more on this theme | more from this thinker
Full Idea
We can give an exhaustive account of the identity of the empty set and its singleton without mentioning infinite sets, and it might be possible to defend the view that one natural number depends on its predecessor but not vice versa.
Gist of Idea
There may be a one-way direction of dependence among sets, and among natural numbers
Source
Øystein Linnebo (Structuralism and the Notion of Dependence [2008], V)
Book Ref
-: 'The Philosophical Quarterly' [-], p.73
A Reaction
Linnebo uses this as one argument against mathematical structuralism, where the small seems to depend on the large. The view of sets rests on the iterative conception, where each level is derived from a lower level. He dismisses structuralism of sets.
| 14083 | Structuralism is right about algebra, but wrong about sets [Linnebo] |
| 14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo] |
| 14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo] |
| 14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo] |
| 14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo] |
| 14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo] |
| 14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo] |
| 14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo] |
| 14091 | There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo] |