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Situating the debate on “geometrical algebra” within the framework of premodern algebra. (English) Zbl 1379.01006

The year 2016 has seen two papers addressed, after a long silence, to the debate regarding the validity of the “geometrical algebra” hypothesis put forward by one generation of historians of mathematics and demolished by another one, in the most forceful language by S. Unguru [Arch. Hist. Exact Sci. 15, No. 1, 67–114 (1975; Zbl 0325.01002)]. Both are devoted to a reappraisal of the validity of the hypothesis. While V. Blåsjö [ibid. 70, No. 3, 325–359 (2016; Zbl 1379.01005)] defines the geometrical algebra hypothesis in precise terms, finding two aspects of it, and criticizes point by point the critiques of the detractors of the hypothesis, the authors of the paper under review find that a compromise is possible by criticizing both sides of the debate for its uncritical view of the concept of algebra, which both sides consider as “a single well-defined category”. By proposing that “algebra should rather be seen as a broad spectrum of ideas”, and by introducing the notion of “premodern algebra” (which is found to have two characteristics: it was a method of solving problems and was a part of arithmetic), the authors look for a comparative analysis that may justify the “geometrical algebra” label in the premodern sense at Book II of Euclid’s Elements, the alternative proofs proposed by Heron, and the extant solutions by three mathematicians, Diophantus, al-Sulamī, and al-Khwārizmī to an arithmetical problem mathematically associated with II.5.

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01A20 History of Greek and Roman mathematics

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