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Three Infinities in Early Modern Philosophy

Mind 128 (512):1117-1147 (2019)

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  1. A Note on Leibniz’s Argument Against Infinite Wholes.Mark van Atten & Mark Atten - 2015 - In Mark Atten (ed.), Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Cham: Springer Verlag. pp. 121-129.
    Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...)
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  • Mind-Body Interaction and Metaphysical Consistency: A Defense of Descartes.Eileen O' Neill - 1987 - Journal of the History of Philosophy 25 (2):227.
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  • Mind-body interaction and metaphysical consistency: A defense of Descartes.Eileen O'Neill - 1987 - Journal of the History of Philosophy 25 (2):227-245.
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  • In Good Company? On Hume’s Principle and the Assignment of Numbers to Infinite Concepts.Paolo Mancosu - 2015 - Review of Symbolic Logic 8 (2):370-410.
    In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...)
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  • Locke on Supposing a Substratum. Szabo - 2000 - Locke Studies 31:11-42.
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  • Innate Idea and the Infinite: The Case Of Locke and Descartes. Rogers - 1995 - Locke Studies 26:49-68.
    Pierre Gassendi, who did not like nonsense, said of the idea of infinity: ‘if someone calls something "infinite" he attributes to a thing which he does not grasp a label which he does not understand’. Gassendi’s is a harsh judgement for, surely, we all do quite cheerfully and successfully use the concept of infinity, and in a variety of contexts. Yet if Gassendi’s judgement is too hard it is easy enough to have sympathy with his claim. For it is a (...)
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  • Where do our ideas come from.Robert Merrihew Adams - 1975 - In Stephen P. Stich (ed.), Innate Ideas. Berkeley, CA, USA: University of California Press. pp. 71--87.
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  • Locke on Consciousness, Personal Identity and the Idea of Duration.Gideon Yaffe - 2011 - Noûs 45 (3):387-408.
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  • On infinite size.Bruno Whittle - 2015 - Oxford Studies in Metaphysics 9:3-19.
    This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The chapter challenges this, arguing (...)
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  • A Note on Leibniz's Argument Against Infinite Wholes.Mark van Atten - 2011 - British Journal for the History of Philosophy 19 (1):121-129.
    Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...)
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  • An Essay Concerning Human Understanding.H. R. Smart - 1925 - Philosophical Review 34 (4):413.
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  • The Ontic and the Iterative: Descartes on the Infinite and the Indefinite.Anat Schechtman - 2018 - In Igor Agostini, Richard T. W. Arthur, Geoffrey Gorham, Paul Guyer, Mogens Lærke, Yitzhak Y. Melamed, Ohad Nachtomy, Sanja Särman, Anat Schechtman, Noa Shein & Reed Winegar (eds.), Infinity in Early Modern Philosophy. Cham: Springer Verlag. pp. 27-44.
    Descartes’s metaphysics posits a sharp distinction between two types of non-finitude, or unlimitedness: whereas God alone is infinite, numbers, space, and time are indefinite. The distinction has proven difficult to interpret in a way that abides by the textual evidence and conserves the theoretical roles that the distinction plays in Descartes’s philosophy—in particular, the important role it plays in the causal proof for God’s existence in the Meditations. After formulating the interpretive task, I criticize extant interpretations of the distinction. I (...)
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  • Substance and Independence in Descartes.Anat Schechtman - 2016 - Philosophical Review 125 (2):155-204.
    Descartes notoriously characterizes substance in two ways: first, as an ultimate subject of properties ; second, as an independent entity. The characterizations have appeared to many to diverge on the definition as well as the scope of the notion of substance. For it is often thought that the ultimate subject of properties need not—and, in some cases, cannot—be independent. Drawing on a suite of historical, textual, and philosophical considerations, this essay argues for an interpretation that reconciles Descartes's two characterizations. It (...)
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  • The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.
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  • Locke and the Intuitionist Theory of Number.Richard Aaron & Philip Walters - 1965 - Philosophy 40 (153):197 - 206.
    The Purpose of this paper is to ask how far Locke can be said to have anticipated modern theories of number, particularly the intuitionist theory of Brouwer and Heyting. It has in mind Mr Edward E. Dawson's statement that Locke's account of number was not merely ‘a good effort in his own day’ but that ‘what Locke had to say really was quite fundamental, and a good deal of modern mathematics assumes his position, either explicitly or implicitly’. Mr Dawson thinks (...)
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  • A Tale of Two Thinkers, One Meeting, and Three Degrees of Infinity: Leibniz and Spinoza (1675–8).Ohad Nachtomy - 2011 - British Journal for the History of Philosophy 19 (5):935-961.
    The article presents Leibniz's preoccupation (in 1675?6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ?Leibniz's Problem? and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ?Spinoza's solution? is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish (...)
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  • Leibniz's Philosophy of Logic and Language.Fabrizio Mondadori & Hide Ishiguro - 1975 - Philosophical Review 84 (1):140.
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  • New Essays on Human Understanding.R. M. Mattern - 1984 - Philosophical Review 93 (2):315.
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  • Philosophy of mathematics and mathematical practice in the seventeenth century.Paolo Mancosu (ed.) - 1996 - New York: Oxford University Press.
    The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...)
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  • Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?Paolo Mancosu - 2009 - Review of Symbolic Logic 2 (4):612-646.
    Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...)
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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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  • Infinity and Continuity in Ancient and Medieval Thought.John Longeway - 1985 - Philosophical Review 94 (2):263.
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  • Leibniz on mathematics and the actually infinite division of matter.Samuel Levey - 1998 - Philosophical Review 107 (1):49-96.
    Mathematician and philosopher Hermann Weyl had our subject dead to rights.
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  • Infinity and continuity in ancient and medieval thought.Norman Kretzmann (ed.) - 1982 - Ithaca, N.Y.: Cornell University Press.
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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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  • Galileo and Leibniz: Different Approaches to Infinity.Eberhard Knobloch - 1999 - Archive for History of Exact Sciences 54 (2):87-99.
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  • Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes.Douglas Michael Jesseph - 1998 - Perspectives on Science 6 (1):6-40.
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  • Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism.Samuel Levey - 2008 - In Douglas Jesseph & Ursula Goldenbaum (eds.), Infinitesimal Differences: Controversies Between Leibniz and His Contemporaries. Walter de Gruyter.
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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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  • Leibniz on Infinite Numbers, Infinite Wholes, and Composite Substances.Adam Harmer - 2014 - British Journal for the History of Philosophy 22 (2):236-259.
    Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz’s argument against the world soul relies on his rejection of infinite number, and, as such, (...)
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  • The Infinite.Janet Folina & A. W. Moore - 1991 - Philosophical Quarterly 41 (164):348.
    Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.
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  • Locke and Descartes.Lisa Downing - 2015 - In Matthew Stuart (ed.), A Companion to Locke. Hoboken, NJ, USA: Wiley. pp. 100–120.
    In this chapter, John Locke's anti‐Cartesian stances on the difference between body and space, on whether the soul always thinks, on the possibility of thinking matter, all connect back to the basic opposition to Cartesian overreaching in regard to essences. The chapter presents a summary of Locke's anti‐Cartesianism, which seems to fit with his own representation of his Cartesian inheritance, which, notoriously, is that it is minimal, consisting only in anti‐scholasticism. The only acknowledgment that Locke wishes to give Descartes is (...)
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  • Locke’s Newtonianism and Lockean Newtonianism.Lisa J. Downing - 1997 - Perspectives on Science 5 (3):285-310.
    I explore Locke’s complex attitude toward the natural philosophy of his day by focusing on Locke’s own treatment of Newton’s theory of gravity and the presence of Lockean themes in defenses of Newtonian attraction/gravity by Maupertuis and other early Newtonians. In doing so, I highlight the inadequacy of an unqualified labeling of Locke as “mechanist” or “Newtonian.”.
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  • Locke on number and infinity.Edward E. Dawson - 1959 - Philosophical Quarterly 9 (37):302-308.
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  • Infinite Accumulations and Pantheistic Implications.Laurence Carlin - 1997 - The Leibniz Review 7:1-24.
    Throughout his early writings, Leibniz was concerned with developing an acceptable account of God's relationship to the created world. In some of these early writings, he endorsed the idea that this relationship was similar to the human soul's relationship to the body. Though he eventually came to reject this idea, theanima mundi thesis remained the topic of several essays and correspondences during his career, culminating in the correspondence with Clarke. At first glance,Leibniz's discussions of this thesis may seem less important (...)
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  • Who’s Afraid of Infinite Numbers?Gregory Brown - 1998 - The Leibniz Review 8:113-125.
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  • Who’s Afraid of Infinite Numbers?Gregory Brown - 1998 - The Leibniz Review 8:113-125.
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  • Leibniz on Wholes, Unities, and Infinite Number.Gregory Brown - 2000 - The Leibniz Review 10:21-51.
    One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...)
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  • Leibniz on Wholes, Unities, and Infinite Number.Gregory Brown - 2000 - The Leibniz Review 10:21-51.
    One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...)
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  • Leibniz's mathematical argument against a soul of the world.Gregory Brown - 2005 - British Journal for the History of Philosophy 13 (3):449 – 488.
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  • Infinity.José A. Benardete - 1964 - Oxford,: Clarendon Press.
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  • The Leibniz-Des Bosses Correspondence. [REVIEW]Philip Beeley - 2008 - The Leibniz Review 18:193-204.
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  • The Leibniz-Des Bosses Correspondence.Philip Beeley - 2008 - The Leibniz Review 18:193-204.
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  • The Leibniz-Des Bosses Correspondence. [REVIEW]Philip Beeley - 2008 - The Leibniz Review 18:193-204.
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  • Leibniz on Infinite Number, Infinite Wholes, and the Whole World.Richard Arthur - 2001 - The Leibniz Review 11:103-116.
    Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It (...)
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  • Leibniz on Infinite Number, Infinite Wholes, and the Whole World.Richard Arthur - 2001 - The Leibniz Review 11:103-116.
    Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It (...)
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  • Infinite Number and the World Soul; in Defence of Carlin and Leibniz.Richard Arthur - 1999 - The Leibniz Review 9:105-116.
    In last year’s Review Gregory Brown took issue with Laurence Carlin’s interpretation of Leibniz’s argument as to why there could be no world soul. Carlin’s contention, in Brown’s words, is that Leibniz denies a soul to the world but not to bodies on the grounds that “while both the world and [an] aggregate of limited spatial extent are infinite in multitude, the former, but not the latter, is infinite in respect of magnitude and hence cannot be considered a whole”. Brown (...)
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  • The Hypercategorematic Infinite.Maria Rosa Antognazza - 2015 - The Leibniz Review 25:5-30.
    This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss the issue of whether the (...)
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  • Locke on measurement.Peter R. Anstey - 2016 - Studies in History and Philosophy of Science Part A 60:70-81.
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  • Cavalieri's method of indivisibles.Kirsti Andersen - 1985 - Archive for History of Exact Sciences 31 (4):291-367.
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