[en] Recent Leibniz scholarship has sought to gauge which
foundational framework provides the most successful account of
the procedures of the Leibnizian calculus (LC). While many scholars
(e.g., Ishiguro, Levey) opt for a default Weierstrassian framework,
Arthur compares LC to a non-Archimedean framework SIA
(Smooth Infinitesimal Analysis) of Lawvere–Kock–Bell. We analyze
Arthur’s comparison and find it rife with equivocations and
misunderstandings on issues including the non-punctiform nature
of the continuum, infinite-sided polygons, and the fictionality of
infinitesimals. Rabouin and Arthur claim that Leibniz considers
infinities as contradictory, and that Leibniz’ definition of incomparables
should be understood as nominal rather than as semantic.
However, such claims hinge upon a conflation of Leibnizian notions
of bounded infinity and unbounded infinity, a distinction emphasized
by early Knobloch.
The most faithful account of LC is arguably provided by Robinson’s
framework. We exploit an axiomatic framework for infinitesimal
analysis called SPOT (conservative over ZF) to provide a
formalisation of LC, including the bounded/unbounded dichotomy,
the assignable/inassignable dichotomy, the generalized relation of
equality up to negligible terms, and the law of continuity.
Disciplines :
Mathematics
Author, co-author :
Bair, Jacques ; Université de Liège - ULiège > Ecole de Gestion de l'Université de Liège
Blaszczyk, Piotr; Pedagogical University of Cracow > Institue of Mathematics
Ely, Robert; University of Idaho, Moscow > Deparment of Mathematics
Katz, Mikhail; Bar Ilan University, Israël > Deparment of Mathematics
Kuhlemann; Gottfried Wilhem University, Hannover (Germany)
Language :
English
Title :
Procedures of Leibnizian infinitesimal calculus: an account in three modern frameworks
Alternative titles :
[fr] Procédures du calcul infinitésimal de Leibniz: un compte-rendu de trois cadres contemporains
Publication date :
2021
Journal title :
BSHM Bulletin: Journal of the British Society for the History of Mathematics
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