Full Idea
If Hume's Principle is all we are given, by way of explanation of what the numbers are, the only conclusion to draw would seem to be the structuralists' conclusion, ...studying all systems that satisfy that principle.
Gist of Idea
If Hume's Principle is the whole story, that implies structuralism
Source
David Bostock (Philosophy of Mathematics [2009], 9.A.2)
Book Reference
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.272
A Reaction
Any approach that implies a set of matching interpretations will always imply structuralism. To avoid it, you need to pin the target down uniquely.
Related Idea
Idea 18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]