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Single Idea 14087

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism ]

Full Idea

'Set-theoretic' structuralism rejects deductive nominalism in favour of a background theory of sets, and mathematics as the various structures realized among the sets. This is often what mathematicians have in mind when they talk about structuralism.

Gist of Idea

'Set-theoretic' structuralism treats mathematics as various structures realised among the sets

Source

Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I)

Book Ref

-: 'The Philosophical Quarterly' [-], p.61

A Reaction

This is the big shift from 'mathematics can largely be described in set theory' to 'mathematics just is set theory'. If it just is set theory, then which version of set theory? Which axioms? The safe iterative conception, or something bolder?

Related Ideas

Idea 14084 Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo]

Idea 14085 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo]

Idea 14086 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo]

The 32 ideas from Øystein Linnebo

23448 Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo]
23441 Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo]
23442 Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo]
23443 The axioms of group theory are not assertions, but a definition of a structure [Linnebo]
23444 To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo]
23445 Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo]
23446 You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo]
23447 In classical semantics singular terms refer, and quantifiers range over domains [Linnebo]
10633 'Some critics admire only one another' cannot be paraphrased in singular first-order [Linnebo]
10634 Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does? [Linnebo]
10635 Second-order quantification and plural quantification are different [Linnebo]
10636 Plural plurals are unnatural and need a first-level ontology [Linnebo]
10637 Ordinary speakers posit objects without concern for ontology [Linnebo]
10638 A pure logic is wholly general, purely formal, and directly known [Linnebo]
10639 Plural quantification may allow a monadic second-order theory with first-order ontology [Linnebo]
10640 Instead of complex objects like tables, plurally quantify over mereological atoms tablewise [Linnebo]
10641 Traditionally we eliminate plurals by quantifying over sets [Linnebo]
10643 We speak of a theory's 'ideological commitments' as well as its 'ontological commitments' [Linnebo]
10778 Can second-order logic be ontologically first-order, with all the benefits of second-order? [Linnebo]
10779 A comprehension axiom is 'predicative' if the formula has no bound second-order variables [Linnebo]
10781 A 'pure logic' must be ontologically innocent, universal, and without presuppositions [Linnebo]
10782 The modern concept of an object is rooted in quantificational logic [Linnebo]
10783 Plural quantification depends too heavily on combinatorial and set-theoretic considerations [Linnebo]
14083 Structuralism is right about algebra, but wrong about sets [Linnebo]
14085 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo]
14084 Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo]
14086 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo]
14087 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo]
14088 An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo]
14090 In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo]
14089 Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo]
14091 There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo]