If we assume existence of a non-material world of ideas that mathematics describes there are some questions that a Platonist has to address.
1) How is the ideal world related to the real one, where mathematics also plays a role?
2) How do we gain access to the ideal world and establish truths about it with "absolute certainty" in mathematics?
Plato's answer to the first question was that the real things imperfectly "imitate" ideal originals, like shadows on a wall. His answer to the second was even more creative. Before birth our soul contemplates the ideal world directly, but forgets the experience upon birth. Interacting with imitations of ideas jogs our memory of them leading to the ideal truths, the process he called anamnesis (unforgetting). While modern Platonists may accept the imitation theory I doubt that many of them would subscribe to anamnesis. Cleared of fantastic elements it essentially equips us with a version of "mindsight", a sixth sense that reveals the ideal world directly, unlike the other five. There is no evidence in the work of mathematicians that we possess such a thing, which is probably why Plato moved his mindsight to pre-birth. And if this mindsight is intuition then it is a rather unreliable source.
On the other hand, if we do not have direct access to the ideal and only reconstruct it from the imperfect reality then there is a problem. Not only can't we be absolutely certain that our reconstructions establish truths about it, we can't even be sure they reflect it at all. Leibniz expanded the imitation theory to "pre-established harmony" between the ideal, the material, and our mind, which exists because "God creates the best and most harmonious world". But this is no better than mindsight. And it gets worse. All we actually have to go on then are our interactions with reality and the process of reconstruction. If we can get to ideas from that the Platonic world and the pre-established harmony are not just speculations, they are unnecessary complications, superfluous like ether in relativity.
I am not very familiar with more recent mathematical Platonism, especially in the 20th century, Stanford article is more about objections to Platonism than arguments in its favor. But it seems to remain popular with mathematicians, perhaps some philosophers too, so I am curious.
How does modern Platonism explain our ability to acquire knowledge about the ideal world? What is the argument for not cutting the Platonic world with Occam's razor?
EDIT: Vow, this is not what I expected. I originally hoped that a Platonist, or someone familiar with modern Platonism, would make the best case for the ideal world while accounting for more recent realizations, like fallibility of intuition and Kant's critisism of metaphysics. But it seems that all answers essentially concede non-existence of the ideal world, and either make an emotion/motivation based arguments for Platonism "in practice only", or reinterpret ideas conceptualistically. I upvoted all answers since they contribute to understanding modern perspectives on Platonism, and accepted the one that comes closest to reproducing something like the ideal realm, albeit radically remade.