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THE SHAPING OF DEDUCTION IN GREEK MATHEMATICS A Study in Cognitive History REVIEL NETZ
published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © Reviel Netz 2004 First published in printed format 1999 ISBN 978-0-511-543296 OCeISBN ISBN 0-521-62279-4 hardback ISBN 0-521-54120-4 paperback
Page 2 and 3: The aim of this book is to explain
Page 4 and 5: ideas in context 51 THE SHAPING OF
Page 8 and 9: To Maya
Page 10 and 11: Contents Preface page xi List of ab
Page 12 and 13: Preface This book was conceived in
Page 14 and 15: Abbreviations greek authors Abbrevi
Page 16 and 17: documentary sources Abbreviation St
Page 18 and 19: Note on the figures As is explained
Page 20 and 21: Introduction This book can be read
Page 22 and 23: Introduction 3 production of ‘nor
Page 24 and 25: Introduction 5 clarify what these m
Page 26 and 27: Introduction 7 is because they are,
Page 28 and 29: A specimen of Greek mathematics Rea
Page 30 and 31: A specimen of Greek mathematics 11
Page 32 and 33: The material implementation of diag
Page 38 and 39: Practices of the lettered diagram 1
Page 50 and 51: Α Τ Ο Ν Υ Η Ξ Ε Practices o
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Practices of the lettered diagram 3
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Τ Π Ζ φ Practices of the letter
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obstructs the development of other,
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Γ Μ Α Ρ Κ Β Μ Α Κ Ρ ∆ P
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Ε Β Ν Λ Η Practices of the let
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Contexts for the emergence of the l
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Summary 67 4 summary Much of the ar
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The origins of the practices 69 Α
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I follow two practices: The origins
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Γ ∆ Ε Β The origins of the pra
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The origins of the practices 75 AB
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The origins of the practices 77 Man
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The implications of the practices 7
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Λ The implications of the practice
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chapter 3 The mathematical lexicon
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Definitions 91 Archimedes and Apoll
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Definitions 93 when a certain prope
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Definitions 95 and no definition of
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Definitions 97 A second issue is th
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Definitions 99 Figure 3.1. Two segm
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Definitions 101 definiens of the co
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The shape of the lexicon 103 ently
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The shape of the lexicon 105 Why do
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The shape of the lexicon 107 Elemen
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The shape of the lexicon 109 At lea
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The shape of the lexicon 111 rectil
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The shape of the lexicon 113 To con
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The shape of the lexicon 115 releva
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The shape of the lexicon 117 Howeve
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The shape of the lexicon 119 former
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Comparative remarks 121 We saw how
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The shaping of the lexicon 123 not
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The shaping of the lexicon 125 regu
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chapter 4 Formulae introduction and
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The Homeric case 129 tools in Homer
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The Homeric case 131 It is markedly
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Greek mathematical formulae 133 2 g
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Greek mathematical formulae 135 Thi
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Greek mathematical formulae 137 23.
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Elements ii contains 12 second-orde
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Greek mathematical formulae 141 We
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This formula signifies a relation b
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Greek mathematical formulae 145 for
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protasis P 73 S 50 C33 P78 ekthesis
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The behaviour of formulae 149 rence
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Formula 31a is a constituent in the
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The behaviour of formulae 153 the s
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The behaviour of formulae 155 certa
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The behaviour of formulae 157 And,
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The behaviour of formulae 159 formu
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The behaviour of formulae 161 total
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one’s text. This expectation may
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Summary: back to the Homeric case 1
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Summary: back to the Homeric case 1
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Starting-points 169 3 1 2 Figure 5.
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Starting-points 171 Deduction, in f
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Ζ Η Α Starting-points 173 Ξ Μ
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Starting-points 175 This clause is
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Α (c) Implicit arguments (d) Hypot
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�στι τ�ς ΓΕ β � ∆Ε
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Starting-points 181 Clearly, there
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Starting-points 183 This relies upo
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eductio argument. The refutation of
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Β Ε Η Ζ Α ∆ Arguments 187 Fi
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Arguments 189 are unworried about i
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Arguments 191 line of reasoning is
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Α Ζ Ν Η Μ Arguments 193 Β ∆
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O 2 ΕΓ 3 1 2 Arguments 195 6 4 5
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Arguments 197 On a wider view still
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all arguments are two-assertion arg
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The structure of proofs 201 The 10-
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1 3 2 The structure of proofs 203 6
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The structure of proofs 205 3.2.3 N
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The structure of proofs 207 smooth,
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The structure of proofs 209 Here is
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The structure of proofs 211 2 4 1 3
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The structure of proofs 213 The pro
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The structure of proofs 215 as in t
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The tool-box 217 Elements i. Def. 1
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The tool-box 219 Elements i.8, for
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The tool-box 221 ‘remembered’ (
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The tool-box 223 Proposition 5 argu
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Elements I Congruence 15 20 4 8 26
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I have not attempted a complete ana
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The tool-box 229 4.4 Accessing the
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The tool-box 231 that the Greek geo
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The tool-box 233 results, I claimed
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Summary 235 Greek mathematics, what
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Summary 237 and Sedley invite Arist
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Summary 239 to Conics i; this is hi
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Introduction and plan of the chapte
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Α ∆ Hints for a solution 243 Ε
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Hints for a solution 245 centre, as
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‘If two circles cut each other, t
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Hints for a solution 249 constructi
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Hints for a solution 251 ‘Active
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Mueller must be right in viewing th
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The Greek solution 255 (‘protasis
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C(x) P(x) C(a) I Say P(a) The Greek
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The Greek solution 259 2.2 The stru
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The Greek solution 261 Archimedes
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transformations consisting in chang
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As suggested by the example above,
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The Greek solution 267 2.5 Conclusi
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Summary 269 To sum up: in arithmeti
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chapter 7 The historical setting in
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The chronology of Greek mathematics
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The chronology of Greek mathematics
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Demography 277 is possible. The imp
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Demography 279 2.1 Class We conside
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Demography 281 should be a mathemat
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Demography 283 the absolute number
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Demography 285 Thrasydaeus and Nauc
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Demography 287 compared to the rich
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Demography 289 are so few philosoph
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Demography 291 I conclude, therefor
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Mathematics within Greek culture 29
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Summary 311 Little wonder, then, th
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The main Greek mathematicians 313 a
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The main Greek mathematicians 315 c
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Bibliography 317 Chemla, K. (1994)
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Bibliography 319 Klein, J. (1934-6/
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Bibliography 321 Russell, B. (1903/
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abaci 63- 4 Aeschylus (astronomer)
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Eudemus of Rhodes 61, 272, 274, 307
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Thales 272, 274, 304 Theaetetus 279
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13 peter novick That Noble Dream Th
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42 g. e. r. lloyd Adversaries and A
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