First on Platonism: it is very difficult to try to summarize a complex philosophical position in one or two paragraphs, and even more difficult to do so when you are not sympathetic with the position. But Clauster should know this, and so he should take extra care when doing so. In particular, his description of Platonism refers to abstract objects as existing in "another dimension of reality in which there are real things called numbers". Dimensions are spatial objects, and the implication is that abstract objects are sitting around somewhere as if the set of even numbers might have been left on top of the set of primes, so that it must be moved to get to the primes.
Platonism is not like this. To a Platonist, numbers and sets are real in this dimension, not some other dimension. When a Platonist says that numbers and sets are real, we don't mean that they are real like chairs and planets; we mean that they are real like numbers and sets. They have no mass, no position, no time. Questions like "Where are they?", "When are they?", "Where did they come from?", "What is going to happen to them?" simply do not apply. Platonism isn't even a subtle philosophical position; it is the natural position. Why is 1+1=2? Well, just because 1 is the sort of thing that when added to itself give you another thing, 2. By calling them things, you are implying that they exist.
Platonism is just the rejection of complex philosophy about what numbers are. Some people, for example, have argued that numbers are just marks on paper and that the rules of mathematics are just arbitrary rules. But then they have to answer questions like this: in the equation 1+1=2, what do the first 1 and the second 1 have in common? On paper they are different marks. And what do they have in common with the other instances of the mark that would be written in the rules? The simple, natural, pre-philosophical answer is just this: the equation isn't about the marks. Rather, the mark 1 represents a thing, a number. And the rules are written to be correct rules that go with the numbers.
Platonism is a form of realism. It is the claim that numbers are real in themselves and cannot be reduced to anything else. This is the opposite of reductionism. Reductionism (of numbers) is the claim that numbers aren't real in themselves; they are in fact something else: marks on paper, thoughts in the head, collections of physical objects, etc. By taking a realist theory and calling it reductionist, Clauser is making the word apply to the very thing that it is supposed to exclude, thereby twisting the word into something entirely different than its common meaning.
What is even more peculiar is that Clauser seems to be a Platonist and not realize it. At least his account of numbers is something that sounds perfectly fine to a Platonist:
We abstract that quantity and set up a symbol system to represent it. And we discover relationships among those quantities. The symbol system is our invention, but we find quantities and their relations in God’s creation.You can't "find" quantities and "discover" relationships among quantities unless the quantities are real. If quantities (that is, numbers) were fictitious then you could invent them and their relationships, but not discover them. To discover relationships among things, the things have to be there to be investigated and understood. They have to be real.