Gregory's sixth operation

Abstract

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians' approach to interpreting James Gregory's expression ultimate terms in his paper attempting to prove the irrationality of pi. Here Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? as well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an "unrigorous" practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory's teacher degli Angeli, and Gregory's books at Venice, in the late 1660s.

17 pages, published in Foundations of Science

References in corpus (12)

• Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
• Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
• Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics
• Interpreting the infinitesimal mathematics of Leibniz and Euler
• Toward a history of mathematics focused on procedures
• Commuting and noncommuting infinitesimals
• Stevin numbers and reality
• Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania
• A non-standard analysis of a cultural icon: The case of Paul Halmos
• Euler's lute and Edwards' oud
• Controversies in the foundations of analysis: Comments on Schubring's Conflicts
• Sum-avoiding sets in groups

Cited by in corpus (8)

• Approaches to analysis with infinitesimals following Robinson, Nelson, and others
• Cauchy's infinitesimals, his sum theorem, and foundational paradigms
• Cauchy, infinitesimals and ghosts of departed quantifiers
• Leibniz's well-founded fictions and their interpretations
• Infinite lotteries, spinners, and the applicability of hyperreals
• Fermat's dilemma: Why did he keep mum on infinitesimals? and the European theological context
• Of pashas, popes, and indivisibles
• Bolzano's Infinite Quantities