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24Structuralism and informal provabilitySynthese 202 (2): 1-26. 2023.Mathematical structuralism can be understood as a theory of mathematical ontology, of the objects that mathematics is about. It can also be understood as a theory of the semantics for mathematical discourse, of how and to what mathematical terms refer. In this paper we propose an epistemological interpretation of mathematical structuralism. According to this interpretation, the main epistemological claim is that mathematical knowledge is purely structural in character; mathematical statements co…Read more
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8Conceptions of Set and the Foundations of Mathematics By Luca IncurvatiAnalysis 81 (1): 184-189. 2021.Conceptions of Set and the Foundations of Mathematics By IncurvatiLucaCambridge University Press, 2020. xvi + 238 pp.
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52Introduction to Special Issue: Foundations of Mathematical StructuralismPhilosophia Mathematica 28 (3): 291-295. 2020.Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formu…Read more
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60The Structuralist Thesis ReconsideredBritish Journal for the Philosophy of Science 70 (4): 1201-1226. 2019.Øystein Linnebo and Richard Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of pos…Read more
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30Individuating Logics: A Category‐Theoretic ApproachThought: A Journal of Philosophy 8 (3): 200-208. 2019.Thought: A Journal of Philosophy, EarlyView.
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51Non-eliminative Structuralism, Fregean Abstraction, and Non-rigid StructuresErkenntnis 86 (1): 113-127. 2018.Linnebo and Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who…Read more
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37Bi-Modal Naive Set TheoryAustralasian Journal of Logic 15 (2): 139-150. 2018.This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper …Read more
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44The Structuralist Thesis ReconsideredBritish Journal for the Philosophy of Science. 2017.Øystein Linnebo and Richard Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They argue that their theory of abstract structures proves a consistent version of the structuralist thesis that positions in abstract structures only have structural properties. They do this by defining a subset of the properties of positions in structures, so-called fundamental properties, and argue that all fundamental properties of pos…Read more
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89Logical anti-exceptionalism and theoretical equivalenceAnalysis 77 (4): 759-767. 2017.Anti-exceptionalism about logic takes logical theories to be continuous with scientific theories. Scientific theories are subject to criteria of theoretical equivalence. This article compares two types of theoretical equivalence – one syntactic and one semantic – in the context of logical anti-exceptionalism, and argues that the syntactic approach leads to undesirable consequences. The anti-exceptionalist should therefore take a semantic approach when evaluating whether logical theories, underst…Read more
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66Logical anti-exceptionalism and theoretical equivalenceAnalysis 77 (4): 768-768. 2017._ doi:10.1093/analys/anx072 _, published: 27 June 2017
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79Set-Theoretic DependenceAustralasian Journal of Logic 12 (3): 159-176. 2015.In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible worlds and give a more sat…Read more
Heslington, York, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
Metaphysics |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
General Philosophy of Science |