Full Idea
A traditional Platonist has to face the possibility that all of the Peano Axioms are false.
Gist of Idea
Platonism must accept that the Peano Axioms could all be false
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 4.7)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.131
A Reaction
This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism!