On the Scope and Force of Indispensability Arguments

Abstract
Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results (of Pour-El and Richards and of the author) are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of "physics".
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Solomon Feferman (1992). Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:442 - 455.
Lieven Decock (2002). Quine's Weak and Strong Indispensability Argument. Journal for General Philosophy of Science 33 (2):231-250.
Geoffrey Hellman (2004). Predicativism as a Philosophical Position. Revue Internationale de Philosophie 3:295-312.
Mark Colyvan (1998). In Defence of Indispensability. Philosophia Mathematica 6 (1):39-62.
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