Manin's platonism

In the latest issue of the Notices of the American Mathematical Society, one can find an interview with eminent mathematician Yuri Manin, who is very highly regarded in the fields of algebraic geometry, number theory, and mathematical physics. He is also a mathematician who possesses a great breadth of cultural and historical knowledge.
Philosophers sometimes criticize mathematicians who express platonist leanings for being philosophically naive, or for not being aware of the latest developments in thought or empirical science. Examples can be found in earlier posts on this blog, where young Fields medalists were rebuked for expressing platonist opinions. In this way the huge number of platonist mathematicians are arrogantly dismissed as not having reflected on their subject maturely and according to current philosophical standards. This is definitely something that Manin cannot be accused of.

The interview reveals that Manin is not afraid of using the 'P'-word (my italics):

What has changed in pure mathematics?
The unique possibility of doing large-scale physi-
cal experiments in mental reality arose. We can
try the most improbable things. More exactly, not
the most improbable things, but things that Euler
could do even without a computer. Gauss could
also do them. But now, what Euler and Gauss could
do, any mathematician can do, sitting at his desk.
So if he doesn’t have the imagination to distin-
guish some features of this Platonic reality, he can
experiment. If some bright idea occurs to him
that something is equal to something else, he can
sit and sit and compute a value, a second value,
a third, a millionth. Not only that. People have
now emerged who have mathematical minds, but
are computer oriented. More precisely, these
sorts of people were around earlier, but, with-
out computers, somehow something was missing.
In a sense, Euler was like that, to the extent that
he was just a mathematician—he was much more
than just a mathematician—but Euler the math-
ematician would have taken to computers passion-
ately. And also Ramanujan, a person who didn’t
even really know mathematics. Or, for instance, my
colleague here at the institute, Don Zagier. He has
a natural and great mathematical mind, which is at
the same time ideally suited to work with comput-
ers. Computers help him study this Platonic reality,
and, I might add, quite effectively.

At this stage it is not clear what Manin believes regarding platonism. He refers to the mathematical universe as both a "mental reality" and (twice) as a "Platonic reality". This mystery of Manin's opinions deepens even further a little later on, when he says

I am an emotional Platonist (not a rational one: there are no rational
arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention.
I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you,
and so on.

This is puzzling for several reasons. First of all, to say that there are no rational arguments of platonism is not true in the most reasonable and common interpretations of these words. There are about as many rational arguments for platonism as there are for most other philosophical opinions. What Manin must have had in mind is the fact that platonism cannot be fully described in terms of today's science (which works in the paradigm of space-time and materialism).
Nevertheless, Manin seems to truly believe that mathematical platonism is a fact since he clearly states that he believes mathematics is discovered, not invented. His belief is reaffirmed in his platonist confession:

Gelfand: Are these properties of the problems
themselves, or is it just that no one is actively inves-
tigating them, for some social reason?
Manin: As a Platonist, I know that this is a
property of the problems themselves, but it is a
property that one cannot recognize at the moment
of formulating the problem. It reveals itself in the
process of historical development.
Partly for this reason, I am not partial to prob-
lems. Solving a problem requires the skill of
finding a detail, but you don’t know what it is a
detail of. As a Platonist, I am partial to complete
programs. A program arises when a great math-
ematical mind sees something as a whole, or not
as a whole, but as something more than a single
detail. But it is seen at first only vaguely.

The conclusion is that Manin has deeply held platonistic beliefs, but admits he has no scientific explanation for how the mathematical realm can exist in the way posited by platonism.