Platonism is the view that objects likes numbers really exist, and they exist outside space & time. Platonists claim that these abstract objects are immaterial, timeless, spaceless, eternal, uncaused, changeless, necessary, non-conscious, non-personal, and causally impotent. ‘Causally impotent’ would mean that abstract objects can’t cause anything. If this seems weird to you, you are not alone.
Some would claim that Platonism is either incoherent or meaningless. ‘Incoherent’ would mean that Platonism is contradictory, and Platonism being ‘meaningless’ would mean that Platonism is not even false. I must confess, I don’t know what an abstract object (as conceived by the Platonist) is supposed to be. They specify what abstract objects are NOT; however, they don’t specify what abstract objects ARE.
Nevertheless, let’s put these concerns aside and see whether Platonism is the best explanation of things like numbers, properties, universals, possible worlds, etc. My contention is that Platonism is not a good explanation: Platonism is a failed hypothesis.
Explanation
But what makes for a strong hypothesis? Well, there are several theoretical virtues, including (but not limited to):
1. Simplicity
2. Explanatory Power
3. Explanatory Scope
4. Testability/Falsifiability
5. Conservatism (i.e. fits with background /prior knowledge)
6. Fruitfulness
Evaluating Platonism
Now, Platonists might object here that Platonism is not a scientific hypothesis/theory. True, but I’m only talking about Platonism as an explanation in general (not necessarily a scientific explanation, specifically). Although, even here, I don’t think Platonism is a good hypothesis.
For one, Platonism does not fit with our background/prior knowledge. In our experience, things that exist are causal entities, but Platonism says that there are things that exist which are causal (non-causal). Not only is this a vice and a theoretical strike against Platonism, but it also means that (all else being equal) Platonism is improbable. And if there’s no independent motivation for thinking that Platonism is true, then it’s ad hoc. If you think Platonism is a good hypothesis to explain things like numbers, that doesn’t mean you have independent reasons to think that Platonism is true.
For another thing, Platonism is not simple. It adds more to our ontology. Not only does it posit an infinite (or near infinite) number of things, but it also posits a new kind of entity. The new kind of entity would be abstract entities. If you can believe it, this would actually be way more absurd than anyone who has ever postulated things like ghosts, unicorns, etc. That’s because the latter would be concrete, whereas the former (abstract objects) would not be. At least, we know concrete things exist, like trees, even if we don’t believe that ghosts exist. But it’s far from obvious that anyone knows that abstract things exist. There’s nothing obviously (or non-obviously) contradictory in denying Platonism, which would make Platonism superfluous…given that there are other accounts of abstract objects that don’t posit an abstract realm.
Thirdly, does Platonism have explanatory power? It certainly doesn’t explain anything in causal terms. In that sense, Platonism seems contrived or dodgy. It’s also dodgy to say something like, “Platonism is explanatory, not causal” and just leave it at that. What type of explanation? Not only would Platonism not be shedding light on anything, but it wouldn’t even shed light on itself. Platonism is certainly not offering any predictions but, at best, retrodictions. Either way, if Platonism is true, then what should we expect to observe/see? I don’t mean that you should literally “see” an abstract object. If the world is exactly the same, whether or not Platonism is true, then what good is Platonism? This would mean that Platonism is not testable. It would also mean that Platonism doesn’t really pass the explanatory scope test if it literally explains everything. If it literally explains everything, then it actually explains nothing.
Finally, Platonism isn’t a fruitful research program. It’s only a “research” program for professional Philosophers/Metaphysicians, where people engage in word games and sophistry. Not only will Platonism never build an iPhone, but it makes no difference at all to our lives. In that case, why should I care whether Platonism is true? And how can we even call something ‘true’ if it doesn’t include practical consequences?
Theism and Platonism
If abstract objects exist, it might be thought that the existence of abstract objects (as conceived by Platonists) would pose problems for theism (because there would be an infinite amount of necessary entities that exist independently of God. Thus, God wouldn’t be unique and have his nature determined by external properties).
Firstly, perhaps there wouldn’t be a problem. If someone thinks that God creating/causing (or destroying) an abstract object is an absurdity (i.e. logical impossibility), then the existence of abstract objects wouldn’t be a problem for theism any more than the fact that God can’t create a square circle. It’s not self-evident that there’s a problem. Secondly, it’s far from apparent that Platonism is true, as this post has argued. Thirdly, a theist could hold that abstract objects are thoughts in the mind of God. With this view, there wouldn’t be independently existing abstract objects. Fourthly, a theist could maintain that God created abstract objects. How is this possible? How could God create objects that don’t stand in causal relations? Plausibly, God’s creation of such abstract objects would have to be non-causal. In other words, the explanation would have to be a non-causal explanation. The Platonists can’t object to this because they say that abstract objects are explanatory and non-causal! The view might seem unmotivated to you, however, that isn’t the case. Theists like Aquinas and Leibniz held that there are such things as non-causal explanations, including when it comes to the topic of God. In fact, under this view, God’s existence is necessary and that necessity is explanatory (and not causal). In addition, even though Aquinas was sympathetic to the views of Aristotle, he still didn’t seem to think God couldn’t create things like abstract objects.1
Notes
1 Presumably, this would also apply to concrete possible worlds that are causally isolated from each other, if they were to exist–i.e. modal realism. This is not to be confused with multi-verse theory, even though they are similar and related. God would be the explanation behind these worlds. Theistic modal realists don’t all agree with each other about everything. Some of them would say that God has counterparts (twins) in each world. The problem here is that, at first glance, it would seem to be a denial of monotheism. Others would say that God stands ‘outside’ all concrete possible worlds. Still, others would argue that God exists in all of these worlds, as conceived by modal realism (this is not to be confused with Plantinga’s view that God exists in all possible “worlds” because Plantinga is not a realist about possible worlds).
Philosophy of Religion blog 
I have always wondered why anyone “bit” on Platonism, ever. It is clear that it is wishful thinking, for a reality in which perfections and absolutes existed, when everything this reality tells us is there are no such things. We invent hypothetical perfections, e.g. whole numbers, to be able to say “See, such things exist!” But they are just mental constructs as much as manticores and unicorns and gods are.
seems as if a great many people want reality to conform to their imagined realities.
I have theorized religion and philosophy as being of the same stripe…the same sort of, as Sellars put it, “…attempt to see how things hang together…”. Simply reduced, followers of Philosophy have DesCartes; Kant; James; Davidson and so on. Followers of Religion have the Bible; Quran and others. I have noted that these studies are often situated in the same university department. Seems a good clue to me.