Grounding and auto-abstraction

Zanetti, Luca

doi:10.1007/s11229-020-02710-3

Cited by 3 publications

(2 citation statements)

References 24 publications

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“…We have made considerable progress, but a final problem identified by Donaldson remains. Even granting our assumptions concerning the grounding of equinumerosity claims, any view that affirms (

{\#\!&lt;}

) faces what has become known as the problem of autoabstraction (Zanetti, 2020). In essence, the problem is simple.…”

Section: The Problem Of Autoabstractionmentioning

confidence: 96%

Abstraction and grounding

deRosset,

Linnebo

2023

Philos Phenomenol Research

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The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one: . The principal aim of this article is to use the notion of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction.

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“…We have made considerable progress, but a final problem identified by Donaldson remains. Even granting our assumptions concerning the grounding of equinumerosity claims, any view that affirms (

{\#\!&lt;}

) faces what has become known as the problem of autoabstraction (Zanetti, 2020). In essence, the problem is simple.…”

Section: The Problem Of Autoabstractionmentioning

confidence: 96%

Abstraction and grounding

deRosset,

Linnebo

2023

Philos Phenomenol Research

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“…See alsoZanetti (2020).7More precisely, Linnebo's argument requires that knowledge is potentially closed under modus ponens, that is, Kϕ and ϕ ! ψ entail that it is at least possible to know that ψ (2018, p. 16).THIN OBJECTS: AN OVERVIEW…”

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confidence: 99%

Thin objects: An overview

Carrara

Zanetti

2023

Theoria

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In Thin objects: an abstractionist account (Oxford University Press, 2018), Øystein Linnebo claims that ‘mathematical objects are thin in the sense that very little is required for their existence’. Linnebo articulates his view in an abstractionist manner: according to Linnebo, the truth of the right‐hand side of a Fregean abstraction principle, which states that two items stand in a given equivalence relation, is sufficient for the truth of its left‐hand side, which states that the same abstract object is associated to both items. This special issue contains nine articles discussing different aspects of Linnebo's book. In this introduction, we provide an opinionated introduction to the Special Issue and an overview of the contributions.

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Collective Abstraction

Litland

2022

The Philosophical Review

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This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations.

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Grounding and auto-abstraction

Cited by 3 publications

References 24 publications

Abstraction and grounding

Abstraction and grounding

Thin objects: An overview

Collective Abstraction

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