How does full-blooded Platonism explain mathematical knowledge through axioms and proofs?

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philosophy of mathematics German philosopher of mathematics, logic, and language Gottlob Frege.

Full-blooded Platonism addresses the challenge of mathematical knowledge by positing that all possible mathematical objects actually exist. This perspective allows mathematicians to acquire knowledge of abstract objects without requiring direct, information-transferring contact with them.1

According to this view, mathematical knowledge is achieved through a two-step methodology that mirrors standard mathematical practice:

Stipulation: Mathematicians formulate axioms to characterize the specific mathematical structures they intend to study. Because full-blooded Platonism asserts that all internally consistent axiom systems accurately describe existing parts of the mathematical realm, the act of laying down axioms serves to stipulate which portion of that realm is being discussed.1
Deduction: Once the structures are defined, mathematicians deduce facts about them by proving theorems from the chosen axioms. Through this process, they discover truths about the mathematical objects they have selected.1

By maintaining that any consistent system corresponds to a real mathematical structure, full-blooded Platonists argue that mathematicians gain genuine knowledge of abstract objects simply by exploring the logical consequences of their stipulated axioms.1