Full Idea
Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects.
Gist of Idea
Arithmetic must allow for the possibility of only a finite total of objects
Source
Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148)
Book Reference
-: 'Journal of Philosophy' [-], p.148
A Reaction
This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system.