Two Fields medalists on platonism

The Fields medal is the most prestigious prize that is awarded to mathematicians. One of the most recent Fields medalists, Andrei Okounkov was interviewed in the EMS Newsletter. Among other things, he didn't hesitate to express the following in reply to a question:

Much of your work has deep connections to physics. Does
that mean that you find it essential that mathematics is
related to the natural word, or that you would even think
of it as subservient to the other natural sciences?

O: When I said “our world” earlier I didn’t mean just the
tangible objects of our everyday experience. Primes are
as real as planets. Or, in the present context, should I say
that celestial bodies are as real as primes?

What I hope will slowly emerge from various posts on this blog is that a large majority of the best mathematical minds see mathematical platonism as a natural and convincing position. The skepticism often found among philosophers and empirical scientists does not take this fact seriously. Either mathematical platonism is true, or mathematics is the only area of human intellectual endeavour where its most prominent practitioners have less understanding about the nature of their subject than people from outside that area.
Here is an excerpt from the same article; an interview with Terrence Tao, who won a Fields medal the same year as Okounkov:

Many mathematicians are Platonists, although many may
not be aware of it, and others would be reluctant to admit
it. A more “sophisticated” approach is to claim that it is
just a formal game. Where do you stand on this issue?

T: I suppose I am both a formalist and a Platonist. On the
one hand, mathematics is one of the best ways we know
to try to formalise thinking and understanding of con-
cepts and phenomena. Ideally we want to deal with these
concepts and phenomena directly, but this takes a lot of
insight and mental training. The purpose of formalism in
mathematics, I think, is to discipline one’s mind (and fil-
ter out bad or unreliable intuition) to the point where
one can approach this ideal. On the other hand, I feel the
formalist approach is a good way to reach the Platonic
ideal. Of course, other ways of discovering mathematics,
such as heuristic or intuitive reasoning, are also impor-
tant, though without the rigour of formalism they are too
unreliable to be useful by themselves.

Two remarks are in order here. First of all, the interviewer obviously didn't find a better word to use than "sophisticated". One must not be led to believe that formalism is somehow deeper, better developed in its details, or somehow the preferred view of sophisticated minds. What was probably meant above was that platonism is the "first" position one naturally develops as a mathematician, and that with later reflection, one might be led to formalism. Obviously this doesn't say anything about the relative merits of truths of the two positions.
Secondly, Tao says he's both a formalist and a platonist. The common meaning of these terms, as used in the philosophical debate, is such that one excludes the other. Mathematical formalism as a philosophical position usually means that mathematics is ontologically seen as formal symbols. Platonism on the other hand posits the independent existence of mathematical objects, beyond the mere symbols, which are modelled by the formal systems used in mathematics. The latter seems to be the position of Tao, and it's a fair bet that most working mathematicians (but not all, of course) would agree.