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  1. Differentials, higher-order differentials and the derivative in the Leibnizian calculus.H. J. M. Bos - 1974 - Archive for History of Exact Sciences 14 (1):1-90.
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  • (1 other version)Leibniz's Philosophy of Logic and Language.Hidé Ishiguro - 1972 - New York: Cambridge University Press.
    This is the second edition of an important introduction to Leibniz's philosophy of logic and language first published in 1972. It takes issue with several traditional interpretations of Leibniz while revealing how Leibniz's thought is related to issues of great interest in current logical theory. For this new edition, the author has added new chapters on infinitesimals and conditionals as well as taking account of reviews of the first edition.
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  • Non-standard Analysis.Gert Heinz Müller - 2016 - Princeton University Press.
    Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...)
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  • Ian Hacking, Why Is There Philosophy of Mathematics at All? [REVIEW]Max Harris Siegel - forthcoming - Mind 124.
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  • Why is there Philosophy of Mathematics AT ALL?Ian Hacking - 2011 - South African Journal of Philosophy 30 (1):1-15.
    Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...)
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  • Stevin Numbers and Reality.Karin Usadi Katz & Mikhail G. Katz - 2012 - Foundations of Science 17 (2):109-123.
    We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.
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  • (2 other versions)Ontological relativity.W. V. O. Quine - 1968 - Journal of Philosophy 65 (7):185-212.
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  • What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
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  • The Wake of Berkeley's Analyst: Rigor Mathematicae?David Sherry - 1987 - Studies in History and Philosophy of Science Part A 18 (4):455.
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  • Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond. [REVIEW]Mikhail G. Katz & David Sherry - 2013 - Erkenntnis 78 (3):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 to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...)
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  • Leibniz on the Parallel Postulate and the Foundations of Geometry: The Unpublished Manuscripts.Vincenzo De Risi - 2016 - New York/London: Birkhäuser.
    This book offers a general introduction to the geometrical studies of Gottfried Wilhelm Leibniz and his mathematical epistemology. In particular, it focuses on his theory of parallel lines and his attempts to prove the famous Parallel Postulate. Furthermore it explains the role that Leibniz’s work played in the development of non-Euclidean geometry. The first part is an overview of his epistemology of geometry and a few of his geometrical findings, which puts them in the context of the 17th-century studies on (...)
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  • Cauchy's Continuum.Karin U. Katz & Mikhail G. Katz - 2011 - Perspectives on Science 19 (4):426-452.
    One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...)
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  • Interpreting the Infinitesimal Mathematics of Leibniz and Euler.Jacques Bair, Piotr Błaszczyk, Robert Ely, Valérie Henry, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, Patrick Reeder, David M. Schaps, David Sherry & Steven Shnider - 2017 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 48 (2):195-238.
    We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...)
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  • The relation between philosophy of science and history of science.Marx W. Wartofsky - 1976 - In R. S. Cohen, P. K. Feyerabend & M. Wartofsky (eds.), Essays in Memory of Imre Lakatos. Reidel. pp. 717--737.
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  • (2 other versions)Ontological relativity: The Dewey lectures 1969.Willard Van Orman Quine - 1968 - Journal of Philosophy 65 (7):185-212.
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  • Infinitesimals, Imaginaries, Ideals, and Fictions.David Sherry & Mikhail Katz - 2012 - Studia Leibnitiana 44 (2):166-192.
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  • Geometry and analysis in Euler’s integral calculus.Giovanni Ferraro, Maria Rosaria Enea & Giovanni Capobianco - 2017 - Archive for History of Exact Sciences 71 (1):1-38.
    Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his (...)
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  • Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics.José Ferreirós - 2002 - Studia Logica 72 (3):437-440.
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  • A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos.Piotr Błaszczyk, Alexandre Borovik, Vladimir Kanovei, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze & David Sherry - 2016 - Logica Universalis 10 (4):393-405.
    We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...)
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  • Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts.Piotr Błaszczyk, Vladimir Kanovei, Mikhail G. Katz & David Sherry - 2017 - Foundations of Science 22 (1):125-140.
    Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...)
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  • (1 other version)Leibniz's philosophy of logic and language.Hidé Ishiguro - 1990 - New York: Cambridge University Press.
    This is the second edition of an important introduction to Leibniz's philosophy of logic and language first published in 1972. It takes issue with several traditional interpretations of Leibniz (by Russell amongst others) while revealing how Leibniz's thought is related to issues of great interest in current logical theory. For this new edition, the author has added new chapters on infinitesimals and conditionals as well as taking account of reviews of the first edition.
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  • Labyrinth of Thought. A history of set theory and its role in modern mathematics.Jose Ferreiros - 2001 - Basel, Boston: Birkhäuser Verlag.
    Review by A. Kanamori, Boston University (author of The Higher Infinite), review in The Bulletin of Symbolic Logic: “Notwithstanding and braving the daunting complexities of this labyrinth, José Ferreirós has written a magisterial account of the history of set theory which is panoramic, balanced and engaging. Not only does this book synthesize much previous work and provide fresh insights and points of view, but it also features a major innovation, a full-fledged treatment of the emergence of the set-theoretic approach in (...)
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  • (1 other version)Leibniz's Philosophy of Logic and Language.Fabrizio Mondadori & Hide Ishiguro - 1975 - Philosophical Review 84 (1):140.
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  • Galileo’s quanti: understanding infinitesimal magnitudes.Tiziana Bascelli - 2014 - Archive for History of Exact Sciences 68 (2):121-136.
    In On Local Motion in the Two New Sciences, Galileo distinguishes between ‘time’ and ‘quanto time’ to justify why a variation in speed has the same properties as an interval of time. In this essay, I trace the occurrences of the word quanto to define its role and specific meaning. The analysis shows that quanto is essential to Galileo’s mathematical study of infinitesimal quantities and that it is technically defined. In the light of this interpretation of the word quanto, Evangelista (...)
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  • The calculus as algebraic analysis: Some observations on mathematical analysis in the 18th century.Craig G. Fraser - 1989 - Archive for History of Exact Sciences 39 (4):317-335.
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  • Why is There Philosophy of Mathematics at All?Ian Hacking - 2014 - New York: Cambridge University Press.
    This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that (...)
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  • The new science of motion: A study of Galileo's De motu locali.Winifred L. Wisan - 1974 - Archive for History of Exact Sciences 13 (2-3):103-306.
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  • Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania.Tiziana Bascelli, Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, David M. Schaps & David Sherry - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (1):117-147.
    Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...)
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  • Ten Misconceptions from the History of Analysis and Their Debunking.Piotr Błaszczyk, Mikhail G. Katz & David Sherry - 2013 - Foundations of Science 18 (1):43-74.
    The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...)
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  • Leibniz's Philosophy of Logic and Language.Hideko Ishiguro - 1974 - Philosophy East and West 24 (3):376-378.
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  • A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography.Karin Usadi Katz & Mikhail G. Katz - 2012 - Foundations of Science 17 (1):51-89.
    We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.
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  • The Invention of the Decimal Fractions and the Application of the Exponential Calculus by Immanuel Bonfils of Tarascon.George Sarton & Solomon Gandz - 1936 - Isis 25 (1):16-45.
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  • Leibniz's Philosophy of Logic and Language.L. E. Loemker - 1974 - Philosophical Quarterly 24 (95):170-172.
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  • Definite values of infinite sums: Aspects of the foundations of infinitesimal analysis around 1820.Detlef Laugwitz - 1989 - Archive for History of Exact Sciences 39 (3):195-245.
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  • Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond.Mikhail G. Katz, David M. Schaps & Steven Shnider - 2013 - Perspectives on Science 21 (3):283-324.
    Adequality, or παρισóτης (parisotēs) in the original Greek of Diophantus 1 , is a crucial step in Fermat’s method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s collected works (1891, pp. 133–172). The first article, Methodus ad Disquirendam Maximam et Minimam 2 , opens with a summary of an algorithm for finding the maximum or minimum value of an (...)
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  • A Cauchy-Dirac Delta Function.Mikhail G. Katz & David Tall - 2013 - Foundations of Science 18 (1):107-123.
    The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.
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  • Charles Méray et la notion de limite.Pierre Dugac - 1970 - Revue d'Histoire des Sciences 23 (4):333-350.
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  • On Cauchy's notion of infinitesimal.Nigel Cutland, Christoph Kessler, Ekkehard Kopp & David Ross - 1988 - British Journal for the Philosophy of Science 39 (3):375-378.
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  • Leibniz’s syncategorematic infinitesimals.Richard T. W. Arthur - 2013 - Archive for History of Exact Sciences 67 (5):553-593.
    In 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 fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, (...)
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  • Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus.Alexandre Borovik & Mikhail G. Katz - 2012 - Foundations of Science 17 (3):245-276.
    Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...)
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  • Leibniz on The Elimination of Infinitesimals.Douglas M. Jesseph - 2015 - In G.W. Leibniz, Interrelations Between Mathematics and Philosophy. Springer Verlag.
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