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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358 found
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  1. added 2018-09-29
    Cardinality and Acceptable Abstraction.Roy T. Cook & Øystein Linnebo - 2018 - Notre Dame Journal of Formal Logic 59 (1):61-74.
    It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.
  2. added 2018-09-29
    To Be Is to Be an F.Øystein Linnebo - 2005 - Dialectica 59 (2):201-222.
    I defend the view that our ontology divides into categories, each with its own canonical way of identifying and distinguishing the objects it encompasses. For instance, I argue that natural numbers are identified and distinguished by their positions in the number sequence, and physical bodies, by facts having to do with spatiotemporal continuity. I also argue that objects belonging to different categories are ipso facto distinct. My arguments are based on an analysis of reference, which ascribes to reference a richer (...)
  3. added 2018-08-03
    Paradox Without Basic Law V: A Problem with Frege’s Ontology.Adam Rieger - 2002 - Analysis 62 (4):327-330.
  4. added 2018-08-01
    Frege's Theorem in Plural Logic.Simon Hewitt - manuscript
    A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.
  5. added 2018-04-30
    Tuples All the Way Down?Simon Thomas Hewitt - 2018 - Thought: A Journal of Philosophy 7 (3):161-169.
  6. added 2018-03-05
    Composition as Abstraction.Jeffrey Sanford Russell - 2017 - Journal of Philosophy 114 (9):453-470.
    The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.
  7. added 2018-03-01
    Priority, Platonism, and the Metaontology of Abstraction.Michele Lubrano - 2016 - Dissertation, University of Turin
    In this dissertation I examine the NeoFregean metaontology of mathematics. I try to clarify the relationship between what is sometimes called Priority Thesis and Platonism about mathematical entities. I then present three coherent ways in which one might endorse both these stances, also answering some possible objections. Finally I try to show which of these three ways is the most promising.
  8. added 2018-03-01
    Ontologie Neofreghiane.Michele Lubrano - 2016 - Philosophy Kitchen 3 (4):113-125.
    In the present contribution I would like to examine some theories of the ontology of abstract entities that take inspiration from the deep insights of Gottlob Frege. These theories develop in full details some ideas explicitly or implicitly articulated in Frege’s works and try to defend a sophisticated version of Platonism about abstract entities. The review of such theories should allow us to cast light on their merits and their possible flaws and, moreover, to determine which of them is the (...)
  9. added 2018-02-17
    Neo-Logicism and Russell’s Logicism.Kevin C. Klement - 2012 - Russell: The Journal of Bertrand Russell Studies 32 (2):159.
    Most advocates of the so-called “neologicist” movement in the philosophy of mathematics identify themselves as “Neo-Fregeans” (e.g., Hale and Wright): presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature, and when it is, often dismissed as not really logicism at all (in lights of its assumption of axioms of infinity, reducibiity and so on). In this paper I have three aims: firstly, to identify more clearly the primary metaontological (...)
  10. added 2018-02-17
    N Eo-F Regeanism and Q Uantifier V Ariance.Katherine Hawley - 2007 - Aristotelian Society Supplementary Volume 81 (1):233-249.
    In his paper in the same volume, Sider argues that, of maximalism and quantifier variance, the latter promises to let us make better sense of neo-Fregeanism. I argue that neo-Fregeans should, and seemingly do, reject quantifier variance. If they must choose between these two options, they should choose maximalism.
  11. added 2018-01-17
    Tuples All the Way Down?Simon Hewitt - manuscript
    We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the difficulty, (...)
  12. added 2017-09-06
    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 (...)
  13. added 2017-06-01
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper critically evaluates the view according to which number words in argument position retain the meaning they have when occurring as determiners or adjectives. In particular, it argues against syntactic evidence from German given by myself in support of that view.
  14. added 2017-05-12
    Categories for the Neologicist.Shay Allen Logan - 2017 - Philosophia Mathematica 25 (1):26-44.
    Abstraction principles provide implicit definitions of mathematical objects. In this paper, an abstraction principle defining categories is proposed. It is unsatisfiable and inconsistent in the expected ways. Two restricted versions of the principle which are consistent are presented.
  15. added 2017-03-24
    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 (...)
  16. added 2017-03-20
    Introduction to Abstractionism.Philip A. Ebert & Marcus Rossberg - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism. Oxford: Oxford University Press. pp. 3-33.
  17. added 2017-03-14
    A Generic Russellian Elimination of Abstract Objects.Kevin C. Klement - 2015 - Philosophia Mathematica:nkv031.
    In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system (...)
  18. added 2017-02-23
    Introduction to Special Issue: Abstraction Principles.Salvatore Florio - 2017 - Philosophia Mathematica 25 (1):1-2.
    Introduction to a special issue on abstraction principles.
  19. added 2017-02-15
    Erratum To: Frege's Logicism and the Neo-Fregean Project.Matthias Schirn - 2014 - Axiomathes 24 (2):245-245.
    Erratum to: Axiomathes DOI 10.1007/s10516-013-9222-7In the online publication, page 13, line 27, after the sentence “Hence, neo-logicism is doomed to failure.”, the following two sentences were missing:This argument was developed by Robert Trueman in a draft of his paper ‘Sham Names andion’. A revised version of this paper is forthcoming in Philosophia Mathematica under the tile ‘A Dilemma for Neo-Fregeanism’.
  20. added 2017-02-15
    C. Oppius on Julius Caesar.Gavin B. Townend - 1987 - American Journal of Philology 108 (2).
  21. added 2017-02-14
    On Non-Standard Models of Peano Arithmetic.Laureano Luna - 2008 - The Reasoner 2:2.
  22. added 2017-02-14
    DA Gillies, Frege, Dedekind and Peano on the Foundations of Arithmetic Reviewed By.S. K. Thomason - 1984 - Philosophy in Review 4 (3):111-113.
  23. added 2017-02-14
    The Opening Up of a Field of Science by Abstraction and Synthesis.M. D. Stafleu - 1980 - Philosophia Reformata 45:47-76.
  24. added 2017-02-13
    Anxiety and Abstraction in Nineteenth-Century Mathematics.Jeremy J. Gray - 2004 - Science in Context 17 (1-2):23-47.
  25. added 2017-02-13
    Julius Caesar and the Number 2.Jill Dieterle - 1997 - Electronic Journal of Analytic Philosophy 5.
  26. added 2017-02-13
    Linear Arithmetic Desecsed.John K. Slaney, Robert K. Meyer & Greg Restall - 1996 - Logique Et Analyse 39:379-388.
  27. added 2017-02-13
    Abstraction and Concept Formation.V. Hope & Anatol Pikas - 1968 - Philosophical Quarterly 18 (72):276.
  28. added 2017-02-12
    “Bare Life” Political Order and the Specter of Antisocial Being in Shakespeare's Julius Caesar.Daniel Juan Gil - 2007 - Common Knowledge 13 (1):67-79.
  29. added 2017-02-10
    Dummett, Brouwer and the Metaphysics of Mathematics.Eric P. Tsui-James - 1998 - Grazer Philosophische Studien 55:143-168.
    Although Brouwer is well known for his Intuitionistic philosophy of mathematics, a constructivist philosophy which calls for restricted use of certain logical principles, there is much less awareness of the well-developed metaphysical basis which underlies those restrictions. In the first half of this paper I outline a basic interpretation of Brouwer's metaphysics, and then in the second half consider the compatibility of that metaphysics with Dummett's argument for a principled non-metaphysical approach to intuitionism. I conclude that once the variously misleading (...)
  30. added 2017-02-10
    Abduction, Generalization, and Abstraction in Mathematical Problem Solving.Vic Cifarelli - 1998 - Semiotics:97-113.
  31. added 2017-02-09
    The Critique of Empiricist Accounts of Abstraction.Arthur David Smith - unknown
  32. added 2017-02-08
    Philosophy of Arithmetic.Dale Jacquette - 2005 - Review of Metaphysics 59 (2):428-431.
  33. added 2017-02-07
    Frege's Theory of Real Numbers.Peter M. Simons - 1987 - History and Philosophy of Logic 8 (1):25--44.
    Frege's theory of real numbers has undeservedly received almost no attention, in part because what we have is only a fragment. Yet his theory is interesting for the light it throws on logicism, and it is quite different from standard modern approaches. Frege polemicizes vigorously against his contemporaries, sketches the main features of his own radical alternative, and begins the formal development. This paper summarizes and expounds what he has to say, and goes on to reconstruct the most important steps (...)
  34. added 2017-02-07
    Book Review: Wittgenstein on the Foundations of Mathematics Crispin Wright. [REVIEW]Charles F. Kielkopf - 1981 - Philosophy of Science 48 (3):503-5.
  35. added 2017-02-02
    4:00 P.M., F Sep 20.Harvey Friedman - manuscript
    There are many familiar theorems whose proofs use methods which are in some appropriate sense substantially more "abstract" than its statement. Some particularly well known examples come from the use of complex variables in number theory. Sometimes such abstraction can be removed - for example by the "elementary proof of the prime number theorem" - and sometimes no appropriate removal is known. The interest in removing abstraction typically varies, with no agreed upon criteria for appropriateness. E.g., the removal might sacrifice (...)
  36. added 2017-02-02
    Agent Intellect and Phantasms. On the Preliminaries of Peripatetic Abstraction.Leen Spruit - 2004 - Poznan Studies in the Philosophy of the Sciences and the Humanities 82 (1):125-146.
    This paper discusses some aspects of the controversies regarding the operation of the agent intellect on sensory images. I selectively consider views developed between the 13th century and the beginning of the 17th century, focusing on positions which question the need for a (distinct) agent intellect or argue for its essential "inactivity" with respect to phantasms. My aim is to reveal limitations of the Peripatetical framework for analyzing and explaining the mechanisms involved in conceptual abstraction. The first section surveys developments (...)
  37. added 2017-01-31
    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.
  38. added 2017-01-29
    Abstraction and the Empiriological Method: Comment.Mark Heath - 1952 - Proceedings and Addresses of the American Philosophical Association 26:50.
  39. added 2017-01-28
    A Philosophical Inquiry Into the Concept of Number.Joongol Kim - 2004 - Dissertation, University of Notre Dame
    The dissertation is an inquiry into the ontology and epistemology of numbers. As regards the former, the Fregean conception of numbers as objects and the Russellian conception of numbers as higher-level entities are both critically examined. A conception of numbers as modes of existence , that is, ways or manners in which things exist, is introduced and defended instead. As regards the latter, the basic concepts of arithmetic are explicated in terms of pure logic alone, and all the truths of (...)
  40. added 2017-01-28
    Ostension and the Problem of Qualitative Abstraction.Ralph Lance Factor - 1970 - Dissertation, University of Georgia
  41. added 2017-01-28
    WEINBERG, Julius.-"Abstraction, Relation and Induction: Three Essays in the History of Thought". [REVIEW]Rita Gupta - 1968 - Philosophy 43:395.
  42. added 2017-01-27
    Arithmetic with Fusions.Jeff Ketland & Thomas Schindler - 2016 - Logique Et Analyse 234:207-226.
    In this article, the relationship between second-order comprehension and unrestricted mereological fusion (over atoms) is clarified. An extension PAF of Peano arithmetic with a new binary mereological notion of “fusion”, and a scheme of unrestricted fusion, is introduced. It is shown that PAF interprets full second-order arithmetic, Z_2.
  43. added 2017-01-27
    A Dilemma for Neo-Fregeanism.Robert Trueman - 2014 - Philosophia Mathematica 22 (3):361-379.
    Neo-Fregeans need their stipulation of Hume's Principle — $NxFx=NxGx \leftrightarrow \exists R (Fx \,1\hbox {-}1_R\, Gx)$ — to do two things. First, it must implicitly define the term-forming operator ‘Nx…x…’, and second it must guarantee that Hume's Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume's Principle, and argue that while one sort of stipulation fixes a meaning for ‘Nx…x…’ and the other guarantees the truth of Hume's Principle, neither does both.
  44. added 2017-01-27
    Chapter 3: Objectivism and Realism in Frege's Philosophy of Arithmetic.Philip Hugly & Charles Sayward - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 90:73-101.
  45. added 2017-01-27
    On Abstraction and Concretization.Izabella Nowakowa - 2005 - Poznan Studies in the Philosophy of the Sciences and the Humanities 88:137-146.
  46. added 2017-01-26
    Numbers, Reference, and Abstraction.Colin Cheyne - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 90 (1):291-315.
  47. added 2017-01-26
    Hale Bob and Wright Crispin. The Reason's Proper Study: Essays Toward a Neo-Fregean Philosophy of Mathematics. Oxford University Press, New York. 2001, 472 Pp. [REVIEW]Gabriel Uzquiano - 2006 - Bulletin of Symbolic Logic 12 (2):291-294.
  48. added 2017-01-26
    Bob Hale and Crispin Wright. The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics.Neil Tennant - 2003 - Philosophia Mathematica 11 (2):226-240.
  49. added 2017-01-26
    Generality Above Abstraction: The General Expressed in Terms of the Paradigmatic in Mathematics in Ancient China.Karine Chemla - 2003 - Science in Context 16 (3).
  50. added 2017-01-25
    Full Abstraction for Reduced ML.Andrzej S. Murawski & Nikos Tzevelekos - 2013 - Annals of Pure and Applied Logic 164 (11):1118-1143.
    We present the first effectively presentable fully abstract model for Starkʼs Reduced ML, a call-by-value higher-order programming language featuring integer-valued references. The model is constructed using techniques of nominal game semantics. Its distinctive feature is the presence of carefully restricted information about the store in plays, combined with conditions concerning the participantsʼ ability to distinguish reference names. We show how it leads to an explicit characterization of program equivalence.
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