Gregory's sixth operation

TL;DR

This paper examines the historiography of mathematics through the lens of ontology and contingency, analyzing Gregory's work on infinite series and its interpretation within modern infinitesimal frameworks.

Contribution

It offers a novel analysis of early modern mathematical procedures, linking historical practices to modern infinitesimal theories and exploring social and religious influences.

Findings

Gregory's 'ultimate terms' relate to infinite series and irrationality of pi.

Modern infinitesimal theories closely proxy early procedures.

Historical suppression of Gregory's works was influenced by social and religious factors.

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.