Full Idea
Hume's Principle can't tell us what a cardinal number is (this is one lesson of Benacerraf's well-known problem). An infinity of pairs of sets could actually be the number two (not just the simplest sets).
Clarification
See Idea 8302 for Hume's Principle
Gist of Idea
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
Source
report of Paul Benacerraf (What Numbers Could Not Be [1965]) by E.J. Lowe - The Possibility of Metaphysics 10.3
Book Reference
Lowe,E.J.: 'The Possibility of Metaphysics' [OUP 2001], p.215
A Reaction
The drift here is for numbers to end up as being basic, axiomatic, indefinable, universal entities. Since I favour patterns as the basis of numbers, I think the basis might be in a pre-verbal experience, which even a bird might have, viewing its eggs.
Related Idea
Idea 8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]