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  1. Introduction to the New Edition.Andrew Aberdein - 2006 - In The Elements: Books I-XIII by Euclid. Barnes & Noble.
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  2. Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus.Fabio Acerbi - 2010 - Science in Context 23 (2):151-186.
  3. Mathematical Deduction by Induction.Christy Ailman - 2013 - 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. Wittgenstein on Some Questions in Foundations of Mathematics.Alice Ambrose - 1955 - Journal of Philosophy 52 (8):197-214.
  5. Bertrand Russell's Theory of Numbers, 1896–1898.I. H. Anellis - 1987 - Epistemologia 10 (2):303-322.
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  6. 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]Irving H. Anellis - 2009 - Transactions of the Charles S. Peirce Society 45 (3):pp. 456-464.
  7. Thematic Files-Mathematics and Knowledge in the Renaissance->: Science and Mathematics According to 16th-Century Commentators of Proclus.Annarita Angelini - 2006 - Revue d'Histoire des Sciences 59 (2):265.
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  8. Mathematics as a Science of Quantities.Hippocrates George Apostle, Arnold M. Adelberg & Elizabeth A. Dobbs - 1991
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  9. Alan Turing's Systems of Logic: The Princeton Thesis.Andrew W. Appel (ed.) - 2012 - 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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  10. Imagination in Mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), The Routledge Handbook of Philosophy of Imagination. Routledge. pp. 463-477.
  11. Purity in Arithmetic: Some Formal and Informal Issues.Andrew Arana - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 315-336.
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  12. L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
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  13. Frank Plumpton Ramsey.Brad Armendt - 2005 - In Sahotra Sarkar & Jessica Pfeifer (eds.), The Philosophy of Science: An Encyclopedia. Routledge. pp. 671-681.
    On the work of Frank Ramsey, emphasizing topics most relevant to philosophy of science.
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  14. A Case For The Utility Of The Mathematical Intermediates.H. S. Arsen - 2012 - 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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  15. Euclid's "Elements" and its Prehistory.Benno Artmann - 1991 - Apeiron 24 (4):1 - 47.
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  16. Euclid's Elements and its Prehistory.Benno Artmann - 1991 - Apeiron 24 (4):1-48.
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  17. 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]Shigeyuki Atarashi - 2014 - Philosophia Mathematica 23 (1):148-152.
  18. Hume on Mathematics.R. F. Atkinson - 1960 - Philosophical Quarterly 10 (39):127-137.
    „My sole purpose in this paper is to try and correct what I take to be a common misinterpretation of Hume’s opinions on mathematics. I shall not enquire whether he was right or wrong in holding these opinions. Nor shall I offer opinions of my own.“.
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  19. On the Concepts of Function and Dependence.André Bazzoni - 2015 - 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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  20. Hermann Weyl's Later Philosophical Views: His Divergence From Husserl.John Bell - manuscript
    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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  21. Bolzano's Logic.Jan Berg - 1962 - Stockholm, Almqvist & Wiksell.
  22. Contemporary State of Research on Bolzano.K. Berka - 1976 - Filosoficky Casopis 24 (5):705-720.
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  23. Bernard Bolzano.Karel Berka - 1981 - Horizont.
  24. Frege on Consistency and Conceptual Analysis.Patricia A. Blanchette - 2007 - 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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  25. David Hilbert: Philosophy, Epistemology, and the Foundations of Physics. [REVIEW]Katherine Brading - 2014 - Metascience 23 (1):97-100.
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  26. Frege's Basic Law V and Cantor's Theorem.Manuel Bremer - manuscript
    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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  27. The Role of Mathematics in Plato's Dialectic.Robert Sherrick Brumbaugh - 1942 - Chicago: Chicago University Press.
  28. Hintikka Et Sandu Versus Frege in Re Arbitrary Functions.John P. Burgess - 1993 - 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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  29. Eudoxos Versus Dedekind.Piotr Błaszczyk - 2007 - Filozofia Nauki 2.
    All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. In the (...)
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  30. One, Two, Three… A Discussion on the Generation of Numbers in Plato’s Parmenides.Florin George Calian - 2015 - New Europe College:49-78.
    The paper argues for considering at least one of the arguments from the second part of the Parmenides, namely the argument of the generation of numbers, as being platonically genuine.
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  31. The Role of Epistemological Models in Veronese's and Bettazzi's Theory of Magnitudes.Paola Cantù - 2010 - 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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  32. Aristotle's Prohibition Rule on Kind-Crossing and the Definition of Mathematics as a Science of Quantities.Paola Cantù - 2010 - 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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  33. Grassmann’s Epistemology: Multiplication and Constructivism.Paola Cantù - 2010 - 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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  34. General Introduction.Paola Cantù & Schlaudt - 2013 - 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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  35. Life and Works of Giovanni Vailati.Paola Cantù & De Zan Mauro - 2009 - 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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  36. Review: Pierobon, Kant Et les Mathématiques: La Conception Kantienne des Mathématiques[REVIEW]Emily Carson - 2006 - 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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  37. Metaphysics, Mathematics and the Distinction Between the Sensible and the Intelligible in Kant's Inaugural Dissertation.Emily Carson - 2004 - 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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  38. Kant on the Method of Mathematics.Emily Carson - 1999 - Journal of the History of Philosophy 37 (4):629-652.
  39. Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, Trans. London and New York: Continuum, 2011. [REVIEW]P. Cassou-Nogues - 2013 - Philosophia Mathematica 21 (3):411-416.
    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.
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  40. Signs, Figures and Time: Cavaillès on “Intuition” in Mathematics.Pierre Cassou-Nogués - 2006 - 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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  41. Images and Logic of the Light Cone: Tracking Robb’s Postulational Turn in Physical Geometry.Jordi Cat - 2016 - Revista de Humanidades de Valparaíso 4 (8):39-100.
    Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research (...)
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  42. Logic and Philosophy of Mathematics in the Early Husserl.Stefania Centrone - 2010 - 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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  43. Kurt Gödel: Philosophical Explorations: History and Theory.Daniele Chiffi - 2012 - Aracne.
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  44. Ørsteds „Gedankenexperiment“: eine Kantianische Fundierung der Infinitesimalrechnung? Ein Beitrag zur Begriffsgeschichte von ‚Gedankenexperiment' und zur Mathematikgeschichte des frühen 19. Jahrhunderts.Daniel Cohnitz - 2008 - Kant-Studien 99 (4):407-433.
  45. Appendix: How to Read Grundgesetze.Roy T. Cook - 2013 - In Gottlob Frege (ed.), Basic Laws of Arithmetic, Derived Using Concept-Script: Volumes I & Ii. Oxford University Press.
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  46. Peirce, Fallibilism, and the Science of Mathematics.Elizabeth F. Cooke - 2003 - 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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  47. Formalizing Euclid’s First Axiom.John Corcoran - 2014 - 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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  48. Semantic Arithmetic: A Preface.John Corcoran - 1995 - 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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  49. REVIEW OF 1988. Saccheri, G. Euclides Vindicatus (1733), Edited and Translated by G. B. Halsted, 2nd Ed. (1986), in Mathematical Reviews MR0862448. 88j:01013.John Corcoran - 1988 - 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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  50. Corcoran Recommends Hambourger on the Frege-Russell Number Definition.John Corcoran - 1978 - 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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