That shapes do not metaphysically exist , or if they do, there is only one metaphysical shape which is beyond perception and just within the grasp of cognition.
How absurd is it to state that shapes do not objectively exist! I can look at myself and in my environs and see a multiplicity of shapes: spherical eyes, near-hexagonal palms (try measuring the rough shape of your palm and it should come back hexagonal) rectangular pieces of paper, etc etc. Everything we see is bounded within a shape. So shapes exist, that much is axiomatic. But is this recollection of the external objective or subjective, that is are these shapes out-there, things-in-themselves, as Kant would put it, or phenomenological appearance, or more simply perspectives.
What would even prompt us to doubt the objective reality of shapes? Let us think about how any given perspective of any shape CAN actually change the shape we are observing. Imagine for fun’s sake, and definitely not thousands of years of empirical data, that we are living on a rotating sphere floating in space. When I look at this sphere from one perspective say a bounded field, an opening within a forest whose ends reach to the horizon it would appear to me that this gigantic shape is not spherical but rectangular. You might say that I would still note the curvature of the earth, so maybe from this perspective the shape is actually a noodly, amorphous rectangle, a non-Euclidean shape. Either way from this perspective the object’s global perspective differs from its manifested perspective. Is this shape a sphere or a rectangle, or something in between? If it is sphere why does it appear rectangular, if it is a rectangle, why does it appear a sphere?
Spheronimus: The Earth is a sphere, because we call a shape by its entirety, not its diminished localized perspective
Rectanglothon: The Earth is a rectangle because we call a shape by its manifested reality, not its ideality.
Spheronimus: How can you believe the Earth is a rectangle? You are stark raving mad!
Rectanglothon: You will see brother that I do not hold this belief in its crude form. But please play along with me. From what dimension of reality, are you viewing reality?
Spheronimus: What do you mean?
Rectanglothon: Is a sphere always a sphere?
Spheronimus: Now you are talking in tautological nonsense.
Rectanglothon: You give me little benefit of the doubt, no matter. When I look at a sphere from four dimensions, what shape is it?
Spheronimus: It is an unextended circle. Ah now I see what you mean. I can agree that changing from a lower dimension to a higher will change the perspective so that what is in reality a sphere will now look a circle.
Rectanglothon: But my brother, you hold on too dearly to your own preconceptions of reality. Let me ask you this, is it certain that the 3-dimensional perspective is the correct one, or merely the one the gods have bestowed upon us. As an esteemed mathematician, you must know that reality extends countably infinitely in every direction. Spacetime exists on an unseeable four-dimensional framework. And so too must a countably infinite number of scientific models and theories belong to the mystical realm beyond three dimensions.
Now if we will be honest men, we will state that our three dimensional perspective is arbitrary. I have heard a story from the Syracusans that Zeus, in his beneficence, wished to give heaven to man, and placed it within a four-dimensional reality. But when man heard this, he demanded more than his station, asking for heaven to be placed one higher dimension, in the 5th dimension. Because of his greed and arrogance, Zeus cursed man to be bound, as Prometheus was to the rock, so was man bound to the third dimension. All in all, there is no prima facie mathematical reason for a three dimensional reality to be the ultimate base-reality perspective.
Back to our question. What has happened to the sphere, how has he transmuted from a sphere to a circle, by merely changing the perspective. And more than that, if we can change its shape by changing our perspective, what is the ultimate correct perspective?
Spheronimus: Ah Rectanglothon, it pains me to hear this. Not because it is wrong, but because I have spent so much time on the fruitless science of geometry! Please speak more.
Rectanglothon: Nothing can come from nothing Spheronimus! How did a sphere become a circle, as though we were magicians and alchemists.
Spheronimus: End mine and humanity’s suffering and reveal all.
Rectanglothon: There must, as a matter of axiom and speculation, be a single metaphysical shape of actually existing objects, the universal, base-reality shape, whose partial perspective produces mathematical shape. Only a shape which transcends, and connects all mathematical dimensions of reality, can produce shapes which change according to perspective.
Mathematics is the science of definition my friend. It relates to boundaries and limitations. Open your eyes wider and you can cognize the infinite and boundless. When we humans are endowed by the gods with 3-dimensional perception, the metaphysical shape conceals itself, and reveals only 3 dimensions of its infinite nature. The essence of a shape is impenetrable in its entirety, but penetrable in part.
Thus brother, the answer is that the metaphysical shape never changes, only the observer and the perspective change. Something is a square pursuant to the observer’s position, or perceptional capabilities. Likewise for the triangle and all other shapes.
Thus we may think and speculate that higher dimensional beings would see us as mere blobs, our entire 3-d world as a sort of chessboard. Spheronimus if reality is a cave of illuminated shadow figures, we must state that Plato, in his divine wisdom, saw but a infinitesimal of an infinitesimal, that the true analogy must be something like the light being used was defective, the shadows on the wall distractions, and an infinity of infinities hidden everywhere.
Spheronimus: I can but barely grasp the meaning of all this. But what do you think is the metaphysical shape, or the God shape as we might call it.
Rectanglothon: I cannot say for certain, and again this is but speculation and my own encounters with the realm of the divine, but it can have no shape. That which produces all shapes must by necessity not be a shape, but something antecedent to it. It is pointless to try to describe such an object, or essence, other than to label it God’s essence, because all our language, mathematics and geometry rely essentially on definition, circumscription and parametrizing. It is the infinite potential; able to morph into anything, everything, and anything possibly imaginable.
I have read in that ancient Jewish religion that the Lord was asked by his prophet Moses to see His face. And the Lord replied that none can see His face and live. From this I interpret an understanding that the infinite is not to be grasped by the finite by its very nature, and that death or a return to infinity is the only means by which we may see the Lord’s essence undiluted by finitude.
Spheronimus: Ah what a fruitless science is this then! Why should I bother myself with these mad but beautiful ravings, when I can never apply it to the world as I could with Euclid’s supreme science.
Rectanglothon: My friend, there is a brutal but efficient nation now conquering the world called the Romans. They build roads and bridges and aqueducts through the old science. If you would be a builder, follow them and their crude understanding of shapes. If you seek truth, and the divine order, follow me.
We may return to the original question. Is the earth a sphere or a rectangle? It is neither, both and everything more since its true shape is metaphysically infinite. It only appears to us, finite three dimensional beings, in certain shapes from certain perspective. Therefore you and I are both partially correct, and both wrong.
“It would seem I am in need of a name change” mused Rectanglothon, deciding to call himself Panthon thereafter.
Spheronimus and Panthon leave, and head towards the nearest bar to drink themselves into a metaphysical stupor.