If the Platonic world exists how would we know?

Question

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.

Answer 1

The Platonic realm exists in the Platonic sense because it has been clearly conceived of as Platonic object, itself. "The realm of ideas" is just the idea of the collection of all ideas, which one automatically has if one has ideas and then expresses ideas about the nature of ideas. The question is whether the way that idea exists qualifies as existence.

The question is akin to asking whether the word 'definition' has a definition. Of course it does. But if you did not already know what a definition was, how could that one define anything?

Similarly this all 'exists' if our definition of existing is as naive as the notion of defining has to be to someone who would write a definition of 'definition' as the first entry in a lexicon. But criticism at that level of naivete is just bullying, not thought.

We have to work into that definition from the outside, and we have no choice but to start from a naive idea of definition, or of idea. To force later problematic results back onto the original consideration is just circular. We have ideas, whether we want to or not. So 'how do we gain access to ideas' is not a real question, unless it is about the process of realization, rather than about access.

For me, the argument for 'not cutting' this concept out of our thought is that this mode of thought is inescapable. It is the one we fall into with the habits of childhood, which is how most of us approach most problems we meet fresh. We need more sophisticated ways of reigning in childish impulses, but we should not lose them, as they are the basis of our thinking, and always will be.

Answer 2

I can't speak for other modern Platonists, but I can give you my perspective:

When interpreting Plato, I find it a mistake to take him too literally. According to his view of the world, the capital T Truth wasn't something that could ever be completely captured in ordinary language. All of his writing should be viewed as primarily metaphorical, aimed at helping people discover the Real for themselves, rather than as an actual attempt at capturing or defining the Real.

I would imagine philosophers and mathematicians who continue to find Platonism compelling do so for the same reason they always have. Some aspect of their work begins to convince them that there must be a deeper level of reality than the one available to our ordinary senses, and it seems to align in a profound sense with the deeper level of reality described by Plato, even if it isn't a perfect match in the details.

Thus, I would consider myself at the least in sympathy with Platonism, even though I don't have a belief in an Ideal Realm of the Forms as described in Plato's dialogs.

Answer 3

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?

Let's look at the second part of your question first.

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Occam's razor is a guiding principle which we have formulated in the belief that it accurately reflects a necessary feature of the landscape of Plato's ideal world. We don't expect this world to included redundant or unnecessary forms just as we would not expect this world to contain poorly formed ideals. We expect this world to be exactly what is necessary for its ideal formulation and nothing else.

Importantly here, we expect the ideal formulation of Occam's razor to be found in Plato's ideal world. It would therefore appear to be in some ways ironic (not to mention disingenuous) to start hacking away at Plato with one of his own forms.

Plato's ideal world must stand or fall on its own. In this regard, one has to say that it appears to fall. I'm sure the Stanford arguments referred to in your comments do a perfectly good job of rejecting Plato's ideal world.

Yet mathematics contains many examples of how simple ideas and elementary arguments can lead to profound, beautiful, and even shocking results. Here, mathematicians often believe they have glimpsed an ideal form. It feels very real. So although Plato's vision of a unique, well-define world of ideals appears to be inconsistent, the notion of an ideal form does not seem to be problematic in a given context.

This more selective acceptance of Plato's vision is consistent with the views expressed in your question and highlights how engrained Plato's ideas are in our own modern view of mathematics. One is happy to accept the ideal expressed by Occam's razor for example, while perhaps feeling uncomfortable with the full implications of an idealized mathematical world.

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Regarding the first part of your question concerning our ability to acquire knowledge of the ideal world, as we agreed in my answer to your previous question, we can never really be certain if our formalization of a particular theory or (non-trivial) ideal is either correct or fundamental. We can be guided by principles like Occam’s razor or we can appeal to aesthetics and experience, but none of these techniques can provide certainty. Ultimately we can never know. On the plus side, the remarkable utility of our mathematical theories tells us that even if we are creating emergent theories rather than fundamental, ideal-world theories, what we are doing has real value, including intellectual and artistic value (if that's not too airy-fairy), and we are guided by Plato's vision in this regard.

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Perhaps the best we can hope for is that our mathematical universe is in some sort of entanglement with a restricted form of Plato's vision.

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EDIT Sep 23

I have been tempted to ask this, but my lack of maturity has made me reluctant since it may be a rather sophomoric point.

Our formalization of (classical) logic may not be ideal but our theorems are surely valid.

One could argue that since arithmetic cannot be both complete and consistent we have big problems with Plato. Even with a complete and consistent form, such as Euclidean Geometry, we have problems. I shall spare you the details and choose a more economically expressed issue.

Plato's world must itself be an ideal form. That is to say, it must, by conception, be a member of itself. (Boy, I really am getting sophomoric now.) This leads to all of the obvious paradoxes associated with self-reference, rendering Plato's world either inconsistent or incomplete.

Answer 4

2) How do we gain access to the ideal world and establish truths about it with "absolute certainty" in mathematics?

Since the words "absolute certainty" appear in quotes in the previous sentence, I will not address that aspect in my answer. Access to the ideal world is based on the neural nets in our brain already operating automatically with a good (but not perfect) grasp of logic - Boolean Algebra - 1st, 2nd, and higher orders of Predicate Logics - intuition being dependent on reasoning by Analogies, Fuzzy Logic, etc. As the brain processes a heretofore unknown-to-it Platonic truth, it also employs the rudiments of what we now call the basic control structures in computer languages: if-else statements, non-deterministic looping structure, as it (the brain) sequences through their application over time. (This does not all have to be cerebral, but can, and usually does involve eye-hand use in jotting down symbols as a means of external memory enhancing/relieving cerebral memory.)

In short, the access to the Platonic mathematical realm is deterministic, as per the Church/Turing Thesis. Where nondeterminism resides in the Platonic realm, we regard them as paradoxes, hypotheses, etc. - including this statement here as well.