David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Erkenntnis 53 (1-2):63-96 (2000)
The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics 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 betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted.
|Keywords||Philosophy Philosophy Epistemology Ethics Logic Ontology|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Andrew Arana (2010). Proof Theory in Philosophy of Mathematics. Philosophy Compass 5 (4):336-347.
Reinhard Kahle (2003). Universes Over Frege Structures. Annals of Pure and Applied Logic 119 (1-3):191-223.
Kentaro Sato (2014). Forcing for Hat Inductive Definitions in Arithmetic. Mathematical Logic Quarterly 60 (4-5):314-318.
Similar books and articles
Thomas Strahm (1997). Polynomial Time Operations in Explicit Mathematics. Journal of Symbolic Logic 62 (2):575-594.
Solomon Feferman, The Proof Theory of Classical and Constructive Inductive Definitions. A 40 Year Saga, 1968-2008.
Peter Aczel, Harold Simmons & S. S. Wainer (eds.) (1992). Proof Theory: A Selection of Papers From the Leeds Proof Theory Programme, 1990. Cambridge University Press.
Raphael van Riel (2010). Identity-Based Reduction and Reductive Explanation. Philosophia Naturalis 47 (1-2):183-219.
David J. Pym (2004). Reductive Logic and Proof-Search: Proof Theory, Semantics, and Control. Oxford University Press.
Solomon Feferman (1993). What Rests on What? The Proof-Theoretic Analysis of Mathematics. In J. Czermak (ed.), Philosophy of Mathematics. Hölder-Pichler-Tempsky 1--147.
Added to index2009-01-28
Total downloads19 ( #196,940 of 1,907,527 )
Recent downloads (6 months)8 ( #90,604 of 1,907,527 )
How can I increase my downloads?