Your comment was very coherent, and absolutely right in identifying a need for clarity concerning what is meant by a "platonic realm". Perhaps I wasn't clear enough in the article. Anyway, of the two options you mention it is the former: a completely distinct, transcendent, non-physical realm.
This definition raises the problem that you and I have mentioned: apparent difficulties with empirical investigation. But I would like to ask you what you mean by "prove scientifically". Are you working in a Popperian scheme? The trouble with doing so is that it itself is more of an over-arching classification (science/not-science), or perhaps a working definition, rather than an actual established, provable fact, or even something open to empirical investigation. In fact, to the extent that it is open to empirical study, it is plainly falsified in the working lives of almost every scientist. Alternatives, such as Quine's holism, are perhaps closer to the everyday experience of most scientists.
Either way, both schemes may allow for an empirical test of the platonic hypothesis, if I may call it that. Because after all, although the platonic realm is distinct and independent of the physical realm, the one influences the other (I'm talking from within the hypothesis, you understand). So it is possible that someone may be able to come up with a (falsifiable?) statement about the existence of the platonic realm which can be investigated with empirical tools of the physical realm.
As for the second option you mention, that the platonic realm is a kind of shorthand for a quasi-empirical science of mathematics (a science of pattern, essentially), this is not what I meant. It is a perspective which has its advocates, but which few mathematicians or mathematical scientists find appealing (this doesn't make it wrong, of course). There are serious implications of such a viewpoint for the entire scientific program.