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The History of Mathematical Proof in Ancient Traditions

614 Pages · 2012 · 5.71 MB · English

  • The History of Mathematical Proof in Ancient Traditions

    Th e History of Mathematical Proof in Ancient Traditions


    Th is radical, profoundly scholarly book explores the purposes and


    nature of proof in a range of historical settings. It overturns the


    view that the fi rst mathematical proofs were in Greek geometry and


    rested on the logical insights of Aristotle by showing how much of


    that view is an artefact of nineteenth-century historical scholarship.


    It documents the existence of proofs in ancient mathematical writ-


    ings about numbers, and shows that practitioners of mathematics in


    Mesopotamian, Chinese and Indian cultures knew how to prove the


    correctness of algorithms, which are much more prominent outside


    the limited range of surviving classical Greek texts that historians have


    taken as the paradigm of ancient mathematics. It opens the way to


    providing the fi rst comprehensive, textually based history of proof.


    Jeremy Gray, Professor of the History of Mathematics, Open University


    ‘Each of the papers in this volume, starting with the amazing


    “Prologue” by the editor, Karine Chemla, contributes to nothing less


    than a revolution in the way we need to think about both the sub-


    stance and the historiography of ancient non-Western mathematics,


    as well as a reconception of the problems that need to be addressed if


    we are to get beyond myth-eaten ideas of “unique Western rationality”


    and “the Greek miracle”. I found reading this volume a thrilling intel-


    lectual adventure. It deserves a very wide audience.’


    Hilary Putnam, Cogan University Professor Emeritus, Harvard


    University


    karine chemla is Senior Researcher at the CNRS (Research


    Unit SPHERE, University Paris Diderot, France), and a Senior Fellow


    at the Institute for the Study of the Ancient World at New York


    University. She is also Professor on a Guest Chair at Northwestern


    University, Xi‘an, as well as at Shanghai Jiaotong University and Hebei


    Normal University, China. She was awarded a Chinese Academy of


    Sciences Visiting Professorship for Senior Foreign Scientists in 2009. Th e History of Mathematical


    Proof In Ancient Traditions



    Edited by karine chemla 林力娜 cambridge university press


    Cambridge, New York, Melbourne, Madrid, Cape Town,


    Singapore, São Paulo, Delhi, Mexico City


    Cambridge University Press


    Th e Edinburgh Building, Cambridge CB2 8RU, UK


    Published in the United States of America by Cambridge University Press, New York


    www.cambridge.org


    Information on this title: www.cambridge.org/9781107012219


    © Cambridge University Press 2012


    Th is publication is in copyright. Subject to statutory exception


    and to the provisions of relevant collective licensing agreements,


    no reproduction of any part may take place without the written


    permission of Cambridge University Press.


    First published 2012


    Printed in the United Kingdom at the University Press, Cambridge


    A catalogue record for this publication is available from the British Library


    ISBN 9781107012219 Hardback


    Cambridge University Press has no responsibility for the persistence or


    accuracy of URLs for external or third-party internet websites referred to in


    this publication, and does not guarantee that any content on such websites is,


    or will remain, accurate or appropriate. Contents


    List of fi gures [ix]


    List of contributors [xii]


    Note on references [xiv]


    Acknowledgements [xv]


    Prologue Historiography and history of mathematical proof:


    a research programme [1]


    Karine Chemla


    part i views on the historiography


    of mathematical proof


    Shaping ancient Greek mathematics: the critical editions of Greek


    texts in the nineteenth century


    1 Th e Euclidean ideal of proof in Th e Elements and philological


    uncertainties of Heiberg’s edition of the text [69]


    bernard vitrac


    2 Diagrams and arguments in ancient Greek mathematics: lessons


    drawn from comparisons of the manuscript diagrams with those


    in modern critical editions [135]


    ken saito and nathan sidoli


    3 Th e texture of Archimedes’ writings: through Heiberg’s veil [163]


    reviel netz


    Shaping ancient Greek mathematics: the philosophers’ contribution


    4 John Philoponus and the conformity of mathematical


    proofs to Aristotelian demonstrations [206]


    orna harari


    Forming views on the ‘Others’ on the basis of mathematical proof


    5 Contextualizing Playfair and Colebrooke on proof and


    demonstration in the Indian mathematical tradition


    (1780–1820) [228]


    dhruv raina v vi Contents


    6 Overlooking mathematical justifi cations in the Sanskrit tradition:


    the nuanced case of G. F. W. Th ibaut [260]


    agathe keller


    7 Th e logical Greek versus the imaginative Oriental: on the


    historiography of ‘non-Western’ mathematics during the


    period 1820–1920 [274]


    françois charette


    part ii history of mathematical proof in


    ancient traditions: the other evidence


    Critical approaches to Greek practices of proof


    8 Th e pluralism of Greek ‘mathematics’ [294]


    g. e. r. lloyd


    Proving with numbers: in Greece


    9 Generalizing about polygonal numbers in ancient Greek


    mathematics [311]


    ian mueller


    10 Reasoning and symbolism in Diophantus: preliminary


    observations [327]


    reviel netz


    Proving with numbers: establishing the correctness of algorithms


    11 Mathematical justifi cation as non-conceptualized practice: the


    Babylonian example [362]


    jens høyrup


    12 Interpretation of reverse algorithms in several Mesopotamian


    texts [384]


    christine proust


    13 Reading proofs in Chinese commentaries: algebraic proofs in an


    algorithmic context [423]


    karine chemla


    14 Dispelling mathematical doubts: assessing mathematical


    correctness of algorithms in Bhāskara’s commentary on the


    mathematical chapter of the Āryabhatīya [487]


    ˙


    agathe keller Contents vii


    Th e later persistence of traditions of proving in Asia: late evidence


    of traditions of proof


    15 Argumentation for state examinations: demonstration in


    traditional Chinese and Vietnamese mathematics [509]


    alexei volkov


    Th e later persistence of traditions of proving in Asia: interactions of


    various traditions


    16 A formal system of the G ougu method: a study on Li Rui’s


    Detailed Outline of Mathematical Procedures for the Right-Angled


    Triangle [552]


    tian miao


    Index [574]


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