Full Idea
The existence of the natural numbers is not a matter of pure logic - it cannot be proved in pure logic. It can be proved in second-order logic plus Hume's principle. Truths of arithmetic are not logic - they depend on the nature of natural numbers.
Gist of Idea
Add Hume's principle to logic, to get numbers; arithmetic truths rest on the nature of the numbers
Source
Bob Hale (Necessary Beings [2013], 07.4)
Book Reference
Hale,Bob: 'Necessary Beings' [OUP 2013], p.177
A Reaction
Hume's principles needs entities which can be matched to one another, so a certain ontology is needed to get neo-logicism off the ground.