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Full Idea
It is not clear how the view that natural numbers are purely intra-structural 'objects' can be squared with the widespread use of numerals outside purely arithmetical contexts.
Gist of Idea
The structural view of numbers doesn't fit their usage outside arithmetical contexts
Source
B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26)
Book Ref
Hale,B/Wright,C: 'The Reason's Proper Study' [OUP 2003], p.15
A Reaction
I don't understand this objection. If they refer to quantity, they are implicitly cardinal. If they name things in a sequence they are implicitly ordinal. All users of numbers have a grasp of the basic structure.
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright] |
| 10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright] |
| 10626 | Objects just are what singular terms refer to [Hale/Wright] |
| 10630 | Abstracted objects are not mental creations, but depend on equivalence between given entities [Hale/Wright] |
| 10627 | Many conceptual truths ('yellow is extended') are not analytic, as derived from logic and definitions [Hale/Wright] |
| 10631 | If 'x is heterological' iff it does not apply to itself, then 'heterological' is heterological if it isn't heterological [Hale/Wright] |
| 10629 | If structures are relative, this undermines truth-value and objectivity [Hale/Wright] |
| 10628 | The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright] |