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  1. Frege’s Theory of Real Numbers: A Consistent Rendering.Francesca Boccuni & Marco Panza - forthcoming - Review of Symbolic Logic:1-44.
    Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.
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  2. Frege on the introduction of real and complex numbers by abstraction and cross-sortal identity claims.Matthias Schirn - 2023 - Synthese 201 (6):1-18.
    In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers in (...)
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  3. Mr. Frege, The Platonist.Daniel Sierra - 2021 - Logiko-Filosofskie Studii 2 (Vol 19):136-144.
    Even though Frege is a major figure in the history of analytic philosophy, it is not surprising that there are still issues surrounding his views, interpreting them, and labeling them. Frege’s view on numbers is typically termed as ‘Platonistic’ or at least a type of Platonism (Reck 2005). Still, the term ‘Platonism’ has views and assumptions ascribed to it that may be misleading and leads to mischaracterizations of Frege’s outlook on numbers and ideas. So, clarification of the term ‘Platonism’ is (...)
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  4. Reflections on Frege’s Theory of Real Numbers†.Peter Roeper - 2020 - Philosophia Mathematica 28 (2):236-257.
    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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  5. Frege’s Constraint and the Nature of Frege’s Foundational Program.Marco Panza & Andrea Sereni - 2019 - Review of Symbolic Logic 12 (1):97-143.
    Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...)
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  6. Neologicism for Real – Are We There Yet?Rafal Urbaniak - 2019 - In Bartłomiej Skowron (ed.), Contemporary Polish Ontology. Berlin: De Gruyter. pp. 181-204.
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  7. 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. Boston: De Gruyter. pp. 25-95.
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  8. Frege's Approach to the Foundations of Analysis (1874–1903).Matthias Schirn - 2013 - History and Philosophy of Logic 34 (3):266-292.
    The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ?theory of quantity? (?Größenlehre?) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section (...)
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  9. (1 other version)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. (1 other version)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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  11. 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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  12. 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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  13. 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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  14. (1 other version)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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  15. (1 other version)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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  16. Frege's theory of real numbers.M. Dummett - 1995 - In William Demopoulos (ed.), Frege's philosophy of mathematics. Cambridge: Harvard University Press. pp. 388--404.
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  17. 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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  18. Frege's conception of numbers as objects.Crispin Wright - 1983 - [Aberdeen]: Aberdeen University Press.