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The History of Mathematical Proof in Ancient Traditions

Edited by : Karine Chemla, Centre National de la Recherche Scientifique (CNRS), Paris

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first 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 19th century historical scholarship. It documents the existence of proofs in ancient mathematical writings 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 first comprehensive, textually based history of proof.

Jeremy Gray

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 substance 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 intellectual adventure. It deserves a very wide audience.

Hilary Putnam, Cogan University Professor Emeritus, Harvard University


Prologue : historiography and history of mathematical proof : a research program : Karine Chemla

             Part I. Views on the Historiography of Mathematical Proof :

  1. The Euclidean ideal of proof in The Elements and philological uncertainties of Heiberg’s edition of the text : Bernard Vitrac
  2. Diagrams and arguments in ancient Greek mathematics : lessons drawn from comparisons of the manuscript diagrams with those in modern critical editions : Ken Saito and Nathan Sidoli
  3. The texture of Archimedes’ arguments : through Heiberg’s veil : Reviel Netz
  4. John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations : Orna Harari
  5. Contextualising Playfair and Colebrooke on proof and demonstration in the Indian mathematical tradition (1780–1820) : Dhruv Raina
  6. Overlooking mathematical justifications in the Sanskrit tradition : the nuanced case of G. F. Thibaut : Agathe Keller
  7. The logical Greek versus the imaginative Oriental : on the historiography of ’non-Western’ mathematics during the period 1820–1920 : François Charette

    Part II. History of Mathematical Proof in Ancient Traditions : The Other Evidence :

  8. The pluralism of Greek ’mathematics’ : Geoffrey Lloyd
  9. Generalizing about polygonal numbers in ancient Greek mathematics : Ian Mueller
  10. Reasoning and symbolism in Diophantus : preliminary observations : Reviel Netz
  11. Mathematical justification as non-conceptualized practice : the Babylonian example : Jens Høyrup
  12. Interpretation of reverse algorithms in several Mesopotamian texts : Christine Proust
  13. Reading proofs in Chinese commentaries : algebraic proofs in an algorithmic context : Karine Chemla
  14. Dispelling mathematical doubts : assessing mathematical correctness of algorithms in Bhaskara’s commentary on the mathematical chapter of the Aryabhatıya : Agathe Keller
  15. Argumentation for state examinations : demonstration in traditional Chinese and Vietnamese mathematics : Alexei Volkov
  16. A formal system of the Gougu method – a study on Li Rui’s detailed outline of mathematical procedures for the right-angled triangle : Tian Miao.

 :: Cambridge University Press, July 2012
 :: 612pages
 :: 93 b/w illus. 29 tables
 :: ISBN:9781107012219