Let us suppose now that the man in Flatland begins talking to the sphere, and he says to the sphere: "What is it like to be a sphere? The sphere says, "I'll tell you what it's like; draw a circle on your floor." This is not easy for the man in Flatland to do. His perception of a circle is a constantly curving line that returns to its origin, but he cannot see the entire circle at once. He can only see the side of the circle facing him. The only way he could see a whole circle would be to be inside the circle, and if he got inside he could never get out. People in Flatland commit suicide by drawing circles around themselves that they can never get out of. Because of this it takes along time for him to draw the circle. The sphere is most impatient with all this because he could have done it instantly. Finally the circle is completed and the sphere says, "Now what I want you to do is to rotate the circle! What he has in mind is that the man in Flatland will rotate the circle about its diameter producing a sphere, but what the man in Flatland does is to rotate the circle about its circumference, spinning it like a record on a record player. "No, no--rotate it the third way,' says the sphere.” There is no third way you fool," cries out the man in Flatland, and for him this is true. There is no third way, no up and down in a thickness direction, and absolutely no way for him to comprehend what the sphere is talking about or what the sphere is. The only thing that he can understand is the world or dimension in which he lives.
I think we are like that, like Flatlanders. We live in our day-to-day three-dimensional world thinking that this is all there is. The perception of those living in Flatland limits their interpretation and awareness of a three-dimension world. Then an idea might come to our mind. Is there a existence of a four-dimensional world in which God, who is in a higher dimension than we are, a God who has the same kind of relationship to us which the sphere had to Flatland (and much more), has indeed touched our little "Flatland," so to speak, and in violation of all of our laws of science created matter out of nothing.
In the allegory of the cave, Plato states there are four stages of knowledge development: Imagining, Belief, Thinking, and Perfect Intelligent. Imaging is at the lowest level of development. For example, take the shadows. If you know a shadow is something real then you are beyond the state of imagination which implies that a person is unaware of observation and amounts to illusion and ignorance. Belief is the next stage of developing knowledge. This stage is more advanced than imagining because it’s based more firmly on reality. But just because we can actually see the object and not just its shadow, doesn’t mean we know all there is to know about the object. And just as the prisoners mistake the world of the shadows as the truest form of reality, so do the non-philosophers when they deny the existence of any stages of reality above Imaging and Belief.
In the next stage, Thinking stands for the power of the mind to take properties from a visible object and applying them. Thinking is the visible object but also the hypotheses. Thinking still doesn’t give us all the information we crave and we still ask why. So for Plato, the last stage of developing knowledge is Perfect Intelligence. In this stage, hypotheses are no longer present because of its limitations. This stage is like the world outside of the cave, where the real objects that create the shadows exist. And when the prisoners, upon leaving the cave, will be forced to abandon their old ways of understanding their world, so the person will realizes the existence of Thinking stage and Perfect Intelligence stage
Plato concludes that the real world is not what we see but what we understand or feel in an intelligible world. For Plato, knowledge gained through the senses (empirical experience) is no more than opinion. Knowledge gained through philosophical reasoning is certain. The allegory of the cave makes a contrast between people who see only appearances and mistakes them for the truth, and those who really do see the truth.
There are similarities in both books that the perception limits their interpretation and awareness of us, really affecting our way of reasoning. We live in our day-to-day three-dimensional world thinking that this is all there is. In the cave, Plato describes a narrow vision of shackled prisoners seated in a dark cave facing the wall. Chained also by their necks, the prisoners can only look forward and see only shadows. These shadows are produced by men, with shapes of objects or men’s, walking in front of a fire behind the prisoners, Plato states that for the prisoners, reality is only the mere shadows thrown onto the wall. So as we can see, only those living in a higher dimension (in Flatland) or in a world outside of the cave, a world of real objects can interpret the world correctly. Some of the prisoners may deny the existence of a world outside the cave and challenge anyone who claimed otherwise. The chains, in the allegory of the cave and the Flatland indicate the lack of freedom of those locked into an understanding of the world that is entirely reliant on unreflective sense-perception. Remember that for Plato, the more one knows the more one is able to act in accordance with the forms, with what is actually true and best. Knowledge is thus both liberating, as well as the necessary and sufficient path to happiness. Thus the individuals in chains are ignorant, and this ignorance of the true nature of reality is what is keeping them in the cave or in the Flatland.
