A History of Mathematics: From Mesopotamia to Modernity

296 Pages·2005·1.891 MB·English

by  Luke Hodgkin

A History of Mathematics
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A History of Mathematics
From Mesopotamia to Modernity
Luke Hodgkin
1
3
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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.
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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