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The Existence (and Non-existence) of Abstract Objects

In Frege's Theorem. Oxford University Press (2011)

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  1. 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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  • 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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  • 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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  • 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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  • Books of Essays.Guillermo E. Rosado Haddock - 2017 - Philosophia Mathematica: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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  • 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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