484 karine chemla
operating on quantities such as a + b/c ( huan, baochu, lianchu ) on the other
hand. In that way, these terms are introduced at the beginning of the book. 83
Th ese concluding remarks lead to a whole range of questions, which we
shall formulate as a conclusion to the chapter. How was the correctness
of algorithms for multiplying and dividing quantities with fractions
approached elsewhere, and what connections did this concern have with
the kind of ‘algebraic proof in an algorithmic context’ discussed here? Is
there a historical relationship between the proofs we examined and the
overall history of algebraic proofs? If there exists some relationship, did the
proofs that were devised in ancient China actually play a historical part in
this process? It is clear, I believe, that the history of mathematical proof still
has many new territories to explore.Acknowledgements
Th is chapter has benefi ted from the discussions of the working group
‘History and historiography of mathematical proof in ancient traditions’,
which the research group REHSEIS convened at Columbia University in
Paris, at Reid Hall, between March and June 2002 thanks to the generosity
of the Maison des sciences de l’homme and Columbia University. It is a
pleasure to express my gratitude to all the participants in this workshop and
the sponsors who made it possible.Bibliography
Chemla , K. ( 1991 ) ‘ Th eoretical aspects of the Chinese algorithmic tradition (fi rst
to third century) ’, Historia Scientiarum 42 : 75–98 (+ errata in the following
issue).
( 1992 ) ‘ Les fractions comme modèle formel en Chine ancienne ’, in Histoire de
fractions, fractions d’histoire , ed. P. Benoit and J. Ritter. Basel : 188 –207.
( 1996 ) ‘ Positions et changements en mathématiques à partir de textes chinois
des dynasties Han à Song-Yuan. Quelques remarques ’, Extrême-Orient,
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( 1997/8 ) ‘ Fractions and irrationals between algorithm and proof in ancient
China ’, Studies in History of Medicine and Science n.s. 15 : 31 –54.83 O n l y fan will be introduced at the beginning of Chapter 2.