Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

@article{Katz2013LeibnizsIT,
  title={Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond},
  author={Mikhail G. Katz and David Sherry},
  journal={Erkenntnis},
  year={2013},
  volume={78},
  pages={571-625}
}
Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others… 
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References

SHOWING 1-10 OF 199 REFERENCES
Leibniz’s syncategorematic infinitesimals
AbstractIn contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are
Leibniz on The Elimination of Infinitesimals
My aim in this paper is to consider Leibniz’s response to concerns raised about the foundations of his differential calculus, and specifically with his doctrine that infinitesimals are “fictions,”
George Berkeley, The analyst (1734)
THE TENSION BETWEEN INTUITIVE INFINITESIMALS AND FORMAL MATHEMATICAL ANALYSIS
Abstract: we discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating
Infinitesimal Differences: Controversies between Leibniz and his Contemporaries
The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This
Meaning in Classical Mathematics: Is it at Odds with Intuitionism?
We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a
On Cauchy's Notion of Infinitesimal
The historical literature of nineteenth century mathematical analysis contains much discussion of Cauchy's conception of the continuum, and of the nature of his infinitesimals in particular. In a
Leibniz's laws of continuity and homogeneity
We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus
The practice of reason : Leibniz and his controversies
1. Foreword 2. Abbreviations 3. Contributors 4. 1. The principle of continuity and the 'paradox' of Leibnizian mathematics (by Serfati, Michel) 5. 2. Geometrization or mathematization: Christiaan
Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century
TLDR
Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period with a sophisticated picture of the subtle dependencies between technical development and philosophical reflection.
...
...