Foundations of Science

, Volume 23, Issue 1, pp 133–144 | Cite as

Gregory’s Sixth Operation

  • Tiziana Bascelli
  • Piotr Błaszczyk
  • Vladimir Kanovei
  • Karin U. Katz
  • Mikhail G. KatzEmail author
  • Semen S. Kutateladze
  • Tahl Nowik
  • David M. Schaps
  • David Sherry


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.


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



M. Katz was partially supported by the Israel Science Foundation Grant No. 1517/12.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Tiziana Bascelli
    • 1
  • Piotr Błaszczyk
    • 2
  • Vladimir Kanovei
    • 3
    • 4
  • Karin U. Katz
    • 5
  • Mikhail G. Katz
    • 5
    Email author
  • Semen S. Kutateladze
    • 6
  • Tahl Nowik
    • 5
  • David M. Schaps
    • 7
  • David Sherry
    • 8
  1. 1.Lyceum Gymnasium “F. Corradini”ThieneItaly
  2. 2.Institute of MathematicsPedagogical University of CracowCracowPoland
  3. 3.IPPIMoscowRussia
  4. 4.MIITMoscowRussia
  5. 5.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  6. 6.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  7. 7.Department of Classical StudiesBar Ilan UniversityRamat GanIsrael
  8. 8.Department of PhilosophyNorthern Arizona UniversityFlagstaffUSA

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