Full Idea
Hume's Principle gives a criterion of identity for numbers, but it is obvious that many other things satisfy that criterion. The simplest example is probably the numerals (in any notation, decimal, binary etc.), giving many different interpretations.
Gist of Idea
Many things will satisfy Hume's Principle, so there are many interpretations of it
Source
David Bostock (Philosophy of Mathematics [2009], 9.A.2)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.271
Related Ideas
Idea 18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
Idea 18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]