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The Existence (and Non-existence) of Abstract Objects

In Richard G. Heck (ed.), Frege's theorem. New York: Clarendon Press (2011)

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  1. Two-Sorted Frege Arithmetic is Not Conservative.Stephen Mackereth & Jeremy Avigad - 2022 - Review of Symbolic Logic 16 (4):1199-1232.
    Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. (...)
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  • Of Marriage and Mathematics: Inferentialism and Social Ontology.James Henry Collin - 2023 - Topoi 42 (1):247-257.
    The semantic inferentialist account of the social institution of semantic meaning can be naturally extended to account for social ontology. I argue here that semantic inferentialism provides a framework within which mathematical ontology can be understood as social ontology, and mathematical facts as socially instituted facts. I argue further that the semantic inferentialist framework provides resources to underpin at least some aspects of the objectivity of mathematics, even when the truth of mathematical claims is understood as socially instituted.
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  • Plural Ancestral Logic as the Logic of Arithmetic.Oliver Tatton-Brown - 2024 - Review of Symbolic Logic 17 (2):305-342.
    Neo-Fregeanism aims to provide a possible route to knowledge of arithmetic via Hume’s principle, but this is of only limited significance if it cannot account for how the vast majority of arithmetic knowledge, accrued by ordinary people, is obtained. I argue that Hume’s principle does not capture what is ordinarily meant by numerical identity, but that we can do much better by buttressing plural logic with plural versions of the ancestral operator, obtaining natural and plausible characterizations of various key arithmetic (...)
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  • On the Philosophical Significance of Frege’s Constraint.Andrea Sereni - 2019 - Philosophia Mathematica 27 (2):244–275.
    Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...)
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  • The Caesar Problem — A Piecemeal Solution.J. P. Studd - 2023 - Philosophia Mathematica 31 (2):236-267.
    The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.
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  • Cantor's Abstractionism and Hume's Principle.Claudio Ternullo & Luca Zanetti - 2021 - History and Philosophy of Logic 43 (3):284-300.
    Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is (...)
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  • For Better and for Worse. Abstractionism, Good Company, and Pluralism.Andrea Sereni, Maria Paola Sforza Fogliani & Luca Zanetti - 2023 - Review of Symbolic Logic 16 (1):268-297.
    A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good (...)
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  • (1 other version)Frege’s Theorem: AN INTRODUCTION.Richard Heck Jr - 2003 - Manuscrito 26 (2):471-503.
    Frege's work was largely devoted to an attempt to argue that the'basic laws of arithmetic' are truths of logic. That attempt had both philosophical and formal aspects. The present note offers an introduction to both of these, so that readers will be able to appreciate contemporary discussions of the philosophical significance of 'Frege's Theorem'.
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  • Abstraction and Four Kinds of Invariance.Roy T. Cook - 2017 - Philosophia Mathematica 25 (1):3–25.
    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 (...)
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  • Robert Lorne Victor Hale FRSE May 4, 1945 – December 12, 2017.Roy T. Cook & Stewart Shapiro - 2018 - Philosophia Mathematica 26 (2):266-274.
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  • Abstractionism and Mathematical Singular Reference.Bahram Assadian - 2019 - Philosophia Mathematica 27 (2):177-198.
    ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...)
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  • The Strength of Abstraction with Predicative Comprehension.Sean Walsh - 2016 - Bulletin of Symbolic Logic 22 (1):105–120.
    Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence (...)
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  • The Limits of Reconstructive Neologicist Epistemology.Eileen S. Nutting - 2018 - Philosophical Quarterly 68 (273):717-738.
    Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...)
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  • Books of Essays.Guillermo E. Rosado Haddock - 2017 - Philosophia Mathematica 25 (2):nkw34.
    Guillermo E. Rosado Haddock, ed. Husserl and Analytic Philosophy. Berlin/Boston: Walter de Gruyter GmbH, 2016. ISBN 978-3-11-049655-0 ; 978-3-11-049737-3 ; 978-3-11-049418-1. Pp. viii + 338.
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  • What did Frege take Russell to have proved?John Woods - 2019 - Synthese 198 (4):3949-3977.
    In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...)
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  • Paolo Mancosu.*Abstraction and Infinity. [REVIEW]Roy T. Cook & Michael Calasso - 2019 - Philosophia Mathematica 27 (1):125-152.
    MancosuPaolo.* *ion and Infinity. Oxford University Press, 2016. ISBN: 978-0-19-872462-9. Pp. viii + 222.
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  • Neologicism, Frege's Constraint, and the Frege‐Heck Condition.Eric Snyder, Richard Samuels & Stewart Shapiro - 2018 - Noûs 54 (1):54-77.
    One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...)
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  • Quantity evaluations in Yudja: judgements, language and cultural practice.Suzi Lima & Susan Rothstein - 2020 - Synthese 197 (9):3851-3873.
    In this paper we explore the interpretation of quantity expressions in Yudja, an indigenous language spoken in the Amazonian basin, showing that while the language allows reference to exact cardinalities, it does not generally allow reference to exact measure values. It does, however, allow non-exact comparison along continuous dimensions. We use this data to argue that the grammar of exact measurement is distinct from a grammar allowing the expression of exact cardinalities, and that the grammar of counting and the grammar (...)
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  • (1 other version)Cardinals, Ordinals, and the Prospects for a Fregean Foundation.Eric Snyder, Stewart Shapiro & Richard Samuels - 2018 - Royal Institute of Philosophy Supplement 82:77-107.
    There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...)
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