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The role and development of geometric analysis and synthesis in ancient Greece and Medieval Islam. (English) Zbl 1011.01003

Suppes, Patrick (ed.) et al., Ancient and medieval traditions in the exact sciences. Essays in memory of Wilbur Knorr. Papers of the Wilbur Knorr memorial conference, Stanford, CA, USA, March 13–14, 1998. Stanford, CA: CSLI Publications (ISBN 1-57586-273-5/hbk; 1-57586-274-3/pbk). CSLI Lecture Notes 112, 1-31 (2000).
This article deals with the problem of mathematical analysis and synthesis as it applied to Geometry. In terms of the transmission of scientific ideas across cultures this problem remains to be one of the most interesting problems that was originally formulated in the Greek mathematical tradition, and was later inherited by the mathematicians working within the Islamic domain. The article uses several examples from Greek and Arabic traditions to illustrate first the use of these mathematical processes in Greek and Arabic, and then tries to apply technical philological techniques in order to determine the terminology that was used to distinguish between the various applications of those processes. As usual unexpected results eventually surfaced after such an investigation, most important of which is the conclusion marshaled by the authors to detect the subtlety in the nuanced employment of such notions by the Arabic writing authors in a subject that was undeniably Greek. While they admit that the concepts and the procedures, confused and confusing as they were at times in the Greek tradition, continued to be inspired by the Greek tradition, nevertheless those same processes were subjected to three major modifications in the Arabic tradition. Firstly, analysis by knowns was brought very much to the forefront. Secondly, such analysis established pure existence arguments as mathematical arguments important in their own right. Finally, analysis brought attention to the problem of heuristics and stimulated lively reflection on the nature of mathematical problem solving. More than anything else this example of transmission demonstrates very well the very nature of the process itself.
For the entire collection see [Zbl 0964.00010].

MSC:

01A20 History of Greek and Roman mathematics
01A30 History of mathematics in the Golden Age of Islam
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