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  1. Andrew Aberdein (2014). Commentary On: Begoña Carrascal's "The Practice of Arguing and the Arguments: Examples From Mathematics". In Dima Mohammed & Marcin Lewinski (eds.), Virtues of argumentation: Proceedings of the 10th International Conference of the Ontario Society for the Study of Argumentation (OSSA), May 22–25, 2013. OSSA
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  2. Bahareh Afshari & Michael Rathjen (2010). A Note on the Theory of Positive Induction, {{Rm ID}^*_1}. Archive for Mathematical Logic 49 (2):275-281.
    The article shows a simple way of calibrating the strength of the theory of positive induction, ${{\rm ID}^{*}_{1}}$ . Crucially the proof exploits the equivalence of ${\Sigma^{1}_{1}}$ dependent choice and ω-model reflection for ${\Pi^{1}_{2}}$ formulae over ACA 0. Unbeknown to the authors, D. Probst had already determined the proof-theoretic strength of ${{\rm ID}^{*}_{1}}$ in Probst, J Symb Log, 71, 721–746, 2006.
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  3. Alan Ross Anderson (1958). Mathematics and the "Language Game". Review of Metaphysics 11 (3):446 - 458.
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  4. Toshiyasu Arai (2007). On the Consistency Proofs. Journal of the Japan Association for Philosophy of Science 34 (2):91-99.
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  5. Leslie Armour & Suzie Johnston (1999). The Ethics of the Infinite. Maritain Studies/Etudes Maritainiennes 15.
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  6. Richard A. Arms (1919). The Relation of Logic to Mathematics. The Monist 29 (1):146-152.
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  7. K. Ashton (1972). [Inverted Triangle]-Structures. Auckland, N.Z.,University of Auckland, Dept. Of Mathematics.
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  8. Andrea Asperti & Jeremy Avigad, Zen and the Art of Formalization.
    N. G. de Bruijn, now professor emeritus of the Eindhoven University of Technology, was a pioneer in the field of interactive theorem proving. From 1967 to the end of the 1970’s, his work on the Automath system introduced the architecture that is common to most of today’s proof assistants, and much of the basic technology. But de Bruijn was a mathematician first and foremost, as evidenced by the many mathematical notions and results that bear his name, among them de Bruijn (...)
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  9. Mark Atten (2003). Critical Studies/Book Reviews. [REVIEW] Philosophia Mathematica 11 (2):241-244.
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  10. Eric Audureau (2005). Relativité restreinte et cosmologie relativiste chez K. Gödel. Kairos 26:133-162.
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  11. Marvin Barsky (1969). Two General Methods of Extending Mathematical Theory Creative Process in Mathematics. Philosophia Mathematica (1-2):22-27.
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  12. Jon Barwise (1999). Critical Studies / Book Reviews. Philosophia Mathematica 7 (2):238-240.
  13. Jon Barwise (ed.) (1977). Handbook of Mathematical Logic. North-Holland.
  14. O. Bradley Bassler (1997). The Principles of Mathematics Revisited. [REVIEW] Review of Metaphysics 51 (2):424-425.
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  15. Robert J. Baum (1973). Philosophy and Mathematics, From Plato to the Present. San Francisco,Freeman, Cooper.
  16. Edward Beach (2006). Hegel's Misunderstood Treatment of Gauss in the Science of Logic: Its Implications for His Philosophy of Mathematics. Idealistic Studies 36 (3):191-218.
    This essay explores Hegel’s treatment of Carl Friedrich Gauss’s mathematical discoveries as examples of “Analytic Cognition.” Unfortunately, Hegel’s main point has been virtually lost due to an editorial blunder tracing back almost a century, an error that has been perpetuated in many subsequent editions and translations.The paper accordingly has three sections. In the first, I expose the mistake and trace its pervasive influence in multiple languages and editions of the Wissenschaftder Logik. In the second section, I undertake to explain the (...)
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  17. JC Beall (2005). Review of Stewart Shapiro (Ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. [REVIEW] Notre Dame Philosophical Reviews 2005 (9).
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  18. JC Beall & David Ripley (2003). Review of Paradox and Paraconsistency. [REVIEW] Notre Dame Philosophical Reviews.
    When physicists disagree as to whose theory is right, they can (if we radically idealize) form an experiment whose results will settle the difference. When logicians disagree, there seems to be no possibility of resolution in this manner. In Paradox and Paraconsistency John Woods presents a picture of disagreement among logicians, mathematicians, and other “abstract scientists” and points to some methods for resolving such disagreement. Our review begins with (very) short sketches of the chapters. Following the sketches, we respond to (...)
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  19. Ali Behboud (2006). Steve Russ. The Mathematical Works of Bernard Bolzano. Oxford: Oxford University Press, 2004. Pp. XXX + 698. Isbn 0-19-853930-. [REVIEW] Philosophia Mathematica 14 (3):352-362.
    In his book on The Mathematics of Great Amateurs Coolidge starts the chapter on Bolzano saying that he included Bolzano because it seemed interesting to him ‘that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have thought so far into the deepest problems of a science which he never taught in a professional capacity’ [Coolidge, 1990, p. 195]. In fact, considering Bolzano's poor health and his enormous productivity in his ‘professional (...)
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  20. E. A. Beliaev & Vasilii Iakovlevich Perminov (1981). Filosofskie I Metodologicheskie Problemy Matematiki. Izd-Vo Moskovskogo Universiteta.
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  21. Eric Temple Bell (1934). The Search for Truth. Baltimore, the Williams & Wilkins Company.
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  22. Gary D. Bell, A Characterization Of Mathematics.
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  23. John Bell, Hermann Weyl: Mathematician-Philosopher.
    MATHEMATICS AND PHILOSOPHY ARE CLOSELY LINKED, and several great mathematicians who were at the same time great philosophers come to mind— Pythagoras, Descartes and Leibniz, for instance. One great mathematician of the modern era in whose thinking philosophy played a major role was Hermann Weyl (1885–1955), whose work encompassed analysis, number theory, topology, differential geometry, relativity theory, quantum mechanics, and mathematical logic. His many writings are informed by a vast erudition, an acute philosophical awareness, and even, on occasion, a certain (...)
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  24. John Bell, Lectures on the Foundations of Mathematics.
    THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world— explicating such facts presents philosophy with some of its subtlest problems. Let me begin by reminding you of some celebrated past attempts made by philosophers and mathematicians to explicate the nature of mathematics.
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  25. John Bell, Oppositions and Paradoxes in Mathematics and Philosophy John L. Bell Abstract.
    In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.
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  26. John Bell, The Philosophy of Mathematics.
    THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...)
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  27. John L. Bell (2005). Oppositions and Paradoxes in Mathematics and Philosophy. Axiomathes 15 (2):165-180.
    In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.
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  28. Paul Benacerraf (1964). Philosophy of Mathematics. Englewood Cliffs, N.J.,Prentice-Hall.
    The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.
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  29. Paul Benacerraf & Hilary Putnam (eds.) (1983). Philosophy of Mathematics: Selected Readings. Cambridge University Press.
    The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, (...)
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  30. Ermanno Bencivenga (2006). Mathematics and Poetry. Inquiry 49 (2):158 – 169.
    Since Descartes, mathematics has been dominated by a reductionist tendency, whose success would seem to promise greater certainty: the fewer basic objects mathematics can be understood as dealing with, and the fewer principles one is forced to assume about these objects, the easier it will be to establish a secure foundation for it. But this tendency has had the effect of sharply limiting the expressive power of mathematics, in a way that is made especially apparent by its disappointing applications to (...)
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  31. Roseanne Benn & Rob Burton (1996). Mathematics: A Peek Into the Mind of God? Philosophy of Mathematics Education Journal 9.
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  32. H. H. Benson (2012). The Problem is Not Mathematics, but Mathematicians: Plato and the Mathematicians Again. Philosophia Mathematica 20 (2):170-199.
    I argue against a formidable interpretation of Plato’s Divided Line image according to which dianoetic correctly applies the same method as dialectic. The difference between the dianoetic and dialectic sections of the Line is not methodological, but ontological. I maintain that while this interpretation correctly identifies the mathematical method with dialectic, ( i.e. , the method of philosophy), it incorrectly identifies the mathematical method with dianoetic. Rather, Plato takes dianoetic to be a misapplication of the mathematical method by a subset (...)
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  33. Arthur Fisher Bentley (1932). Linguistic Analysis of Mathematics. Bloomington, Ind.,The Principia Press, Inc..
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  34. K. Berka (1980). Bolzano, B Philosophy of Mathematics. Filosoficky Casopis 28 (4):559-589.
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  35. Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
    An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...)
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  36. Evert Willem Beth (1965). Mathematical Thought an Introduction to the Philosophy of Mathematics. D. Reidel Pub. Co.
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  37. S. Bhave (1986). Some Neglected Problems in the Philosophy of Mathematics. Indian Philosophical Quarterly 13 (3-4):319.
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  38. Giuseppe Biancani, Bartolomeo Cochi & Geronimo Tamburini (1615). Aristotelis Loca Mathematica Ex Vniuersis Ipsius Operibus Collecta Et Explicata. Aristotelicae Videlicet Expositionis Complementum Hactenus Desideratum. Accessere de Natura Mathematicarum Scientiarum Tractatio Atq[Ue] Clarorum Mathematicorum Chronologia. Apud Bartholomaeum Cochium. ... Sumptibus Hieronymi Tamburini.
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  39. Robert Black (2000). Proving Church's Thesis. Philosophia Mathematica 8 (3):244--58.
    Arguments to the effect that Church's thesis is intrinsically unprovable because proof cannot relate an informal, intuitive concept to a mathematically defined one are unconvincing, since other 'theses' of this kind have indeed been proved, and Church's thesis has been proved in one direction. However, though evidence for the truth of the thesis in the other direction is overwhelming, it does not yet amount to proof.
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  40. Eric Blaire, C. P. Ormell & Mathematics Applicable Group (1992). New Thinking About the Nature of Mathematics.
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  41. Francesca Boccuni (2006). Mathematics and Cognition: Some Objections to a Cognitive Foundation for Mathematics. The Baltic International Yearbook of Cognition, Logic and Communication 2.
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  42. Luka Boršić (2008). Measure – From Mathematics to Ethics. Filozofska Istrazivanja 27 (4):751-764.
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  43. Jonathan Borwein (2008). (Implications of Experimental Mathematics Jor the Philosoj) Hij of Mathematics1. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America 33.
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  44. David Bostock (2012). Aristotle's Philosophy of Mathematics. In Christopher Shields (ed.), The Oxford Handbook of Aristotle. OUP Usa 465.
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  45. David Bostock (2009). Philosophy of Mathematics: An Introduction. Wiley-Blackwell.
    Finally the book concludes with a discussion of the most recent debates between realists and nominalists.
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  46. Clare Boucher (2000). The Six Blind Men and the Elephant: A Traditional Indian Story. Candlewick Press.
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  47. Genevieve Boulet (unknown). Philosophy and the Teaching of Mathematics. Eidos: The Canadian Graduate Journal of Philosophy 10.
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  48. Georges Bouligand & Jean Desgranges (1949). Le Déclin des Absolus Mathématico-Logiques. Société d'Édition d'Enseignement Supérieur.
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  49. John Bowers (1988). Invitation to Mathematics.
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  50. Eva Brann (2011). Jacob Klein's Two Prescient Discoveries. New Yearbook for Phenomenology and Phenomenological Philosophy 11:144-153.
    I present two of Jacob Klein’s chief discoveries from a perspective of peculiar fascination to me: the enchanting (to me) contemporaneous significance, the astounding prescience, and hence longevity, of his insights. The first insight takes off from an understanding of the lowest segment of the so-called DividedLine in Plato’s Republic. In this lowest segment are located the deficient beings called reflections, shadows, and images, and a type of apprehension associatedwith them called by Klein “image-recognition” (εἰκασία). The second discovery involves a (...)
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