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Full Idea
That 'two dogs are more than one' is clearly true, but its truth doesn't depend on the existence of dogs, as is seen if we consider 'two unicorns are more than one', which is true even though there are no unicorns.
Gist of Idea
Arithmetic doesn’t simply depend on objects, since it is true of fictional objects
Source
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.2)
Book Ref
-: 'Philosophical Review 114' [Phil Review 2005], p.215
A Reaction
This is an objection to crude empirical accounts of arithmetic, but the idea would be that there is a generalisation drawn from objects (dogs will do nicely), which then apply to any entities. If unicorns are entities, it will be true of them.
| 9998 | What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber] |
| 10000 | We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] |
| 10001 | An adjective contributes semantically to a noun phrase [Hofweber] |
| 10002 | '2 + 2 = 4' can be read as either singular or plural [Hofweber] |
| 10003 | Why is arithmetic hard to learn, but then becomes easy? [Hofweber] |
| 10004 | Our minds are at their best when reasoning about objects [Hofweber] |
| 10005 | Arithmetic doesn’t simply depend on objects, since it is true of fictional objects [Hofweber] |
| 10006 | First-order logic captures the inferential relations of numbers, but not the semantics [Hofweber] |
| 10008 | Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber] |
| 10007 | Quantifiers for domains and for inference come apart if there are no entities [Hofweber] |