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Full Idea
According to 'in re' structuralism, a statement that appears to be about numbers is a disguised generalization about all natural-number sequences; the numbers are bound variables, not singular terms.
Clarification
'In re' means the structures are in the entities, rather than preceding them
Gist of Idea
Number statements are generalizations about number sequences, and are bound variables
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 5.3.4)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.166
A Reaction
Any theory of anything which comes out with the thought that 'really it is a variable, not a ...' has my immediate attention and sympathy.
| 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] |