The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis.

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)

Mathematical texts raise particular dilemmas for the translator. With its arm’s-length relation to verbal expression and long-standing “mathematics is written for mathematicians” ethos, mathematics lends itself awkwardly to textually centered analysis. Otherwise sound standards of historical scholarship can backfire when rigidly upheld in a mathematical context. Mathematically inclined historians have had more faith in a purported empathic sixth sense—and there is a case to be made that this is how mathematical authors have generally expected their works to be read—but it (...) is difficult to pin down exact evidentiary standards for this supposed instinct. This essay urges that both of these points of view, for all the tension between them, be kept in the historian’s toolbox. It illustrates these considerations with a case study from the Ptolemaic astronomical tradition on computing lunar model parameters from eclipse data. (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)