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9951
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It appears that numbers are adjectives, but they don't apply to a single object
[Frege, by George/Velleman]
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Full Idea:
Numbers as adjectives appear to attribute a property - but to what? Superficially it seems to be to the objects themselves, as it makes sense to say that a plague is 'deadly', but not that it is 'ten'.
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From:
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
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A reaction:
Surely they could be adjectival if they were properties of groups? Groups can be 'numerous', or 'more than a hundred', or 'too many for this taxi'.
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9952
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Numerical adjectives are of the same second-level type as the existential quantifier
[Frege, by George/Velleman]
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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'.
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From:
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
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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.
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8637
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The number 'one' can't be a property, if any object can be viewed as one or not one
[Frege]
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Full Idea:
How can it make sense to ascribe the property 'one' to any object whatever, when every object, according as to how we look at it, can be either one or not one?
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From:
Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §30)
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A reaction:
This remark seems to point to numbers being highly subjective, but the interest of Frege is that he then makes out a case for numbers being totally objective, despite being entirely non-physical in nature. How do they do that?
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9999
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For science, we can translate adjectival numbers into noun form
[Frege]
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Full Idea:
We want a concept of number usable for science; we should not, therefore, be deterred by everyday language using numbers in attributive constructions. The proposition 'Jupiter has four moons' can be converted to 'the number of Jupiter's moons is four'.
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From:
Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §57)
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A reaction:
Critics are quick to point out that this could work the other way (noun-to-adjective), so Frege hasn't got an argument here, only an escape route. How about the verb version ('the moons of Jupiter four'), or the adverb ('J's moons behave fourly')?
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9903
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Number words are not predicates, as they function very differently from adjectives
[Benacerraf]
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Full Idea:
The unpredicative nature of number words can be seen by noting how different they are from, say, ordinary adjectives, which do function as predicates.
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From:
Paul Benacerraf (What Numbers Could Not Be [1965], II)
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A reaction:
He points out that 'x is seventeen' is a rare construction in English, unlike 'x is happy/green/interesting', and that numbers outrank all other adjectives (having to appear first in any string of them).
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9620
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Empiricists base numbers on objects, Platonists base them on properties
[Brown,JR]
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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.
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From:
James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
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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).
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10000
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We might eliminate adjectival numbers by analysing them into blocks of quantifiers
[Hofweber]
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Full Idea:
Determiner uses of number words may disappear on analysis. This is inspired by Russell's elimination of the word 'the'. The number becomes blocks of first-order quantifiers at the level of semantic representation.
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From:
Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §2)
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A reaction:
[compressed] The proposal comes from platonists, who argue that numbers cannot be analysed away if they are objects. Hofweber says the analogy with Russell is wrong, as 'the' can't occur in different syntactic positions, the way number words can.
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