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Single Idea 9952
[filed under theme 6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
]
Full Idea
A numerical adjective forms part of a predicate of second-level, needing supplementation from the first level (F). So the second-level predicate is of the same type as the existential quantifier, and can be called a 'numerical quantifier'.
Gist of Idea
Numerical adjectives are of the same second-level type as the existential quantifier
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.24
A Reaction
This seems like a highly plausible account of how numbers work in language, but it leaves you wondering what the ontological status of a quantifier is. I presume platonic heaven is not full of elite entities called quantifiers, marshalling the others.
The
14 ideas
with the same theme
[numbers as properties, rather than objects]:
9113
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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
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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
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We might eliminate adjectival numbers by analysing them into blocks of quantifiers
[Hofweber]
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