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Single Idea 10178

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism ]

Full Idea

It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.

Gist of Idea

Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)

A Reaction

[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.

The 13 ideas with the same theme [structuralism denying real objects or real structures]:

12329 If mathematics is a logic of the possible, then questions of existence are not intrinsic to it [Badiou]
8698 Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
9557 Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
10263 Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
8922 Maybe mathematical objects only have structural roles, and no intrinsic nature [Hellman]
10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
9925 Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen]
10192 We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
10168 Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
10178 Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
8696 Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
8695 Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]