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Mathematical Internal Realism

In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam. Berlin, Germany: De Gruyter. pp. 157-182 (2022)

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  1. Securing Arithmetical Determinacy.Sebastian G. W. Speitel - 2024 - Ergo: An Open Access Journal of Philosophy 11.
    The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues (...)
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  • Internal Categoricity, Truth and Determinacy.Martin Fischer & Matteo Zicchetti - 2023 - Journal of Philosophical Logic 52 (5):1295-1325.
    This paper focuses on the categoricity of arithmetic and determinacy of arithmetical truth. Several ‘internal’ categoricity results have been discussed in the recent literature. Against the background of the philosophical position called internalism, we propose and investigate truth-theoretic versions of internal categoricity based on a primitive truth predicate. We argue for the compatibility of a primitive truth predicate with internalism and provide a novel argument for (and proof of) a truth-theoretic version of internal categoricity and internal determinacy with some positive (...)
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  • The iterative conception of function and the iterative conception of set.Tim Button - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory. Palgrave.
    Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...)
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  • Categoricity by convention.Julien Murzi & Brett Topey - 2021 - Philosophical Studies 178 (10):3391-3420.
    On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem (...)
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  • Internalism and the Determinacy of Mathematics.Lavinia Picollo & Daniel Waxman - 2023 - Mind 132 (528):1028-1052.
    A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably (...)
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  • On what there is in particular.Jonathan D. Payton - 2023 - Analysis 83 (1):70-79.
    Quine says that ontology is about what there is, suggesting that to be ontologically committed to Fs is to be committed to accepting a sentence which existentially quantifies over Fs. Kit Fine argues that this gets the logical form of some ontological theses wrong. Fine is right that some ontological theses cannot be rendered simply as ‘There are Fs’. But the root of the problem has yet to be recognized, either by Fine or by his critics. Sometimes to adopt an (...)
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