Full Idea
Hume's Principle will not do as an implicit definition because it makes a positive claim about the size of the universe (which no mere definition can do), and because it does not by itself explain what the numbers are.
Gist of Idea
Hume's Principle is a definition with existential claims, and won't explain numbers
Source
David Bostock (Philosophy of Mathematics [2009], 9.A.2)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.274
Related Idea
Idea 18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]