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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]