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2000 Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint
Crispin Wright
Notre Dame J. Formal Logic 41(4): 317-334 (2000). DOI: 10.1305/ndjfl/1038336879


We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals by Abstraction" program differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis.


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Crispin Wright. "Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint." Notre Dame J. Formal Logic 41 (4) 317 - 334, 2000.


Published: 2000
First available in Project Euclid: 26 November 2002

zbMATH: 1014.03012
MathSciNet: MR1963485
Digital Object Identifier: 10.1305/ndjfl/1038336879

Primary: 00A30
Secondary: 03A05

Keywords: abstraction principles , analysis , Frege , structuralism

Rights: Copyright © 2000 University of Notre Dame


Vol.41 • No. 4 • 2000
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