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Full Idea
Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated.
Gist of Idea
Structuralism focuses on relations, predicates and functions, with objects being inessential
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.82
A Reaction
The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661.
Related Ideas
Idea 7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
Idea 8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
| 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] |