A Comparison of Flatland and Plato's Parable of the Cave

A Comparison of Flatland and Plato's Parable of the Cave

A Story by Stein

This was an essay I had to write for a summer math class I took during the summer of 2008. I consider this to be one of my most unique challenges while in school.


Edwin Abbott’s Flatland and Plato’s Parable of the Cave are regarded as two of the greatest works of their time. They both have the same underlying theme with an important message to the reader. That message is revealed as the two stories unfold with characters that live in a world that they consider their own, much like ours. While these characters are led to think and believe that the only world that exists is theirs, they are eventually forced into the realization that a greater world exists, whether it be the outside of a cave or the vast world of a third dimension. The translation of Plato’s Parable of the Cave that I have chosen to use for my comparison is by Benjamin Jowett. 

            One of the first comparisons made when reading Plato’s Parable of the Cave and Flatland is when Glaucon says to Socrates “you have shown me a strange image, and they are strange prisoners.” Socrates responds by saying “like ourselves.” This exchange takes place when Socrates is explaining to Glaucon how the cave looks and how it is laid out. While in Flatland no dialogue takes place that could be similar to that, one must ask, what exactly does Plato mean? Is Plato speaking of a prison in which everyone in the world is failing to expand their minds and believe that there is a greater element to the world in which we live? Clearly the comparison could be made from that statement to Flatland. If you recall when the Square talks of the nature of Flatland and how they travel about and what they see, you remember that in Flatland they can only see in two dimensions. But when the Square is confronted by the Sphere, he refuses to believe that there is a greater world than his; that a third dimension doesn't exist. The comparison could also be comparable to when Jesus was healing people and nobody but the disciples had seen it. Obviously nobody believed that Jesus could heal the sick. Jesus also said that his father (God) had a heaven. A similarity could be made between that and the sphere saying that a third dimension existed. The third dimension would appear to be above Flatland which would be comparable to the Christian belief that heaven is above earth among the stars and planets.

            Another comparison can be made when Socrates says “to them truth would literally be nothing but the shadows of the images.” He is talking about the prisoners in the cave and how the only thing that they can see is the shadows of the puppets on the wall that stands in front of them, which is the only thing they have known since they have been in the cave. When you compare this to Flatland, one is reminded of how the good people of Flatland can only see things in the second dimension as the Square explains about the penny on the table and how moving to the level of the table you can better understand how Flatland people see (Sect 1.20-28). Flatlanders have only known two dimensions in their lives and refuse to believe that a third dimension exists, as stated earlier. In both situations, the prisoners in the cave and the Flatlanders, there is a greater world beyond their own, they just don’t know it yet.

            Socrates also says, “Suppose once more, that he is reluctantly dragged up a steep and rugged ascent and held fast until he’s forced into the presence of the sun himself. Is he not likely to be pained and irritated? When he approaches the light his eyes will be dazzled and he will not be able to see anything at all of what are now called his realities.” Comparisons can clearly be made here between Flatland and The Cave. What Socrates is explaining here is that when the prisoner is forced to leave the cave, he will first realize that the shadows aren't reality. His eyes will be irritated because the only thing he has ever known was darkness and now being forced into the light he will not be used to it. The prisoner will realize that once he has left the cave that the real world isn't a world of darkness, but rather a world of sunlight. He will realize that shadows don’t inhabit the real world, but rather human beings. And he will realize that he is now in what is and what will be his new reality, his new world. The same can be said about the Square in Flatland.  The Square refuses to believe that the existence of a third dimension is possible. Although he has had an alien visitor in the Sphere, who tries for several moments to convince the Square that his young grandson was correct and that a third dimension does exist, for which the Sphere came, the Square still refuses to believe that it is possible for a third dimension to exist. Finally, the Sphere realizes that he cannot use reason with the Square and he pulls the Square into the third dimension where the Square becomes disoriented and “an unspeakable horror seized me” (Section 16.26-18.1).  The Square says that “there was darkness; then a dizzy sickening sensation of sight that was not like seeing” which was when he was pulled from Flatland to the third dimension, otherwise known as Spaceland (Section 18.2).  Obviously in this case the Square has been forced into the third dimension similar to the way that the prisoner was forced out of the cave. Both the prisoner and the square are having reactions to their new view of the world; both are trying to comprehend their new realities. Another comparison can be made even from this.

            In Plato’s Parable of the Cave, Socrates says to Glaucon, “he will see him in his own proper place, and not in another; and he will contemplate him as he is.” The point to be made here is that the prisoner, who for so long could only see shadows and believed for so long that those shadows were the true beings of his world, sees for the first time that he is a human being and that he is not a shadow and that shadows don’t inhabit his world. Simply, he is seeing himself as he truly is. The same can be said about the square in Flatland. When the square was pulled into the third dimension by the Sphere, the square was able to see things in the third dimension. Even more interesting is that the square can see the world that he once thought only existed as his reality. The square says “I looked down below, and with my physical eye all that domestic individuality which I had hitherto merely inferred with the understanding” (Section 18.35). The square is explaining to the reader that he can see the world that is Flatland. The square says that he can see his grandsons and sons asleep in their rooms (Section 18.38), and he can see all of Flatland as he and the sphere fly around. The square finally believes and understands what the sphere was trying to explain to him. While the prisoner and square have different meanings to their new realities, they are essentially the same idea, that both lived in worlds that they thought were the only realities and once they were forced out of their worlds, they came to realize the true realities they once thought didn't exist.

            Socrates also explains to Glaucon, “and when he remembered his old habitation, and the wisdom of the cave and his fellow-prisoners, do you not suppose that he would felicitate himself on the change, and pity them?” Socrates is speaking of the prisoner that has left the cave, once he realizes the reality of his new world, he will think back to the other prisoners and have pity on them since they would have no idea that the new reality that he now inhabits, exists. Similar is the situation that the square finds himself in Flatland. After the square has been pulled from Flatland into the third dimension by the sphere, he can see all things in a way that is countrymen cannot. We see that both the prisoner and the square would have pity on their own kind though, since they are experiencing the riches of a new reality while the other prisoners will forever be forced to live within their imprisoned minds.

            When Socrates says “the prison-house is the world of sight, the light of the fire is the sun, and you will not misapprehend me if you interpret the journey upwards to be the ascent of the soul into the intellectual world according to my poor belief,” he is saying that what we see and what our visions are, can be imprisoning. Imagine being a prisoner in the cave and only being able to see shadows on a wall and for a lifetime believing that those shadows of puppets were reality. Or that you’re an inhabitant of Flatland and the only world that exists to you is a world of two dimensions. Doesn't that seem imprisoning? Another comparison has yet to be made though. The final words of Socrates statement talk about the “ascent of the soul into the intellectual world.” A clear and concise comparison can be made here. When you think of this statement in reference to Plato’s The Parable of the Cave, the prisoner in the cave that ascends out of the cave is ascending into the intellectual world. Everything the prisoner has known in the cave, the darkness, the shadows, is no longer his reality. The intellectual world is represented by the outside of the cave, what is real. When you compare this to Flatland, it is a similar experience for the square. Obviously we know that Flatland is a two dimensional world. When the square is visited by the sphere and is eventually pulled into the third dimension by the sphere, the square is making the ascent to the intellectual world. But, the square’s true ascent into the intellectual world is when he finally views the third dimension for himself and sees what the third dimension looks like that he finally has made the ascent into the intellectual world. One could even make the argument that the square hasn't fully made the ascent into the intellectual world until he expresses his intense desire for knowledge to the sphere about the third dimension and a possible fourth dimension that he has finally made the full ascent into the intellectual world (Section 19.101-19.200).

            Socrates then states “anyone who has common sense will remember that the bewilderment of the eyes are of two kinds, and arise from two causes, either from coming out of the light or from going into the light.” Almost immediately a comparison can be made to Flatland, but first what was Plato trying to say about the cave? Is he saying that prisoners of the cave lacked common sense? What Plato is saying here is that the prisoner of the cave that has finally been able to come out into the light is confused and is trying to figure out what is real and what is not, although the shadows we know as real, the prisoner of the cave believed that the world that we call reality is actually a world of fantasy. Or perhaps Plato is saying that the prisoners that are still in the cave are being betrayed by their eyes, leaving them puzzled and confused as to what they believe is reality, because they of course have seen shadows on a wall for so long they can only believe what they are seeing, whether they are coming out of the cave into the world of reality that is lighted by the sun, or by a prisoner that is too stubborn to believe that anything beyond the cave is reality, and that the darkness and the shadows are the true world. Comparing this to Flatland, one can see that this is comparable to when the square and the sphere were arguing about a third dimension. Clearly the square was puzzled when he first was forced into the third dimension, and he could not believe what was happening. But this is also comparable to when the square refused to believe that a third dimension existed and even when the sphere refused to admit that a fourth dimension existed after the square was becoming curious and wanted to know more. These are both prime examples of what Plato was trying to explain with that quote.

            One of the final statements that is made in Plato’s The Parable of the Cave, is when Socrates says to Glaucon, “our argument shows that the power and capacity of learning exists in the soul already; and that just as the eye was unable to turn from the darkness to light without the whole body, so too the instrument of knowledge can only by the movement of the whole soul be turned from the world of becoming into that of being.” What Plato meant by this statement was that as human beings we are curious to learn. It’s our natural instincts to be a curious being and want to learn more. But in some cases curiosity isn't enough. Some things seem so outlandish, so unbelievable that it seems like more of a fantasy than a reality. It doesn't just take the thought of something to turn us into believers, it takes facts, and that is what Plato means by this. Take the square for example. He didn't believe in a third dimension until he was finally in it and could see for himself that it existed. The latter part of the statement also suggests that we as humans must truly see something before we can believe in it. Just because it isn't there doesn't mean that it doesn’t exist. Naturally humans are curious, but they are also hard to convince. The world of becoming is the path and journey that we take into that level of believing and the world of being is once we have finally accepted the truth and are content with it. No matter what the journey is that we choose to take whether it is a religious journey or a simple fact finding journey, they are all the same, and all require the same process of becoming to being.

            Many lessons can be taken from these two stories. Although at first they seem simple, they are actually works that, even after all of these years, have been used as academic tools to show society and different cultures that the ascent from our caves of comfort into the real world of academic thought and progression is necessary. In both of these works we watched two characters expand their minds, even in the most simplistic ways, and they both provide the basis of the same academic thought that will continue for the next century and beyond. 

© 2013 Stein

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Added on June 23, 2013
Last Updated on September 14, 2013
Tags: Academic, Math, Essay, Plato



Pittsburgh, PA

I am a father and a 2009 graduate of Clarion University of Pennsylvania with a degree in Mass Media Arts, Journalism and Communication Studies more..


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