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  1. Andrew Aberdein (2006). Introduction to the New Edition. In The Elements: Books I-XIII by Euclid. Barnes & Noble.
  2. Alice Ambrose (1955). Wittgenstein on Some Questions in Foundations of Mathematics. Journal of Philosophy 52 (8):197-214.
  3. Irving H. Anellis (2009). Review of D. M. Gabbay and J. Woods (Eds.), Handbook of the History of Logic, Volume 3: The Rise of Modern Logic From Leibniz to Frege. [REVIEW] Transactions of the Charles S. Peirce Society 45 (3):pp. 456-464.
  4. Andrew W. Appel (ed.) (2012). Alan Turing's Systems of Logic: The Princeton Thesis. Princeton University Press.
  5. H. S. Arsen (2012). A Case For The Utility Of The Mathematical Intermediates. Philosophia Mathematica 20 (2):200-223.
    Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...)
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  6. R. F. Atkinson (1960). Hume on Mathematics. Philosophical Quarterly 10 (39):127-137.
  7. John Bell, Hermann Weyl's Later Philosophical Views: His Divergence From Husserl.
    In what seems to have been his last paper, Insight and Reflection (1954), Hermann Weyl provides an illuminating sketch of his intellectual development, and describes the principal influences—scientific and philosophical—exerted on him in the course of his career as a mathematician. Of the latter the most important in the earlier stages was Husserl’s phenomenology. In Weyl’s work of 1918-22 we find much evidence of the great influence Husserl’s ideas had on Weyl’s philosophical outlook—one need merely glance through the pages of (...)
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  8. Patricia A. Blanchette (2007). Frege on Consistency and Conceptual Analysis. Philosophia Mathematica 15 (3):321-346.
    Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent (...)
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  9. Manuel Bremer, Frege's Basic Law V and Cantor's Theorem.
    The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...)
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  10. Robert Sherrick Brumbaugh (1942). The Role of Mathematics in Plato's Dialectic. Chicago.
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  11. John P. Burgess (1993). Hintikka Et Sandu Versus Frege in Re Arbitrary Functions. Philosophia Mathematica 1 (1):50-65.
    Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.
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  12. Paola Cantù (2010). Aristotle's Prohibition Rule on Kind-Crossing and the Definition of Mathematics as a Science of Quantities. Synthese 174 (2):225 - 235.
    The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...)
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  13. Paola Cantù (2010). Grassmann’s Epistemology: Multiplication and Constructivism. In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context.
    The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity conditions required (...)
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  14. Paola Cantù (2010). The Role of Epistemological Models in Veronese's and Bettazzi's Theory of Magnitudes. In M. D'Agostino, G. Giorello, F. Laudisa, T. Pievani & C. Sinigaglia (eds.), New Essays in Logic and Philosophy of Science. College Publications.
    The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way epistemology (...)
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  15. Paola Cantù & Schlaudt (2013). General Introduction. Philosophia Scientiæ 17 (17-1).
    1 The epistemology of Otto Hölder This special issue is devoted to the philosophical ideas developed by Otto Hölder (1859-1937), a mathematician who made important contributions to analytic functions and group theory. Hölder’s substantial work on the foundations of mathematics and the general philosophical conception outlined in this work are, however, still largely unknown. Up to the present, philosophical interest in Hölder’s work has been limited to his axiomatic formulation of a theory of..
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  16. Paola Cantù & De Zan Mauro (2009). Life and Works of Giovanni Vailati. In Arrighi Claudia, Cantù Paola, De Zan Mauro & Suppes Patrick (eds.), Life and Works of Giovanni Vailati. CSLI Publications.
    The paper introduces Vailati’s life and works, investigating Vailati’s education, the relation to Peano and his school, and the interest for pragmatism and modernism. A detailed analysis of Vailati’s scientific and didactic activities, shows that he held, like Peano, a a strong interest for the history of science and a pluralist, anti-dogmatic and anti-foundationalist conception of definitions in mathematics, logic and philosophy of language. Vailati’s understanding of mathematical logic as a form of pragmatism is not a faithful interpretation of Peano’s (...)
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  17. Emily Carson (2006). Review: Pierobon, Kant Et les Mathématiques: La Conception Kantienne des Mathématiques [Kant and Mathematics: The Kantian Conception of Mathematics]. [REVIEW] Philosophia Mathematica 14 (3):370-378.
  18. Emily Carson (2004). Metaphysics, Mathematics and the Distinction Between the Sensible and the Intelligible in Kant's Inaugural Dissertation. Journal of the History of Philosophy 42 (2):165-194.
    In this paper I argue that Kant's distinction in the Inaugural Dissertation between the sensible and the intelligible arises in part out of certain open questions left open by his comparison between mathematics and metaphysics in the Prize Essay. This distinction provides a philosophical justification for his distinction between the respective methods of mathematics and metaphysics and his claim that mathematics admits of a greater degree of certainty. More generally, this illustrates the importance of Kant's reflections on mathematics for the (...)
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  19. Emily Carson (1999). Kant on the Method of Mathematics. Journal of the History of Philosophy 37 (4):629-652.
  20. P. Cassou-Nogues (2013). Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, Trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (Pbk); 978-1-44114656-4 (Hbk); 978-1-44114433-1 (Pdf E-Bk); 978-1-44114654-0 (Epub E-Bk). Pp. Xlii + 310. [REVIEW] Philosophia Mathematica 21 (3):411-416.
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  21. Pierre Cassou-Nogués (2006). Signs, Figures and Time: Cavaillès on “Intuition” in Mathematics. Theoria 21 (1):89-104.
    This paper is concerned with Cavaillès’ account of “intuition” in mathematics. Cavaillès starts from Kant’s theory of constructions in intuition and then relies on various remarks by Hilbert to apply it to modern mathematics. In this context, “intuition” includes the drawing of geometrical figures, the use of algebraic or logical signs and the generation of numbers as, for example, described by Brouwer. Cavaillès argues that mathematical practice can indeed be described as “constructions in intuition” but that these constructions are not (...)
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  22. Stefania Centrone (2010). Logic and Philosophy of Mathematics in the Early Husserl. Springer.
    This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ...
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  23. Daniele Chiffi (2012). Kurt Gödel: Philosophical Explorations: History and Theory. Aracne.
  24. Daniel Cohnitz (2008). Ørsteds „Gedankenexperiment“: eine Kantianische Fundierung der Infinitesimalrechnung? Ein Beitrag zur Begriffsgeschichte von ‚Gedankenexperiment' und zur Mathematikgeschichte des frühen 19. Jahrhunderts. Kant-Studien 99 (4):407-433.
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  25. Elizabeth F. Cooke (2003). Peirce, Fallibilism, and the Science of Mathematics. Philosophia Mathematica 11 (2):158-175.
    In this paper, it will be shown that Peirce was of two minds about whether his scientific fallibilism, the recognition of the possibility of error in our beliefs, applied to mathematics. It will be argued that Peirce can and should hold a theory of fallibilism within mathematics, and that this position is more consistent with his overall pragmatic theory of inquiry and his general commitment to the growth of knowledge. But to make the argument for fallibilism in mathematics, Peirce's theory (...)
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  26. F. M. Cornford (1932). Mathematics and Dialectic in the Republic VI.-VII. (I.). Mind 41 (161):37-52.
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  27. F. M. Cornford (1932). Mathematics and Dialectic in the Republic VI.-VII. (II.). Mind 41 (162):173-190.
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  28. John Crossley (2008). Review of E. Menzler-Trott, Logic's Lost Genius: The Life of Gerhard Gentzen. [REVIEW] Australasian Journal of Logic 6:83-86.
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  29. Dirk Van Dalen (1995). Hermann Weyl's Intuitionistic Mathematics. Bulletin of Symbolic Logic 1 (2):145 - 169.
  30. O. Darrigol (2003). Number and Measure: Hermann Von Helmholtz at the Crossroads of Mathematics, Physics, and Psychology. Studies in History and Philosophy of Science Part A 34 (3):515-573.
    In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics (Grassmann / Du Bois), on the possibility of quantitative psychology (Fechner / Kries, Wundt / Zeller), and on the meaning of temperature measurement (Maxwell, Mach). Late nineteenth-century scrutinisers of the foundations of mathematics (Dedekind, Cantor, Frege, Russell) made little of Helmholtz's essay. Yet it inspired two mathematicians with an eye on (...)
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  31. Graciela De Pierris (2012). Hume on Space, Geometry, and Diagrammatic Reasoning. 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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  32. Shannon Dea (2006). &Quot;merely a Veil Over the Living Thought&Quot;: Mathematics and Logic in Peirce's Forgotten Spinoza Review. Transactions of the Charles S. Peirce Society 42 (4):501-517.
    This paper considers Peirce's striking remarks about mathematics in a little-known review of Spinoza's Ethics within the larger context of his philosophy of mathematics. It argues that, for Peirce, true mathematical reasoning is always at the vanguard of thought, and resists logical demonstration. Through diagrammatic thought and her pre-theoretical innate faculty of logica utens, the great mathematician is able to see a theorem as true long before the logical apparatus necessary to demonstrate its truth exists. For Peirce, true (theoremic) mathematical (...)
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  33. William Demopoulos (2013). Logicism and its Philosophical Legacy. Cambridge University Press.
    Frege's analysis of arithmetical knowledge -- Carnap's thesis -- On extending 'empiricism, semantics and ontology' to the realism-instrumentalism controversy -- Carnap's analysis of realism -- Bertrand Russell's The analysis of matter: its historical context and contemporary interest (with Michael Friedman) -- On the rational reconstruction of our theoretical knowledge -- Three views of theoretical knowledge -- Frege and the rigorization of analysis -- The philosophical basis of our knowledge of number -- The 1910 Principia's theory of functions and classes -- (...)
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  34. M. Detlefsen (1998). Walter van Stigt. Brouwer's Intuitionism. Amsterdam: North-Holland Publishing Co., 1990. Pp. Xxvi + 530. ISBN 0-444-88384-3 (Cloth). [REVIEW] Philosophia Mathematica 6 (2):235-241.
  35. Michael Detlefsen (1995). Review of J. Folina, Poincare and the Philosophy of Mathematics. [REVIEW] Philosophia Mathematica 3 (2):208-218.
  36. Michael Detlefsen (1993). Poincaré Vs. Russell on the Rôle of Logic in Mathematicst. Philosophia Mathematica 1 (1):24-49.
    In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This (...)
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  37. Émilie du Châtelet & Lydia Patton (2014). On the Divisibility and Subtlety of Matter. In L. Patton (ed.), Philosophy, Science, and History. Routledge. 332-42.
    Translation for this volume by Lydia Patton of Chapter 9 (pages 179-200) of Émilie du Châtelet's Institutions de Physique (Foundations of Physics). Original publication date 1750. Paris: Chez Prault Fils.
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  38. Scott Edgar, Hermann Cohen's Principle of the Infinitesimal Method and its History: A Rationalist Interpretation.
    This paper defends a Leibnizian rationalist interpretation of Hermann Cohen’s Principle of the Infinitesimal Method and its History (1883). The first half of the paper identifies Cohen’s various different philosophical aims in the PIM. It argues that they are unified by the fact that Cohen’s arguments for addressing those aims all depend on a single shared premise. That linchpin premise is the claim that mathematical natural science can represent individual objects only if it also represents infinitesimal magnitudes. The second half (...)
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  39. William Bragg Ewald (2005). From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. OUP Oxford.
    Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives to show (...)
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  40. William Bragg Ewald & William Bragg Ewald (2005). From Kant to Hilbert Volume 2. OUP Oxford.
    Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics--algebra, geometry, number theory, analysis, logic and set theory--with narratives to show (...)
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  41. José Ferreiros Domínguez (1992). Sobre los orígenes de la Matemática abstracta. Theoria 7 (1-2):473-498.
    Dedekind used to refer to Riemann as his main model concerning mathematical methodology, particularly regarding the use of abstract notions as a basis for mathematical theories. So, in passages written in 1876 and 1895 he compared his approach to ideal theory with Riemann’s theory of complex functions. In this paper, I try to make sense of those declarations, showing the role of abstract notions in Riemann’s function theory, its influence on Dedekind, and the importance of the methodological principle of avoiding (...)
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  42. Juliet Floyd (2010). On Being Surprised: Wittgenstein on Aspect-Perception, Logic, and Mathematics. In William Day & Víctor J. Krebs (eds.), Seeing Wittgenstein Anew. Cambridge University Press.
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  43. Juliet Floyd (2004). Wittgenstein on Philosophy of Logic and Mathematics. Graduate Faculty Philosophy Journal 25 (2):227-287.
    A survey of Wittgenstein's writings on logic and mathematics; an analytical bibliography of contemporary articles on rule-following, social constructivism, Wittgenstein, Godel, and constructivism is appended. Various historical accounts of the nature of mathematical knowledge glossed over the effects of linguistic expression on our understanding of its status and content. Initially Wittgenstein rejected Frege's and Russell's logicism, aiming to operationalize the notions of logical consequence, necessity and sense. Vienna positivists took this to place analysis of meaning at the heart of philosophy, (...)
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  44. Juliet Floyd (2001). Prose Versus Proof: Wittgenstein on Gödel, Tarski and Truth. Philosophia Mathematica 9 (3):280-307.
    A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's semantical (...)
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  45. Juliet Floyd (1995). On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle. In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. 373-426.
  46. Juliet Floyd (1991). Wittgenstein on 2, 2, 2 ...: The Opening of Remarks on the Foundations of Mathematics. Synthese 87 (1):143 - 180.
  47. Juliet Floyd & Hilary Putnam (2000). A Note on Wittgenstein's "Notorious Paragraph" About the Gödel Theorem. Journal of Philosophy 97 (11):624-632.
  48. Brandon Fogel (2009). Review of Hermann Weyl, Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics. [REVIEW] Notre Dame Philosophical Reviews 2009 (11).
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  49. Giambattista Formica (2013). Da Hilbert a von Neumann: La Svolta Pragmatica Nell'assiomatica. Carocci.
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  50. Sébastien Gandon (2008). Which Arithmetization for Which Logicism? Russell on Relations and Quantities in The Principles of Mathematics. History and Philosophy of Logic 29 (1):1-30.
    This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible 'by logical principles from logical principles' does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV-V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...)
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