more from this thinker | more from this text
Full Idea
According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines.
Gist of Idea
Structuralist says maths concerns concepts about base objects, not base objects themselves
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
This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world.
| 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] |