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A History of Mathematics From Mesopotamia to Modernity

296 Pages · 2007 · 4.15 MB · English

  • A History of Mathematics From Mesopotamia to Modernity

    A History of Mathematics This page intentionally left blank A History of Mathematics


    From Mesopotamia to Modernity


    Luke Hodgkin


    1 3


    GreatClarendonStreet,OxfordOX26DP


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    ©OxfordUniversityPress,2005


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    Firstpublished2005


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    ISBN0–19–852937–6(Hbk) 978–0–19–852937–8


    1 3 5 7 9 10 8 6 4 2 Preface


    This book has its origin in notes which I compiled for a course on the history of mathematics at


    King’s College London, taught for many years before we parted company. My major change in


    outlook (which is responsible for its form) dates back to a day ten years ago at the University of


    Warwick, whenIwascomparingnotesonteachingwiththelateDavidFowler. Heexplainedhis


    own history of mathematics course to me; as one might expect, it was detailed, scholarly, and


    encouragedstudentstodoresearchof theirown,particularlyontheGreeks.ItoldhimthatIgave


    what I hoped was a critical account of the whole history of mathematics in a series of lectures,


    trying to go beyond what they would find in a textbook. David was scornful. ‘What’, he said,


    ‘doyoumeanthatyoustandupinfrontof thosestudentsandtellstories?’ Ihadtoacknowledge


    thatIdid.


    David’sapproachmeantthatstudentsshouldbetaughtfromthestartnottoacceptanystoryat


    facevalue,andtobeinterestedinquestionsratherthannarrative.It’scertainlydesirableasregards


    theGreeks,andit’sagoodapproachingeneral,evenif itmaysometimesseemtoodifficultandtoo


    purist.Ihopehewouldnotbetoohardonmyattemptsatacompromise.Theaimsof thebookin


    this, itsultimateform, aresetoutintheintroduction; briefly, Ihopetointroducestudentstothe


    history,orhistoriesof mathematicsasconstructionswhichwemaketoexplainthetextswhichwe


    have,andtorelatethemtoourownideas.Suchconstructionsareoftencontroversial,andalways


    provisional;butthatisthenatureof history.


    TheoriginalimpulsetowritecamefromDavidRobinson,mycollaboratoronthecourseatKing’s,


    whosuggested(unsuccessfully)thatIshouldturnmycoursenotesintoabook;andprovidentially


    fromAlisonJonesof theOxfordUniversityPress, whoturnedupatKing’swhenIwasataloose


    endandaskedif Ihadabooktopublish.Iproducedaproposal;shepersuadedthepresstoaccept


    itandkeptmewriting.Withoutherconstantfeedbackandinvolvementitwouldneverhavebeen


    completed.


    Iamgratefultoanumberof friendsforadviceandencouragement. JeremyGrayreadanearly


    draft and promoted the project as a referee; the reader is indebted to him for the presence of


    exercises.GeoffreyLloydgaveexpertadviceontheGreeks;Iamgratefulforallof it,evenif Ionly


    paidattentiontosome.JohnCairns,FelixPiraniandGervaseFletcherreadpartsof themanuscript


    and made helpful comments; various friends and relations, most particularly Jack Goody, John


    Hope,JessicaHinesandSamandJoeGoldHodgkinexpressedawishtoseethefinishedproduct.


    Finally, I’m deeply grateful to my wife Jean who has supported the project patiently through


    writingandrevision.Toher,andtomyfatherThomaswhoIhopewouldhaveapproved,thisbook


    isdedicated. This page intentionally left blank Contents


    Listof figures xi


    PictureCredits xiv


    Introduction 1


    Whythisbook? 1


    Ontexts,andonhistory 2


    Examples 5


    Historicismand‘presentism’ 6


    Revolutions,paradigms,andallthat 8


    Externalversusinternal 10


    Eurocentrism 12


    1. Babylonianmathematics 14


    1. Onbeginnings 14


    2. Sourcesandselections 17


    3. Discussionof theexample 20


    4. Theimportanceof number-writing 21


    5. Abstractionanduselessness 24


    6. Whatwentbefore 27


    7. Someconclusions 30


    AppendixA.Solutionof thequadraticproblem 30


    Solutionstoexercises 31


    2. Greeksand‘origins’ 33


    1. PlatoandtheMeno 33


    2. Literature 35


    3. Anexample 36


    4. Theproblemof material 39


    5. TheGreekmiracle 42


    6. Tworevolutions? 44


    7. Drowningintheseaof Non-identity 45


    8. Onmodernizationandreconstruction 47


    9. Onratios 49


    AppendixA.FromtheMeno 51


    AppendixB.Onpentagons,goldensections,andirrationals 52


    Solutionstoexercises 54 viii Contents


    3. Greeks,practicalandtheoretical 57


    1. Introduction,andanexample 57


    2. Archimedes 60


    3. HeronorHero 63


    4. Astronomy,andPtolemyinparticular 66


    5. OntheunculturedRomans 69


    6. Hypatia 71


    AppendixA.FromHeron’sMetrics 73


    AppendixB.FromPtolemy’sAlmagest 75


    Solutionstoexercises 76


    4. Chinesemathematics 78


    1. Introduction 78


    2. Sources 80


    3. Aninstanthistoryof earlyChina 80


    4. TheNineChapters 82


    5. Countingrods—whoneedsthem? 85


    6. Matrices 88


    7. TheSongdynastyandQinJiushao 90


    8. On‘transfers’—when,andhow? 95


    9. Thelaterperiod 98


    Solutionstoexercises 99


    5. Islam,neglectanddiscovery 101


    1. Introduction 101


    2. Onaccesstotheliterature 103


    3. Twotexts 106


    4. Thegoldenage 108


    5. Algebra—theorigins 110


    6. Algebra—thenextsteps 115


    7. Al-Samaw’alandal-Ka¯sh¯i 117


    8. Theusesof religion 123


    AppendixA.Fromal-Khwa¯rizm¯i’salgebra 125


    AppendixB.Tha¯bitibnQurra 127


    AppendixC.Fromal-Ka¯sh¯i,TheCalculator’sKey,book4,chapter7 128


    Solutionstoexercises 130


    6. Understandingthe‘scientificrevolution’ 133


    1. Introduction 133


    2. Literature 134


    3. Scholasticsandscholasticism 135


    4. Oresmeandseries 138


    5. Thecalculatingtradition 140


    6. Tartagliaandhisfriends 143


    7. Onauthority 146 Contents ix


    8. Descartes 149


    9. Infinities 151


    10. Galileo 153


    AppendixA 155


    AppendixB 156


    AppendixC 157


    AppendixD 158


    Solutionstoexercises 159


    7. Thecalculus 161


    1. Introduction 161


    2. Literature 163


    3. Theprioritydispute 165


    4. TheKeralaconnection 167


    5. Newton,anunknownwork 169


    6. Leibniz,aconfusingpublication 172


    7. ThePrincipiaanditsproblems 176


    8. Thearrivalof thecalculus 178


    9. Thecalculusinpractice 180


    10. Afterword 182


    AppendixA.Newton 183


    AppendixB.Leibniz 185


    AppendixC.FromthePrincipia 186


    Solutionstoexercises 187


    8. Geometriesandspace 189


    1. Introduction 189


    2. Firstproblem:thepostulate 194


    3. Spaceandinfinity 197


    4. Sphericalgeometry 199


    5. Thenewgeometries 201


    6. The‘time-lag’question 203


    7. Whatrevolution? 205


    AppendixA.Euclid’spropositionI.16 207


    AppendixB.Theformulaeof sphericalandhyperbolictrigonometry 209


    AppendixC.FromHelmholtz’s1876paper 210


    Solutionstoexercises 210


    9. Modernityanditsanxieties 213


    1. Introduction 213


    2. Literature 214


    3. Newobjectsinmathematics 214


    4. Crisis—whatcrisis? 217


    5. Hilbert 221


    6. Topology 223


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