The Applicability of Mathematics and the Indispensability Arguments
DOI:
https://doi.org/10.20416/lsrsps.v3i1.313Keywords:
Philosophy of Mathematics, Indispensability Argument, Mathematical Explanation, Applicability of Mathematics, Metaphysics, Mathematical RealismAbstract
In this paper I will take into examination the relevance of the main indispensability arguments (Quine’s and Colyvan’s, Putnam’s, and explanatory indispensability argument) for the comprehension of the applicability of mathematics. I will conclude not only that none of these indispensability arguments are of any help for understanding mathematical applicability, but also that these arguments rather require a preliminary analysis of the problems raised by the applicability of mathematics in order to avoid some tricky difficulties in their formulations. As a consequence, we cannot any longer consider the applicability problems as subordinate to ontological ones: no ontological stance on mathematical entities (or truths) can offer an easy road to the comprehension of the applicability of mathematics.
References
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QUINE, Willard V. O. 1957. The scope and language of science. British Journal for the History of Philosophy, VIII(29), 1–17.
QUINE, Willard V. O. 1961. From a Logical Poin of View. 2nd (revised) ed. Cambridge: Harvard University Press.
QUINE, Willard V. O. 1981. Theories and Things. Cambridge: Harvard University Press
SMART, John J. C. 1963. Philosophy and Scientific Realism. New York: Routledge and Kegan Paul.
STEINER, Mark. 1998. The Applicability of Mathematics as a Philosophical Problem. Cambridge: Harvard University Press.
WIGNER, Eugene. 1960. The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14. Reprinted in WIGNER, Eugene. 1967. Symmetries and Reflections. Bloomington : Indiana University Press.
BAKER, Alan 2009. Mathematical explanation in science. British Journal for the Philosophy of Science, 60, 611–633.
BATTERMAN, Robert W. 2002. The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence. New York: Oxford University Press.
BATTERMAN, Robert W. 2010. On the explanatory role of mathematics in empirical science. The British Journal for the Philosophy of Science, 61(1), 1–25.
BIGELOW, John. 1988. The Reality of Numbers: A Physicalist’s Philosophy of Mathematics. Oxford: Clarendon Press.
CHEYNE, Colin, PIGDEN, Charles R. 1996. Pythagorean powers or a challenge to platonism. Australasian Journal of Philosophy, 74(4), 639– 645.
COLYVAN, Mark. 1999. Confirmation theory and indispensability. Philosophical Studies, 96, 1–19.
COLYVAN, Mark. 2001a. The Indispensability of Mathematics. Oxford: Oxford University Press.
COLYVAN, Mark. 2001b. The miracle of applied mathematics. Synthese, 127, 265–277.
COLYVAN, Mark. 2002. Mathematics and aesthetic considerations in science. Mind, 111, 69–74.
COLYVAN, Mark. 2010. There is no easy road to nominalism. Mind, 119, 285–306.
DALY, Chris, LANGFORD, Simon. 2009. Mathematical explanation and indispensability arguments. Philosophical Quarterly, (59), 641–658.
DAVIES, Paul C. W. 1992. The Mind of God. London: Penguin Book.
FIELD, Hartry. 1980. Science Without Numbers. Princeton: Princeton University Press.
HELLMAN, Geoffrey. 1989. Mathematics Without Numbers. Oxford : Oxford University Press.
KITCHER, Philip. 1984. The Nature of Mathematical Knowledge. New York : Oxford University Press.
LIGGINS, David. 2008. Quine, Putnman and the ‘Quine-Putnam’ indispensability argument. Erkenntnis, 68, 113–127.
LYON, Aidan, COLYVAN, Mark. 2008. The explanatory power of phase spaces. Philosophia Mathematica, 16, 227–243.
MADDY, Penelope. 1990. Realism in Mathematics. Oxford: Clarendon Press.
MADDY, Penelope. 1992. Indispensability and practice. Journal of Philosophy, 89(6), 275–89.
MADDY, Penelope. 1997. Naturalism in Mathematics. Oxford: Oxford University Press.
MADDY, Penelope. 2007. Second Philosophy. Oxford: Oxford University Press.
MELIA, Joseph. 2000. Weaseling away the indispensability argument. Mind, 109, 455–479.
MELIA, Joseph. 2002. Response to Colyvan. Mind, 111, 75–79.
PANZA, Marco, SERENI, Andrea. 2015. On the indispensable premises of the indispensability argument. In LOLLI, Gabriele, PANZA, Marco, VENTURI, Giorgio (eds), From Logic to Practice. Italian Studies in the Philosophy of Mathematics. Vol. 308 of Boston Studies in the Philosophy and History of Science. Springer. 241–276.
PENROSE, Roger. 1990. The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics. London: Vintage.
PINCOCK, Christopher. 2011a. Mathematical explanations of the rainbow. Studies in History and Philosophy of Modern Physics, 42(1), 13–22.
PINCOCK, Christopher. 2011b. On Batterman’s “On the explanatory role of mathematics in empirical science” ’. British Journal for the Philosophy of Science, 62, 211–217.
PINCOCK, Christopher. 2012. Mathematics and Scientific Representation. Oxford: Oxford University Press.
PUTNAM, Hilary. 1965. Craig’s theorem. Journal of Philosophy, 62(10), 251– 260. Reprinted in (Putnam 1975), pp.228-236.
PUTNAM, Hilary. 1975. Mathematics, Matter and Method: Philosophical Papers. Cambridge: Cambridge University Press.
PUTNAM, Hilary. 1979a. Philosophy of logic. In Mathematics Matter and Method: Philosophical Papers Vol. 1. 2nd edn. Cambridge: Cambridge University Press.
PUTNAM, Hilary (ed.). 1979b. Mathematics, Matter and Method: Philosophical Papers, Vol. 1. 2nd edn. Cambridge: Cambridge University Press.
QUINE, Willard V. O. 1957. The scope and language of science. British Journal for the History of Philosophy, VIII(29), 1–17.
QUINE, Willard V. O. 1961. From a Logical Poin of View. 2nd (revised) ed. Cambridge: Harvard University Press.
QUINE, Willard V. O. 1981. Theories and Things. Cambridge: Harvard University Press
SMART, John J. C. 1963. Philosophy and Scientific Realism. New York: Routledge and Kegan Paul.
STEINER, Mark. 1998. The Applicability of Mathematics as a Philosophical Problem. Cambridge: Harvard University Press.
WIGNER, Eugene. 1960. The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1), 1–14. Reprinted in WIGNER, Eugene. 1967. Symmetries and Reflections. Bloomington : Indiana University Press.
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Published
2016-12-22
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Ginammi, Michele. 2016. “The Applicability of Mathematics and the Indispensability Arguments”. Lato Sensu: Revue De La Société De Philosophie Des Sciences 3 (1). https://doi.org/10.20416/lsrsps.v3i1.313.
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Proceedings of the 2014 congress (metaphysics of science)
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