IN the context of a discussion about Platonic philosophy, the 'higher plane of being' is the domain of forms. — Wayfarer
Hmm. I thought you were referring to a realm of meaning, value, wisdom and consciousness rather than a realm of mathematical abstracta.
I'd say that Plato's forms are easy to understand in terms of constraints or immanent limitations. They are the shapes, the structuration, that stand at the edge of material possibility. And this connects to the initial discussion about information/entropy.
So the realm that maths inhabits is the zeroed realm where dimensionality and energy have gone to their effective material limit. A code, for example, is dimensionality constrained to a 1D sequence that is then composed of 0D points. The natural numbers are just such a structure - with the addition of the points being arranged in an ordinal sequence. The code contains a message in that it points the temporal direction for acts of counting.
So everything about mathematics can be understood as a limit description on dimensioned materiality. There is a world of action and direction. Then there is the antithesis of the "realm" which is the emergent limit on actions and directions. The reduction of actions and directions produces this ghostly space of the zero-d - an infinite wasteland of discrete points, which can then be semiotically imbued with private meanings.
Once entropic existence is reduced to a set of bare marks, then the marks can take on unlimited meaning within a new level of semiotic mechanism. That is, us humans can describe the structure of the Cosmos in terms of constructive patterns. We can build systems of constraint using our mathematical templates - our ideas about triangles, numbers, manifolds, and so forth.
So Plato's realm is what springs up at the edge of material existence. It is the "flattened" view of the whole produced by dimensional constraint going to its extreme. Collapse dimensionality and energetics - actions with directions - and you wind up with patterns of marks that can be used semiotically to encode the world just collapsed.
Is a triangle real? Well our concept of a triangle certainly encodes the core facts of spatial geometry. We can throw away nearly everything - all entropic irregularity or actual dimensionality - to arrive at a limit state description in terms of a number of sides, a sum of internal angles, a quantification of a compact surface in terms of its ultimate simplicity.
Where does this triangle exist? Well, in our minds, in our habits of conception. But also it "exists" in the world as a particular ideal limit - a constraint on 2D dimensionality using the least number of 1D edges and 0D vertices. So it doesn't really exist in the world as limits are where existence finally ceases to exist. There are no perfect triangles in a materially real world, just their asymptotically close approximations.
So my point is - connecting again to the OP - is that the "higher plane" really exists in the semiotic view. There is an epistemic cut that divides reality into its entropic material sphere and its semiotic informational sphere.
Existence or being can be accounted for as the constraint on potential. In the beginning, the Cosmos would have been unlimited materiality - just pure unbounded fluctuation. An infinity or chaos of action and direction. In expressing every possible action and direction, this vagueness would have been no kind of action or direction in any proper sense at all.
Then out of this "everything goes" conflict, constraints would have to emerge. All the conflicts would start to cancel each other out, leaving only what counts as the simplest harmonies or resonances. In quantum cosmology terms, this is exactly the path integral or sum over histories approach. Order must emerge from chaos. Free action must still find its long-run equilibrium balance.
And so the dimensionality of initial cosmic chaos would be reduced. It would collapse towards the definite three spatial directions and the one collective temporal dimension we experience. The maths of symmetry and symmetry breaking are particularly good for describing this natural self-simplifying tendency. Physics is deeply mathematical because symmetry maths encodes the greatest possible states of simplicity. Symmetry maths explains why the goal of the Cosmos is to become as reduced in direction and action as it can get - the ultimate imperative that is a Heat Death.
But in this story of entropification - the slide towards greatest equilibrium simplicity - is then to be found the other side of the coin, the informational realm that emerges ever more strongly as dimensionality is flattened and simplified. As the Universe heads towards the closest it can get to zero-d constraint, that produces the new possibility of negentropic semiosis. You can get the counter-action of regulating the material world through zero-d systems of symbols, marks or codes.
Plato's realm comes alive in our hands. Ideas about numbers, triangles, and other abstracta, can be turned from being the deadened limits on materiality to the formative constraints we place on still lively materiality. As long as there is a little heat left in the Universe, we can mine it, regulate it, for some privately created meaning or purpose.
Of course, overall, our human-centric semiosis or negentropy has to be entropic. We must produce waste heat whenever we do work. But for us, Plato's realm is a higher plane of being in that it is a place from which we can actually act and give direction. It encodes the physics of the world in a way that is real, but also in a way that is causally reversed in that we are imagining the patterns as not emergent but constructed. We reframe the entropic truth of the world in a way that is technologically convenient.
Now you may see that as a spiritual move. Plato's realm is somehow accessing something divine or actually transcendent.
But the semiotic story is only of a faux transcendence - an epistemic cut. We extract a story about limits so that we can impose those limits on nature through acts of entropic construction. If I want to build things, I can have in mind a kitbag of ideal shapes, like triangles, cubes, planes, etc.
Plato's higher plane exists in our imagination as the entropic world turned around on itself. So it "really exists" as being a realm of physical limits or ideal constraints. But it is really a mirror-land in that it is this physical realm imagined in terms of the constraints being constructable. The causality is flipped around from being top-down to bottom-up.
(Although of course the causal relation between the forms and the physics is precisely what created all those Platonic ontic puzzles - the allegory of the cave. It was clear that the constructive approach of actual mathematics was in conflict with the fact that in nature, limitations are emergent. Platonic debate recognised the disjunct, but failed to resolve it. Hence the dualism that has bedeviled the subject ever since.)