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Full Idea
Perhaps, instead of objects, numbers are associated with properties of objects. Basing them on objects is strongly empiricist and uses first-order logic, whereas the latter view is somewhat Platonistic, and uses second-order logic.
Clarification
Second-order logic qunatifies over properties as well as objects
Gist of Idea
Empiricists base numbers on objects, Platonists base them on properties
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.49
A Reaction
I don't seem to have a view on this. You can count tomatoes, or you can count red objects, or even 'instances of red'. Numbers refer to whatever can be individuated. No individuation, no arithmetic. (It's also Hume v Armstrong on laws on nature).
| 9113 | Just as unity is not a property of a single thing, so numbers are not properties of many things [William of Ockham] |
| 9624 | Numbers are a very general property of objects [Mill, by Brown,JR] |
| 9951 | It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman] |
| 9952 | Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman] |
| 11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege] |
| 8637 | The number 'one' can't be a property, if any object can be viewed as one or not one [Frege] |
| 9999 | For science, we can translate adjectival numbers into noun form [Frege] |
| 14465 | Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell] |
| 9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
| 18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
| 13873 | Treating numbers adjectivally is treating them as quantifiers [Wright,C] |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| 10000 | We might eliminate adjectival numbers by analysing them into blocks of quantifiers [Hofweber] |