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)

A form of context-appropriate verificationism is proposed that distinguishes between scientific theories as evolving systems of ideas and operationally-specified, testable formal-empirical models. Theories undergo three stages (modes): a formative, exploratory, heuristic phase of theory conception, a developmental phase of theory-pruning and refinement, and a mature, rigorous phase of testing specific, explicit models. The first phase depends on Feyerabendian open possibility, the second on theoretical plausibility and internal coherence, and the third on testability (falsifiability, predictive efficacy). Multiple perspectives produce variety necessary (...) for theory formation, whereas explicit agreement on evaluative criteria is essential for testing. Hertzian observer-mechanics of empirical-deductive scientific models are outlined that use semiotic operations of measurement/evaluation, computation, and physical action/construction. If models can be fully operationalized, then they can be intersubjectively verified (tested) irrespective of metaphysical, theoretical, value-, or culture-based disagreements. Verificationism can be expanded beyond simple predictive efficacy to incorporate testing for pragmatic, functional efficacy in engineering, medicine, and design contexts. Such a more open, pragmatist, operationalist, epistemically-constructivist perspective is suggested in which verification is contingent on the type of assertion (e.g., heuristic, analytic, empirical, pragmatic), its intended purpose, degree and reliability of model-based evidence, and existence of alternate, competing predictive models. Suggestions for epistemological hygiene amidst the world-wide pandemic of misinformation and propaganda are offered. (shrink)