TY - JOUR

T1 - Gregory’s Sixth Operation

AU - Bascelli, Tiziana

AU - Błaszczyk, Piotr

AU - Kanovei, Vladimir

AU - Katz, Karin U.

AU - Katz, Mikhail G.

AU - Kutateladze, Semen S.

AU - Nowik, Tahl

AU - Schaps, David M.

AU - Sherry, David

N1 - Publisher Copyright: © 2016, Springer Science+Business Media Dordrecht.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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 π. 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.

AB - 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 π. 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.

KW - Convergence

KW - Gregory’s sixth operation

KW - Infinite number

KW - Law of continuity

KW - Transcendental law of homogeneity

KW - Gregory's sixth operation

UR - http://www.scopus.com/inward/record.url?scp=85006833539&partnerID=8YFLogxK

U2 - 10.1007/s10699-016-9512-9

DO - 10.1007/s10699-016-9512-9

M3 - Article

AN - SCOPUS:85006833539

VL - 23

SP - 133

EP - 144

JO - Foundations of Science

JF - Foundations of Science

SN - 1233-1821

IS - 1

ER -