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Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics

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Book cover Logicism, Intuitionism, and Formalism

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References

  1. Awodey, S. [2004], “An answer to Hellman’s question: Does category theory provide a framework for mathematical structuralism?”, Philosophia Mathematica (3) 12, 54–64.

    Article  MATH  MathSciNet  Google Scholar 

  2. Benacerraf, P. [1965], “What numbers could not be”, Philosophical Review 74, 47–73; reprinted in [4].

    Article  MathSciNet  Google Scholar 

  3. Bernays, P. [1967], “Hilbert, David” in The encyclopedia of philosophy, Volume 3, edited by P. Edwards, New York, Macmillan publishing company and The Free Press, 496–504.

    Google Scholar 

  4. Benacerraf, P., and H. Putnam [1983], Philosophy of mathematics, second edition, Cambridge, Cambridge University Press.

    MATH  Google Scholar 

  5. Blanchette, P. [1996], “Frege and Hilbert on consistency”, Journal of Philosophy 93, 317–336.

    Article  MathSciNet  Google Scholar 

  6. Coffa, A. [1986], “From Geometry to tolerance: sources of conventionalism in nineteenth-century geometry” in From quarks to quasars: Philosophical problems of modern physics, University of Pittsburgh Series, Volume 7, Pittsburgh, Pittsburgh University Press, 3–70.

    Google Scholar 

  7. Frege, G. [1884], Die Grundlagen der Arithmetik, Breslau, Koebner; The foundations of arithmetic, translated by J. Austin, second edition, New York, Harper, 1960.

    Google Scholar 

  8. Frege, G. [1971], On the foundations of geometry and formal theories of arithmetic, translated by Eikee-Henner W. Kluge, New Haven, Connecticut, Yale University Press.

    Google Scholar 

  9. Frege, G. [1976], Wissenschaftlicher Briefwechsel, edited by G. Gabriel, H. Hermes, F. Kambartel, and C. Thiel, Hamburg, Felix Meiner.

    Google Scholar 

  10. Frege, G. [1980], Philosophical and mathematical correspondence, Oxford, Basil Blackwell.

    MATH  Google Scholar 

  11. Hale, B. [1987], Abstract objects, Oxford, Basil Blackwell.

    Google Scholar 

  12. Hellman, G. [1989], Mathematics without numbers, Oxford, Oxford University Press.

    MATH  Google Scholar 

  13. Hellman, G. [1996], “Structuralism without structures”, Philosophia Mathematica (III) 4, 100–123.

    MATH  MathSciNet  Google Scholar 

  14. Hellman, G. [2003], “Does category theory provide a framework for mathematical structuralism?”, Philosophia Mathematica (3) 11, 129–157.

    MATH  MathSciNet  Google Scholar 

  15. Hilbert, D. [1899], Grundlagen der Geometrie, Leipzig, Teubner; Foundations of geometry, translated by E. Townsend, La Salle, Illinois, Open Court, 1959.

    Google Scholar 

  16. Hilbert, D. [1925], “Über das Unendliche”, Mathematische Annalen 95, 161–190; translated as “On the infinite”, [25], 369–392 and [4], 183–201.

    Article  MATH  MathSciNet  Google Scholar 

  17. Hilbert, D. [1935], Gesammelte Abhandlungen, Dritter Band, Berlin, Julius Springer.

    Google Scholar 

  18. Husserl, E. [1900], Logische Untersuchen 1, Halle a.d. S., M. Niemeyer, translated as Logical Investigations, by J. N. Findlay, Amherst, New York, Humanity Books, 2000.

    Google Scholar 

  19. McLarty, C. [1993], “Numbers can be just what they have to”, Nous 27, 487–498.

    MathSciNet  Google Scholar 

  20. McLarty, C. [2004], “Exploring categorical structuralism”, Philosophia Mathematica (3) 12, 37–53.

    Article  MATH  MathSciNet  Google Scholar 

  21. Resnik, M. [1997], Mathematics as a science of patterns, Oxford, Oxford University Press.

    MATH  Google Scholar 

  22. Shapiro, S. [1991], Foundations without foundationalism: A case for second-order logic, Oxford, Oxford University Press.

    MATH  Google Scholar 

  23. Shapiro, S. [1996], “Space, number, and structure: A tale of two debates”, Philosophia Mathematica (3) 4 (1996), 148–173.

    MATH  MathSciNet  Google Scholar 

  24. Shapiro, S. [1997], Philosophy of mathematics: structure and ontology, New York, Oxford University Press.

    MATH  Google Scholar 

  25. Van Heijenoort, J. [1967], From Frege to Gödel, Cambridge, Massachusetts, Harvard University Press.

    MATH  Google Scholar 

  26. Wilson, M. [1993], “There’s a hole and a bucket, dear Leibniz”, Midwest Studies in Philosophy 18, 202–241.

    Article  Google Scholar 

  27. Wright, C. [1983], Frege’s conception of numbers as objects, Aberdeen, Aberdeen University Press.

    MATH  Google Scholar 

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Shapiro, S. (2009). Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics. In: Lindström, S., Palmgren, E., Segerberg, K., Stoltenberg-Hansen, V. (eds) Logicism, Intuitionism, and Formalism. Synthese Library, vol 341. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8926-8_18

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