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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. Jonathan E. Adler (1984). Abstraction is Uncooperative. Journal for the Theory of Social Behaviour 14 (2):165–181.
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  2. Aldo Antonelli, Logicism, Quantifiers, and Abstraction.
    With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.
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  3. 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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  4. 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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  5. G. Aldo Antonelli & Robert C. May (2005). Frege's Other Program. Notre Dame Journal of Formal Logic 46 (1):1-17.
    Frege’s logicist program requires that arithmetic be reduced to logic. Such a program has recently been revamped by the “neo-logicist” approach of Hale & Wright. Less attention has been given to Frege’s extensionalist program, according to which arithmetic is to be reconstructed in terms of a theory of extensions of concepts. This paper deals just with such a theory. We present a system of second-order logic augmented with a predicate representing the fact that an object x is the extension of (...)
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  6. Gordon Barnes (2002). Hale’s Necessity: It’s Indispensable, But is It Real? Disputatio 13:3 - 10.
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  7. 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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  8. 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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  9. John L. Bell (1994). Fregean Extensions of First‐Order Theories. Mathematical Logic Quarterly 40 (1):27-30.
    It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.
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  10. Jean-Pierre Belna (2006). Objectivity and the Principle of Duality: Paragraph 26 of Frege's Foundations of Arithmetic. Revue d'Histoire des Sciences 59 (2):319.
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  11. 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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  12. 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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  13. George Boolos & Richard G. Heck (1998). Die Grundlagen der Arithmetik, 82-3. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. 407-28.
    This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in (...)
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  14. 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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  15. 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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  16. 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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  17. Howard Burdick (1974). On the Problems of Abstraction and Concretion. Noûs 8 (3):295-297.
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  18. John P. Burgess (2012). Frege’s Theorem by Richard G. Heck, Jr. Journal of Philosophy 109 (12):728-732.
  19. John P. Burgess (2012). Richard G. Heck, Jr.: Frege’s Theorem. [REVIEW] Journal of Philosophy 109 (12):728-733.
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  20. 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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  21. Louis Caruana, Abstraction and the Environment.
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  22. Howard Caygill (2006). From Abstraction to Wunsch: The Vocabulaire Européen des Philosophies. Radical Philosophy 138:10-14.
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  23. 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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  24. Philip Ebert with Roy T. Cook, Critical Notice of Fine’s “Limits of Abstraction”.
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  25. 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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  26. R. T. Cook (2012). Conservativeness, Stability, and Abstraction. British Journal for the Philosophy of Science 63 (3):673-696.
    One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable (...)
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  27. 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.
  28. Roy T. Cook (2009). Hume's Big Brother: Counting Concepts and the Bad Company Objection. Synthese 170 (3):349 - 369.
    A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...)
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  29. 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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  30. 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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  31. 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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  32. Roy T. Cook & Philip Ebert (2004). The Limits of Abstraction (Book Review). British Journal for the Philosophy of Science 55 (4):791-800.
    Critical Notice of The Limits of abstraction by Kit Fine, Oxford: Clarendon Press, 2002, pp.216. ISBN 9780191567261.
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  33. 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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  34. Francis A. Cunningham (1958). A Theory on Abstraction in St. Thomas. Modern Schoolman 35 (4):249-270.
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  35. C. Daly & D. 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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  36. 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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  37. Gérard Deledalle (1950). Réflexions Sur l'Abstraction Et la Nature de L'Abstrait. À Propos de la Philosophie de J. Laporte. Revue Philosophique De Louvain 48 (17):63-89.
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  38. 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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  39. William Demopoulos (2006). The Neo-Fregean Program in the Philosophy of Arithmetic. In Emily Carson & Renate Huber (eds.), Intuition and the Axiomatic Method. Springer. 87--112.
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  40. 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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  41. William Demopoulos (ed.) (1995). Frege's Philosophy of Mathematics. Harvard University Press.
  42. 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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  43. Albert Dondeyne (1938). L'abstraction. Revue Néo-Scolastique de Philosophie 41 (57):5-20.
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  44. Michael Dummett (1973). Frege's Way Out: A Footnote to a Footnote. Analysis 33 (4):139 - 140.
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  45. 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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  46. 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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  47. 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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  48. 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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  49. 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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  50. Richard L. Epstein (2013). Mathematics as the Art of Abstraction. In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer. 257--289.
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