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

ArgumentThe 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 toElem.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.

The Ancient Greek World

This chapter focuses on the mathematicians of Ancient Greece; more specifically, on the elements of geometrical algebra present in the works of Euclid and Apollonius, as well as the propositions of perhaps the greatest of the ancient mathematicians—Archimedes. Only fragmentary documentation exists of the actual beginnings of mathematics in Greece, though the concept and necessity of proofs in mathematics might have come about due to the unique climate of argument and debate fostered in Ancient Greek society. In fact, most of these early developments took place in Athens, one of the richest of the Greek states at the time and one where public life was especially lively and discussion particularly vibrant.

GEOMETRIC ALGEBRAS AND EXTENSORS

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors [Formula: see text] and the theory of its deformations leading to metric geometric algebras [Formula: see text] and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in [Formula: see text] with easier ones in [Formula: see text] (e.g. a noticeable relation between the Hodge star operators associated to G and GE). Several useful examples are worked in details for the purpose of transmitting the "tricks of the trade".