 If we add Hume’s Principle to second-order logic, then we can analytically derive (second-order) Peano Arithmetic. (The essentials of the argument are found already in Frege 1884.) It is a central thesis of Neo-Fregeanism that Hume’s Principle is an implicit definition of the functional expression ‘the number of’ (see Hale and Wright 2001). If this thesis can be defended then logicism about arithmetic can be sustained while foregoing Frege’s explicit (and inconsistent) definition. However, the neo-Fregean thesis is in conflict with the traditional account of definitions, for Hume’s principle violates both Conservativeness and Eliminability. The principle allows one to prove, for arbitrary \(n\), that there are at least \(n\) objects. (A related application aims to sustain the analyticity of a geometry through the idea that the axioms of geometry are implicit definitions of geometrical concepts such as “point” and “line.” Here, too, there is conflict with the traditional account, for Conservativeness and Eliminability are violated.)