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  1. The Logic of Relatives.Charles S. Peirce - 1897 - The Monist 7 (2):161-217.
  • Peano's Axioms and Models of Arithmetic.Solomon Feferman - 1957 - Journal of Symbolic Logic 22 (3):306-306.
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  • Infinitesimals as an Issue of Neo-Kantian Philosophy of Science.Thomas Mormann & Mikhail Katz - 2013 - Hopos: The Journal of the International Society for the History of Philosophy of Science (2):236-280.
    We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...)
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  • Perceiving the Infinite and the Infinitesimal World: Unveiling and Optical Diagrams in Mathematics. [REVIEW]Lorenzo Magnani & Riccardo Dossena - 2005 - Foundations of Science 10 (1):7-23.
    Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example, they actually reveal the behavior of a real function not “close to” a point (as in the standard limit theory) but “in” the point. We are interested (...)
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  • Early Delta Functions and the Use of Infinitesimals in Research.Detlef Laugwitz - 1992 - Revue d'Histoire des Sciences 45 (1):115-128.
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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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  • Effective Moduli From Ineffective Uniqueness Proofs. An Unwinding of de La Vallée Poussin's Proof for Chebycheff Approximation.Ulrich Kohlenbach - 1993 - Annals of Pure and Applied Logic 64 (1):27-94.
    Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Annals of Pure and Applied Logic 64 27–94.We consider uniqueness theorems in classical analysis having the form u ε U, v1, v2 ε Vu = 0 = G→v 1 = v2), where U, V are complete separable metric spaces, Vu is compact in V and G:U x V → is a constructive function.If is proved by arithmetical means from analytical assumptions x (...)
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  • Leibniz's Rigorous Foundation of Infinitesimal Geometry by Means of Riemannian Sums.Eberhard Knobloch - 2002 - Synthese 133 (1-2):59 - 73.
    In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...)
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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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  • 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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  • 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 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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  • 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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  • The Fan Theorem and Unique Existence of Maxima.Josef Berger, Douglas Bridges & Peter Schuster - 2006 - Journal of Symbolic Logic 71 (2):713 - 720.
    The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.
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  • Mlle Camille Bos.[author unknown] - 1907 - Revue Philosophique de la France Et de l'Etranger 64:660-660.
     
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  • An Omniscience Principle, the König Lemma and the Hahn‐Banach Theorem.Hajime Ishihara - 1990 - Mathematical Logic Quarterly 36 (3):237-240.
  • An Omniscience Principle, the König Lemma and the Hahn-Banach Theorem.Hajime Ishihara - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (3):237-240.
  • Mathematical Constructivism in Spacetime.Geoffrey Hellman - 1998 - British Journal for the Philosophy of Science 49 (3):425-450.
    To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...)
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  • The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of Non-Archimedean Systems of Magnitudes.Philip Ehrlich - 2006 - Archive for History of Exact Sciences 60 (1):1-121.
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  • Sequences of Real Functions on [0, 1] in Constructive Reverse Mathematics.Hannes Diener & Iris Loeb - 2009 - Annals of Pure and Applied Logic 157 (1):50-61.
    We give an overview of the role of equicontinuity of sequences of real-valued functions on [0,1] and related notions in classical mathematics, intuitionistic mathematics, Bishop’s constructive mathematics, and Russian recursive mathematics. We then study the logical strength of theorems concerning these notions within the programme of Constructive Reverse Mathematics. It appears that many of these theorems, like a version of Ascoli’s Lemma, are equivalent to fan-theoretic principles.
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  • Constructive Functional Analysis.D. S. Bridges & Peter Zahn - 1982 - Journal of Symbolic Logic 47 (3):703-705.
  • 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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  • Foundations of Constructive Analysis.John Myhill - 1972 - Journal of Symbolic Logic 37 (4):744-747.
  • Constructive Analysis.Errett Bishop & Douglas Bridges - 1987 - Journal of Symbolic Logic 52 (4):1047-1048.
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  • Brouwer's Fan Theorem and Unique Existence in Constructive Analysis.Josef Berger & Hajime Ishihara - 2005 - Mathematical Logic Quarterly 51 (4):360-364.
    Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. Furthermore, (...)
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  • What Numbers Could Not Be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
  • 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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  • Nonstandard Analysis and Constructivism?Frank Wattenberg - 1988 - Studia Logica 47 (3):303 - 309.
    The purpose of this paper is to investigate some problems of using finite (or *finite) computational arguments and of the nonstandard notion of an infinitesimal. We will begin by looking at the canonical example illustrating the distinction between classical and constructive analysis, the Intermediate Value Theorem.
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  • Hidden Lemmas in Euler's Summation of the Reciprocals of the Squares.Curtis Tuckey & Mark McKinzie - 1997 - Archive for History of Exact Sciences 51 (1):29-57.
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  • Cauchy et Bolzano.H. Sinaceur - 1973 - Revue d'Histoire des Sciences 26 (2):97-112.
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  • Zeno's Metrical Paradox Revisited.David M. Sherry - 1988 - Philosophy of Science 55 (1):58-73.
    Professor Grünbaum's much-discussed refutation of Zeno's metrical paradox turns out to be ad hoc upon close examination of the relevant portion of measure theory. Although the modern theory of measure is able to defuse Zeno's reasoning, it is not capable of refuting Zeno in the sense of showing his error. I explain why the paradox is not refutable and argue that it is consequently more than a mere sophism.
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  • Unique Solutions.Peter Schuster - 2006 - Mathematical Logic Quarterly 52 (6):534-539.
    It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. (...)
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  • A Nonstandard Proof of a Lemma From Constructive Measure Theory.David A. Ross - 2006 - Mathematical Logic Quarterly 52 (5):494-497.
    Suppose that fn is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that equation imageT fn < T f0. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that equation imagefn < f0. This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves (...)
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  • Non-Standard Analysis.Abraham Robinson - 1961 - North-Holland Publishing Co..
    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 (...)
  • A Definable Nonstandard Model Of The Reals.Vladimir Kanovei & Saharon Shelah - 2004 - Journal of Symbolic Logic 69 (1):159-164.
    We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals.
     
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  • Foundations of Constructive Analysis.Errett Bishop - 1967 - New York, NY, USA: Mcgraw-Hill.
    This book, Foundations of Constructive Analysis, founded the field of constructive analysis because it proved most of the important theorems in real analysis by constructive methods. The author, Errett Albert Bishop, born July 10, 1928, was an American mathematician known for his work on analysis. In the later part of his life Bishop was seen as the leading mathematician in the area of Constructive mathematics. From 1965 until his death, he was professor at the University of California at San Diego.
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  • A Primer of Infinitesimal Analysis.John L. Bell - 1998 - Cambridge University Press.
    This is the first elementary book to employ the concept of infinitesimals.
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  • Constructivism in Mathematics: An Introduction.A. S. Troelstra - 1988 - Elsevier.
    Provability, Computability and Reflection.
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  • Continuity and Infinitesimals.John L. Bell - unknown
    The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...)
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  • Mathematics Through Diagrams: Microscopes in Non-Standard and Smooth Analysis.R. Dossena & L. Magnani - 2007 - In L. Magnani & P. Li (eds.), Model-Based Reasoning in Science, Technology, and Medicine. Springer. pp. 193--213.
     
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  • Relationships Between Constructive, Predicative and Classical Systems of Analysis.Solomon Feferman - unknown
    Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to (...)
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  • Zeno.[author unknown] - 2006 - International Studies in Philosophy Monograph Series:77-81.
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  • Peirce's Clarifications of Continuity.Jérôme Havenel - 2008 - Transactions of the Charles S. Peirce Society 44 (1):pp. 86-133.
    This article aims to demonstrate that a careful examination of Peirce's original manuscripts shows that there are five main periods in Peirce's evolution in his mathematical and philosophical conceptualizations of continuity. The aim of this article is also to establish the relevance of Peirce's reflections on continuity for philosophers and mathematicians.
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