Indispensability Without Platonism

In Alexander Bird, Brian Ellis & Howard Sankey (eds.), Properties, Powers, and Structures: Issues in the Metaphysics of Realism. New York, USA: Routledge. pp. 81-97 (2012)
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Abstract
According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage.
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