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  1. John Alfred Henry Anderson (1974). Mathematics, the Language Concepts. Stanley Thornes (Publishers) Ltd..
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  2. Irving H. Anellis (2010). Joong Fang (1923–2010). Philosophia Mathematica 18 (2):137-143.
    (No abstract is available for this citation).
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  3. Irving H. Anellis (1993). Letters. Philosophia Mathematica 1 (1):71-73.
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  4. Irving H. Anellis (1987). Book-Review. Philosophia Mathematica (1):110-116.
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  5. Irving H. Anellis (1987). Report on the Thirteenth Annual Meeting of the Canadian Society for History and Philosophy of Mathematics. Philosophia Mathematica (2):211-223.
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  6. Irving H. Anellis (1987). The Conference on the History of Mathematics. Philosophia Mathematica (1):123-125.
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  7. W. S. Anglin (1996). Mathematics, a Concise History and Philosophy. Springer.
    This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed explanations of the important mathematical procedures (...)
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  8. W. S. Anglin (1991). Mathematics and Value. Philosophia Mathematica 6 (2):145-173.
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  9. Hippocrates George Apostle (1952). Aristotle's Philosophy of Mathematics. [Chicago]University of Chicago Press.
  10. David Auerbach (1992). How to Say Things with Formalisms. In Michael Detlefsen (ed.), Proof, logic, and formalization. Routledge. 77--93.
  11. David D. Auerbach (1985). Intensionality and the Gödel Theorems. Philosophical Studies 48 (3):337--51.
  12. Jeremy Avigad, Philosophy of Mathematics.
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  13. Jeremy Avigad, Philosophy of Mathematics: 5 Questions.
    In 1977, when I was nine years old, Doubleday released Asimov on Numbers, a collection of essays that had first appeared in Isaac Asimov’s Science Fiction and Fantasy column. My mother, recognizing my penchant for science fiction and mathematics, bought me a copy as soon as it hit the bookstores. The essays covered topics such as number systems, combinatorial curiosities, imaginary numbers, and π. I was especially taken, however, by an essay titled “Varieties of the infinite,” which included a photograph (...)
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  14. Steve Awodey & A. W. Carus (2010). Gödel and Carnap. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  15. Jody Azzouni (1999). Comments on Shapiro. Journal of Philosophy 96 (10):541 - 544.
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  16. Matthias Baaz (ed.) (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press.
    Machine generated contents note: Part I. Historical Context - Gödel's Contributions and Accomplishments: 1. The impact of Gödel's incompleteness theorems on mathematics Angus Macintyre; 2. Logical hygiene, foundations, and abstractions: diversity among aspects and options Georg Kreisel; 3. The reception of Gödel's 1931 incompletabilty theorems by mathematicians, and some logicians, to the early 1960s Ivor Grattan-Guinness; 4. 'Dozent Gödel will not lecture' Karl Sigmund; 5. Gödel's thesis: an appreciation Juliette C. Kennedy; 6. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on (...)
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  17. Mark Balaguer (2002). Review: Stewart Shapiro, Thinking About Mathematics. The Philosophy of Mathematics. [REVIEW] Bulletin of Symbolic Logic 8 (1):89-91.
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  18. A. G. Barabashev (1988). On the Impact of the World Outlook on Mathematical Creativity. Philosophia Mathematica (1):1-20.
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  19. A. G. Barabashev, S. S. Demidov & M. I. Panov (1987). Regularities and Modern Tendencies of the Development of Mathematics. Philosophia Mathematica (1):32-47.
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  20. Alexei G. Barabashev (1986). The Philosophy of Mathematics in U.S.S.R. Philosophia Mathematica (1-2):15-25.
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  21. Stephen Francis Barker (1964). Philosophy of Mathematics. Englewood Cliffs, N.J.,Prentice-Hall.
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  22. Jeffrey A. Barrett (1995). Review of I. Ekeland, The Broken Dice, and Other Mathematical Tales of Chance. [REVIEW] Philosophia Mathematica 3 (3):310-313.
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  23. Marvin Barsky (1969). Two General Methods of Extending Mathematical Theory Creative Process in Mathematics. Philosophia Mathematica (1-2):22-27.
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  24. Jon Barwise (1999). Critical Studies / Book Reviews. Philosophia Mathematica 7 (2):238-240.
  25. Jon Barwise & H. Jerome Keisler (eds.) (1977). Handbook of Mathematical Logic. North-Holland Pub. Co..
  26. Robert J. Baum (1973). Philosophy and Mathematics, From Plato to the Present. San Francisco,Freeman, Cooper.
  27. 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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  28. 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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  29. 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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  30. 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.
  31. Eric Temple Bell (1934). The Search for Truth. Baltimore, the Williams & Wilkins Company.
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  32. 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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  33. 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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  34. 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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  35. 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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  36. 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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  37. 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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  38. 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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  39. 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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  40. 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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  41. Arthur Fisher Bentley (1932). Linguistic Analysis of Mathematics. Bloomington, Ind.,The Principia Press, Inc..
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  42. 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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  43. 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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  44. 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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  45. Clare Boucher (2000). The Six Blind Men and the Elephant: A Traditional Indian Story. Candlewick Press.
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  46. 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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  47. J. R. Brown (2003). Vladimir Tasic. Mathematics and the Roots of Postmodern Thought. Philosophia Mathematica 11 (2):244-245.
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  48. J. R. Brown (1994). John D. Barrow, Pi in the Sky: Counting, Thinking, and Being. Philosophia Mathematica 2 (3):251-251.
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  49. James Robert Brown (2008). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Routledge.
    1. Introduction : the mathematical image -- 2. Platonism -- 3. Picture-proofs and Platonism -- 4. What is applied mathematics? -- 5. Hilbert and Gödel -- 6. Knots and notation -- 7. What is a definition? -- 8. Constructive approaches -- 9. Proofs, pictures and procedures in Wittgenstein -- 10. Computation, proof and conjecture -- 11. How to refute the continuum hypothesis -- 12. Calling the bluff.
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  50. James Robert Brown (1999). Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures. Routledge.
    Philosophy of Mathematics is clear and engaging, and student friendly The book discusses the great philosophers and the importance of mathematics to their thought. Among topics discussed in the book are the mathematical image, platonism, picture-proofs, applied mathematics, Hilbert and Godel, knots and notation definitions, picture-proofs and Wittgenstein, computation, proof and conjecture.
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