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  1. H. Leitgeb A. Hieke (ed.) (2009). Reduction – Abstraction – Analysis. Ludwig Wittgenstein Society.
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  2. Bird Alexander (1997). The Logic in Logicism. Dialogue 36.
    Frege's logicism consists of two theses: the truths of arithmetic are truths of logic; the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G are equinumerous is (...)
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  3. Alice Ambrose (1933). A Controversy in the Logic of Mathematics. Philosophical Review 42 (6):594-611.
  4. Irving H. Anellis (1987). Russell and Engels: Two Approaches to a Hegelian Philosophy of Mathematics. Philosophia Mathematica (2):151-179.
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  5. Irving H. Anellis (1987). Russell's Earliest Interpretation of Cantorian Set Theory, 1896–1900. Philosophia Mathematica (1):1-31.
  6. G. A. Antonelli (2010). Notions of Invariance for Abstraction Principles. Philosophia Mathematica 18 (3):276-292.
    The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the philosophical significance (...)
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  7. G. Aldo Antonelli (2012). Review of Frege's Theorem. [REVIEW] International Studies in the Philosophy of Science 26 (2):219-222.
  8. Timothy Bays (2006). Review of John Burgess, Fixing Frege. [REVIEW] Notre Dame Philosophical Reviews 2006 (6).
  9. Timothy Bays (2000). The Fruits of Logicism. Notre Dame Journal of Formal Logic 41 (4):415-421.
    You’ll be pleased to know that I don’t intend to use these remarks to comment on all of the papers presented at this conference. I won’t try to show that one paper was right about this topic, that another was wrong was about that topic, or that several of our conference participants were talking past one another. Nor will I try to adjudicate any of the discussions which took place in between our sessions. Instead, I’ll use these remarks to make (...)
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  10. Paul Benacerraf (1960). Logicism, Some Considerations. Dissertation, Princeton University
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  11. Alexander Bird (1997). The Logic in Logicism. Dialogue 36 (2):341--60.
    Frege's logicism consists of two theses: the truths of arithmetic are truths of logic; the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G are equinumerous is (...)
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  12. Kenneth Blackwell, Nicholas Griffin & Bernard Linsky (eds.) (2011). Principia Mathematica at 100. Bertrand Russell Research Centre.
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  13. George Boolos (1990). The Standard of Equality of Numbers. In Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press 261--77.
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  14. George S. Boolos (ed.) (1990). Meaning and Method: Essays in Honor of Hilary Putnam. Cambridge University Press.
    This volume is a report on the state of philosophy in a number of significant areas.
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  15. Andrew Boucher, Who Needs (to Assume) Hume's Principle?
    Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.
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  16. Andrew Boucher, Who Needs (to Assume) Hume's Principle? July 2006.
    In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege (...)
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  17. Otavio Bueno (2001). Logicism Revisited. Principia 5 (1-2):99-124.
    In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue (...)
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  18. John Burgess, Logicism: A New Look.
    Adapated from talks at the UCLA Logic Center and the Pitt Philosophy of Science Series. Exposition of material from Fixing Frege, Chapter 2 (on predicative versions of Frege’s system) and from “Protocol Sentences for Lite Logicism” (on a form of mathematical instrumentalism), suggesting a connection. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.
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  19. John Burgess, Mending the Master.
    Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege’s writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well informed overview provided by the first of its three chapters. Specialists will find the book an (...)
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  20. Rudolf Carnap (1983). The Logicist Foundations of Mathematics. In Paul Benacerraf & Hilary Putnam (eds.), Philosophy of Mathematics: Selected Readings. Cambridge University Press 41--52.
  21. R. T. Cook (2012). RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. Xiv + 307. Philosophia Mathematica 20 (3):346-359.
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  22. Roy T. Cook (2013). Patricia A. Blanchette. Frege's Conception of Logic. Oxford University Press, 2012. ISBN 978-0-19-926925-9 (Hbk). Pp. Xv + 256. [REVIEW] Philosophia Mathematica (1):nkt029.
  23. Roy T. Cook (ed.) (2007). The Arché Papers on the Mathematics of Abstraction. Springer.
    Unique in presenting a thoroughgoing examination of the mathematical aspects of the neo-logicist project (and the particular philosophical issues arising from these technical concerns).
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  24. Roy T. Cook & Philip A. Ebert (2005). Abstraction and Identity. Dialectica 59 (2):121–139.
    A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
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  25. Boudewijn de Bruin (2008). Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number. Philosophia Mathematica 16 (3):354-373.
    Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...)
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  26. William Demopoulos (2013). Logicism and its Philosophical Legacy. Cambridge University Press.
    Frege's analysis of arithmetical knowledge -- Carnap's thesis -- On extending 'empiricism, semantics and ontology' to the realism-instrumentalism controversy -- Carnap's analysis of realism -- Bertrand Russell's The analysis of matter: its historical context and contemporary interest (with Michael Friedman) -- On the rational reconstruction of our theoretical knowledge -- Three views of theoretical knowledge -- Frege and the rigorization of analysis -- The philosophical basis of our knowledge of number -- The 1910 Principia's theory of functions and classes -- (...)
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  27. William Demopoulos (ed.) (1995). Frege's Philosophy of Mathematics. Harvard University Press.
  28. William Demopoulos & Peter Clark (2005). The Logicism of Frege, Dedekind, and Russell. In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press 129--165.
  29. William Demopoulus & William Bell (1993). Frege's Theory of Concepts and Objects and the Interpretation of Second-Order Logict. Philosophia Mathematica 1 (2):139-156.
    This paper casts doubt on a recent criticism of Frege's theory of concepts and extensions by showing that it misses one of Frege's most important contributions: the derivation of the infinity of the natural numbers. We show how this result may be incorporated into the conceptual structure of Zermelo- Fraenkel Set Theory. The paper clarifies the bearing of the development of the notion of a real-valued function on Frege's theory of concepts; it concludes with a brief discussion of the claim (...)
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  30. Michael Detlefsen (1992). Poincaré Against the Logicians. Synthese 90 (3):349 - 378.
    Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of (...)
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  31. Michael A. E. Dummett (1991). Frege: Philosophy of Mathematics. Harvard University Press.
    In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...
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  32. P. A. Ebert (2011). Guillermo E. Rosado Haddock. A Critical Introduction to the Philosophy of Gottlob Frege. Aldershot, Hampshire, and Burlington, Vermont: Ashgate Publishing, 2006. Isbn 978-0-7546-5471-1. Pp. X+157. [REVIEW] Philosophia Mathematica 19 (3):363-367.
    Guillermo E. Rosado Haddock's critical introduction to the philosophy of Gottlob Frege is based on twenty-five years of teaching Frege's philosophy at the University of Puerto Rico. It developed from an earlier publication by Rosado Haddock on Frege's philosophy which was, however, available only in Spanish. This introduction to Frege is meant to steer a path between the two main approaches to Frege studies: on the one hand, we have interpretations of Frege which portray him as a neo-Kantian and thus (...)
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  33. Philip A. Ebert & Marcus Rossberg (2009). Ed Zalta's Version of Neo-Logicism: A Friendly Letter of Complaint. In Hannes Leitgeb & Alexander Hieke (eds.), Reduction – Abstraction – Analysis. Ontos 11--305.
    In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
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  34. Philip A. Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415 - 441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  35. Philip Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415-441.
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  36. Matti Eklund (2009). Bad Company and Neo-Fregean Philosophy. Synthese 170 (3):393 - 414.
    A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on (...)
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  37. Fernando Ferreira & Kai F. Wehmeier (2002). On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze. Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...)
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  38. José Ferreirós (2009). Hilbert, Logicism, and Mathematical Existence. Synthese 170 (1):33 - 70.
    David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...)
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  39. Gottlob Frege (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Northwestern University Press.
    § i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...
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  40. Gottlob Frege, Philip A. Ebert & Marcus Rossberg (eds.) (2013). Basic Laws of Arithmetic. Oxford University Press.
    This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik, with introduction and annotation. The importance of Frege's ideas within contemporary philosophy would be hard to exaggerate. He was, to all intents and purposes, the inventor of mathematical logic, and the influence exerted on modern philosophy of language and logic, and indeed on general epistemology, by the philosophical framework within which his technical contributions were conceived and developed has been so deep that he has a strong case (...)
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  41. S. Gandon & B. Halimi (2013). Introduction: Logicism Today. Philosophia Mathematica 21 (2):129-132.
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  42. Sébastien Gandon (2008). Which Arithmetization for Which Logicism? Russell on Relations and Quantities in The Principles of Mathematics. History and Philosophy of Logic 29 (1):1-30.
    This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible 'by logical principles from logical principles' does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV-V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: Russell's logicism (...)
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  43. Bonnie Gold & Roger Simons (eds.) (2008). Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America.
    This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to ...
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  44. N. Griffin (2013). Bernard Linsky. The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition. Cambridge: Cambridge University Press, 2011. ISBN 978-1-107-00327-9. Pp. Vii + 407. [REVIEW] Philosophia Mathematica 21 (3):403-411.
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  45. Bob Hale (2000). Reals by Abstraction. 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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  46. Bob Hale (1999). Frege's Philosophy of Mathematics. Philosophical Quarterly 49 (194):92–104.
  47. William H. Hanson (1990). Second-Order Logic and Logicism. Mind 99 (393):91-99.
    Some widely accepted arguments in the philosophy of mathematics are fallacious because they rest on results that are provable only by using assumptions that the con- clusions of these arguments seek to undercut. These results take the form of bicon- ditionals linking statements of logic with statements of mathematics. George Boolos has given an argument of this kind in support of the claim that certain facts about second-order logic support logicism, the view that mathematics—or at least part of it—reduces to (...)
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  48. William S. Hatcher (1982). The Logical Foundations of Mathematics. Pergamon Press.
  49. Richard Heck (2011). Ramified Frege Arithmetic. Journal of Philosophical Logic 40 (6):715-735.
    Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.
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  50. Richard Heck (2011). A Logic for Frege's Theorem. In Frege's Theorem. Oxford University Press
    It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...)
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