should all be grateful for what it has done for the world. This book will make you appreciate that.

Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon

[Maurizio Codogno · Rating 2 out of 5]

Secondo me il titolo scelto nella traduzione italiana di questo libro è fuorviante. Sì, qualcosa sulla storia dei simboli matematici si trova, anche se ad esempio Più per meno diviso di Peppe Liberti ne ha di più. Ma quello di cui Mazur vuole parlare è in realtà della filosofia dei simboli matematici, o per meglio dire di come l'uso di simboli scelti in modo opportuno possa illuminare ("enlighten", come da titolo originale) la comprensione dei concetti matematici. Delle tre parti in cui il libro è composto, la più riuscita è la seconda, sull'era moderna: la prima, con la nascita delle cifre e dei sistemi posizionali, mi è parsa confusa mentre la terza sulla fisiologia del cervello relativa alla comprensione dei simboli, è forse un po' fuori posto. Più preoccupanti sono le numerose ripetizioni nel testo, e in qualche caso - come nel caso dell'umbro Livero de l'abbecho - anche contraddizioni. L'impressione che ho avuto è che Mazur abbia scritto il testo a spizzichi e bocconi nel corso di vari anni e non abbia poi provveduto a rileggerlo e asciugarlo, cosa che avrebbe favorito la lettura. La traduzione di Paolo Bartesaghi è scorrevole, anche se mi ha lasciato perplesso vedere all'inizio che il Webster da un dizionario è diventato una persona; ci sono infine parecchi refusi, alcuni dei quali forse già nell'originale, a giudicare dai ringraziamenti di Mazur al suo traduttore in ceco.

[Jim · Rating 2 out of 5]

Middle section on the rise of algebraic symbols was fascinating. The first section on the origin of our number system was interesting but dragged - probably too long. Last section was odd speculation.

[Jeremy Sawruk · Rating 4 out of 5]

Discusses development of numeric and algebraic notation, but does not cover much material after that.

[Mills College Library]

510.148 M4768 2014

[JJ · Rating 5 out of 5]

This book is not for the faint of heart. Mazur presents a well organized and in depth view on mathematical symbols, their origins, and cognitive implications. Though not strictly mathematical, reading it requires some mathematical background but mostly just paying close attention. Recommended for the mathematics aficionado.