Benacerraf's Dilemma and Informal Mathematics.
The Review of Symbolic Logic, 2
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This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerraf’s dilemma. The account builds upon Georg Kreisel’s work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a Fregean account
of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles.
References:Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70(19), 661–679.
Boolos, G. (1983). The iterative conception of set. In Benacerraf, P., and Putnam, H., editors. Philosophy of Mathematics (second edition). Cambridge, UK: Cambridge University Press, pp. 486–502. Originally published in the Journal of Philosophy, 68(1971), 215–232.
Carnap, R. (1950). Empiricism, semantics and ontology. Revue International de Philosophie, 4, 20–40. Reprinted in Meaning and Necessity: A Study in Semantics and Modal Logic (second edition). Chicago, IL: University of Chicago Press, 1956.
Dummett, M. (1978). Realism. In Truth and Other Enigmas. Cambridge, MA: Harvard University Press, 145–165.
Dummett, M. (1981). Frege: Philosophy of Language (second edition). London: Duckworth.
Dummett, M. (1991). Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press.
Field, H. (1989). Realism, Mathematics & Modality. New York, NY: Basil Blackwell Ltd.
Field, H. (1994). Are our logical notions highly indeterminate? In French, P. A., Uehling, T. E., and Wettstein, H. K., editors. Midwest Studies in Philosophy, Philosophical Naturalism, Vol. XIX. Notre Dame, IN: University of Notre Dame Press, pp. 391–429.
Field, H. (1998). Do we have a determinate conception of finiteness and natural number? In Schirn, M., editor. The Philosophy of Mathematics Today. New York, NY: Oxford, pp. 99–130.
Frege, G. (1980). The Foundations of Arithmetic (second revised edition). Evanston, IL: Northwestern University Press.
Hacker, P. (2006). Passing by the naturalistic turn: On quine’s cul-de-sac. Philosophy, 81(2), 231–253.
Hale, B., & Wright, C. (2002). Benacerraf’s dilemma revisited. European Journal of Philosophy, 10(1), 101–129.
Isaacson, D. (2004). Quine and logical positivism. In Gibson, R. F., editor. The Cambridge Companion to Quine. New York, NY: Cambridge University Press, pp. 214–269.
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. New York, NY: Humanities Press, pp. 138–186.
Kreisel, G., & Krivine, J.-L. (1971). Element of Mathematical Logic. Amsterdam, The Netherlands: North-Holland.
Lavers, G. (2008). Carnap, formalism, and informal rigour. Philosophia Mathematica, 16(1), 4–24.
Parsons, C. (1983). What is the iterative conception of set? In Benacerraf, P., and Putnam, H., editors. Philosophy of Mathematics. Cambridge, UK: Cambridge University Press, second edition, pp. 503–529. Originally published in the Proceedings of the 5th International Congress on Logic, Methodology and Philosophy of Science 1975, Part I: Logic, Foundations of Mathematics and Computability Theory. Dordrecht, the Netherlands. Butts, R., and Hintikka, J., editors. Reidel, D. (1977).
Potter, M. (2000). Reason’s Nearest Kin. Oxford: Oxford University Press.
Wang, H. (1983). The concept of set. In Benacerraf, P., and Putnam, H., editors. Philosophy of Mathematics. Cambridge, UK: Cambridge University Press, second edition, pp. 530–570. Originally published in H. Wang From Mathematics to Philosophy. London: Routledge and Kegan Paul 1974, pp. 181–223.
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