About this topic
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. The Troubled History of Abstraction.Ignacio Angelelli - 2005 - Logical Analysis and History of Philosophy 8.
    For centuries abstraction was understood as an operation according to which, from a given phenomenon, something is kept, but something else is not paid attention to, is ”abstracted from”. This notion of abstraction not only has been rejected by the mainstream of analytic philosophy and logic as worthless psychologism but, moreover, largely replaced by a new conception of abstraction in which the allegedly ”psychological” feature of ”not paying attention to”, or ”abstracting from”, is no longer visible. Psychologism has been overcome, (...)
  2. Adventures of Abstraction.Ignacio Angelelli - 2004 - Poznan Studies in the Philosophy of the Sciences and the Humanities 82 (1):11-35.
  3. Frege and Abstraction.Ignacio Angelelli - 1984 - Philosophia Naturalis 21 (2/4):453-471.
  4. Logicism, Quantifiers, and Abstraction.Aldo Antonelli - manuscript
    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.
  5. Notions of Invariance for Abstraction Principles.G. A. Antonelli - 2010 - 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 (...)
  6. A Note on Induction, Abstraction, and Dedekind-Finiteness.G. Aldo Antonelli - 2012 - 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.
  7. Review of Frege's Theorem[REVIEW]G. Aldo Antonelli - 2012 - International Studies in the Philosophy of Science 26 (2):219-222.
  8. Perceptual Abstraction and Art.Rudolf Arnheim - 1947 - Psychological Review 54 (2):66-82.
  9. Hale’s Necessity: It’s Indispensable, But is It Real?Gordon Barnes - 2002 - Disputatio 13:3 - 10.
  10. Some Measurement-Theoretic Concerns About Hale's ‘Reals by Abstraction';.Vadim Batitsky - 2002 - 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 (...)
  11. Review of John Burgess, Fixing Frege[REVIEW]Timothy Bays - 2006 - Notre Dame Philosophical Reviews 2006 (6).
  12. The Fruits of Logicism.Timothy Bays - 2000 - 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 (...)
  13. Fregean Extensions of First‐Order Theories.John L. Bell - 1994 - 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.
  14. Objectivity and the Principle of Duality: Paragraph 26 of Frege's Foundations of Arithmetic.Jean-Pierre Belna - 2006 - Revue d'Histoire des Sciences 59 (2):319.
  15. The Logic in Logicism.Alexander Bird - 1997 - 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 (...)
  16. The Breadth of the Paradox.Patricia Blanchette - 2016 - Philosophia Mathematica 24 (1):30-49.
    This essay examines Frege's reaction to Russell's Paradox and his views about the grounding of existence claims in mathematics. It is argued that Frege's strict requirements on existential proofs would rule out the attempt to ground arithmetic in. It is hoped that this discussion will help to clarify the ways in which Frege's position is both coherent and significantly different from the neo-logicist position on the issues of: what's required for proofs of existence; the connection between models, consistency, and existence; (...)
  17. Logicism Reconsidered.Patricia A. Blanchette - 1990 - Dissertation, Stanford University
    This thesis is an examination of Frege's logicism, and of a number of objections which are widely viewed as refutations of the logicist thesis. In the view offered here, logicism is designed to provide answers to two questions: that of the nature of arithmetical truth, and that of the source of arithmetical knowledge. ;The first objection dealt with here is the view that logicism is not an epistemologically significant thesis, due to the fact that the epistemological status of logic itself (...)
  18. Abstraction, Relation, and Induction: Three Essays in the History of Thought.John F. Boler & Julius R. Weinberg - 1967 - Philosophical Review 76 (3):394.
  19. Strukturalizm jako alternatywa dla platonizmu w filozofii matematyki.Izabela Bondecka-Krzykowska - 2004 - 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 (...)
  20. Die Grundlagen der Arithmetik, 82-3.George Boolos & Richard G. Heck - 1998 - In Matthias Schirn (ed.), Bulletin of Symbolic Logic. Clarendon Press. pp. 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 (...)
  21. Logic and Arithmetic.David Bostock - 1981 - Noûs 15 (4):551-559.
  22. Against Angles and the Fregean-Cantorian Theory of Number.Andrew Boucher - unknown
    How-many numbers, such as 2 and 1000, relate or are capable of expressing the size of a group or set. Both Cantor and Frege analyzed how-many number in terms of one-to-one correspondence between two sets. That is to say, one arrived at numbers by either abstracting from the concept of correspondence, in the case of Cantor, or by using it to provide an out-and-out definition, in the case of Frege.
  23. Who Needs (to Assume) Hume's Principle?Andrew Boucher - manuscript
    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.
  24. Who Needs (to Assume) Hume's Principle? July 2006.Andrew Boucher - manuscript
    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 (...)
  25. Question Disputée Sur l'Abstraction d'Un Concept Unique.O. Boulnois - 1993 - Archives d'Histoire Doctrinale et Littéraire du Moyen Âge 60.
  26. Logicism Revisited.Otavio Bueno - 2001 - 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 (...)
  27. On the Problems of Abstraction and Concretion.Howard Burdick - 1974 - Noûs 8 (3):295-297.
  28. Frege’s Theorem by Richard G. Heck, Jr. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-732.
  29. Richard G. Heck, Jr.: Frege’s Theorem. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-733.
  30. Review: The Limits of Abstraction by Kit Fine. [REVIEW]John P. Burgess - 2003 - Notre Dame Journal Fo Formal Logic 44:227-251.
  31. Abstraction and the Environment.Louis Caruana - unknown
  32. From Abstraction to Wunsch: The Vocabulaire Européen des Philosophies.Howard Caygill - 2006 - Radical Philosophy 138:10-14.
  33. Generality Above Abstraction: The General Expressed in Terms of the Paradigmatic in Mathematics in Ancient China.Karine Chemla - 2003 - Science in Context 16 (3).
  34. Numbers, Reference, and Abstraction.Colin Cheyne - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 90 (1):291-315.
  35. Abduction, Generalization, and Abstraction in Mathematical Problem Solving.Vic Cifarelli - 1998 - Semiotics:97-113.
  36. Abstraction in Computer Science.Timothy Colburn & Gary Shute - 2007 - 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 (...)
  37. Iteration One More Time.R. Cook - 2003 - 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 (...)
  38. Conservativeness, Stability, and Abstraction.R. T. Cook - 2012 - 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 (...)
  39. RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. Xiv + 307.R. T. Cook - 2012 - Philosophia Mathematica 20 (3):346-359.
  40. Kit Fine, The Limits of Abstraction Oxford, Clarendon Press, 2002, Cloth 18.99/US $25.00 ISBN: 0-19-924618-1.R. T. Cook - 2004 - British Journal for the Philosophy of Science 55 (4):791-800.
  41. Abstraction and Four Kinds of Invariance.Roy T. Cook - forthcoming - Philosophia Mathematica:nkw014.
    Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume’s Principle, and the comparative lack (...)
  42. Necessity, Necessitism, and Numbers.Roy T. Cook - 2016 - Philosophical Forum 47 (3-4):385-414.
    Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s arguments either (...)
  43. Frege's Cardinals and Neo-Logicism.Roy T. Cook - 2016 - Philosophia Mathematica 24 (1):60-90.
    Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding (...)
  44. New Waves on an Old Beach: Fregean Philosophy of Mathematics Today.Roy T. Cook - 2009 - In Ø. Linnebo O. Bueno (ed.), New Waves in Philosophy of Mathematics.
  45. Hume's Big Brother: Counting Concepts and the Bad Company Objection.Roy T. Cook - 2009 - 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 (...)
  46. Aristotelian Logic, Axioms, and Abstraction.Roy T. Cook - 2003 - 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 (...)
  47. The State of the Economy: Neo-Logicism and Inflationt.Roy T. Cook - 2002 - 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 (...)
  48. Frege's Recipe.Roy T. Cook & Philip A. Ebert - 2016 - Journal of Philosophy 113 (7):309-345.
    In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and (...)
  49. Abstraction and Identity.Roy T. Cook & Philip A. Ebert - 2005 - 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.
  50. Kit Fine, the Limits of Abstraction Oxford, Clarendon Press, 2002, Cloth £18.99/US $25.00 ISBN: 0-19-924618-. [REVIEW]Roy T. Cook & Philip A. Ebert - 2004 - 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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