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.
    – nwr
    Sep 22 '14 at 22:02
  • @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
    Sep 23 '14 at 0:42
  • 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.
    – nwr
    Sep 23 '14 at 0:46
  • 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. Sep 23 '14 at 16:38
  • @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
    Sep 23 '14 at 20:03

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.

  • Your argument looks like a variation on the ontological argument for God's existence. It's known to be a fallacy, just because we can conceive an idea of something doesn't mean that it exists, existence can't be established through definitions. We can even conceive ideas of logical nonsense like the set of all sets.
    – Conifold
    Sep 20 '14 at 0:13
  • But one cannot introduce the notion of set without immediately creating the idea of the set of all sets. One must work dialectically with humans, there is no way to start from clarity. Plato's philosophy is inconsistent or incomplete. So is every system that can support arithmetic, according to Goedel. What is the point of fighting human nature to the point that we lose track of our roots?
    – user9166
    Sep 20 '14 at 0:19
  • I don't believe that 'exists' has a single definition. One needs a collection of different ontologies for different purposes, and Platonism is among the most useful. The ontological argument for God's existence is not a fallacy, it is an expression of the human experience of hierarchy and parentood. God is as real as your expectation someone will take care of you. Does that expectation deserve no expression because it is not always true?
    – user9166
    Sep 20 '14 at 0:24
  • God being real may or may not be true, the point is that the ontological argument does nothing to tell us either way. It makes sense that something so non-trivial can't be established by a logical trick.
    – Conifold
    Sep 20 '14 at 1:12
  • Or, more from my point of view, that something so trivial might as well be established that way. God's existence has no effect that the idea of God's existence would not already have had. Platonism is a 'game' in the sense of Wittgenstein and the search for a consistent ontology in the mathematical sense is a different sort of game. Applying the standards of one to the other is not a source of questions, it is a way of indicating lack of trust.
    – user9166
    Sep 20 '14 at 19:11

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.

  • Why not go with Kant then? He replaces unreachable ideas with synthetic a priori that come from ourselves, but can only be reconstructed a posteriori from our perceptions. This gives a deeper level and explains "absolute certainty", while avoiding the access problem and the multiplication of entities.
    – Conifold
    Sep 19 '14 at 23:59
  • @Conifold going with Plato and going with Kant are not that different as the 19th century neo-Kantians recognized... but both are committed to a world of ideas that is somewhat inaccessible to our knowledge.
    – virmaior
    Sep 20 '14 at 14:39
  • @Conifold Kant's version is too well-defined. It isn't open ended enough to truly serve as a gateway onto a wider world. It's actually the flaws and inconsistencies in Plato's theories that are compelling. Many of them, I would argue are placed there deliberately, to ensure you don't end your search prematurely. Sep 21 '14 at 2:05
  • @Chris Sunami I actually find parts of Kant very obscure and open ended, but I liked how his theory demystifies some of Plato's and removes some inconsistencies, while keeping the valuable parts, objective ideas and necessity. His main flaw is that synthetic a priori are unchangeable, like Plato's ideas, whereas in reality they seem to evolve and get refined over time. I thought Husserl tried to fix that but couldn't really understand him.
    – Conifold
    Sep 22 '14 at 20:03
  • @Conifold I understand that. But for those who do continue to prefer Plato to Kant, it's probably precisely the parts of Plato that Kant refines out that they value. Sep 22 '14 at 20:26

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.

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.

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.

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.

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.

  • 1
    Incompleteness is not a problem, it applies to provable statements in effectively axiomatized systems stronger than arithmetic. So all Godel's theorem says is that all truths about ideal world can't be proved from a well defined list of axioms, which you'd expect of something so vast anyway. As for self-reference, ideas are not extensional like sets, so paradoxes won't go through. There is infinite regress (idea of idea of idea, etc.), but one can dismiss such iterations as meaningless, hence not expressing valid ideas.
    – Conifold
    Sep 24 '14 at 17:41
  • @Conifold D'oh! Apparently I have not fully understood the concept of extensionality. Your comment has certainly made it more clear, and it highlights something which should have been clear to me at this stage since I do at least (vaguely) understand the problems of unrestricted collectivization.
    – nwr
    Sep 24 '14 at 18:39

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.

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