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A Brief History of Mathematics for Curious Minds This page intentionally left blank
A Brief History of Mathematics for Curious Minds Krzysztof R. Apt Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands
MIMUW, University of Warsaw, Poland
Published by
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Library of Congress Cataloging-in-Publication Data
Names: Apt, Krzysztof R., 1949– author.
Title: A brief history of mathematics for curious minds / Krzysztof R. Apt, Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands, MIMUW, University of Warsaw, Poland.
Description: New Jersey : World Scientific, [2024] | Includes bibliographical references and index.
Identifiers: LCCN 2023034145 | ISBN 9789811280443 (hardcover) | ISBN 9789811281495 (paperback) | ISBN 9789811280450 (ebook for institutions) | ISBN 9789811280467 (ebook for individuals)
Subjects: LCSH: Mathematics--History--Popular works.
Classification: LCC QA21 .A65 2024 | DDC 510.9--dc23/eng/20231016
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Preface Mathematicsispartofourculturalheritage,justashowphilosophy,art,history,andgreatworksofliteratureare.Yet,evenwithalampinbroaddaylight, onecanhardlyfindanintellectualwhowouldbeabletosayanythingmeaningfulabouttheachievementsofmathematicians,letaloneawell-educated person.Mosthistorytextbooksdevotehardlyanyattentiontomathematics andmathematicians.1
Iwouldliketoofferhereashortandaccessibleaccountofthehistoryof mathematics,writtenforanintelligentlayman,hopingthatitwillallowhimor hertobetterappreciateitsbeauty,relevance,andplaceinhistory.And,asyou willsee,thecastofcharactersisatleastasfascinatingasthatofphilosophers orwriters.
Itisnaturaltoassumethatmathematicsoriginatedbecauseoftheneedfor counting.Moreambitioustasks,suchasthedeterminationofdistancesand areas,andalsooftaxes,required calculating.Soonafterorperhapsatthesame time,mathematicswasusedtodealwithmorechallengingproblemsposedby astronomyandphysics.DuringtheRenaissance,mathematicswasalready appliedinoptics,mechanics,geography,cartography,andeveninpainting (intheconceptofperspective).Astimewenton,moreandmoreusesfor mathematicswerefound.Forinstance,asearlyas1762,insurancecompanies startedtousemathematicsindeterminingtheirpremiums.2
Sincethen,mathematicshascontributedtoprogressinalmostallsciences,includingpsychology(statisticalanalysis),sociology(networkanalysis), medicine(e.g.,geometryandcalculusareusedinCTscantechnologyandin virology),chemistry(e.g.,graphtheory),crystallography(algebra),economics (e.g.,calculusandgametheory),politicalscience(votingtheory),meteorology (dynamicalsystems),biology(e.g.,calculus),computerscience(e.g.,mathematicallogic,combinatorics),linguistics(e.g.,formalgrammars),geosciences
(Fourieranalysisandwaveletanalysis),andevenhistory(radiocarbondating useslogarithms).SomemathematicianswonfortheirworksNobelPrizesin physics,economics,andmedicine.
Therearetwocompetingwaysoftellingthestoryofmathematics.Oneis throughachronologicalaccount,introducingmathematicianswhosuccessively enterthehalloffame.Theotheristodiscusstheareasofmathematicsand thenotionstheyfocuson.Ideally,toreconcilebothapproachesabookshould nothavethecustomarylinearformbutinsteadconsistofagridformedby theentries(mathematician/area)combiningbothapproaches.Butthen,the outcomewouldbeahandbookinsteadofabook.
Ichosethechronologicalapproachhere,though,inmyenthusiasm,Ioccasionallyjumpforward,pursuingtheimpactofanidea.Indoingso,Imay beguiltyoftoomanydigressions,buthopefully,theywillallowthereaderto appreciatetheboldnessofsomeoftheearlyachievements.
Itisnoteasytoprovideanaccessibleaccountofthehistoryofmathematics, foratleasttworeasons.Oneisthatmathematicsisavastsubjectthatis continuouslygrowing.Thismakesitdifficulttowriteaboutitinacompetent way.Theotheristhatstartingfromthe16thcentury,weenteraperiodin whichmathematiciansintroducemoreandmoretechnicalconcepts.Injargon usedbymathematicians,‘technical’usuallymeans‘difficulttoexplain’.
Suchconcepts,tomentionafew,includefunctions,relations,operations, real(numbers),complex(numbers),groups,rings,fields,filters,kernels,lattices,grids,spaces,modules,orders,matrices,faces,graphs,trees,interpretations,transformations,andmodels.Alltheseconcepts,justlikeadjectives, e.g.,ideal,rational,irrational,open,closed,dense,sparse,cardinal,anddiscrete,havecompletelydifferentmeaningsoutsideofmathematics.Inparticular,themathematicalconceptofcommutinghasnothingtodowithtraveling andtreesinmathematicsmaybeinfiniteandusuallygrowdownwards.
Otherconceptssuchasline,induction,dimension,vector,probability,tautology,continuous(change),orlimitseemtomeanthesamebothwithinand outsideofmathematics,butmathematicsoffersprecisedefinitionsofthem, which,incidentally,haveoftenbeenbornoutofcenturiesoflongstruggles, discussions,andoccasionalanimosities.
Acaseapartistheconceptofinfinity,whichhasbeentroublingmathematicians(andphilosophers)sincethetimeoftheGreeks.In1784,theBerlin Academyofferedaprizefora“clearandprecisetheoryofwhatistheinfinitein mathematics”.Itwasnotsatisfiedwithanyoftheentries.3 Theproblemsassociatedwithinfinitycontinuedtobothermathematiciansofthe19thcentury andpromptedthemtosubstantiallyredefinethebasicnotionsofanimportant
Preface
branchofmathematics.Yet,in1925,DavidHilbert,aGermanmathematician, wrote:“Ifwepaycloseattention,wefindthattheliteratureofmathematicsis repletewithabsurditiesandinanities,whichcanusuallybeblamedontheinfinite.”Hefurtherwrote(italicsintheoriginal):“[ ] thedefinitiveclarification ofthe natureoftheinfinite hasbecomenecessary,notmerelyforthespecial interestsoftheindividualsciencesbutforthe honorofthehumanunderstandingitself.”4 Apparently,thishonorhasnotbeensavedyet—in2013,Scientific Americanrananarticle Disputeoverinfinitydividesmathematicians thatdiscussestwocompetingtheoriesofinfinitesetsoverwhichlogicianscurrently debate.5
Yetanotherdifficultyinpresentingthehistoryofmathematicstoaneducatedlaymanisthatthedrivingforcebehindmathematicsis generalization Overcenturies,mathematicianshavegeneralizedalmostallpossibleconcepts totheextenttheyevenoccasionallyquestiontheirusefulness.
Thesimplestexampleofageneralizationistheconceptofa number TheBabyloniansacceptednumberssuchas 1, 2, 3 ,aswellas unitfractions (forexample, 1 6 ).TheGreeksstudiedspecialnumbers,called prime numbers andalsobegrudginglyagreedtoadmit irrationalnumbers,suchas √2.TheChineseandIndiansintroducednegativenumbers,theusualfractions,andenricheduswiththenumberzero.Thiswayweobtained integers ..., 3, 2, 1, 0, 1, 2, 3,...,—andrealnumbers,(reals,forshort),e.g., 5 1 3 , √3, 0, 1 1 2 ,or √2
However,inthe16thcentury,outoftheworkdoneonsolvingthird-degree equations(forinstance, x3 15x 4=0),thereemerged complexnumbers,a conceptthatgoesbeyondsecondaryschoolmathematics.Thenwealsohave transcendentalnumbers, p-adicnumbers,and algebraicnumbers.Totopitoff GeorgCantorintroduced infinitenumbers (called cardinals,ashorthandfora cardinalnumber)inthelate19thcentury.Sointheend,theword‘number’ isalittlebitambiguousinmathematics,tosaytheleast.6
Inwhatfollowsbya number,Ishallusuallymean1,2,3,etc.;so,whatis calledinmathematicalparlancea naturalnumber.Anexcellentwaytolosethe readeristodelveintothenotionthatanaturalnumbershouldnotbetaken forgranted.SoofcourseIshallnotdothat.
Anyinformalaccountofthehistoryofmathematicsiseasilypronetocriticism.Mathematicianswillcomplainthatseveralimportantareasofmathematicsandmathematiciansarenotmentioned,whilea‘generalreader’willsoonbe lostifoneassumesfromhimorherknowledgeofmathematicsatthesecondary schoollevel.Tosolvethisdilemmaandtokeeptheaccountinformal,Imoved alltechnicalcommentsandreferencestofootnotes.Further, 32 appendices
x ABriefHistoryofMathematicsforCuriousMinds
providemoredetailedinformationandselectedproofs,somelessknownbutall atanelementarylevel.Finally,attheend,Ilistsomenovelsandfilmsabout mathematiciansandrecommendvariousbooksonthehistoryofmathematics thatcanbeofinteresttoa‘generalreader’.
Acknowledgments IwouldliketothankJoséMaríaAlmirawhoreadtheinitialmanuscriptand providedseveralusefulcommentsthatledtovariousimprovementsinthepresentation.IalsoprofitedfromseveralusefulsuggestionsmadebyAlmaApt, JanHeering,HelenaStockmann,NickTrefethen,RonalddeWolf,thelate MaartenvanEmden,andthethreeanonymousreferees.
Thisbookcouldnothavebeenwrittenwithoutaccesstothewonderful CWIlibrary.Iammostgratefultoitsstaff,inparticularitsdirectorVera Sarkol,andalsoRobvanRooijen,forhavingorderedseveralbooksonthe historyofmathematicsthatwerehighlyrelevant.Further,Iwouldliketo thankMagdalenaKyclerandPiotrSitekfortheirexpertproductionofseveral drawingsusingtheTikzpackage.MyspecialappreciationgoestoMs.LaiFun KwongofWorldScientificforamostefficientandsmoothcooperation,andto GregoryLeeforhisremarkablythoroughdeskcopyediting.
ThisbookissetusingthefbbLATEXpackagethatprovidesafreeBembo-like fontdesignedbyMichaelSharpe.
AllproceedsfromthisbookwillbedonatedtoAmnestyInternational.
Notes
1NotableexceptionsareD.J.Boorstin, TheDiscoverers:AHistoryofMan’sSearchto KnowHisWorldandHimself,Vintage,1985,andP.Watson, Ideas:AHistoryofThought andInvention,fromFiretoFreud,Weidenfeld&Nicholson,2005.
2K.Devlin, TheUnfinishedGame:Pascal,Fermat,andtheSeventeenth-CenturyLetter thatMadetheWorldModern,BasicBooks,2008.
3D.Bressoud, ARadicalApproachtoRealAnalysis,TheMathematicalAssociationof America,p.52,2007.
4D.Hilbert,OntheInfinite,in:J.vanHeijenoort, FromFregetoGödel,HarvardUniversity Press,pp.367–392,1967.
5N.Wolchover,Disputeoverinfinitydividesmathematicians, ScientificAmerican, 3December2013, https://www.scientificamerican.com/article/infinity-logiclaw/.Originallypublishedin QuantaMagazine.Forarecentbookonthesubject,see I.Stewart, Infinity:aVeryShortIntroduction,OxfordUniversityPress,2017.
6Considerthisremark:“Todayitisnolongerthateasytodecidewhatcountsasa ‘number’.”fromF.Q.Gouvêa,FromNumberstoNumberSystems,in:T.Gowers,I.Leader, andJ.Barrow-Green(eds.), ThePrincetonCompaniontoMathematics,PrincetonUniversity Press,pp.77–82,2008.
Appendices
1AproofofThales’theorem..................
3Thebrokenbambooproblem.................
4Irrationalityof √2 andtheproportionsoftheA4papersheet
5Threeproofsthatthereareinfinitelymanyprimenumbers.
6Eratosthenes’estimateoftheEarth’scircumference.....
7AproofofPtolemy’stheorem.................
8Al-Biruni’scomputationoftheradiusoftheEarth......
13Aconstructionofaregularpentagon.............
22CondorcetparadoxandBorda’scount............
Chapter1 FromtheBeginningsto 6thCenturyBCE Tallymarks Itmakessensetoassumethatthefirstinstanceofmathematicalactivitywas counting.Thereisevidencethateven birdsareabletodistinguishsmallnumbers,sayfourfromfive.Earlyhumans coulddobetter,ofcourse.
Inthe1970s,theso-called LebomboBone,ababoon’slegbonewith29 tallymarks,wasdiscoveredinSwaziland.Onespeculatesthatthisnumber hastodowiththefactthatittakestheMoon29daystocircletheEarth. Infact,thefirstcivilizationsusedalunarcalendar.TheLebomboBoneis theoldestknownmathematicalobject,estimatedtobeatleast43,000years old.Itsageexceedsthatofanapproximately30,000-year-oldwolfbonewith 55tallymarks,excavatedin1937intheCzechRepublic.Asmallattemptat organizingtheresultinginformationwasmadeasthemarksweredividedinto groupsof5.
Anotherbonewithtallymarks,approximately20,000yearsold,became,in somesense,aboneofcontention.Calledthe Ishangobone,itwasdiscovered in1950inwhatisnowtheDemocraticRepublicoftheCongo.Itcontains 168notchesarrangedindifferentpatternsinthreecolumns.Manytheories havebeenputforwardtoexplainthemeaningoftheusedarrangements.In particular,oneclaimsthatitrecordsdifferentphasesoftheMoon,whileanother claimsthatitconstitutesaprecursorofwriting.Determiningwhichtheoryis rightisdifficult,giventhelackofadditionalevidence.1
Still,inmodernterminology,thesebonesrepresentcountingin unarynotation astheyrelyonasinglesymbol,amark.Progresswasachievedby inventingnotationsthatcouldrelyonmoresymbols.
Babylonians
Twoancientculturessubstantiallycontributedtodeveloping mathematics.OneofthemwastheBabyloniancivilization,
namedafterthecityofBabyloninIraq.Itemergedaround2000BCE.The principalsourceofinformationonitisthecollectionofclaytablets.Afterthe scribeswroteonthem(fromlefttoright)thetabletsweredriedorbaked. Thisprocessensuredthatseveralsurviveduntiltoday.Thetabletsofgreatest mathematicalinterestwerewrittenintheperiod1800–1600BCE,inthetimes whentheextensivelawcodeofHammurabiwasestablished.
TheBabyloniansdevelopedanumbersystemthathad59differentsymbols forthenumbers1to59.Inmodernterminology,thisisasystemwithbase60 with0missing.Themotivationforsuchapeculiarchoiceisnotclear.Ithas beensuggestedthatitaroseasamergeroftwoothernumbersystems.
Babyloniannumerals.
TheabovelistingofBabyloniannumeralsshowsthatthesymbolsusedare notindependent:thoserepresentingnumberslargerthan10arepairsoftwo symbols.Forexample,thesymbolfor37is ����,whichisacombinationofthe symbolsfor30and7: �� and ��
Thiswayofcountingwithbase60hassurviveduntiltoday,asseenin ourdivisionofhoursinto60minutesandminutesinto60seconds.Also,our divisionofadayinto24hoursandacircleinto360degreesgoesbacktothe Babylonians.
TheBabyloniannumbersystemwas positional,whichmeansthatasymbol representsa different valuedependingonthepositionitstands.Largernumbers werewritteninthispositionalnotationusingtheabove59numerals,justlike wedoinourdecimalsystem.Spacingwasusedinsteadofthedigitzero,but this,ofcourse,couldcauseambiguities.2
In1854twotabletsfromabout2000BCEwerefoundthatcontainedsquares andcubesofnumbersupto59and32,respectively.Thisshowshowimpressive
thecalculatingskillsoftheBabylonianswere.Todividethenumbers,they realizedthat a b = a · 1 b ,whichbroughtthemtheideaofconstructingtables oftheunitfractions(i.e.,fractionsoftheform 1 b )andrecordingthemintheir base60arithmetic.3
Somecomplicationsarosewithfractionslike 1 7 ,whichcouldnotbewritten insuchaform.Ascribewasawareoftheproblem,admittingthat“anapproximationisgivensince7doesnotdivide”.4 Ofcourse,asimilarproblemarises inthedecimalnotation.
TheBabyloniansusedmathematicsforaccounting,trade,timerecording, astronomicobservations,andsurveyingpurposes.Preservedclaytabletsrecord solutionsofsimplemathematicalproblemsthatinvolveoneortwovariablesthat ledtowhatwecall linearequations.Hereisanexampleproblemfromatext foundduringexcavationsinIraqin1949; qa isaweightunit.5
Ifsomebodyasksyouthus:IfIaddtothetwo-thirdsofmytwothirdsahundredqaofbarley,theoriginalquantityissummedup. Howmuchistheoriginalquantity?6
However,nogeneralmethodswereproposed,andeachproblemwassolved individually.AnothertabletshowedthattheBabyloniansdeterminedthevalue of π,theratioofthecircumferenceofthecircleanditsdiameter,as 3 1 8 ,i.e., 3.125.OtherdiscoveredtabletsrevealedthattheBabylonianswerealsoableto solvespecific quadratic (i.e.,second-degree)equationsandevengaveatryat somespecific third-degree equations.Hereisanexampleofatypicalproblem thatledtoaquadraticequation:7
Thelengthofarectangleexceedsitswidthby7.Itsareais60. Finditslengthandwidth.8
AnotherimportantsourceconfirmingthesophisticationofBabylonian mathematicsis Plimpton322,asmallclaytabletmeasuring12.7centimeters by8.8centimeters(soitwouldfitintoanA6-sizedenvelope).Itwaspurchased in1922for $10byanAmericanpublisher,GeorgeArthurPlimpton,froman antiquarian.ItcomesfromIraqandisestimatedtohavebeenwrittenaround 1800BCE.In1945,twomathematiciansfoundoutthatthetabletprovided informationabout15so-calledPythagoreantriples,i.e.,thenumbers a, b,and c,suchthat a2 + b2 = c2 (forinstance3,4,and5,as 32 +42 =52).9
Egyptians EssentiallyinparallelwiththeBabylonians,theEgyptiancivilizationdeveloped.TheEgyptiansrecordedtheirwritingon papyriusinghieroglyphs.Theusualwritingdirectionwasfromrighttoleft.
ThePlimpton322tablet.10
OurknowledgeofEgyptianmathematicsstemsfromtwomanuscripts.Thefirst isthe RhindMathematicalPapyrus fromapproximately1650BCE.Itcontains materialcopiedfromanoriginalthatisabout200yearsolder.Thepapyrus waspurchasedin1858byaScottishantiquary,HenryRhind,andacquired afterhisdeathbytheBritishMuseuminLondon.Itisabout536centimeters by32centimeters,butwithsomepartsmissing.Byaremarkablecoincidence, someusefulmissingfragmentswerefoundhalfacenturylaterinthedeposits ofaNewYorkmuseum.
Thepapyrusisprobablytoday’sequivalentofamathematicstextbook.It includes84problemsconcernedwithdivisions,multiplication,andhandlingof fractions.Anexampleofaproblemtackledishowtodivide6loavesamong 10men.Theansweris 1 2 + 1 10 ,astheEgyptians,likeBabylonians,usedunit fractions(withthesingleexceptionof 2 3 )exclusively.Also,moreadvanced problemsweredealtwith;theseamountedtosolvinglinearequationsinone variable.Forinstance,anotherproblemcallsforfindingthesizeofascoop thatrequires 3 1 3 tripstoawelltofillone heqat (anancientEgyptianunitof volume).Inmodernterminology,thisissimplytheequation 3 1 3 x =1,with theanswer x = 3 10
11
TheRhindMathematicalPapyrus. Thepapyrusalsocontainsgeometryproblems.Inparticular,itprovideda methodofcalculating π,thevalueofwhichwasdeterminedas 256 81 ,i.e.,about 3.16.Oneoftheproblemscontainsthefollowingpuzzle:
Sevenhouseshavesevencatsthateacheatssevenmicethateach eatssevengrainsofbarley.Eachbarleygrainwouldhaveproduced sevenheqatsofgrain.
Thetaskistodeterminehowmanythingsaredescribed.Thisisclearlyan exerciseinaddingupconsecutivepowers:theansweris 7+72 +73 +74 +75 = 19, 607
So,thisisaprecursortothefollowingfamousEnglishriddlethatdates backtothe18thcentury(thoughtheanswerishere:justone):
AsIwasgoingtoSt.Ives, Imetamanwithsevenwives, Eachwifehadsevensacks, Eachsackhadsevencats, Eachcathadsevenkits:
Kits,cats,sacks,andwives, HowmanyweretheregoingtoSt.Ives?
Thesecondmanuscriptisthe MoscowMathematicalPapyrus,purchased inEgyptbyaRussianEgyptologistaround1892.Itisnowpartofacollection ofthePushkinStateMuseumofFineArtsinMoscow,henceitsname.The papyrusisabout5.5meterslongandbetween4and7centimeterswideandis approximatelyfrom1850BCE.
Itcontains25problemsconcernedwitharithmeticandthecomputationof areasandvolumes.Inparticular,itprovidesacorrectformulaforcomputingthe volumeofatruncatedpyramid,likeapyramid‘underconstruction’withthetop partmissing.ItisthemostremarkableachievementofEgyptianmathematics weknowof.
Torepresentnumbers,theEgyptiansemployedasystemthatuseddifferent symbolsforeachpowerof10,allthewayupto1,000,000:
Numberswerethenwrittenbyrepeatingthesesymbolstheappropriate numberoftimes,sometimesdisplayedinmorerows.Armedwiththisknowledge,thereadercaneasilydecipherinformationwrittenonsomeoldartifacts.
AfragmentofthemaceheadofKingNarmer.12
Forexample,in1898,amaceheadfromabout3100BCEwasfoundinEgypt thatbelongedtoKingNarmer.ItisondisplayintheAshmoleanMuseumin Oxford.Theaboverepresentationofitsfragmentshowsthenumber400,000 (thelowerpartunderthesignofanox)writtenas
(intwocolumns),thenumber1,422,000(underandnexttothesignofagoat) writtenas
andtotheright,thenumber120,000(underthesignofaprisoner(asitting manwithhisarmstiedbehindhisback))writtenas
ThismaceheadshowsthattheEgyptianswerealreadyusingtheirnumber systemmorethan5,000yearsago.
Finally,torepresentfractions,asystemeventuallyemergedaccordingto whichtheunitfraction 1 n waswrittenbyputtingthesymbol rover n Forexample, 1 15 waswrittenas
TheNileanditsfloodingswerecrucialfortheEgyptians.Consequently, theydevelopedasolarcalendarearlyontopredictthem.Iteventuallyreplaced alunarcalendar.Intheircalendar,ayearhad365daysandwasdividedinto 12monthsof30days,with5additionaldaysattheend.Duetothelackof leapyears,thecalendaryeardriftedfartherandfartherawayfromthesolar yearandafter1,460years,madeafullcircle,whichmusthavehappenedmore thanonce.ThisproblemwaseventuallyaddressedbyJuliusCaesarduringthe 1stcenturyBCE.
Notes
1G.G.Joseph, TheCrestofthePeacock:Non-EuropeanRootsofMathematics,Princeton UniversityPress,3rdedition,pp.33–35,2011.
2Forexample,62(so 1 601 +2)isrepresentedas ���� ,while3,602(so 1 602 +0 601 +2) isrepresentedby ����
3Anexamplemayhelptounderstandthisidea.Thedecimalnotationusesbase10,so thefraction 1 25 iswrittenas 0.04,since 1 25 = 0 10 + 4 102 .IntheBabylonians’approach,base 60isused,so 1 25 wasrecordedasthesequenceoftwonumerals�������� representing2and24, since 1 25 =2 · 1 60 +24 · 1 602
4J.J.O’ConnorandE.F.Robertson,AnoverviewofBabylonianmathematics, http: //www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html,2000.
5G.G.Joseph,op.cit.,p.153.
6Itleadstoalinearequation 2 3 2 3 x +100= x,withthesolution x =180
7G.G.Joseph,op.cit.,p.154.
8Itcanbeformalizedbytheequations y = x +7 and xy =60.Theyleadtotheequation x(x +7)=60,whichexpandstothequadraticequation x2 +7x =60,withthesolution x =5
9V.J.KatzandK.H.Parshall, TamingtheUnknown:AHistoryofAlgebrafromAntiquity totheEarlyTwentiethCentury,PrincetonUniversityPress,2014.
10Courtesy:PublicDomain,WikipediaCommons.
11Courtesy:PublicDomain,WikipediaCommons.
12Courtesy:PublicDomain,WikipediaCommons.
Chapter2 TheGreeks (From6thCenturyBCEto 5thCenturyCE) AncientGreekmathematicscoversaperiodofmorethan1,200years.Atthat time,therewasnodivisionofscienceintodisciplines.So,mathematicianswere oftenalsophilosophers,astronomers,orengineers,whoappliedtheirmathematicalknowledgeandideastothesedisciplines.But—crucially—theyalso studiedmathematicsforitsownsake.Theirfocuswasonnumbertheory, geometry,logic,and,occasionally,algebra.Thisbroughtthemtoastudyof problemsthatwentfarbeyondtheneedsofdailylife.Severaloftheirresults hadnodirectuse,but,asweshallsoonsee,blamingmathematiciansfortheir lackofconcernwithapplicationscanbeshortsighted.
Thalesandtrigonometry WebeginouraccountofGreekmathematics with ThalesofMiletus (c.625–545BCE), asuccessfuloilmerchantandoneofthewisemenofhistimes.Thalesis consideredthetraditional‘fatherofphilosophy’;hebelievedthatinnature, everythingoriginatesfromwater.Hecanalsobeviewedasthe‘fatherof trigonometry’,theareaofmathematicsconcernedwiththestudyoftriangles. Studentsinsecondaryschoolslearnthe Thales’theorem,whichisconcerned withatriangleinscribedinacircle.ThistheoremisdiscussedinAppendix 1
Thaleswasabrilliantastronomer(inparticular,heapparentlypredicteda solareclipse)andamathematicianwhowasabletocomputetheheightofthe pyramidsandthedistanceofshipsatseafromtheshore.
Trigonometrygaveriseto triangulation,amethodofcreatingprecisemaps byconstructingsuccessivetriangles.Itwasinventedinthe16thcenturybya Dutchmathematician, GemmaFrisius (1508–1555).Sixtyyearslater,another Dutchman, WillebrordSnelvanRoyen (1580–1626),betterknownas WillebrordSnellius,vastlyimproveduponitbyproposingtomeasuretheangles insteadofthesidesofthetriangles.1 Inspiteofthisimprovement,producing
thefirstprecisemapsofcountriesorcontinentswasnotatriflingmatter.For instance,inthe18thcenturyittooktheItalian-FrenchCassinifamilysome50 yearstoproducethefirstreliablemapofFrance.2
PythagorasandthePythagoreanTheorem
AcontemporaryofThales, PythagorasofSamos (c.570–c.500BCE),iscreditedwiththefamous Pythagoreantheorem.It statesthatinaright-angledtriangle,withthesidelengths a,b,and c,where c isthelengthofthehypotenuse,wehave a2 + b2 = c2.Pythagoraswasso delightedwiththisdiscoverythatheapparently“offeredahundredoxentothe Muses”tothankfortheinspiration.3
ThealreadydiscussedPlimpton322 tabletseemstoindicatethatspecific casesofthisresultwerealreadyknown totheBabylonians.Also,theEgyptiansknewthatatrianglewithsides3, 4,and5wasright-angled.Theproof ofthePythagoreantheoremisgivenin Euclid’s Elements,abookfromaround the3rdcenturyBCE,whichIshalldiscussshortly.However,Euclid’sproofis notsostraightforward.
OneofthesimplestproofswaspublishedinashortarticlebyanAmerican President,JamesAbramGarfield,who wasinofficeforonly199days—hedied in1881afterbeingshotbyanassassin.HisproofispresentedinAppendix 2.Garfieldconcludedhisarticlewiththisremark:“Wethinkitsomethingon whichthemembersofbothhousescanunitewithoutdistinctionofparty.”4
Pythagoreantheorem:
Bynow,thereareplentyofproofsofthePythagoreantheorem.Already, in1940,acollectionofnolessthan370proofsappeared.5 Newproofskeep appearing.In2016,aWorldBankeconomistKaushikBasupublishedoneina papersuccinctlytitled AnewandratherlongproofofthePythagorastheorem bywayofapropositiononisoscelestriangles 6
PythagorasisconsideredtobethefirstpersonwhoconcludedthattheEarth wasasphere,asopposedtoaflatcirculardisc,aswasbelievedamongothers likeHomer.AsstatedbyBertrandRussell:“Pythagoras[...]wasintellectually oneofthemostimportantmenthateverlived,bothwhenhewaswiseand whenhewasunwise.[...]Hefoundedareligion,ofwhichthemaintenets
werethetransmigrationofsoulsandthesinfulnessofeatingbeans.”7 Butthe schoolhefoundedinsouthernItalywasalsoconcernedwithphilosophyand mathematics.
Oneofthesymbolsusedbytheschoolwasthe pentagram,afive-pointed starformedbydrawingthediagonalsofa regularpentagon 8 ThePythagoreanswerefascinatedbynumbersandbelievedthattheuniversecouldbefully understoodintheirterms.Apparently,amottooftheirschoolwas“Allis number”.9
Theyrealizedthatthesumsofsuccessiveoddnumbersformconsecutive squares,so 1+3=22 , 1+3+5=32 , 1+3+5+7=42,andsoon.The Pythagoreansalsoinventedtheconcept ofa perfectnumber.Thesearethe numbersthatareequaltothesumof theirproperdivisors.Forexample,6 isperfectsince1+2+3=6andsois 28,since1+2+4+7+14=28.They attributedmysticalpropertiestothese perfectnumbers.Thisideapersisted forsometime.Forexample,duringthe 5thcentury,St.Augustineclaimedthat Godcreatedtheworldin6daysbecause 6isaperfectnumber.Perfectnumbers giverisetooneoftheoldestopenproblemsinmathematics:areallperfect numberseven?Alsoitisnotknownwhetherthereareinfinitelymanyperfect numbers.
Apentagram.
ThePythagoreansalsoattachedimportancetomusicanddiscoveredthat musicalintervalsbetweennotescouldbeexpressedasnumbers.Inparticular, Pythagorasdiscoveredthathalvingthelengthofastringwouldproduceatone exactlyanoctavehigherwhenstruckorplucked.
π andsomeotherirrationalnumbers ThePythagoreanswerefamiliar withfractionslike 1 3 or 2 5 ,but, totheirhorror,theydiscoveredthatothernumbersexistedaswell.They found,forexample,thatthediagonalofasquarewhosesizesare1couldn’tbe expressedasafraction.Nowadayswewritethisnumberas √2 andcallitthe ‘squarerootof2’.
Wehappentoencounterthisnumberdaily:itistheratioofthesidesofan
ABriefHistoryofMathematicsforCuriousMinds
A4papersheet.Theratio √2 isnotaccidental;itensuresthatanA5paper sheet,obtainedbyfoldinganA4papersheetinhalfalongitslongeredge,has thesameproportionsastheoriginalA4papersheet.
Thisideaofusingtheratio √2 isrelativelyrecent: ItisduetoGermanphysicistGeorgChristophLichtenberg,whoproposeditonlyin1786.
Fractions,suchas 2 5 or 5 2 ,arenowadayscalled rationalnumbers,whilenumbersthatcannotbeexpressedasfractionsarecalled irrationalnumbers.So, √2 isanirrationalnumber.Asimpleproofisgivenin Appendix 4,whereitisalsoexplainedwhy √2 isthe ratioofthesidesoftheA4paper.
TheA4papersheet.
Anotherexampleofanirrationalnumberisthealreadydiscussed π.Itiscelebratedinmanyways,for instanceina1976poembyPolishpoetWisławaSzymborska,aNobelPrizelaureate,whichstartswith:
Theadmirablenumberpi: threepointonefourone. Allthefollowingdigitsarealsoinitial, fiveninetwobecauseitneverends.
In2009,inabidforstrongerscienceeducation,theUSHouseofRepresentativesapprovedaresolutiondesignating14March2009asNationalPiDay. (TheAmericanwayofwriting14Marchis3/14.)Sincethen,14Marchis internationallycelebratedasPiDay.
Computingthevalueof π hasfascinatedmathematicianseversinceantiquityandisarecurringthemethroughoutthehistoryofmathematics,oneto whichIshallreturnmorethanonce.TheGreekssubstantiallyimprovedupon theBabylonianandEgyptianestimates.Inthe3rdcenturyBCE,Archimedes useda96-gon(aregularpolygonwith96sides)toapproximatethecircumferenceofacircleandconcludedthat 3 10 71 <π< 3 10 70 ,i.e., 3.1408 <π < 3.1428 ... ,whichgivesacorrectapproximationtotwodecimalplaces.A substantiallybetterapproximation 355 113 wasfoundinthe5thcenturybyChinesemathematician ZuChongzhi andwasnotimproveduponforthenext900 years.Thiseasy-to-rememberfraction(thinkintermsofdividingthesequence ofsixdigits113355intotwoequalparts)yieldsacorrectapproximationof π tosixdecimalplaces,whichisthebestonecanachieveusingafractionwith sixdigits.
TheDutch-Germanmathematician LudolphvanCeulen (1540–1610)
spentamajorpartofhislifecalculating π to35decimalplaces.Hewas soproudofthisachievementthatheaskedtoengravethisapproximationon histombstone.(Thiswouldbeadifficultrequestforthemathematiciansresponsibleforthecurrentbest-knownapproximationof π,asitisgoesupto morethan100trillion(1014)decimalplaces.)ABritishamateurmathematician,WilliamShanks,wassomewhatlessfortunate.Inthe19thcentury,he spent15yearscalculating π to707decimalplaces,butaswaslaterdiscovered, madeamistakeatthe528thplace.Provingthat π isirrationalwasahardnut tocrackandwasestablishedonlyinthe18thcentury.10
AnotherirrationalnumberknowntotheGreekswasthe goldenratio.It appearsinmanynaturalconstructions.Forexample,asexplainedinAppendix 12,thisistheratioofthelengthsofadiagonalandasideinapentagram. Severalastonishingapplicationsofthegoldenratiowerefoundinmusic,nature, painting,andarchitecture,tonameafew.
Aristotleandthefoundationsoflogic
Oneofthetoweringfigures ofGreekhistorywas Aristotle (384–322BCE),ateacherofAlexandertheGreat.For20years,hewasa studentatPlato’sschool,the Academy,andlaterfoundedhisownschool, the Lyceum.Aristotlewroteonavastarrayofsubjects,inparticular,politics, ethics,andphysics.Hewasalsothefirsthistorianofphilosophy,providinga systematicaccountoftheworksofpreviousphilosophers.Someofhiswork canbeseenastheoriginsoflinguistics,thestudyofhumanlanguages.11 His maincontributiontomathematicswasinprovidingthefoundationsof logic.In hiswork PriorAnalytics,Aristotlearguedthatlogicalargumentsshouldbeconstructedusing syllogisms,inferencesthatallowonetoconcludenewstatements fromthealreadyestablishedones.Anexampleisthefamiliardeduction:
EveryGreekisahuman. Everyhumanismortal. Therefore,everyGreekismortal.
Aristotleproposed192syllogismsintotal,butthemoreknown
Allmenaremortal. Socratesisaman. Therefore,Socratesismortal.
isnotamongthem,sinceaccordingtohislogic,onecouldnotmakespecific statements(likeaboutSocrates).Thissyllogismseemstohavebeeninvented onlyintheHighMiddleAges,probablybyWilliamofOckham,a13th-century
Oxfordscholar(furtherdiscussedinChapter 4)knownforthe‘Ockham’sRazor’ principle,whocameupwith1,368newsyllogisms.12
AccordingtoAristotle,theonlywaytoattainvalidscientificknowledgeisby meansofalogicalargumentbasedonsyllogisms,startingwithtruestatements, whichareeither postulates thatholdforaparticularscienceor axioms that alwayshold.Thisapproachliesatthebasisofscientificreasoning.
Inthe3rdcenturyBCE,Aristotle’sapproachtologicwastakenfurtherby theStoics,whoadded connectives tologic,suchasconjunctionandnegation. Thisallowedthem,incontrasttoAristotle,tostudycomplexpropositions.The modusponens rule(fromtwopremises: A,and A implies B,infer B)isdue toStoics.Theirapproachtologiccanbeconsideredastheoriginofwhatis nowcalled propositionallogic.
Thisviewoflogicwastakenup2,000yearslaterbyGottfriedWilhelm Leibniz.Inturn,asIshallexplainlater,thereisadirect(thoughlengthy)path thatleadsfromLeibniztocomputers.So,inasense,weareallnowprofiting fromAristotle’sinsights.
Euclid’s Elements ThePythagoreantheoremisjustoneexampleofthe manytheoremsthatcanbefoundinthe Elements,a treatiseby Euclid (c.325–c.265BCE),aGreekmathematicianfromAlexandria,inwhichhecompiledallmathematicalknowledgeknowntotheGreeks atthattime.Itconsistsof13volumes(calledbooks)containing465theorems fromgeometryandnumbertheory.ItsEnglishtranslationexceeds500pages.13
The Elements areunbearablydryanddifficulttodigest.Soitissurprising thatitisoneofthemostinfluentialbooksinhistory.Also,almostnothingis knownaboutitsauthor.Thestoryofhowthe Elements becameavailableto ustellsalotabouttheconvolutedwaysofhumancivilization.Thebookwas translatedaround800fromGreekintoArabicinBaghdadbytheIslamicmathematicianal-HajjajibnMatar,duringtheperiodcalledtheIslamicGoldenAge. Then,around1120,anEnglishmonkcalledAdelardofBath,aphilosopherand akeentravelerthroughsouthernEurope,translateditfromArabicintoLatin. Some140yearslater,CampanusofNovara,anItalianmathematician,used Adelard’ssecondtranslationtoproduceanimprovedandannotatedversion. ThisversionwasfirstprintedinVenicein1482.Thefirsttranslationofthe Elements directlyfromGreekintoLatinwascarriedoutonlyin1505byan ItalianBartolomeoZamberti.Thisversionwas,inturn,translatedintoEnglish in1570bySirHenryBillingsley,thelaterlordmayorofLondon.Theoriginal versionsofthe Elements stillexist.Forexample,oneislocatedintheBodleian LibraryinOxford.14
Throughouttheagesthe Elements haveinspiredandinfluencedfamous artists,philosophers,scientists,andpoliticians.Euclid(togetherwithPythagoras,Socrates,Plato,Aristotle,andotherwell-knownGreekfigures)appears inthefamousRenaissancepainting, TheSchoolofAthens (1509–1510),by Raphael.Euclidisthepersonleaningforwardatthebottomrightofthepainting.
Thediscoveryofthe Elements bytheEnglishphilosopherThomasHobbes heavilyinfluencedhisthinkingandconvincedhimthatgeometrywasthekey tothestudyofnature.16 Dutchphilosopher’sBaruchSpinozamainwork, Ethics,waswritteninastyleinspiredbyEuclid’s Elements. 17 IsaacNewton, anuncommonlysolemnperson,apparentlylaughedwhenanassistantasked himwhatbenefittheremightbeinstudyingEuclid.18 BertrandRussellstarted tostudythe Elements atthetenderageof11,withhisbrotherashistutor. Hecommentedonit:“Thiswasoneofthegreateventsofmylife,asdazzling asfirstlove.Ihadnotimaginedthattherewasanythingsodeliciousinthe world.”19
Giventhelengthofthetreatise,itisnowonderthatPtolemy,therulerof
Euclidin TheSchoolofAthens. 15
Egypt,askedEuclidwhethertherewasaneasierwaytounderstandgeometry thanbystudyingthe Elements.Euclidapparentlyreplied:“Thereisnoroyal roadtogeometry.”20
Thefirstfourbooksofthe Elements discussgeometryandculminatein theconstructionofaregularpentagonthatbuildsuponallgeometryresults establishedsofar.RobinHartshornewroteaboutitinhisguidedreadingofthe Elements:“Ifthereissuchathingasbeautyinamathematicalproof,Ibelieve thatthisproofofEuclid’sfortheconstructionoftheregularpentagonsets thestandardforabeautifulproof.”21 InAppendix 13,Iprovideaparticularly simpleconstructionfoundinthe19thcentury.
TheGreeksheldgeometryinhighesteem.Platohadthesaying“Letnobody ignorantofgeometryenterhere”inscribedovertheentrancetohisschool.In TheRepublic,heoutlinedacurriculumcalledthe quadrivium thatconsistedof fourstudies:arithmetic,geometry,astronomy,andmusic.Itwaswidelyused intheMiddleAges.
Inanotherbook, Timaeus,Platointroducedfive Platonicsolids,whichare regularandconvexpolyhedra(defined asthree-dimensionalobjectsenclosed byidenticalflatfaces,theverticesof whichlieonasphere).Inthe13th (andfinal)bookofthe Elements,itwas provedthatnootherPlatonicsolidsexist.AproofcanbefoundinAppendix 21
Thisstrikingresultwasdescribedby HermannWeyl,aprominentGerman mathematicianofthe20thcentury,as “oneofthemostbeautifulandsingular discoveriesinthewholehistoryofmathematics”.23
Platonicsolids.22
Euclidattemptedtoderiveallhistheoremsaboutgeometryfromashort listofintuitivepostulatesandaxioms,likethestatements“allrightanglesare equal”or“thewholeisgreaterthanthepart”.Itwasnoticedonlyattheend ofthe19thcenturythathedidnotcompletelysucceedsinceheoverlooked someunstatedassumptions.24
OneofEuclid’saxioms,the5thone,states(inan equivalent,modernform)thatonaplane,givena straightlineandapointnotlyingonit,onecandrawthroughthepoint
Euclid’s5thaxiom exactlyonestraightlineparalleltothegivenline,asinthefollowingdrawing:
The5thaxiomofEuclideangeometry.
Overthecenturies,mathematicianstriedinvaintoderivethisaxiomfrom theotherEuclideanaxioms.Thematter,asweshallseelater,wasonlyclarified inthe19thcentury,whichgaveriseto non-Euclideangeometry,ofwhicha variantwasusedinthe20thcenturyinEinstein’sgeneralrelativitytheory.
Primenumbers The Elements alsocontainsseveralimportantresults aboutnumbers,inparticularabout primenumbers Thesearenaturalnumbersgreaterthan1thataredivisibleonlyby1and themselves.Forinstance3,7,and23areprimenumberswhile12,21,and 733,055,621arenot(thelatterbeingtheproductoftwoprimenumbers,27,073 and27,077).Euclidproved,inparticular,thatthereareinfinitelymanyprime numbers.ThreeproofsofthisresultcanbefoundinAppendix 5
Primenumbershavealwayskepttheattentionofmathematicians,but studyingthemseemslikeauselessoccupation.However,in1978, RonRivest, AdiShamir,and LeonardAdleman proposedwhatisnowcalledtheRSA cryptosystem,whichiswidelyusedinsecuredatatransmission,forinstance,in creditcardpaymentsovertheInternet.Thissystemisbasedonthefactthat multiplyingtwoprimenumbersiseasy,butdecomposingaproductofthem (liketheabovementioned733,055,621)intoprimenumberscantakealong time.
Primenumbersaredeceptiveintheirsimplicity.Infact,thereexistsimple tostateproblemsaboutthemthatremainopen.Themostfamousexample isthe Goldbachconjecture from1742,whichstatesthateveryevennumber largerthan2isasumoftwoprimenumbers.
Threeclassicalproblems TheGreeksarealsoresponsibleforthesocalled threeclassicalproblems thataskforthe followingconstructionsusingonlyarulerandacompass:
• Doublingthecube:constructingacubewiththedoublevolumeofa givenone,
