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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

[structuralism denying real objects or real structures]

13 ideas
If mathematics is a logic of the possible, then questions of existence are not intrinsic to it [Badiou]
Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
Maybe mathematical objects only have structural roles, and no intrinsic nature [Hellman]
Is there is no more to structures than the systems that exemplify them? [Shapiro]
Number statements are generalizations about number sequences, and are bound variables [Shapiro]
Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen]
We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]