Notre Dame Journal of Formal Logic

Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint

Crispin Wright

Abstract

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.

Article information

Source
Notre Dame J. Formal Logic Volume 41, Number 4 (2000), 317-334.

Dates
First available in Project Euclid: 26 November 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1038336879

Digital Object Identifier
doi:10.1305/ndjfl/1038336879

Mathematical Reviews number (MathSciNet)
MR1963485

Zentralblatt MATH identifier
1014.03012

Subjects
Primary: 00A30: Philosophy of mathematics [See also 03A05]
Secondary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
abstraction principles analysis Frege structuralism

Citation

Wright, Crispin. Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint. Notre Dame J. Formal Logic 41 (2000), no. 4, 317--334. doi:10.1305/ndjfl/1038336879. http://projecteuclid.org/euclid.ndjfl/1038336879.


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