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Neo-Fregean foundations for real analysis: Some reflections on Frege’s constraint. (English) Zbl 1014.03012
Summary: 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 S. Shapiro in his contribution to this volume [ibid. 335-364 (2000; Zbl 1014.03013); see below], 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.

MSC:
 03A05 Philosophical and critical aspects of logic and foundations 00A30 Philosophy of mathematics 03F35 Second- and higher-order arithmetic and fragments 03B30 Foundations of classical theories (including reverse mathematics) 26A03 Foundations: limits and generalizations, elementary topology of the line
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References:
 [1] Boolos, G., Logic, Logic, and Logic , Harvard University Press, Cambridge, 1998. · Zbl 0955.03008 [2] Boolos, G., ”The consistency of Frege’s Foundations of Arithmetic ”, pp. 3–20 in On Being and Saying: Essays in Honor of Richard Cartwright , edited by J. Thomson, The MIT Press, Cambridge, 1987. Reprinted in [?], pp. 183–201. · Zbl 0972.03503 [3] Boolos, G., ”The standard of equality of numbers”, pp. 261–77 in Meaning and Method: Essays in Honor of Hilary Putnam , edited by G. Boolos, Cambridge University Press, Cambridge, 1990. Reprinted in ?, pp. 202–19. · Zbl 0972.03504 [4] Boolos, G., and R. G. Heck, Jr., ” Die Grundlagen der Arithmetik $$\S\S$$82–3”, pp. 407–28 in The Philosophy of Mathematics Today (Munich, 1993) , edited by M. Schirn, Oxford University Press, New York, 1998. Reprinted in [?], pp. 315–38. · Zbl 0972.03510 [5] Dummett, M., Frege: Philosophy of Mathematics , Harvard University Press, Cambridge, 1991. [6] Frege, G., Die Grundlagen der Arithmetik , Wilhelm Koebner, Breslau, 1884. Translated by J. L. Austin as The Foundations of Arithmetic , Blackwell, Oxford, 1959. [7] Frege, G., Grundgesetze der Arithmetik I , Hildescheim, Olms, 1893. · JFM 25.0101.02 [8] Hale, B., ”Reals by abstraction”, Philosophia Mathematica , vol. 8 (2000), pp. 100–23. Reprinted in [?], pp. 399–420. · Zbl 0968.03010 [9] Hale, B., ”Abstraction and set theory”, Notre Dame Journal of Formal Logic , vol. 41 (2000), pp. 379–98. · Zbl 1014.03014 [10] Hale, B., and C. Wright, The Reason’s Proper Study , Clarendon Press, Oxford, 2001 · Zbl 1005.03006 [11] Heck, R. G., Jr., ”Finitude and Hume’s Principle”, Journal of Philosophical Logic , vol. 26 (1997), pp. 589–617. · Zbl 0885.03045 [12] Resnik, M., Mathematics as a Science of Patterns , The Clarendon Press, New York, 1997. · Zbl 0905.03004 [13] Shapiro, S., ”Frege meets Dedekind: A neo-logicist treatment of real analysis”, Notre Dame Journal of Formal Logic , vol. 41 (2000), pp. 335–64. · Zbl 1014.03013 [14] Shapiro, S., Philosophy of Mathematics. Structure and Ontology , Oxford University Press, New York, 1997. · Zbl 0897.00004 [15] Wright, C., Frege’s Conception of Numbers as Objects , Aberdeen University Press, Aberdeen, 1983. · Zbl 0524.03005
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