Leibniz's well-founded fictions and their interpretations

J. Bair; P. Blaszczyk; R. Ely; P. Heinig; M. Katz

doi:10.15330/ms.49.2.186-224

Leibniz used the term fiction in conjunction with infinitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contrasts with that of Ishiguro and Arthur. Leibniz's own views, expressed in his published articles and correspondence, led Bos to distinguish between two methods in Leibniz's work: (A) one exploiting classical `exhaustion' arguments, and (B) one exploiting inassignable infinitesimals together with a law of continuity. Of particular interest is evidence stemming from Leibniz's work Nouveaux Essais sur l'En- tendement Humain as well as from his correspondence with Arnauld, Bignon, Dagincourt, Des Bosses, and Varignon. A careful examination of the evidence leads us to the opposite conclusion from Arthur's. We analyze a hitherto unnoticed objection of Rolle's concerning the lack of justification for extending axioms and operations in geometry and analysis from the ordinary domain to that of infinitesimal calculus, and reactions to it by Saurin and Leibniz. A newly released 1705 manuscript by Leibniz (Puisque des personnes. . . ) currently in the process of digitalisation, sheds light on the nature of Leibnizian inassignable infinitesimals. In a pair of 1695 texts Leibniz made it clear that his incomparable magnitudes violate Euclid's Definition V.4, a.k.a. the Archimedean property, corroborating the non-Archimedean construal of the Leibnizian calculus.

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	Leibniz's well-founded fictions and their interpretations
Author	J. Bair¹, P. Blaszczyk², R. Ely³, P. Heinig⁴, M. Katz⁵ J.Bair@ULiege.be¹, pb@up.krakow.pl², ely@uidaho.edu³, heinig@ma.tum.de⁴, katzmik@macs.biu.ac.il⁵ 1) HEC-ULG, University of Liege, Belgium; 2) Institute of Mathematics, Pedagogical University of Cracow, Poland; 5) Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
Abstract	Leibniz used the term fiction in conjunction with infinitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contrasts with that of Ishiguro and Arthur. Leibniz's own views, expressed in his published articles and correspondence, led Bos to distinguish between two methods in Leibniz's work: (A) one exploiting classical `exhaustion' arguments, and (B) one exploiting inassignable infinitesimals together with a law of continuity. Of particular interest is evidence stemming from Leibniz's work Nouveaux Essais sur l'En- tendement Humain as well as from his correspondence with Arnauld, Bignon, Dagincourt, Des Bosses, and Varignon. A careful examination of the evidence leads us to the opposite conclusion from Arthur's. We analyze a hitherto unnoticed objection of Rolle's concerning the lack of justification for extending axioms and operations in geometry and analysis from the ordinary domain to that of infinitesimal calculus, and reactions to it by Saurin and Leibniz. A newly released 1705 manuscript by Leibniz (Puisque des personnes. . . ) currently in the process of digitalisation, sheds light on the nature of Leibnizian inassignable infinitesimals. In a pair of 1695 texts Leibniz made it clear that his incomparable magnitudes violate Euclid's Definition V.4, a.k.a. the Archimedean property, corroborating the non-Archimedean construal of the Leibnizian calculus.
Keywords	Archimedean property; assignable vs inassignable quantity; Euclid's Definition V.4; infinitesimal; law of continuity; law of homogeneity; logical fiction; Nouveaux Essais; pure fiction; quantifier-assisted paraphrase; syncategorematic; transfer principle
DOI	doi:10.15330/ms.49.2.186-224
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Pages	186-224
Volume	49
Issue	2
Year	2018
Journal	Matematychni Studii
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