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Situating the Debate on “Geometrical Algebra” within the Framework of Premodern Algebra

  • Michalis Sialaros (a1) and Jean Christianidis (a2)

The aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

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Len J. Berggren 1984. “History of Greek Mathematics: A Survey of Recent Research.” Historia Mathematica 11:394410.

Alain Bernard . 2003. “Ancient Rhetoric and Greek Mathematics: A Response to a Modern Historiographical Dilemma.” Science in Context 16:391412.

Viktor Blåsjö . 2014. “A Critique of the Modern Consensus in the Historiography of Mathematics.” Journal of Humanistic Mathematics 4:113123.

Jean Christianidis , and Jeffrey A. Oaks . 2013. “Practicing Algebra in Late Antiquity: The Problem-Solving of Diophantus of Alexandria.” Historia Mathematica 40:127163.

Leo Corry . 2013. “Geometry and Arithmetic in the Medieval Traditions of Euclid's Elements: A View from Book II.” Archive for History of Exact Sciences 67:637705.

Hans Freudenthal . 1977. “What Is Algebra and What Has It Been in History.” Archive for the History of Exact Sciences 16:189200.

Michael N. Fried 2014The Discipline of History and the ‘Modern Consensus in the Historiography of Mathematics’.” Journal of Humanistic Mathematics 4:124136.

David H. Fowler 1980. “Book II of Euclid's Elements and a Pre-Eudoxean Theory of Ratio.” Archive for History of Exact Sciences 22:536.

Ivor Grattan-Guinness . 1996. “Numbers, Magnitudes, Ratios, and Proportions in Euclid's Elements: How Did He Handle Them?” Historia Mathematica 23:355375.

Anthony Lo Bello , ed. 2009. The Commentary of al-Nayrizi on Books II-IV of Euclid's Elements of Geometry. Boston: Brill.

Jeffrey A. Oaks 2007. “Medieval Arabic Algebra as an Artificial Language.” Journal of Indian Philosophy 35:543575.

Jeffrey A. Oaks 2009. “Polynomials and Equations in Arabic Algebra.” Archive for History of Exact Sciences 63:169203.

Jeffrey A. Oaks 2010a. “Equations and Equating in Arabic Mathematics.” Archives Internationales d'Histoire des Sciences 60:265298.

Jeffrey A. Oaks , and Haitham M. Alkhateeb . 2005. “Māl, Enunciations, and the Prehistory of Arabic Algebra.” Historia Mathematica 32:400425.

Jeffrey A. Oaks , and Haitham M. Alkhateeb . 2007. “Simplifying Equations in Arabic Algebra.” Historia Mathematica 34:4561.

Árpad Szabó . 1978. The Beginnings of Greek Mathematics. Translated by A. M. Ungar. Dordrecht/Boston: Reidel. Originally published in German as Anfänge der griechischen Mathematik. Budapest: Akadémiai Kiadó, 1969.

Sabetai Unguru . 1975. “On the Need to Rewrite the History of Greek Mathematics.” Archive for History of Exact Sciences 15:67114.

Sabetai Unguru . 1979. “History of Ancient Mathematics: Some Reflections on the State of the Art.” Isis 70:555565.

Bartel L. Van der Waerden 1930–1. Moderne Algebra. Berlin: Verlag von Julius Springer.

Bartel L. Van der Waerden 1975. “Defence of a Shocking Point of View.” Archive for History of Exact Sciences 15:199210.

André Weil . 1978. “Who Betrayed Euclid?Archive for History of Exact Sciences 19:9193.

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Science in Context
  • ISSN: 0269-8897
  • EISSN: 1474-0664
  • URL: /core/journals/science-in-context
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