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  1. Analytical Reasoning and Problem-Solving in Diophantus’s Arithmetica : Two Different Styles of Reasoning in Greek Mathematics.Jean Christianidis - 2021 - Philosophia Scientiae 25:103-130.
    Over the past few decades, the question regarding the proper understanding of Diophantus’s method has attracted much scholarly attention. “Modern algebra”, “algebraic geometry”, “arithmetic”, “analysis and synthesis”, have been suggested by historians as suitable contexts for describing Diophantus’s resolutory procedures, while the category of “premodern algebra” has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus’s modus operandi within the conceptual framework of the ancient analysis (...)
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  • Case for the Irreducibility of Geometry to Algebra†.Victor Pambuccian & Celia Schacht - 2022 - Philosophia Mathematica 30 (1):1-31.
    This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...)
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