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Full Idea
The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us.
Gist of Idea
Modal structuralism can only judge possibility by 'possible' models
Source
comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.229
A Reaction
I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities?
| 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] |