Abstract
Empiricism has traditionally had great difficulty in making sense of mathematics. The problems with Mill’s empiricism in this regard, as exposed by Frege, are well known. In this century, Carnap’s logical empiricism turned on the claim that all mathematical truths are really analytic, i.e., true solely in virtue of linguistic meanings, a claim that cannot withstand scrutiny in light of Gödel’s work and its aftermath. Quine’s naturalistic empiricism has sought to do better. The insight that testing of scientific theories is holistic, involving substantial bodies of propositions, suggested that even purely mathematical assumptions, essential in both the formulation of (say) physical theories and in deduction of testable consequences, can gain empirical support or confirmation indirectly, in a manner analogous to the way in which highly theoretical physical postulates can. (The locus classicus is [20].) There is no need to claim that, say, abstract set-existence axioms are really analytic, nor is there any need to invoke a special intuitive faculty for grasping mathematical objects or propositions. Mathematics can be seen as continuous with the sciences, and mathematical epistemology promises to be brought within a thorough-going naturalistic framework.
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References
Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.
Boolos, G., ‘Nominalist Platonism’, Philosophical Review, 94 (1985), 327–44.
Bridges, D. S., Constructive Functional Analysis, Pitman, London, 1979.
Burgess, J., Hazen, A., and Lewis, D., Appendix on Pairing,in [17].
Chihara, C., Ontology and the Vicious Circle Principle, Cornell University Press, Ithaca, NY, 1973.
Colyvan, M., ‘In Defence of Indispensability’, forthcoming.
Earman, J., Bayes or Bust?, MIT Press, Cambridge, MA, 1992.
Feferman, S., ‘Infinity in Mathematics: Is Cantor Necessary?’ Philosophical Topics, 17 (2) (1989), 23–45.
Field, H., Science without Numbers, Princeton University Press, Princeton, NJ, 1980.
Gödel, K., ‘What is Cantor’s Continuum Problem?’ in P. Benacerraf and H. Putnam (eds), Philosophy of Mathematics, 2d edn, Cambridge University Press, Cambridge, U.K., 1983, pp. 470–485.
Hellman, G., ‘Logical Truth by Linguistic Convention’, in L. Hahn and P. A. Schilpp (eds), The Philosophy of W.V. Quine, Open Court, La Salle, IL, 1986, pp. 189–205.
Hellman, G., Mathematics without Numbers: Towards a Modal-Structural Interpretation, Oxford University Press, Oxford, 1989.
Hellman, G., ‘Constructive Mathematics and Quantum Mechanics’, J. Philos. Logic, 22 (1993), 221–248.
Hellman, G., ‘Real Analysis without Classes’, Philos. Math. (3), 2 (1994), 228–250.
Hellman, G., ‘Structuralism without Structures’, Philos. Math. (3), 4 (1996), 100–123.
Hempel, C. G., Aspects of Scientific Explanation, Free Press, New York, 1965.
Lewis, D., Parts of Classes, Blackwell, Oxford, 1991.
Maddy, P., ‘Indispensability and Practice’, J. Philos., 89 (1992), 275–289.
Parsons, C., ‘Quine on the Philosophy of Mathematics’, in L. Hahn and P. A. Schilpp (eds), The Philosophy of W.V. Quine, Open Court, La Salle, IL, 1986, pp. 369–395.
Quine, W. V., ‘Two Dogmas of Empiricism’, in From a Logical Point of View, 2nd edn, Harper, New York, 1961, pp. 20–46.
Resnik, M., ‘Scientific vs. Mathematical Realism: The Indispensability Argument’, Philos. Math. (3), 3 (1995), 166–174.
Simpson, S., ‘Subsystems of Z2 and Reverse Mathematics’, in G. Takeuti (ed.), Proof Theory, 2nd edn, North-Holland, Amsterdam, Appendix, pp. 432–446.
Smorynski, C., ‘The Varieties of Arboreal Experience’, Math. Well., 4 (1982), 182–189.
Sober, E., ‘Mathematics and Indispensability’, The Philosophical Review, 102 (1) (1993), 35–37.
Yu, Xiaokang, ‘Radon—Nikodym Theorem is Equivalent to Arithmetic Comprehension’, Content Math., 106 (1990), 289–297.
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Hellman, G. (1999). Some Ins and Outs of Indispensability: A Modal-Structural Perspective. In: Cantini, A., Casari, E., Minari, P. (eds) Logic and Foundations of Mathematics. Synthese Library, vol 280. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2109-7_2
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