I never understood what is meant by the Platonic world. I always considered abstract objects like mathematical objects as mental stuff. So when Penrose writes this quote for me this seems more an argument for mathematical objects being innate and mental stuff:
I mean except for being quite specific, this isn't different than the way internalists like Chomsky treat linguistic concepts:
It's not surrprising that Chomsky thinks the two are related:
Some simple evo-devo theses: how true might they be for language?
To be honest, the more I read on this stuff, the more I'm persuaded by both the internalist and nativist view.