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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.

Regarding your edit, if you are looking for someone to defend the indefensible, then philosophy is a good place to start looking. Just don't expect a clear defence. I wish I could have been more helpful. Maybe in the future, when I have a more mature view of the subject, I will be in a better position to understand the issues more fully so as to mount a reasonable defence. – Nick – 2014-09-22T22:02:42.073

@Nick R I don't think it's indefensible though, long traditions of thought are usually flexible enough to deal with new objections. Even I can think of ways to make the ideal realm more palatable, and I am not sympathetic to Platonism. There is work of Husserl I heard intriguing things about, but he seems too dense and technical for me to understand. I definitely believe that Platonism captures some non-trivial aspects of mathematical discovery, and there has to be a modern philosophical expression for them. – Conifold – 2014-09-23T00:42:38.893

I'm all for better understanding Plato. My current "sophomore" status is a considerable hurdle when reading someone like Husserl, but it looks like a good place to stick my nose at a later date. Cheers. – Nick – 2014-09-23T00:46:29.570

Perhaps the lack of expected responses is itself an answer. I would ventured that those who remain sympathetic to Plato are typically interested in a quite different aspect of his thought than you are. However, I could be wrong. It might be helpful if you could elaborate on what you personally consider to be the "non-trivial aspects of mathematical discovery" captured by Platonism. – Chris Sunami supports Monica – 2014-09-23T16:38:00.837

@Chris Sunami The sense that mathematical truths are discovered rather than constructed although mathematical theories are apparently constructed through a cascade of abstractions with the foundation in experience, the "stability" of conclusions despite the changes in the experience itself, the process of refining intuition of the abstract ("anamnesis"), which is individual but not conventionally psychological because it is meant to explore something universal. – Conifold – 2014-09-23T20:03:07.090

It's hard to describe those aspects without embracing Platonism itself, which is exactly why it remains so appealing, but I think that a comprehensive philosophy of mathematics has to reduce to Platonism "effectively" as a correct "naive" perspective of a mathematician, like quantum mechanics reduces to the classical one. – Conifold – 2014-09-23T20:05:28.037

@Conifold I have added a short edit at the bottom of my answer, if you dare.... – Nick – 2014-09-24T05:01:03.403

@Conifold Plato had a wealth of amazing insights into fields as diverse as mathematics, art and politics. The question is do we view him as a [mathematical] genius who unfortunately had some nutty ideas, or did he have all his insights because of his nutty ideas? If you're more sympathetic to the former than the latter, I'm not sure there's much more in the way of profitable rehabilitation and demystification of Plato's mathematics that wasn't already performed centuries ago. I would venture that Plato himself considered his math to be the finger pointing at the star, not the star itself. – Chris Sunami supports Monica – 2014-09-25T13:36:43.630