Putnam on Mathematics as Modal Logic

Linnebo, Øystein

doi:10.1007/978-3-319-96274-0_14

Outstanding Contributions to Logic

2018

DOI: 10.1007/978-3-319-96274-0_14

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Putnam on Mathematics as Modal Logic

Øystein Linnebo

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Cited by 4 publications

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“… 35 Linnebo (2018a, pp. 262–6) offers good reasons to suggest that Putnam should have considered a potentialist modality. …”

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confidence: 99%

Level Theory, Part 2: Axiomatizing the Bare Idea of a Potential Hierarchy

Button¹

2021

Bull. symb. log

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Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with Level Theory, as developed in Part 1.What we need to do is to replace the language of time and activity by the more bloodless language of potentiality and actuality.Parsons [1977: 293] Potentialists, such as Charles Parsons, Øystein Linnebo, and James Studd, think that the concept of set is importantly modal. Put thus, potentialism is a broad church; different potentialists will disagree on the precise details of the relevant modality. 1 My aim is shed light on potentialism, in general, using Level Theory, LT, as introduced in Part 1.I start by formulating Potentialist Set Theory, PST. This uses a tensed logic to formalize the bare idea of a 'potential hierarchy of sets'. 2 Though PST is extremely minimal, it packs a surprising punch (see § §1-4).In the vanilla version of PST, we need not assume that time is linear. However, if we make that assumption, then the resulting theory is almost definitionally equivalent to LT, its non-modal counterpart (see § §5-8).

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“… 35 Linnebo (2018a, pp. 262–6) offers good reasons to suggest that Putnam should have considered a potentialist modality. …”

mentioning

confidence: 99%

Level Theory, Part 2: Axiomatizing the Bare Idea of a Potential Hierarchy

Button¹

2021

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Nominalism and Mathematical Objectivity

Luo

2022

Axiomathes

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We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, and we argue that all these strategies are untenable.

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On Putnam’s Proof of the Impossibility of a Nominalistic Physics

Barrett

2020

Erkenn

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Putnam on Mathematics as Modal Logic

Cited by 4 publications

References 17 publications

Level Theory, Part 2: Axiomatizing the Bare Idea of a Potential Hierarchy

Level Theory, Part 2: Axiomatizing the Bare Idea of a Potential Hierarchy

Nominalism and Mathematical Objectivity

On Putnam’s Proof of the Impossibility of a Nominalistic Physics

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