Papers by Øystein Linnebo
Routledge eBooks, Dec 29, 2020
Philosophia Mathematica, 2021
Modal logic has been used to analyze potential infinity and potentialism more generally. However,... more Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms.
Noûs
According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes t... more According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
Noûs
According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes t... more According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
The notion of definiteness has played a fundamental role in the early developments of set
theory... more The notion of definiteness has played a fundamental role in the early developments of set
theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different
forms of definiteness. First, a condition can be definite in the sense that, given any object, either the
condition applies to that object or it does not. We call this intensional definiteness. Second, a condition
or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has
been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns
whether an intension applies to objects considered one by one, extensional definiteness concerns the
totality of objects to which the intension applies. We discuss how these two forms of definiteness
admit of precise mathematical analyses. We argue that two main types of explication of extensional
definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based
on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation.
We submit that the two notions of extensional definiteness that emerges from our investigation enable
us to identify and understand some of the most important fault lines in the philosophy and foundations
of mathematics.
The British Journal for the Philosophy of Science, 2009
Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstra... more Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction ...
Philosophia Mathematica, 2020
What is the relation between some things and the set of these things? Mathematical practice does ... more What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.
Mind, 2012
Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain... more Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain superfluous restrictions are removed'. The aim of this paper is to develop a clearer understanding of Gödel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.
Notre Dame Journal of Formal Logic, 2018
It is widely thought that the acceptability of an abstraction principle is a feature of the cardi... more It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but analyze the interesting idea on which it is based, namely that an acceptable abstraction has to 'generate' the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.
Philosophical Studies, 2020
Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discuss... more Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discussion focuses on the book’s novel account of extensive magnitudes and numbers. First, I further develop and defend Peacocke’s argument against nominalistic approaches to magnitudes and numbers. Then, I argue that his view is more Aristotelian than Platonist because reified magnitudes and numbers are accounted for via corresponding properties and these properties’ application conditions, and because the mentioned objects have a “shallow nature” relative to the corresponding properties. The result is an asymmetric conception of abstraction, which contrasts with the neo-Fregeans’ but has important tenets in common with an approach that I have recently developed.
The Philosophical Quarterly, 2014
ABSTRACT If numbers were identified with any of their standard set-theoretic realizations, then t... more ABSTRACT If numbers were identified with any of their standard set-theoretic realizations, then they would have various non-arithmetical properties that mathematicians are reluctant to ascribe to them. Dedekind and later structuralists conclude that we should refrain from ascribing to numbers such ‘foreign’ properties. We first rehearse why it is hard to provide an acceptable formulation of this conclusion. Then we investigate some forms of abstraction meant to purge mathematical objects of all ‘foreign’ properties. One form is inspired by Frege; the other by Dedekind. We argue that both face problems.
Philosophia Mathematica, 2011
Does category theory provide a foundation for mathematics that is autonomous with respect to the ... more Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer 'nature' than is preserved under isomorphism, then such an approach will be inadequate.
Noûs, 2017
The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle... more The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
Erkenntnis
Ordinarily, the order in which some objects are attached to a scale does not affect the total wei... more Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well known by mathematicians but has so far eluded philosophical scrutiny.
Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2013
ABSTRACT Quantum mechanics tells us that states involving indistinguishable fermions must be anti... more ABSTRACT Quantum mechanics tells us that states involving indistinguishable fermions must be antisymmetrized. This is often taken to mean that indistinguishable fermions are always entangled. We consider several notions of entanglement and argue that on the best of them, indistinguishable fermions are not always entangled. We also present a simple but unconventional way of representing fermionic states that allows us to maintain a link between entanglement and non-factorizability.
The Review of Symbolic Logic
In the literature, predicativism is connected not only with the Vicious Circle Principle but also... more In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
Philosophical Issues
Dummetts notion of indefinite extensibility is influential but obscure. The notion figures centra... more Dummetts notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaningtheoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.
Philosophia Mathematica, 2006
Nous
Plural logic is widely assumed to have two important virtues: ontological innocence and determina... more Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard (Henkin) interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.
Norsk filosofisk tidsskrift
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Papers by Øystein Linnebo
theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different
forms of definiteness. First, a condition can be definite in the sense that, given any object, either the
condition applies to that object or it does not. We call this intensional definiteness. Second, a condition
or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has
been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns
whether an intension applies to objects considered one by one, extensional definiteness concerns the
totality of objects to which the intension applies. We discuss how these two forms of definiteness
admit of precise mathematical analyses. We argue that two main types of explication of extensional
definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based
on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation.
We submit that the two notions of extensional definiteness that emerges from our investigation enable
us to identify and understand some of the most important fault lines in the philosophy and foundations
of mathematics.
theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different
forms of definiteness. First, a condition can be definite in the sense that, given any object, either the
condition applies to that object or it does not. We call this intensional definiteness. Second, a condition
or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has
been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns
whether an intension applies to objects considered one by one, extensional definiteness concerns the
totality of objects to which the intension applies. We discuss how these two forms of definiteness
admit of precise mathematical analyses. We argue that two main types of explication of extensional
definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based
on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation.
We submit that the two notions of extensional definiteness that emerges from our investigation enable
us to identify and understand some of the most important fault lines in the philosophy and foundations
of mathematics.