Question: How badly does Platonic Realism interfere with science, secularism, evolutionary biology, etc.?
The scientistic response to Platonic Realism (PR) is that it violates scientific principles - empirical observability and verifiability (I'll leave "falsifiability" aside for now), testability, explanatory power, and other related concepts.
It's safe to say that PR is anti-scientific because it violates a central principle of cognition as such: induction. A fully philosophically-worked-out theory of induction wasn't made possible until the 20th century; that theory itself, of course, would have to be inductively-based. Induction is axiomatic-level; to try to deny it is to reaffirm it. Plato's Realism by its very nature is not inductively-based, but pure floating abstraction in the most applicable and fitting sense. By consequence it has no explanatory power (philosophically, this irresolvable problem is stated in such terms as "problem of partaking," a nice precursor to Cartesian "mind-body problem" given the basic principle involved: the commonsense-impossible "interaction" between the supernatural and the natural). By being an intrinsically floating theory, it cannot be reduced to the perceptual, hence the reason there is no testability, observability, or verifiability.
The question that initially occurred to me in the context of this thread was whether PR is empirically false given the established theory of evolution. The question came to mind because in evolution there are no Eternal Forms, but always transitional forms, and that the form of Man (in scientific terms: human DNA) was not actualized until some point in time, which means the form of Man is contingent, finite, etc. I'm guessing - without having given the matter much thought at all yet - that there is a proposed workaround of some sort to shelter PR from this particular refutation. After all, we might simply say that it is arbitrary, referencing beings entirely beyond the realm of the empirical, making it empirically unverifiable as well as unfalsifiable.
And adept advocate of PR might say that its not being "empirical" is not a philosophical problem, and that philosophy needn't be beholden to science (rather than vice versa). But the more adept philosopher than that, will say that PR is nonetheless arbtirary and fails to be inductive.
Aristotle is in a weird limbo-area on all this. His philosophical method was admirably inductive, but he was still limited by his own variant of Plato's Realism, also known as Moderate Realism. The first, first question I asked myself in the context of this thread was, "What was holding Aristotle back from positing what Darwin did over 2,000 years later? Was he lacking in the relevant empirical observations?" (I'll note here that Ayn Rand did not given an opinion on the theory of evolution when the subject came up. My best guess is that she did not regard herself as someone in a position - such as that of a biologist - to render a verdict on the matter. However, biology was one of Aristotle's chief areas of focus.) The supposed positing of the seed of evolutionary theory would be based on a straightforward observation of similarity between man and very similar animals - it might well have to involve higher primates. But if Form is thought to be eternal and unchanging, then the very notion of humans and other primates having a common ancestor might not even occur.
I'll just note in passing that PR as well as Moderate (Aristotelian-Thomistic) Realism is all-too-convenient cover for creationist views about our origins. Indeed, the notion of there being such a thing as Eternal Form, absent a Creator, is weird, to say the least, and probably incoherent. Aquinas performed a most understandable integration in his context. Let not this inference stand in place of the other (cosmological and ontological) arguments for God's existence, however.
The very problems stated here are the main reason scientists have basically cast off philosophy as useless to their field. Post-Aquinas and the Scholastics - with Bacon and others - modern science, called "natural philosophy" at the time, took on its own form independent of philosophy. At the same time, it has also had remarkable practical success - way more success than the humanities have had, in the meantime. (The Humanities are so screwed up that there may well have been regression since that time.) Some (understandably ignorant) scientists have used this track record of comparative progress as a victory cry for science and a reason to dismiss philosophy.
Here's what happened with the development of science: Well, first, there was Aquinas bringing Aristotle back into the fold, bringing with him a revival of this-worldly concerns. This led to the Renaissance and to the scientific revolution. Plato, Aristotle, and Aquinas, however, would have been of limited use, at best, in formulating the basic methods of the natural sciences. Now, simply as a matter of practical necessity, rational principles of natural philosophy had to be discovered and formulated - and they were. The basic methodological principle? Induction.
Induction is the basic method of learning and cognition. It involves a painstaking process of organizing sensory observations into a coherent generalizable whole, in which later observations and theories build upon the earlier ones. Einstein, for instance, had the same knowledge-base Newton had to work with, and then some. So in some important, crucial and relevant sense, Einstein's theories did not contradict Newton's. Newton wasn't all of a sudden overturned and repudiated; Newton was working from a more limited base of knowledge, is all.
This point might not matter much to the practical scientist - their working methods get results, and that's mainly what matters to them - but it does indicate a proper epistemological approach. Theories can turn out to be wrong; observations cannot; the role of epistemology is to determine what conclusions and theories are warranted given the knowledge-base, such that later conclusions do not contradict earlier ones. (Philosophy's role is to explain in underlying terms how it is that induction is practical; it has something to do with these things called identity and causation, concepts pretty darn well undermined by Pragmatism and plenty other bastardizations of inductivist method to come out of analytic philosophy in the last century-plus. Thanks a lot, Kant.)
The reason that science made leaps and bounds over philosophy in the last few hundred years is that science was based on induction, while philosophy was not. Philosophy, at the hands of Descartes and the rest, floundered; these thinkers failed to identify at root the principles of induction as applied to all areas, including philosophy itself. Their basic anti-inductive psycho-epistemological paradigm, emulated en masse by philosophers to this very day? Rationalism. (And a heaping dose of social metaphysics thrown in for good measure.) That paradigm, however, is about to change, thank Rand.
(Note: I write this without yet having read Harriman's The Logical Leap.)