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  1. Andrew Aberdein (2006). Introduction to the New Edition. In The Elements: Books I-XIII by Euclid. Barnes & Noble
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  2. Fabio Acerbi (2010). Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus. Science in Context 23 (2):151-186.
  3. Christy Ailman (2013). Mathematical Deduction by Induction. Gratia Eruditionis:4-12.
    In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotle’s philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek culture. (...)
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  4. Christy Ailman (2013). Mathematical Deduction by Induction. Gratia Eruditionis:4-12.
    In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotle’s philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek culture. (...)
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  5. Alice Ambrose (1955). Wittgenstein on Some Questions in Foundations of Mathematics. Journal of Philosophy 52 (8):197-214.
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  6. 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.
  7. Andrew W. Appel (ed.) (2012). Alan Turing's Systems of Logic: The Princeton Thesis. Princeton University Press.
    Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing, the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Gödel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. (...)
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  8. Brad Armendt (2005). Frank Plumpton Ramsey. In Sahotra Sarkar & Jessica Pfeifer (eds.), The Philosophy of Science: An Encyclopedia. Routledge 671-681.
    On the work of Frank Ramsey, emphasizing topics most relevant to philosophy of science.
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  9. 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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  10. Shigeyuki Atarashi (2015). Alison Walsh. Relations Between Logic and Mathematics in the Work of Benjamin and Charles S. Peirce. Boston: Docent Press, 2012. ISBN 978-098370046-3 . Pp. X + 314. [REVIEW] Philosophia Mathematica 23 (1):148-152.
  11. R. F. Atkinson (1960). Hume on Mathematics. Philosophical Quarterly 10 (39):127-137.
  12. André Bazzoni (2015). On the Concepts of Function and Dependence. Principia: An International Journal of Epistemology 19 (1):01-15.
    This paper briefly traces the evolution of the function concept until its modern set theoretic definition, and then investigates its relationship to the pre-formal notion of variable dependence. I shall argue that the common association of pre-formal dependence with the modern function concept is misconceived, and that two different notions of dependence are actually involved in the classic and the modern viewpoints, namely effective and functional dependence. The former contains the latter, and seems to conform more to our pre-formal conception (...)
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  13. 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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  14. Jan Berg (1962). Bolzano's Logic. Stockholm, Almqvist & Wiksell.
  15. K. Berka (1976). Contemporary State of Research on Bolzano. Filosoficky Casopis 24 (5):705-720.
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  16. Karel Berka (1981). Bernard Bolzano. Horizont.
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  17. 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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  18. 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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  19. Robert Sherrick Brumbaugh (1942). The Role of Mathematics in Plato's Dialectic. Chicago.
  20. 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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  21. 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 (...)
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  22. 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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  23. 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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  24. 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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  25. 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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  26. Emily Carson (2006). Review: Pierobon, Kant Et les Mathématiques: La Conception Kantienne des Mathématiques. [REVIEW] Philosophia Mathematica 14 (3):370-378.
    This book is a welcome contribution to the literature on Kant's philosophy of mathematics in two particular respects. First, the author systematically traces the development of Kant's thought on mathematics from the very early pre-Critical writings through to the Critical philosophy. Secondly, it puts forward a challenge to contemporary Anglo-Saxon commentators on Kant's philosophy of mathematics which merits consideration.A central theme of the book is that an adequate understanding of Kant's pronouncements on mathematics must begin with the recognition that mathematics (...)
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  27. 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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  28. Emily Carson (1999). Kant on the Method of Mathematics. Journal of the History of Philosophy 37 (4):629-652.
  29. 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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  30. 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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  31. 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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  32. Daniele Chiffi (2012). Kurt Gödel: Philosophical Explorations: History and Theory. Aracne.
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  33. 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.
  34. Roy T. Cook (2013). Appendix: How to Read Grundgesetze. In Gottlob Frege (ed.), Basic Laws of Arithmetic, Derived Using Concept-Script: Volumes I & Ii. Oxford University Press
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  35. 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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  36. John Corcoran (2014). Formalizing Euclid’s First Axiom. Bulletin of Symbolic Logic 20:404-405.
    Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...)
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  37. John Corcoran (1995). Semantic Arithmetic: A Preface. Agora 14 (1):149-156.
    SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad subject which begins when (...)
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  38. John Corcoran (1988). REVIEW OF 1988. Saccheri, G. Euclides Vindicatus (1733), Edited and Translated by G. B. Halsted, 2nd Ed. (1986), in Mathematical Reviews MR0862448. 88j:01013. MATHEMATICAL REVIEWS 88 (J):88j:01013.
    Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...)
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  39. John Corcoran (1978). Corcoran Recommends Hambourger on the Frege-Russell Number Definition. MATHEMATICAL REVIEWS 56.
    It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...)
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  40. John Corcoran (1971). Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value. Journal of Structural Learning 3 (2):1-16.
    1971. Discourse Grammars and the Structure of Mathematical Reasoning II: The Nature of a Correct Theory of Proof and Its Value, Journal of Structural Learning 3, #2, 1–16. REPRINTED 1976. Structural Learning II Issues and Approaches, ed. J. Scandura, Gordon & Breach Science Publishers, New York, MR56#15263. -/- This is the second of a series of three articles dealing with application of linguistics and logic to the study of mathematical reasoning, especially in the setting of a concern for improvement of (...)
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  41. F. M. Cornford (1932). Mathematics and Dialectic in the Republic VI.-VII. (I.). Mind 41 (161):37-52.
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  42. F. M. Cornford (1932). Mathematics and Dialectic in the Republic VI.-VII. (II.). Mind 41 (162):173-190.
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  43. Sorin Costreie (2012). Frege on Identity: The Transition From Begriffsschrift to Über Sinn Und Bedeutung. Logos and Episteme 3 (3):297-308.
    The goal of the paper is to offer an explanation why Frege has changed his Begriffsschrift account of identity to the one presented in Über Sinn und Bedeutung. The main claim of the paper is that in order to better understand Frege’s motivation for the introduction of his distinction between sense and reference, which marks his change of views, one should place this change in its original setting, namely the broader framework of Frege’s fundamental preoccupations with the foundations of arithmetic (...)
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  44. 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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  45. Dirk Van Dalen (1995). Hermann Weyl's Intuitionistic Mathematics. Bulletin of Symbolic Logic 1 (2):145 - 169.
  46. 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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  47. 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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  48. Shannon Dea (2006). "Merely a Veil Over the Living Thought": 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 mathematical thought (...)
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  49. 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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  50. William Demopoulos & Peter Clark (2005). The Logicism of Frege, Dedekind, and Russell. In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press 129--165.
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