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6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / b. Varieties of structuralism

[proposed options for how to understand structuralism]

11 ideas
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]
The main versions of structuralism are all definitionally equivalent [Shapiro]
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo]
'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo]
Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo]
'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo]
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]