Overview
Platonism is synonomous with realism in the philosophy of mathematics. It is a a philosophy that Plato developed in several of his works, in particular, the Republic.
Plato reasoned that there exists a world of perfect abstract objects, which he called forms. Mathematical objects exist in this Platonic realm of forms.
As such, the Platonic position is a realist position, in which mathematical objects really exist. (for whatever that means)