Abstract
Paul Benacerraf’s “What Numbers Could Not Be” (Benacerraf 1965) has dominated thinking in the philosophy of mathematics for almost 50 years.
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References
Benacerraf, P. (1965) What Numbers Could Not Be, The Philosophical Review, 74, 47–73.
Burgess, J. P., & Rosen, G. (1997). A subject with no object: Strategies for nominalistic interpretation of mathematics. Oxford: Oxford UP.
Dummett, M. A. E. (1991). Frege: Philosophy of mathematics. Cambridge, MA: Harvard UP.
Frege, G. [1884] (1960). Die Grundlagen der Arithmetik, Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: Verlag von Wilhelm Koebner [The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number (2nd Ed., J. L. Austin, Trans.). New York: Harper].
Geach, P. (1967). Identity. The review of metaphysics, 21(1), 3–12.
Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Oxford UP.
Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford UP.
Kreisel, G. (1967). Informal rigour and completeness proofs. In I. Lakatos (Ed.), Problems in the philosophy of mathematics (pp. 138–186). Amsterdam: North Holland Publishing Company.
Maddy, P. (1997). Naturalism in mathematics. Oxford: Oxford UP.
Moschovakis, Y. (1994). Notes on set theory. New York: Springer.
Parsons, C. (1990). The structuralist view of mathematical objects. Synthese, 84(3), 303–346.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford: Oxford UP.
Shapiro, S. (2006). Computability, proof and open-texture. In R. Janusz, A. Olszewski, & J. Wokeński (Eds.), Church’s thesis after 70 years (pp. 420–455). Frankfurt: Ontos Verlag.
Takeuti, G. (1954). Construction of the set theory from the theory of ordinal numbers. Journal of the Mathematical Society of Japan, 6, 196–220.
Tappenden, J. (2005). Proof style and understanding in mathematics I: visualization, unification and axiom choice. In P. Mancosu, K. F. Jørgensen, & S. A. Pederson (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 147–207). Dordrecht: Springer.
Waismann, F. (1949). Analytic-synthetic I. Analysis, 10(2), 25–40.
Waismann, F. (1950). Analytic-synthetic II. Analysis, 11(2), 25–38.
Waismann, F. (1951a). Analytic-synthetic III. Analysis, 11(3), 49–61.
Waismann, F. (1951b). Analytic-synthetic IV. Analysis, 11(6), 115–124.
Waismann, F. (1952). Analytic-synthetic V. Analysis, 13(1), 1–14.
Waismann, F. (1953). Analytic-synthetic VI. Analysis, 13(4), 73–89.
Waismann, F. [1945] (1968). Verifiability. In A. Flew (Ed.) Logic and language (pp. 117–144), Oxford: Basil Blackwell.
Weyl, H. (1955). The Concept of a Riemann Surface (G. Maclane, English translation of the 3rd revised German edition of 1913). Reading, MA: Addison-Wesley.
Wilson, M. (1981). The double standard in ontology. Philosophical Studies, 39(4), 409–427.
Wilson, M. (1992). Frege: The royal road from geometry. Noûs, 26(2), 149–180.
Wilson, M. (2006). Wandering significance. Oxford: Oxford UP.
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Shapiro, S. (2016). What Numbers Could Be; What Objects Could Be. In: Pataut, F. (eds) Truth, Objects, Infinity. Logic, Epistemology, and the Unity of Science, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-45980-6_9
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