### Abstract

In relation to a thesis put forward byMarx 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 (Formula presented.). 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 ofclassical 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Foundations of Science |

DOIs | |

State | Accepted/In press - Dec 20 2016 |

### Fingerprint

### Keywords

- Convergence
- Gregory’s sixth operation
- Infinite number
- Law of continuity
- Transcendental law of homogeneity

### ASJC Scopus subject areas

- General
- History and Philosophy of Science

### Cite this

*Foundations of Science*, 1-12. https://doi.org/10.1007/s10699-016-9512-9

**Gregory’s Sixth Operation.** / Bascelli, Tiziana; Błaszczyk, Piotr; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Kutateladze, Semen S.; Nowik, Tahl; Schaps, David M.; Sherry, David M.

Research output: Contribution to journal › Article

*Foundations of Science*, pp. 1-12. https://doi.org/10.1007/s10699-016-9512-9

}

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 M

PY - 2016/12/20

Y1 - 2016/12/20

N2 - In relation to a thesis put forward byMarx 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 (Formula presented.). 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 ofclassical 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 byMarx 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 (Formula presented.). 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 ofclassical 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

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

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

U2 - 10.1007/s10699-016-9512-9

DO - 10.1007/s10699-016-9512-9

M3 - Article

SP - 1

EP - 12

JO - Foundations of Science

JF - Foundations of Science

SN - 1233-1821

ER -