Why does full-blooded Platonism claim that every consistent mathematical structure exists?

Full-blooded Platonism is a philosophical position that asserts all mathematical objects that could possibly exist actually do exist. This view posits that the mathematical realm is exhaustive, meaning that every internally consistent axiom system accurately describes a part of that realm.1

The primary motivation for this claim is to resolve the epistemological challenge of how humans acquire knowledge of abstract objects without direct, information-transferring contact. By maintaining that all consistent structures exist, full-blooded Platonists argue that mathematicians do not need to "discover" which specific structures exist in a vacuum. Instead, when mathematicians formulate axioms to characterize a structure, they are simply stipulating which part of the existing mathematical realm they intend to study.1

Consequently, mathematical knowledge is acquired through the standard methodology of the field: by deducing theorems from those stipulated axioms. Because the structure is guaranteed to exist by virtue of its consistency, the mathematician successfully gains knowledge of an abstract object without requiring a special faculty of intuition or sensory perception.1