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Summary An abstraction principle (AP) allows one to introduce new singular terms by providing appropriate identity conditions. For instance, the most celebrated abstraction principle, called Hume's Principle (HP), introduces numerical terms by saying: "The number of Fs is the same as the number of Gs if and only if Fs and Gs are equinumerous (the relation of equinumerosity is definable in a second-order language without reference to numbers)." The first (and unsuccessful, because inconsistent) attempt at using APs in foundations of mathematics is due to Frege. Neo-Fregeans try to salvage Frege's project. One of the tasks is to show how various mathematical theories can be derived from appropriate APs. Another task is to develop a well-motivated acceptability criterion for APs (given that Frege's Basic Law V leads to contradiction and HP doesn't). The Bad Company objection (according to which there are separately consistent but mutually inconsistent abstraction principles) indicates that mere consistency of an AP is not enough for its acceptability. Finally neo-Fregeans have to develop a philosophically acceptable story explaining why APs can play an important role in the platonist epistemology of mathematics and what role exactly it is. 
Key works Wright 1983 is a seminal book on the topic. The consistency of arithmetic based on Hume's Principle has been proven by Boolos 1987Fine 2002 is a good survey of technical aspects of neologicism. A nice anthology of papers related to the Bad Company problem is vol. 70 no 3 of Synthese edited by Linnebo 2009. A good collection of essays related to neologicism is Hale 2001.
Introductions A good place to start is Zalta 2008 and more focused Zalta 2008 and Tennant 2013. A good introductory paper focused on philosophical motivations is  Cook 2009. A nice introduction to worries surrounding the acceptability criteria of APs is Linnebo 2009.
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  1. 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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  2. G. Aldo Antonelli (2012). A Note on Induction, Abstraction, and Dedekind-Finiteness. Notre Dame Journal of Formal Logic 53 (2):187-192.
    The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
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  3. Vadim Batitsky (2002). Some Measurement-Theoretic Concerns About Hale's ‘Reals by Abstraction';. Philosophia Mathematica 10 (3):286-303.
    Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely (...)
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  4. 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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  5. Alexander Bird (1997). The Logic in Logicism. Dialogue 36 (02):341--60.
    Frege's logicism consists of two theses: (1) the truths of arithmetic are truths of logic; (2) 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 (...)
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  6. Izabela Bondecka-Krzykowska (2004). Strukturalizm jako alternatywa dla platonizmu w filozofii matematyki. Filozofia Nauki 1.
    The aim of this paper is to analyze structuralism as an alternative view to platonism in the philosophy of mathematics. We also try to find out if ontological and epistemological problems of platonism can be avoided by admitting the principles of structuralism. Structuralism claims that mathematical objects are merely positions in structures and have no identity or in general any important features outside these structures. Such view allows to avoid problems of the nature of numbers and other mathematical objects. But (...)
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  7. 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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  8. 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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  9. John P. Burgess (2003). Review: The Limits of Abstraction by Kit Fine. [REVIEW] Notre Dame Journal Fo Formal Logic 44:227-251.
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  10. Timothy Colburn & Gary Shute (2007). Abstraction in Computer Science. Minds and Machines 17 (2):169-184.
    We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction through information hiding is a (...)
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  11. R. Cook (2003). Iteration One More Time. Notre Dame Journal of Formal Logic 44 (2):63--92.
    A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that (...)
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  12. 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.
  13. Roy T. Cook (2009). New Waves on an Old Beach: Fregean Philosophy of Mathematics Today. In Ø. Linnebo O. Bueno (ed.), New Waves in Philosophy of Mathematics.
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  14. Roy T. Cook (2003). Aristotelian Logic, Axioms, and Abstraction. Philosophia Mathematica 11 (2):195-202.
    Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to Neo-logicism so much as a clarification (...)
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  15. Roy T. Cook (2002). The State of the Economy: Neo-Logicism and Inflationt. Philosophia Mathematica 10 (1):43-66.
    In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why this (...)
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  16. Roy T. Cook & Philip Ebert (2004). The Limits of Abstraction (Book Review). British Journal for the Philosophy of Science 55 (4):791-800.
    The Limits ofion, Kit Fine, Oxford: Clarendon Press, 2002, pp.216. ISBN 9780191567261.
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  17. 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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  18. Chris Daly & Simon Langford (2011). Two Anti-Platonist Strategies. Mind 119 (476):1107-1116.
    This paper considers two strategies for undermining indispensability arguments for mathematical Platonism. We defend one strategy (the Trivial Strategy) against a criticism by Joseph Melia. In particular, we argue that the key example Melia uses against the Trivial Strategy fails. We then criticize Melia’s chosen strategy (the Weaseling Strategy.) The Weaseling Strategy attempts to show that it is not always inconsistent or irrational knowingly to assert p and deny an implication of p . We argue that Melia’s case for this (...)
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  19. Chris Daly & David Liggins (2014). Nominalism, Trivialist Platonism and Benacerraf's Dilemma. Analysis 74 (2):224-231.
    In his stimulating new book The Construction of Logical Space, Agustín Rayo offers a new account of mathematics, which he calls ‘Trivialist Platonism’. In this article, we take issue with Rayo’s case for Trivialist Platonism and his claim that the view overcomes Benacerraf’s dilemma. Our conclusion is that Rayo has not shown that Trivialist Platonism has any advantage over nominalism.
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  20. William Demopoulos (2013). Generality and Objectivity in Frege's Foundations of Arithmetic. In Alex Miller (ed.), Logic, Language and Mathematics: Essays for Crispin Wright. Oxford University Press.
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  21. William Demopoulos (2003). Book Symposium: The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics by Bob Hale and Crispin Wright: On the Philosophical Interest of Frege Arithmetic. Philosophical Books 44 (3):220-228.
    The paper considers Fregean and neo-Fregean strategies for securing the apriority of arithmetic. The Fregean strategy recovers the apriority of arithmetic from that of logic and a family of explicit definitions. The neo-Fregean strategy relies on a principle which, though not an explicit definition, is given the status of a stipulation; unlike the Fregean strategy it relies on an extension of second order logic which is not merely a definitional extension. The paper argues that this methodological difference is important in (...)
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  22. William Demopoulos (ed.) (1995). Frege's Philosophy of Mathematics. Harvard University Press.
  23. 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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  24. Michael Dummett (1973). Frege's Way Out: A Footnote to a Footnote. Analysis 33 (4):139 - 140.
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  25. Philip A. Ebert (2008). A Puzzle About Ontological Commitments. Philosophia Mathematica 16 (2):209-226.
    This paper raises and then discusses a puzzle concerning the ontological commitments of mathematical principles. The main focus here is Hume's Principle—a statement that, embedded in second-order logic, allows for a deduction of the second-order Peano axioms. The puzzle aims to put pressure on so-called epistemic rejectionism, a position that rejects the analytic status of Hume's Principle. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the neo-Fregeans, defenders of the analytic status of (...)
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  26. 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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  27. 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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  28. Matti Eklund (2006). Neo-Fregean Ontology. Philosophical Perspectives 20 (1):95–121.
    Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...)
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  29. Kit Fine (2002). The Limits of Abstraction. Oxford University Press.
    Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits of Abstraction breaks new ground both technically and philosophically.
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  30. S. Gandon (2013). Variable, Structure, and Restricted Generality. Philosophia Mathematica 21 (2):200-219.
    From 1905–1908 onward, Russell thought that his new ‘substitutional theory’ provided him with the right framework to resolve the set-theoretic paradoxes. Even if he did not finally retain this resolution, the substitutional strategy was instrumental in the development of his thought. The aim of this paper is not historical, however. It is to show that Russell's substitutional insight can shed new light on current issues in philosophy of mathematics. After having briefly expounded Russell's key notion of a ‘structured variable’, I (...)
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  31. Bob Hale & Crispin Wright (2008). Abstraction and Additional Nature. Philosophia Mathematica 16 (2):182-208.
    What is wrong with abstraction’, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their ‘Hale on Caesar’ and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are: To get the character of this objection into sharper focus; To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and (...)
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  32. Bob Hale (ed.) (2001). The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press.
    Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...)
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  33. Bob Hale (2000). Reals by Abstractiont. 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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  34. Bob Hale (2000). Abstraction and Set Theory. Notre Dame Journal of Formal Logic 41 (4):379--398.
    The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties confronting the (...)
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  35. Bob Hale (1999). Frege's Philosophy of Mathematics. Philosophical Quarterly 49 (194):92–104.
  36. Bob Hale (1997). Grundlagen §64. Proceedings of the Aristotelian Society 97 (3):243–261.
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  37. Bob Hale (1994). Dummett's Critique of Wright's Attempt to Resuscitate Frege. Philosophia Mathematica 2 (2):122-147.
    Michael Dummett mounts, in Frege: Philosophy of Mathematics, a concerted attack on the attempt, led by Crispin Wright, to salvage defensible versions of Frege's platonism and logicism in which Frege's criterion of numerical identity plays a leading role. I discern four main strands in this attack—that Wright's solution to the Caesar problem fails; that explaining number words contextually cannot justify treating them as enjoying robust reference; that Wright has no effective counter to ontological reductionism; and that the attempt is vitiated (...)
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  38. Bob Hale & Crispin Wright (2009). Focus Restored: Comments on John MacFarlane. Synthese 170 (3):457 - 482.
    In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...)
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  39. Bob Hale & Crispin Wright (2001). Introduction. In Bob Hale & Crispin Wrigth (eds.), The Reason's Proper Study. Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press. 1-27.
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  40. Richard Heck, Frege Arithmetic and "Everyday Mathematics&Quot;.
    The purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets some weak but non-trivial arithmetical theories. The weak theories in question are all related to Tarski, Mostowski, and Robinson's R. In saying that the interpretation is "natural", I mean that it relies only upon "definitions" of arithmetical notions that are themselves "natural", that is, that have some claim to be "definitions" in something other than a purely formal sense.
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  41. 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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  42. Richard Heck (2000). Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic 41 (3):187-209.
    Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...)
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  43. Richard Heck (1999). Frege's Theorem: An Introduction. The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  44. Richard Heck (1998). The Finite and the Infinite in Frege's Grundgesetze der Arithmetik. In M. Schirn (ed.), Philosophy of Mathematics Today. OUP.
    Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.
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  45. Richard Heck (ed.) (1997). Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford University Press.
    In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought. Two contributors discuss time, in particular the (...)
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  46. Richard Heck (1997). The Julius Caesar Objection. In R. Heck (ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford University Press. 273--308.
    This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us.
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  47. Richard Heck (1997). Finitude and Hume's Principle. Journal of Philosophical Logic 26 (6):589-617.
    The paper formulates and proves a strengthening of Freges Theorem, which states that axioms for second-order arithmetic are derivable in second-order logic from Humes Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. Finite Humes Principle also suffices for (...)
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  48. Richard Heck (1996). The Consistency of Predicative Fragments of Frege's Grundgesetze der Arithmetik. History and Philosophy of Logic 17 (1):209-220.
    As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell?s Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege?s Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is (...)
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  49. Richard Heck (1995). Definition by Induction in Frege's Grundgesetze der Arithmetik. In W. Demopoulos (ed.), Frege's Philosophy of Mathematics. OUP.
    This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.
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  50. Richard Heck (1995). Frege's Principle. In J. Hintikka (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics. Kluwer.
    This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.
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