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Philosophy of Mathematics

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  1. A. R. A. (1956). Théorie Métamathématique des Idéaux. Review of Metaphysics 9 (4):709-709.
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
     Mathematical Proof
     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
     Logic and Phil of Logic
     Number Theory
     Set Theory
     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
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     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
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     Mathematical Explanation
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  2. Oscar João Abdounur, Vecchio Junior & Jacintho Del (2013). Sobre os números transfinitos. Scientiae Studia 11 (2):417-426.
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
     Mathematical Proof
     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
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     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
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     Predicativism in Mathematics
     Mathematical Naturalism
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     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
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  3. Andrew Aberdein, Commentary On: Begoῆa Carrascal's "The Practice of Arguing and the Arguments: Examples From Mathematics.
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    Epistemology of Mathematics
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     Mathematical Intuition
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
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    Ontology of Mathematics
     Mathematical Fictionalism
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     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
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    Theories of Mathematics
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     Intuitionism and Constructivism
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     Mathematical Naturalism
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
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  4. Andrew Aberdein & Ian J. Dove (eds.) (2013). The Argument of Mathematics. Springer.
    Bartha, P. (2013). Analogical arguments in mathematics. In A. Aberdein & I.J. Dove (Eds), The Argument of Mathematics (pp. 197—236). Dordrecht: Springer. Bourbaki, N. (1968). Elements of mathematics: Theory of sets. Berlin: Springer.
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
     Mathematical Proof
     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
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    Theories of Mathematics
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     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
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  5. Fabio Acerbi (2010). Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus. Science in Context 23 (2):151-186.
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
     Mathematical Proof
     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
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     Geometry
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     Number Theory
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     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
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     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
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  6. W. Ackermann (1954). Solvable Cases of the Decision Problem. Amsterdam, North-Holland Pub. Co..
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
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     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
     Logic and Phil of Logic
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     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
     Logicism in Mathematics
     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
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  7. J. Agassi (1981). Lakatos on proof and on mathematics. Logique Et Analyse 24 (95):437.
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
     Mathematical Proof
     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
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     Geometry
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     Number Theory
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     Areas of Mathematics, Misc
    Theories of Mathematics
     Logicism in Mathematics
     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
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  8. Evandro Agazzi (1978). Non-contradiction et existence en mathématique. Logique Et Analyse 21 (84):459.
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    Epistemology of Mathematics
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     Mathematics and the Causal Theory of Knowledge
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
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     Areas of Mathematics, Misc
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     Predicativism in Mathematics
     Mathematical Naturalism
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     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
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  9. Tryg A. Ager (1976). A Metaphysics of Elementary Mathematics. International Studies in Philosophy 8:196-198.
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    Epistemology of Mathematics
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
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     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
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     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
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    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
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     Predicativism in Mathematics
     Mathematical Naturalism
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     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
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  10. I. A. Akchurin, M. F. Vedenov & Iu V. Sachkov (1966). Methodological Problems of Mathematical Modeling in Natural Science. Russian Studies in Philosophy 5 (2):23-34.
    The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that theory possesses (...)
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    Epistemology of Mathematics
     Apriority in Mathematics
     Mathematics and the Causal Theory of Knowledge
     Mathematical Intuition
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     Revisability in Mathematics
     Visualization in Mathematics
     Phenomenology of Mathematics
     Mathematical Methodology
     Nondeductive Methods in Mathematics
     Epistemology of Mathematics, Misc
    Ontology of Mathematics
     Mathematical Fictionalism
     Mathematical Nominalism
     Mathematical Platonism
     Mathematical Psychologism
     Mathematical Structuralism
     Mathematical Neo-Fregeanism
     Indeterminacy in Mathematics
     Indispensability Arguments in Mathematics
     Numbers
     The Nature of Sets
    Mathematical Truth
     Analyticity in Mathematics
     Axiomatic Truth
     Objectivity Of Mathematics
     Mathematical Truth, Misc
    Set Theory
     The Nature of Sets
     Axioms of Set Theory
     Cardinals and Ordinals
     Set Theory as a Foundation
    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
     Logic and Phil of Logic
     Number Theory
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     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
     Logicism in Mathematics
     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
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  11. Keith Algozin (1981). Whence the Infinite God? Proceedings of the American Catholic Philosophical Association 55:73-83.
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    Epistemology of Mathematics
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    Set Theory
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
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  12. R. B. J. T. Allenby (1997). Numbers and Proofs. Copublished in North, South, and Central America by John Wiley & Sons Inc..
    'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any (...)
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     Visualization in Mathematics
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    Theories of Mathematics
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    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
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  13. D. F. Almeida (2010). Are There Viable Connections Between Mathematics, Mathematical Proof and Democracy? Philosophy of Mathematics Education Journal 25.
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  14. Dennis Almeida (1996). Justifying and Proving in the Mathematics Classroom. Philosophy of Mathematics Education Journal 9.
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  15. Dennis Almeida (1995). Aspects Of Proof: Special Issue Of Educational Studies In Mathematics. [REVIEW] Philosophy of Mathematics Education Journal 8.
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  16. Logique A. Analyse (2002). Mathematics and Fiction II: Analogy Robert Thomas. Logique Et Analyse 45:185.
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  17. Alan Ross Anderson (1958). Mathematics and the "Language Game". Review of Metaphysics 11 (3):446 - 458.
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  18. Bernard A. Anderson & Jeffry L. Hirst (2009). Partitions of Trees and {{Sf ACA}^Prime_{0}}. Archive for Mathematical Logic 48 (3-4):227-230.
    We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem ${{\sf ACA}^\prime_{0}}$ of reverse mathematics.
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  19. John Mueller Anderson (1962). Natural Deduction. Belmont, Calif.,Wadsworth Pub. Co..
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  20. Philip W. Anderson, Contributors.
    Is string theory a futile exercise as physics, as I believe it to be? It is an interesting mathematical specialty and has produced and will produce mathematics useful in other contexts, but it seems no more vital as mathematics than other areas of very abstract or specialized math, and doesn't on that basis justify the incredible amount of effort expended on it.
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  21. I. H. Anellis (1987). Bertrand Russell's Theory of Numbers, 1896–1898. Epistemologia 10 (2):303-322.
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  22. Annarita Angelini (2006). Thematic Files-Mathematics and Knowledge in the Renaissance->: Science and Mathematics According to 16th-Century Commentators of Proclus. Revue d'Histoire des Sciences 59 (2):265.
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  23. Lemanska Anna (2010). Truth and Mathematics (Prawda a Matematyka). Studia Philosophiae Christianae 46 (1):37-54.
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  24. G. Aldo Antonelli (2012). Frege's Theorem. International Studies in the Philosophy of Science 26 (2):219-222.
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  25. H. G. Apostle (1958). Methodological Superiority of Aristotle Over Euclid. Philosophy of Science 25 (2):131-134.
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  26. Hippocrates George Apostle, Arnold M. Adelberg & Elizabeth A. Dobbs (1991). Mathematics as a Science of Quantities. Monograph Collection (Matt - Pseudo).
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  27. Toshiyasu Arai (2007). On the Consistency Proofs. Journal of the Japan Association for Philosophy of Science 34 (2):91-99.
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  28. Mohammad Ardeshir & Bardyaa Hesaam (2008). An Introduction to Basic Arithmetic. Logic Journal of the Igpl 16 (1):1-13.
    We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of , a fragment of PA with (...)
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  29. Bernhard Arens (1985). Die Non-Standard Analysis: Eine Rehabilitierung Des Unendlichkleinen in den Grundlagen der Mathematik. Journal for General Philosophy of Science 16 (1):147-150.
    Summary The historical development of the non-standard analysis is sketched. With the help of this mathematical branch infinite and infinitesimal quantities are placed in an extension of the real numbers and so find their justification. In this way an old mathematical and philosophical problem is solved in the 20th century, but not in such a manner, mathematicians with classical „standard methods thought of.
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  30. Konstantine Arkoudas & Selmer Bringsjord (2007). Computers, Justification, and Mathematical Knowledge. Minds and Machines 17 (2):185-202.
    The original proof of the four-color theorem by Appel and Haken sparked a controversy when Tymoczko used it to argue that the justification provided by unsurveyable proofs carried out by computers cannot be a priori. It also created a lingering impression to the effect that such proofs depend heavily for their soundness on large amounts of computation-intensive custom-built software. Contra Tymoczko, we argue that the justification provided by certain computerized mathematical proofs is not fundamentally different from that provided by surveyable (...)
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  31. Leslie Armour & Suzie Johnston (1999). The Ethics of the Infinite. Maritain Studies/Etudes Maritainiennes 15.
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  32. Richard A. Arms (1919). The Relation of Logic to Mathematics. The Monist 29 (1):146-152.
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  33. A. F. W. Armstrong (1994). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Boston Studies in the Philosophy of Science 156.
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  34. Tatiana Arrigoni (2003). Foundational Instances and Attention to Practices in the Philosophy of Contemporary Mathematics. Rivista di Filosofia Neo-Scolastica 95 (2):199-232.
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  35. K. Ashton (1972). [Inverted Form of Greek Symbol Delta]-Structures, Abstract Algebras and Structural Analysis. Auckland, N.Z.,University of Auckland, Dept. Of Mathematics.
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  36. K. Ashton (1972). [Inverted Triangle]-Structures. Auckland, N.Z.,University of Auckland, Dept. Of Mathematics.
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  37. 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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  38. Mark Atten (2003). Critical Studies/Book Reviews. [REVIEW] Philosophia Mathematica 11 (2):241-244.
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  39. Mark Van Atten (2004). Intuitionistic Remarks on Husserl's Analysis of Finite Number in the Philosophy of Arithmetic. Graduate Faculty Philosophy Journal 25 (2):205-225.
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  40. Karl Egil Aubert (1982). The Role of Mathematics in the Exploration of Reality. Inquiry 25 (3):353 – 359.
    In his well?known paper from 1954, Herbert A. Simon sets out to demonstrate that it is possible, in principle, to make public predictions within the social sciences that will be confirmed by the events. However, Simon's proof by means of the Brouwer fixed?point theorem not only rests on an illegitimate use of continuous variables, it is also founded on the questionable assumption that facts ? even on the level of possibilities ? can be established by purely mathematical means. The ?proof? (...)
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  41. Eric Audureau (2005). Relativité restreinte et cosmologie relativiste chez K. Gödel. Kairos 26:133-162.
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  42. J. Avigad & S. Feferman (1998). Godel's Functional Interpretation. In Samuel R. Buss (ed.), Handbook of Proof Theory. Elsevier.
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  43. Jeremy Avigad, By Dennis E. Hesseling.
    The early twentieth century was a lively time for the foundations of mathematics. This ensuing debates were, in large part, a reaction to the settheoretic and nonconstructive methods that had begun making their way into mathematical practice around the turn of the twentieth century. The controversy was exacerbated by the discovery that overly na¨ıve formulations of the fundamental principles governing the use of sets could result in contradictions. Many of the leading mathematicians of the day, including Hilbert, Henri Poincar´e, ´.
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  44. Jeremy Avigad (2006). Mathematical Method and Proof. Synthese 153 (1):105 - 159.
    On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
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  45. Jeremy Avigad (1998). An Effective Proof That Open Sets Are Ramsey. Archive for Mathematical Logic 37 (4):235-240.
    Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.
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  46. Jeremy Avigad, Kevin Donnelly, David Gray & Paul Raff, A Formally Verified Proof of the Prime Number Theorem.
    The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.
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  47. Jeremy Avigad & Harvey Friedman, Combining Decision Procedures for the Reals.
    We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local'’ decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let $Tadd[QQ]$ be the first-order theory of the real numbers in the language with symbols $0, 1, +, -.
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  48. Arnon Avron, Safety Signatures for First-Order Languages and Their Applications.
    In several areas of Mathematical Logic and Computer Science one would ideally like to use the set F orm(L) of all formulas of some first-order language L for some goal, but this cannot be done safely. In such a case it is necessary to select a subset of F orm(L) that can safely be used. Three main examples of this phenomenon are: • The main principle of naive set theory is the comprehension schema: ∃Z(∀x.x ∈ Z ⇔ A).
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  49. Jody Azzouni (2006). Tracking Reason: Proof, Consequence, and Truth. Oup Usa.
    When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, (...)
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  50. Jody Azzouni (2000). Applying Mathematics. The Monist 83 (2):209-227.
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    Areas of Mathematics
     Algebra
     Analysis
     Category Theory
     Geometry
     Logic and Phil of Logic
     Number Theory
     Set Theory
     Topology
     Areas of Mathematics, Misc
    Theories of Mathematics
     Logicism in Mathematics
     Formalism in Mathematics
     Intuitionism and Constructivism
     Predicativism in Mathematics
     Mathematical Naturalism
     Mathematical Finitism
     Theories of Mathematics, Misc
    History: Philosophy of MathematicsPhil of Mathematics, Miscellaneous
     Explanation in Mathematics
     The Infinite
     The Application of Mathematics
     History of Mathematics
     Mathematical Practice
     Phil of Mathematics, General Works
     Mathematical Explanation
     Phil of Mathematics, Misc
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