Would you really need three realms (mental, physical, platonic) to answer the question?
I am no mathematician but wouldn’t it be a plausible model to assume that the mind (the mental realm) reflects those properties of the physical realm that were important to our survival at some evolutionary time point? This would obviously include rules like cause and effect and the natural numbers. If these inbuilt rules were correct, they could by inference (likewise a mental rule reflecting properties of the physical world) be used to extrapolate more and more of the rule framework of the physical realm. This would also explain why applied mathematics (in the chosen definition) can describe counter-intuitive physics, i.e. it works beyond the limits where the mind correctly reflects the physical world. This model obviously is still platonic but I do not see that it needs three realms.

Daniel