I am currently reading Ian Hacking’s Why is there Philosophy of Mathematics at all, and it is mostly about the contemporary dabate platonism/ nominalism, so I would recommend it as a good place to look for an answer to this question. A crude copy-paste is given below.
Hacking asserts that it is Paul Bernays who introduced the modern idea of 'platonism' and further he writes about the different brands of platonism, noting (p228):
After Bernays had introduced the word 'platonism' to the philosophy of
mathematics, it should have been clear that one should not speak of platonism,
but about platonism restricted to some domain of objects, such as the
class of integers, or any Zermelo-Fraenkel set, or whatever. Perhaps even
something like the class of all ZF sets, in a van Neumann-Godel-Bernays
So a good point to remember is that according to him
Absolute platonism, asserting the existence of all definable mathematical
objects and relations, is untenable. What remain are relative platonisms.
The weakest interesting platonism described by Bernays asserts the existence
of a totality of whole numbers...a twenty-first-century platonist will say: positive integers are abstract objects. Kronecker said that God created them. So he thought they exist. That's a species of platonism - about numbers. He was a very modest platonist, but a platonist all the same.
Such a thought seems never to have crossed Bernays' mind, for he thought of platonism in terms of totalities rather than 'abstract objects'.
So Bernays turns out to be a restricted platonist while
In Bernays' vocabulary Boolos is a cautious platonist. He has no problem
about the totality of whole numbers, but he has many qualms about sets
whose existence is proven within Zermelo-Fraenkel set theory with the axiom
Perhaps there is something of an answer in this heap of quotes