Full Idea
The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers.
Gist of Idea
Numbers do not exist independently; the essence of a number is its relations to other numbers
Source
Stewart Shapiro (Thinking About Mathematics [2000], 10.1)
Book Reference
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.258
A Reaction
This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate.