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  1. added 2020-05-26
    Les Nombres Réels: Frege Critique de Cantor Et de Dedekind/The Real Numbers: Frege's Criticism of Cantor and Dedekind.Jean Pierre Belna - 1997 - Revue d'Histoire des Sciences 50 (1):131-158.
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  2. added 2020-05-20
    Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...)
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  3. added 2020-05-20
    Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint.Crispin Wright - 2000 - Notre Dame Journal of Formal Logic 41 (4):317--334.
    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 byion" program differs by placing additional emphasis upon what I here term Frege's Constraint, (...)
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  4. added 2020-05-20
    Frege's Theory of Real Numbers.Peter M. Simons - 1987 - History and Philosophy of Logic 8 (1):25--44.
    Frege's theory of real numbers has undeservedly received almost no attention, in part because what we have is only a fragment. Yet his theory is interesting for the light it throws on logicism, and it is quite different from standard modern approaches. Frege polemicizes vigorously against his contemporaries, sketches the main features of his own radical alternative, and begins the formal development. This paper summarizes and expounds what he has to say, and goes on to reconstruct the most important steps (...)
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  5. added 2020-05-19
    The Real Numbers: Frege's Criticism of Cantor and Dedekind.Jean-Pierre Belna - 1997 - Revue d'Histoire des Sciences 50 (1).
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  6. added 2020-05-18
    Frege's Theory of Real Numbers.M. Dummett - 1995 - In William Demopoulos (ed.), Frege's Philosophy of Mathematics. Harvard University Press. pp. 388--404.
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  7. added 2020-05-10
    Frege on Quantities and Real Numbers in Consideration of the Theories of Cantor, Russell and Others.Matthias Schirn - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 25-95.
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  8. added 2020-05-05
    Reflections on Frege’s Theory of Real Numbers†.Peter Roeper - forthcoming - Philosophia Mathematica:nkaa014.
    ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.
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  9. added 2020-05-05
    Reals by Abstraction.Bob Hale - 2000 - The Proceedings of the Twentieth World Congress of Philosophy 6:197-207.
    While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about numbers, I explain (...)
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  10. added 2020-05-05
    Les nombres réels : Frege critique de Cantor et de Dedekind/The real numbers : Frege's criticism of Cantor and Dedekind.Jean-Pierre Belna - 1997 - Revue d'Histoire des Sciences 50 (1-2):131-158.
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  11. added 2020-05-04
    Reals by Abstraction.Bob Hale - 2000 - Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...)
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