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The History of Mathematical Proof in Ancient Traditions

The History of Mathematical Proof in Ancient Traditions

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Cambridge University Press, 7/5/2012
EAN 9781107012219, ISBN10: 110701221X

Hardcover, 614 pages, 24.7 x 17.4 x 3.6 cm
Language: English

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 nineteenth-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.

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.

Advance praise: '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, Open University