ArgumentThe Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. (...) It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. (shrink)

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 (...) and to examine the few instances, in the preserved books of the Arithmetica, which might be regarded as linked to practices belonging to the field of analysis. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the (...) synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense. (shrink)