The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

9
Holistic Mathematics

MICHAEL D. RESNIK

By holism I shall mean epistemic or confirmational holism, that is, the thesis that no claim of theoretical science can be confirmed or refuted in isolation but only as part of a system of hypotheses. Duhem's remarks on the law of inertia and similar scientific claims make a strong case for a version of holism restricted to the theoretical branches of natural science;1 but, as several philosophers of mathematics have noted,2 Quine's extension of Duhemian holism to include mathematics and logic is much more problematic. Recently Penelope Maddy3 and Elliott Sober4 have tied difficulties with Quine's holism to his indispensability argument for the existence of mathematical objects. My purpose here is to defend Quine's thesis, or at least a close variant of it.

I will begin by reviewing the arguments for Duhem's and Quine's versions of holism, and then point out some prima-facie objections to Quine's view that do not apply to Duhem's. This will bring me to the worries Maddy and Sober have raised about Quinean holism and his views on confirming mathematical principles. Although I will not dispute their observations about the practice of science, I will point out that a version of the indispensability argument survives them. However, indispensability is separable from holism: the former purports to support realism about mathematical objects and truths; the latter purports to support the empirical testability of mathematics. So I will still have to defend holism.

I will do this by proposing that scientific and mathematical theories and their attendant methodologies can be partially ordered as more or less

I would like to thank Mark Balaguer, Geoffrey Hellman, Penelope Maddy, Bijan Parsia, Stewart Shapiro, and Paul Teller for helpful correspondence and comments. I am also grateful to participants in the Munich conference for useful discussion following my presentation of an earlier version of this paper.

____________________
1
Pierre Duhem, The Aim and Structure of Physical Theory, trans, by Philip P. Wiener ( Princeton, Princeton University Press, 1954).
2
Charles Parsons, "'Quine on the Philosophy of Mathematics'" in The Philosophy of W. V. Quine (Library of Living Philosophers), ed. by Lewis Edwin Hahn and Paul Arthur Schilpp ( La Salle, Illinois, Open Court, 1986). Charles Chihara, Constructability and Mathematical Existence ( Oxford, Clarendon Press, 1990).
3
Penelope Maddy, "'Indispensability and Practice'", Journal of Philosophy, 89 ( 1992), 275-90.
4
Elliot Sober, "'Mathematics and Indispensability'", Philosophical Review, 102 ( 1993), 35-57.

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