Full Idea
According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure.
Gist of Idea
Does someone using small numbers really need to know the infinite structure of arithmetic?
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.254
A Reaction
Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones.