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Single Idea 17906

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers ]

Full Idea

I would disagree with Quine. The explanation of cardinality - i.e. of the use of numbers for 'transitive counting', as I have called it - is part and parcel of the explication of number.

Clarification

'Transitive counting' is counting objects

Gist of Idea

To explain numbers you must also explain cardinality, the counting of things

Source

Paul Benacerraf (What Numbers Could Not Be [1965], I n2)

Book Ref

'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.275

A Reaction

Quine says numbers are just a progression, with transitive counting as a bonus. Interesting that Benacerraf identifies cardinality with transitive counting. I would have thought it was the possession of numerical quantity, not ascertaining it.

Related Idea

Idea 17905 Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]

The 24 ideas from 'What Numbers Could Not Be'

9151 Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
13891 To understand finite cardinals, it is necessary and sufficient to understand progressions [Benacerraf, by Wright,C]
8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
17904 A set has k members if it one-one corresponds with the numbers less than or equal to k [Benacerraf]
9898 We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
17903 Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
9897 The application of a system of numbers is counting and measurement [Benacerraf]
17906 To explain numbers you must also explain cardinality, the counting of things [Benacerraf]
9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
9903 Number words are not predicates, as they function very differently from adjectives [Benacerraf]
9904 The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
9905 Identity statements make sense only if there are possible individuating conditions [Benacerraf]
9906 If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
9908 The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
9909 The number 3 defines the role of being third in a progression [Benacerraf]
9911 Number words no more have referents than do the parts of a ruler [Benacerraf]
9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
9912 There are no such things as numbers [Benacerraf]
8925 Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
9938 How can numbers be objects if order is their only property? [Benacerraf, by Putnam]