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Ideas of Øystein Linnebo, by Text

[Norwegian, fl. 2006, Lecturer at Bristol University, then Birkbeck, London.]

2003 Plural Quantification Exposed
§0 p.71 Can second-order logic be ontologically first-order, with all the benefits of second-order?
§1 p.73 A comprehension axiom is 'predicative' if the formula has no bound second-order variables
§1 p.75 A 'pure logic' must be ontologically innocent, universal, and without presuppositions
§2 p.78 The modern concept of an object is rooted in quantificational logic
§4 p.88 Plural quantification depends too heavily on combinatorial and set-theoretic considerations
2008 Plural Quantification
1 p.2 'Some critics admire only one another' cannot be paraphrased in singular first-order
1.1 p.4 Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does?
2 p.5 Second-order quantification and plural quantification are different
2.4 p.8 Plural plurals are unnatural and need a first-level ontology
2.4 p.9 Ordinary speakers posit objects without concern for ontology
3 p.10 A pure logic is wholly general, purely formal, and directly known
4.4 p.14 Plural quantification may allow a monadic second-order theory with first-order ontology
4.5 p.4 Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
5 p.15 Traditionally we eliminate plurals by quantifying over sets
5.4 p.20 We speak of a theory's 'ideological commitments' as well as its 'ontological commitments'
2008 Structuralism and the Notion of Dependence
Intro p.59 Structuralism is right about algebra, but wrong about sets
1 p.60 'Deductivist' structuralism is just theories, with no commitment to objects, or modality
I p.60 Non-eliminative structuralism treats mathematical objects as positions in real abstract structures
I p.60 'Modal' structuralism studies all possible concrete models for various mathematical theories
I p.61 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets
II p.65 An 'intrinsic' property is either found in every duplicate, or exists independent of all externals
III p.66 Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
III p.68 In mathematical structuralism the small depends on the large, which is the opposite of physical structures
V p.73 There may be a one-way direction of dependence among sets, and among natural numbers
2017 Philosophy of Mathematics
11.1 p.155 Mathematics is the study of all possible patterns, and is thus bound to describe the world
2 p.21 Logical truth is true in all models, so mathematical objects can't be purely logical
3.3 p.44 Game Formalism has no semantics, and Term Formalism reduces the semantics
3.5 p.52 The axioms of group theory are not assertions, but a definition of a structure
4.1 p.56 To investigate axiomatic theories, mathematics needs its own foundational axioms
4.2 p.62 Naïve set theory says any formula defines a set, and coextensive sets are identical
4.6 p.71 You can't prove consistency using a weaker theory, but you can use a consistent theory
7.1 p.101 In classical semantics singular terms refer, and quantifiers range over domains