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  1. Is Hume’s Principle Analytic?Eamon Darnell & Aaron Thomas-Bolduc - forthcoming - Synthese 198 (1):169-185.
    The question of the analyticity of Hume’s Principle is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to be (...)
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  2. Hume's Principle and Entitlement: On the Epistemology of the Neo-Fregean Programme.Nikolaj Jang Lee Linding Pedersen - forthcoming - In Philip Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford University Press.
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  3. A Justification for the Quantificational Hume Principle.Chris Scambler - forthcoming - Erkenntnis:1-16.
    In recent work Bruno Whittle has presented a new challenge to the Cantorian idea that there are different infinite cardinalities. Most challenges of this kind have tended to focus on the status of the axioms of standard set theory; Whittle’s is different in that he focuses on the connection between standard set theory and intuitive concepts related to cardinality. Specifically, Whittle argues we are not in a position to know a principle I call the Quantificational Hume Principle, which connects the (...)
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  4. Hume’s Principle: A Plea for Austerity.Kai Michael Büttner - 2019 - Synthese 198 (4):3759-3781.
    According to Hume’s principle, a sentence of the form ⌜The number of Fs = the number of Gs⌝ is true if and only if the Fs are bijectively correlatable to the Gs. Neo-Fregeans maintain that this principle provides an implicit definition of the notion of cardinal number that vindicates a platonist construal of such numerical equations. Based on a clarification of the explanatory status of Hume’s principle, I will provide an argument in favour of a nominalist construal of numerical equations. (...)
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  5. Debunking, Supervenience, and Hume’s Principle.Mary Leng - 2019 - Canadian Journal of Philosophy 49 (8):1083-1103.
    Debunking arguments against both moral and mathematical realism have been pressed, based on the claim that our moral and mathematical beliefs are insensitive to the moral/mathematical facts. In the mathematical case, I argue that the role of Hume’s Principle as a conceptual truth speaks against the debunkers’ claim that it is intelligible to imagine the facts about numbers being otherwise while our evolved responses remain the same. Analogously, I argue, the conceptual supervenience of the moral on the natural speaks presents (...)
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  6. 7. The Second-Order Idealism of David Hume.William Boos - 2018 - In Metamathematics and the Philosophical Tradition. De Gruyter. pp. 233-305.
  7. Hume's Foundational Project in the Treatise.Miren Boehm - 2016 - European Journal of Philosophy 24 (1):55-77.
    In the Introduction to the Treatise Hume very enthusiastically announces his project to provide a secure and solid foundation for the sciences by grounding them on his science of man. And Hume indicates in the Abstract that he carries out this project in the Treatise. But most interpreters do not believe that Hume's project comes to fruition. In this paper, I offer a general reading of what I call Hume's ‘foundational project’ in the Treatise, but I focus especially on Book (...)
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  8. On the Nature, Status, and Proof of Hume’s Principle in Frege’s Logicist Project.Matthias Schirn - 2016 - In Sorin Costreie (ed.), Early Analytic Philosophy – New Perspectives on the Tradition. Springer Verlag.
    Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and (...)
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  9. 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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  10. A Dilemma for Neo-Fregeanism.Robert Trueman - 2014 - Philosophia Mathematica 22 (3):361-379.
    Neo-Fregeans need their stipulation of Hume's Principle — $NxFx=NxGx \leftrightarrow \exists R (Fx \,1\hbox {-}1_R\, Gx)$ — to do two things. First, it must implicitly define the term-forming operator ‘Nx…x…’, and second it must guarantee that Hume's Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume's Principle, and argue that while one sort of stipulation fixes a meaning for ‘Nx…x…’ and the other guarantees the truth of Hume's Principle, neither does both.
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  11. Hume on the Objects of Mathematics.Charles Echelbarger - 2013 - The European Legacy 18 (4):432-443.
    In this essay, I argue that Hume?s theory of Quantitative and Numerical Philosophical Relations can be interpreted in a way which allows mathematical knowledge to be about a body of objective and necessary truths, while preserving Hume?s nominalism and the basic principles of his theory of ideas. Attempts are made to clear up a number of obscure points about Hume?s claims concerning the abstract sciences of Arithmetic and Algebra by means of re-examining what he says and what he could comfortably (...)
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  12. Filling the Gaps in Hume’s Vacuums.Miren Boehm - 2012 - Hume Studies 38 (1):79-99.
    The paper addresses two difficulties that arise in Treatise 1.2.5. First, Hume appears to be inconsistent when he denies that we have an idea of a vacuum or empty space yet allows for the idea of an “invisible and intangible distance.” My solution to this difficulty is to develop the overlooked possibility that Hume does not take the invisible and intangible distance to be a distance at all. Second, although Hume denies that we have an idea of a vacuum, some (...)
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  13. Hume on Space, Geometry, and Diagrammatic Reasoning.Graciela De Pierris - 2012 - Synthese 186 (1):169-189.
    Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...)
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  14. Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
    This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper (...)
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  15. Standards of Equality and Hume's View of Geometry.Emil Badici - 2011 - Pacific Philosophical Quarterly 92 (4):448-467.
    It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view (...)
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  16. Numbers as Ontologically Dependent Objects Hume’s Principle Revisited.Robert Schwartzkopff - 2011 - Grazer Philosophische Studien 82 (1):353-373.
    Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.
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  17. Hume’s Principle, Beginnings.Albert Visser - 2011 - Review of Symbolic Logic 4 (1):114-129.
    In this note we derive Robinson???s Arithmetic from Hume???s Principle in the context of very weak theories of classes and relations.
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  18. Doubting the Truth of Hume’s Principle.Dušan Dožudić - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (3):269-287.
    Hume’s Principle states that for any two concepts, F and G, the number of Fs is identical to the number of Gs iff the Fs are one-one correlated with the Gs. Backed by second-order logic HP is supposed to be the starting point for the neo-logicist program of the foundations of arithmetic. The principle brings a number of formal and philosophical controversies. In this paper I discuss some arguments against it brought out by Trobok, as well as by Potter and (...)
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  19. Hume’s Big Brother: Counting Concepts and the Bad Company Objection.Roy T. Cook - 2009 - Synthese 170 (3):349 - 369.
    A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...)
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  20. Hume and Frege on Identity.John Perry - 2009 - Philosophical Studies 146 (3):413-423.
  21. Reason, Habit, and Applied Mathematics.David Sherry - 2009 - Hume Studies 35 (1-2):57-85.
    Hume describes the sciences as "noble entertainments" that are "proper food and nourishment" for reasonable beings (EHU 1.5-6; SBN 8).1 But mathematics, in particular, is more than noble entertainment; for millennia, agriculture, building, commerce, and other sciences have depended upon applying mathematics.2 In simpler cases, applied mathematics consists in inferring one matter of fact from another, say, the area of a floor from its length and width. In more sophisticated cases, applied mathematics consists in giving scientific theory a mathematical form (...)
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  22. On the Compatibility Between Euclidean Geometry and Hume’s Denial of Infinite Divisibility.Emil Badici - 2008 - Hume Studies 34 (2):231-244.
    It has been argued that Hume’s denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume’s thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume’s view of geometry is the distinction he draws between a precise and (...)
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  23. Human Identity and Bioethics by David DeGrazia.David B. Hershenov - 2008 - The National Catholic Bioethics Quarterly 8 (4):790-793.
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  24. 3. Geometry as Scientia and as Applied Science: Hume’s Empiricist Account of Geometry.Fred Wilson - 2008 - In The External World and Our Knowledge of It: Hume's Critical Realism, an Exposition and a Defence. University of Toronto Press. pp. 254-305.
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  25. On Some Consequences of the Definitional Unprovability of Hume's Principle.Luca Incurvati - 2007 - In Pierre Joray (ed.), Contemporary Perspectives on Logicism and the Foundations of Mathematics. CDRS.
  26. Artifice and the Natural World: Mathematics, Logic, Technology.James Franklin - 2006 - In K. Haakonssen (ed.), Cambridge History of Eighteenth-Century Philosophy. Cambridge University Press.
    If Tahiti suggested to theorists comfortably at home in Europe thoughts of noble savages without clothes, those who paid for and went on voyages there were in pursuit of a quite opposite human ideal. Cook's voyage to observe the transit of Venus in 1769 symbolises the eighteenth century's commitment to numbers and accuracy, and its willingness to spend a lot of public money on acquiring them. The state supported the organisation of quantitative researches, employing surveyors and collecting statistics to..
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  27. Hume’s Principle and Axiom V Reconsidered: Critical Reflections on Frege and His Interpreters.Matthias Schirn - 2006 - Synthese 148 (1):171 - 227.
    In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction (...)
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  28. Hume on Length, Space, and Geometry.E. Slownik - 2004 - Canadian Journal of Philosophy 34:355-74.
  29. Infinite Divisibility and Actual Parts in Hume’s Treatise.Thomas Holden - 2002 - Hume Studies 28 (1):3-25.
    According to a standard interpretation of Hume’s argument against infinite divisibility, Hume is raising a purely formal problem for mathematical constructions of infinite divisibility, divorced from all thought of the stuffing or filling of actual physical continua. I resist this. Hume’s argument must be understood in the context of a popular early modern account of the metaphysical status of the parts of physical quantities. This interpretation disarms the standard mathematical objections to Hume’s reasoning; I also defend it on textual and (...)
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  30. Hume = Small Hume.Jeffrey Ketland - 2002 - Analysis 62 (1):92–93.
    We can modify Hume’s Principle in the same manner that George Boolos suggested for modifying Frege’s Basic Law V. This leads to the principle Small Hume. Then, we can show that Small Hume is interderivable with Hume’s Principle.
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  31. David Hume's Critique of Infinity.Dale Jacquette - 2001 - Brill.
    The present work considers Hume's critique of infinity in historical context as a product of Enlightenment theory of knowledge, and assesses the prospects of ...
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  32. Is Hume's Principle Analytic?Crispin Wright - 2001 - In Bob Hale & Crispin Wright (eds.), Notre Dame Journal of Formal Logic. Oxford University Press. pp. 307-333.
    This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...)
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  33. The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics.Crispin Wright & Bob Hale - 2001 - Oxford: Clarendon Press.
    Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...)
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  34. Hume’s Finite Geometry: A Reply to Mark Pressman.Lorne Falkenstein - 2000 - Hume Studies 26 (1):183-185.
    In “Hume on Geometry and Infinite Divisibility in the Treatise”, H. Mark Pressman charges that “the geometry Hume presents in the Treatise faces a serious set of problems”. This may well be; however, at least one of the charges Pressman levels against Hume invokes a false dichotomy, and a second rests on a non sequitur.
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  35. On Finite Hume.Fraser Macbride - 2000 - Philosophia Mathematica 8 (2):150-159.
    Neo-Fregeanism contends that knowledge of arithmetic may be acquired by second-order logical reflection upon Hume's principle. Heck argues that Hume's principle doesn't inform ordinary arithmetical reasoning and so knowledge derived from it cannot be genuinely arithmetical. To suppose otherwise, Heck claims, is to fail to comprehend the magnitude of Cantor's conceptual contribution to mathematics. Heck recommends that finite Hume's principle be employed instead to generate arithmetical knowledge. But a better understanding of Cantor's contribution is achieved if it is supposed that (...)
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  36. Bayle, Hume y los molinos de viento.Andrés Páez - 2000 - Ideas Y Valores 49 (113):29-44.
    El análisis de los conceptos de espacio y tiempo es generalmente considerado uno de los aspectos menos satisfactorios de la obra de Hume. Kemp Smith ha demostrado que en esta sección del Tratado Hume estaba respondiendo a los argumentos que Pierre Bayle había utilizado para probar que el razonamiento humano siempre termina refutándose a sí mismo. En este ensayo expongo las falacias en los argumentos de Bayle, las cuales están basadas en una comprensión inadecuada del concepto de extensión. Hume no (...)
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  37. To Err is Humeant.Mark Wilson - 1999 - Philosophia Mathematica 7 (3):247-257.
    George Boolos, Crispin Wright, and others have demonstrated how most of Frege's treatment of arithmetic can be obtained from a second-order statement that Boolos dubbed ‘Hume's principle’. This note explores the historical evidence that Frege originally planned to develop a philosophical approach to numbers in which Hume's principle is central, but this strategy was abandoned midway through his Grundlagen.
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  38. From Inexactness to Certainty: The Change in Hume's Conception of Geometry.Vadim Batitsky - 1998 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 29 (1):1-20.
    Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science (...)
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  39. Is Hume's Principle Analytic?G. Boolos - 1998 - Logic, Logic, and Logic:301--314.
  40. Hume's Philosophy More Geometrico Demonstrata.Marina Frasca‐Spada - 1998 - British Journal for the History of Philosophy 6 (3):455 – 462.
    Don Garrett, Cognition and Commitment in Hume's Philosophy, New York and Oxford, Oxford University Press, 1997, pp. xiv + 270, Hb 40.00 ISBN 0-19-509721-1.
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  41. On the Harmless Impredicativity of N=('Hume's Principle').Crispin Wright - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 339--68.
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  42. Finitude and Hume's Principle.Richard G. Heck Jr - 1997 - Journal of Philosophical Logic 26 (6):589 - 617.
    The paper formulates and proves a strengthening of 'Frege's Theorem', which states that axioms for second-order arithmetic are derivable in second-order logic from Hume's Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. 'Finite Hume's Principle' also suffices (...)
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  43. Hume on Geometry and Infinite Divisibility in the Treatise.H. Mark Pressman - 1997 - Hume Studies 23 (2):227-244.
  44. Hume on the Certainty and Necessity of Arithmetic.Scott W. Gaylord - 1996 - Dissertation, The University of North Carolina at Chapel Hill
    David Hume's central distinction in the Treatise and Enquiry is between relations of ideas and matters of fact. Although most of the attention in the secondary literature is on the latter, I am directly concerned with analyzing Hume's characterization of relations of ideas. In particular, I analyze Hume's attempt to secure certainty and necessity within his empiricist system since these are the defining characteristics of Hume's perfect species of knowledge--arithmetic. The focus, then, is on how Hume justifies the special status (...)
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  45. The Psychologistic Foundations of Hume’s Critique of Mathematical Philosophy.Wayne Waxman - 1996 - Hume Studies 22 (1):123-167.
  46. Achievements and Fallacies in Hume's Account of Infinite Divisibility.James Franklin - 1994 - Hume Studies 20 (1):85-101.
    Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...)
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  47. Hume and the Culture of Science in the Early Eighteenth Century. Barfoot - 1990 - In M. A. Stewart (ed.), Studies in the Philosophy of the Scottish Enlightenment. Oxford University Press. pp. 155.
  48. Hume on Infinite Divisibility.Donald L. M. Baxter - 1988 - History of Philosophy Quarterly 5 (2):133-140.
    Hume seems to argue unconvincingly against the infinite divisibility of finite regions of space. I show that his conclusion is entailed by respectable metaphysical principles which he held. One set of principles entails that there are partless (unextended) things. Another set entails that these cannot be ordered so that an infinite number of them compose a finite interval.
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  49. Kant's Misrepresentations of Hume's Philosophy of Mathematics in the Prolegomena.Mark Steiner - 1987 - Hume Studies 13 (2):400-410.
  50. Coerenza e realtà: la geometria in Hume.Marina Frasca Spada - 1986 - Rivista di Storia Della Filosofia 41 (4):675.
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