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Single Idea 8637

[filed under theme 6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival ]

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?

Gist of Idea

The number 'one' can't be a property, if any object can be viewed as one or not one

Source

Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §30)

Book Ref

Frege,Gottlob: 'The Foundations of Arithmetic (Austin)', ed/tr. Austin,J.L. [Blackwell 1980], p.41

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?

The 14 ideas with the same theme [numbers as properties, rather than objects]:

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]