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

311 Pages·2014·29.22 MB·English

by  Joseph Mazur

“Mazur” — // — : — page  — #
Enlightening Symbols
“Mazur” — // — : — page  — #
Also by Joseph Mazur
EuclidintheRainforest:DiscoveringUniversalTruth
inLogicandMath(2006)
Zeno’sParadox:UnravelingtheAncientMystery
behindtheScienceofSpaceandTime(2007)
What’sLuckGottoDowithIt?TheHistory,Mathematics,
andPsychologyoftheGambler’sIllusion(2010)
Edited
Number:TheLanguageofScience(2007)
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Enlightening Symbols
A Short History of Mathematical Notation
and Its Hidden Powers
Joseph Mazur
PrincetonUniversityPress
PrincetonandOxford
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Mazur,Joseph.
Enlighteningsymbols:ashorthistoryofmathematicalnotationanditshidden
powers/JosephMazur.
pagescm
Includesbibliographicalreferencesandindex.
ISBN----(hardcover:alk.paper).Mathematical
notation—History.I.Title.
QA.M
.(cid:156)—dc

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To my big brother, Barry,
who taught me from 0
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Contents
Introduction ix
Definitions xxi
NoteontheIllustrations xxiii
Part 1 Numerals 1
. CuriousBeginnings 
. CertainAncientNumberSystems 
. SilkandRoyalRoads 
. TheIndianGift 
. ArrivalinEurope 
. TheArabGift 
. LiberAbbaci 
. RefutingOrigins 
Part 2 Algebra 81
. SansSymbols 
. Diophantus’sArithmetica 
. TheGreatArt 
. SymbolInfancy 
. TheTimidSymbol 
. HierarchiesofDignity 
Contents vii
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. VowelsandConsonants 
. TheExplosion 
. ACatalogueofSymbols 
. TheSymbolMaster 
. TheLastoftheMagicians 
Part 3 The Power of Symbols 177
. RendezvousintheMind 
. TheGoodSymbol 
. InvisibleGorillas 
. MentalPictures 
. Conclusion 
AppendixALeibniz’sNotation 
AppendixBNewton’sFluxionofxn 
AppendixCExperiment 
AppendixDVisualizingComplexNumbers 
AppendixEQuaternions 
Acknowledgments 
Notes 
Index 
viii Contents
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Introduction
Amathematician,amusician,andapsychologistwalkedintoabar...
Severalyearsago,beforeIhadanythoughtsofwritingabookonthehistoryof
symbols,IhadaconversationwithafewcolleaguesattheCavaTuracciolo,alittle
winebar inthe villageof Bellagioon LakeComo. Thepsychologist declaredthat
symbolshadbeenaroundlongbeforehumanshadaverballanguage,andthatthey
areattherootsofthemostbasicandprimitivethoughts.Themusicianpointedout
thatmodernmusicalnotationismostlyattributedtooneBenedictinemonkGuido
d’Arezzo,wholivedattheturnofthefirstmillennium,butthatamoreprimitive
formofsymbolnotationgoesalmostasfarbackasPhoenicianwriting.I,themath-
ematician,astonishedmyfriendsbyrevealingthat,otherthannumerals,mathemat-
icalsymbols—evenalgebraicequations—arerelativelyrecentcreations,andthatal-
most all mathematical expressions were rhetorical before the end of the fifteenth
century.
“What?!”thepsychologistsnapped.“Whataboutmultiplication?Youmeanto
tellusthattherewasnosymbolfor‘times’?”
“Notbeforethesixteenth...maybeevenseventeenthcentury.”
“Andequality?Whatabout‘equals’?themusicianasked.
“Notbefore...oh...thesixteenthcentury.”
“ButsurelyEuclidmusthavehadasymbolforaddition,”saidthepsychologist.
“WhataboutthePythagoreantheorem,thatthingaboutaddingthesquaresofthe
sidesofarighttriangle?”
Introduction ix