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Showing papers on "Finitism published in 2008"


Journal ArticleDOI
TL;DR: The correspondence between Paul Bernays and Kurt Godel is one of the most extensive in the two volumes of Godel's collected works devoted to his letters of (primarily) scientific, philosophical and historical interest.
Abstract: 1. Godel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Godel is one of the most extensive in the two volumes of Godel's collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Godel's thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Godel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Godel's Collected Works, pp. 41-

19 citations


Journal ArticleDOI
TL;DR: In this article, it is argued that an underlying extensionalism is the root of SSK's intractable philosophical difficulties, and that a critical philosophy is advocated in its place.
Abstract: The dramatic and ongoing changes in the funding of science have stimulated interest in an economics of scientific knowledge (ESK), which would investigate the effects of these changes on the scientific enterprise. Hands (1994) has previously explored the lessons for such an ESK from the existing precedent of the sociology of scientific knowledge (SSK). In particular, he examines the philosophical problems of SSK and those that any ESK in its image would face. This paper explores this argument further by contending that more recent literature in SSK exposes even deeper philosophical problems than those identified by Hands. Meaning finitism has emerged as the philosophical core of SSK. An examination of the profound problems with this position is used to show that an underlying extensionalism is the root of SSK’s intractable philosophical difficulties, and to illustrate the entirely different approach of a critical philosophy that is advocated in its place. In this way, the project of an ESK is shown to depend upon a critical philosophy.

13 citations


Journal ArticleDOI
TL;DR: A Πı-transfer principle is added to ERNA and it is proved that transfer is ‘too strong’ for finitism by reconsidering Rössler and Jeřábek's conclusions.
Abstract: Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Patrick Suppes and Richard Sommer, who also proved its consistency inside PRA. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes, of which Michal R¨ ossler and Emil Jeˇ r´ abek have recently proposed a weakened version. We add a! 1-transfer principle to ERNA and prove the consistency of the extended theory inside PRA. In this extension of ERNA a " 1-supremum principle 'up-to-infinitesimals', and some well-known calculus results for sequences are deduced. Finally, we prove that transfer is 'too strong' for finitism by reconsidering R¨ ossler and Jeˇ´ conclusions. §1. Introduction. Hilbert's Program, proposed in 1921, called for an axiomatic formalization of mathematics, together with a proof that this axiomatization is consistent. TheconsistencyproofitselfwastobecarriedoutusingonlywhatHilbert called finitary methods. In due time, many characterized Hilbert's informal notion of 'finitary' as that which can be formalized in Primitive Recursive Arithmetic (PRA), proposed in 1923 by Skolem. By G¨ odel's second incompleteness theorem (1931) it became evident that only partial realizations of Hilbert's program are possible. The system proposed by Chuaqui and Suppes, recently adapted by R¨

9 citations


01 Jan 2008

2 citations