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The Stanford Encyclopedia of philosophy states that -

"Because abstract objects are wholly non-spatiotemporal, it follows that they are also entirely non-physical (they do not exist in the physical world and are not made of physical stuff) and non-mental (they are not minds or ideas in minds; they are not disembodied souls, or Gods, or anything else along these lines). In addition, they are unchanging and entirely causally inert — that is, they cannot be involved in cause-and-effect relationships with other objects"

But instantiations of any form belong to that form according to proper platonism, so why aren't the "non physical" forms considered causally effective by modern Platonists?

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    Hello: by 'modern Platonists' do you mean (a) present-day or recent scholars of Plato's philosophy or is your reference to (b) Mathematical Platonists, for example, who adopt or adapt only a portion of Plato's work? – Geoffrey Thomas Nov 24 '19 at 13:44
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    Because abstract objects are not platonic forms, and modern platonists are not Platonists. Plato's animated forms are deemed too magical and metaphorical to be taken at face value today. This said, Plato did not operate with the modern notion of causation, so ascribing "causal powers" even to them is a stretch. – Conifold Nov 24 '19 at 13:46
  • @Conifold Abstract objects are platonic forms according to at least some mathematical platonists, if i am not wrong? – ramseysdream111 Nov 24 '19 at 14:47
  • @GeoffreyThomas I meant mathematical platonists like Gödel and so forth – ramseysdream111 Nov 24 '19 at 14:47
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    If there are Gödel is not one of them, his philosophical source was Husserl. And "mathematical platonism" generally is a very generic idea that there is something objectively existent about mathematical entities that takes little specific from Plato, even when the expression "platonic form" is used. – Conifold Nov 24 '19 at 14:59
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As I understand, this question: a) pertains to modern Platonists' thinking, and b) has a false assumption behind it.

I'm going to take a chance and make an assumption of my own: if I succeed in explaining (b), then (a) becomes a moot point. If, however, you want to know (a) even if it's hypothetical -- you don't need to read any further.

The false assumption (mentioned earlier in the comment by polcott) is about abstract objects exiting in their own realm as the non-physical timeless forms. And in Plato days it was perfectly reasonable to envision this kind of magical cloud storage for concepts. In purely design terms was is way more efficient than the actual implementation. Unfortunately, we cannot ignore the fact that it needs magic to work, not anymore.

The recent advances in Artificial Intelligence accidentally shed the light on the most mysterious aspects of the human psyche -- emotions, intuition, consciousness, qualia -- things that we perceive as magical or supernatural. Now we have a model that explains away the magical stuff.¹

For the abstract objects specifically (and just as the rest of the concepts/ideas we develop), it means each of them is but a piece of information:

that was Google.. last time I checked

We can always think of it as a binary string -- because no matter how it is actually encoded, it can always be translated into the famous sequence of ones and zeros. And those things are very much real, existing as records on some physical media. In this particular case, that physical media is our brains. Every person that has acquired an idea of a square, has their very own copy.

And they are copies -- very often they are actually replicas, as it is the case with the square. Other times every person comes up with their own ideas, as Mary did after seeing red for the first time. And, given the (very observable) fact that we all share, and are parts of the same objective reality, we end up sharing many identical (like ) or similar (like ) or, at times, even predominantly wrong² ideas about it.

Incidentally, it also creates an illusion of every concept existing as a singleton that we all share.

¹ available upon request

² like 'cmon, what do you mean that wasn't funny?!

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    "we have a model that explains away the magical stuff." is not true. Whether you want to belittle qualia as "magical" or not, the hard problem of our experiencing them remains. My guess is that you are thinking of Integrated Information Theory, which one of its founders described as based on the proposition that "consciousness is what information feels like when it reaches a certain level of complexity." That "magic" feeling is known as the Hard Problem. You have not explained it away, merely brought it into sharper focus. – Guy Inchbald Jun 5 at 10:48
  • What is this model that explains away the "magical" stuff? – ramseysdream111 Jun 5 at 14:50
  • @GuyInchbald -- I never heard of Integrated Information Theory and by the sound of it, it takes the opposite approach to explain our minds. In my model, we have two minds and their designs and principles of operation are nothing alike. One is the rational/conscious self.¹ It allows us to understand things by creating and running a c̸o̸m̸p̸u̸t̸e̸r̸ simulation of the world around in our heads. We understand something when we acquire a mental model for it. We become self-aware when we model ourselves as part of that simulation. ...end of part 1 – Yuri Alexandrovich Jun 6 at 0:23
  • ramseysdream111 -- it was too long, so here is a link to a Google doc – Yuri Alexandrovich Jun 6 at 0:36
  • @YuriAlexandrovich That is very old news. You do not address the subjective qualities of experience, of what it feels like to run a self-model within a world model. That, specifically, is known as the Hard Problem. It is far older and more well established than IIT. Where Platonism proposes an Ideal realm, IIT proposes that we consider thoughts as information and you simply ignore the Hard Problem as if it did not exist. – Guy Inchbald Jun 6 at 7:06
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The relationship between abstract objects and platonic forms is obviously difficult, as is the relationship between modern (specifically mathematical) platonists and adherents of Plato's philosophy of forms. While the latter may seem to have a bigger problem denying the causal efficacy of forms (per your argument) I am not sure whether the mathematical platonists (or philosophical platonists modelling their platonism on the mathematical one) cannot also be maneuvered into problems concerning their claim of the causal inefficacy of abstract objects.

https://plato.stanford.edu/entries/abstract-objects/ points out the problem in claiming "that abstract objects are distinctively neither causes nor effects", simply because abstract objects like novels "come into being as a result of human activity." So there seems to be a clear one-way causal efficacy that the mind has in creating abstract objects/artefacts. If you now consider what happens if you read such a work, then (at least from a platonist perspective) the abstract object that is Dante’s Inferno causally interacts with your mind via its instantiation (your copy). So in that sense the causal efficacy of abstract objects (or forms) could be argued for.
I am sure that specifically non-platonists would want to deny that this kind of "interaction" can be considered causal in nature, but I am not quite sure what their main argument would be.

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