Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

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 (implicit diagrammatic reasoning, superposition) 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.

Piero della Francesca

The reputation of the Italian Renaissance painter and mathematician Piero della Francesca (hereafter Piero) has risen dramatically from near oblivion in the Early Modern period to the present when he is judged as nearly equal to Leonardo da Vinci and Michelangelo Buonarroti, especially in regard to technical excellence and writing. In his paintings and his treatises Piero achieved the fullest expression of Quattrocento perspective, which he had derived from his capacious understanding of Greek geometry. Born into a family of artisans-shopkeepers of leather in the Tuscan town of Borgo San Sepolcro (today Sansepolcro) c. 1412, Piero began with several small projects in his native town in the 1430s. He served in Florence as an assistant to Domenico Veneziano in 1439 where he came in contact with the achievements of Masaccio, Donatello, Brunelleschi, and other artists in Florence, though he never became strictly a Florentine painter. He sojourned throughout central Italy in the 1440s and 1450s, receiving commissions in Ferrara, Ancona, Rimini, Arezzo, Rome, and Sansepolcro. After his two sojourns in Rome in the 1450s, Piero became increasingly interested in the representation of classical values and especially Roman architecture and in learning Greek geometry. In the decade of the 1470s he was often in the court of Federico da Montefeltro in Urbino, accepting commissions from the ruler and consulting manuscripts in his library. In the history of mathematics, he is now recognized as an indispensable participant in the revival of Greek geometry and as one of the few individuals in 15th-century Europe with extensive knowledge of both Euclid and Archimedes. In the last three decades of his life (d. 1492) Piero wrote treatises on commercial mathematics and geometry (Trattato d’abaco), perspective in painting (De prospectiva pingendi), and a reflection on several classical procedures and problems in Greek geometry (Libellus de quinque corporibus regularibus), as well as copying the Opera of Archimedes.

Vowels and Consonants

This chapter focuses on the evolution of the vowel–consonant notation. In particular, it discusses François Viète's contribution to algebra through his use of vowels to represent unknowns and consonants to represent known quantities. Viète, a French mathematician, expressed his famous computation for π‎ in proposition II of his Isagoge. Even Christoff Rudolff and Nicolas Chuquet had no proper notation for expressing such an infinite sum of nested square roots. Viète was showing us an intimate link between Greek geometry and algebra, a link from the mathematics of lines, figures, and solids to the underlying channels of symbolic algebra. The chapter also considers Viète's work on what are now called “homogeneous equations” as well as the significance of his lettering system to symbolic algebra.

Advanced Math

This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked with the view that inductive methods are at play when the hypotheses are established. The presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory and undermines the simple distinction between “formal” and “empirical” sciences. The chapter first elaborates on a hypothetical conception of mathematics before discussing the ideas (and ideals) of certainty and objectivity in mathematics. It then considers the modern problems of the continuum that exist in ancient Greek geometry, along with the so-called methodological platonism of modern mathematics and its focus on mathematical objects. Finally, it describes the Axiom of Completeness and the Riemann Hypothesis.