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Does Reductive Proof Theory Have a Viable Rationale?

Solomon Feferman
Erkenntnis (1975-)
Vol. 53, No. 1/2, Concepts of Reduction in Logic and Philosophy (2000), pp. 63-96
Published by: Springer
Stable URL: http://www.jstor.org/stable/20013006
Page Count: 34
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Does Reductive Proof Theory Have a Viable Rationale?
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Abstract

The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal" system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reduction relations between systems are explained and compared, and arguments against proof-theoretic reduction as a "good" reducibility relation are taken up and rebutted.

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