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Single Idea 14086
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
]
Full Idea
The 'modal' version of eliminativist structuralism lifts the deductivist ban on modal notions. It studies what necessarily holds in all concrete models which are possible for various theories.
Gist of Idea
'Modal' structuralism studies all possible concrete models for various mathematical theories
Source
Øystein Linnebo (Structuralism and the Notion of Dependence [2008], I)
Book Ref
-: 'The Philosophical Quarterly' [-], p.60
A Reaction
[He cites Putnam 1967, and Hellman 1989] If mathematical truths are held to be necessary (which seems to be right), then it seems reasonable to include modal notions, about what is possible, in its study.
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 14087
'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [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]
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23441
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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
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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
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Plural plurals are unnatural and need a first-level ontology
[Linnebo]
|
10637
|
Ordinary speakers posit objects without concern for ontology
[Linnebo]
|
10638
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A pure logic is wholly general, purely formal, and directly known
[Linnebo]
|
10639
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Plural quantification may allow a monadic second-order theory with first-order ontology
[Linnebo]
|
10640
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Instead of complex objects like tables, plurally quantify over mereological atoms tablewise
[Linnebo]
|
10641
|
Traditionally we eliminate plurals by quantifying over sets
[Linnebo]
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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]
|
14089
|
Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure
[Linnebo]
|
14090
|
In mathematical structuralism the small depends on the large, which is the opposite of physical structures
[Linnebo]
|
14091
|
There may be a one-way direction of dependence among sets, and among natural numbers
[Linnebo]
|