Reactions to Plato's Republic and Gulliver's Travels

In Book VII of Plato's Republic, Socrates and his buffoon of a sidekick, Glaucon are deep in conversation covering several topics. At the start, Socrates describes his now-famous Allegory of the Cave. This is a very interesting part of the book, especially regarding its parallels to Edwin Abbott Abbott's Flatland. When Socrates begins his explanation of the cave, he describes a group of people shackled to the cave floor. All that these people have ever seen consists of moving shadows. In a sense, they are living in a limited dimensional space at this point. What they can see is basically a plane, and the images that they see are all two dimensional images on the plane. They can see two-dimensional shapes, they can see motion, and occasionally they also hear voices. This is obviously very limited compared to the spacious three-dimensional world which we humans have become quite accustomed to. Eventually, in Socrates' story, one of the inhabitants of the cave is allowed to venture out into the real world--the world of three vibrant dimensions. The shock of "realizing" the real world is great for the cavemen. They are in much pain due to the sun. Eventually, however, they become accustomed enough to see their shadows, then their colored reflections, then actual three-dimensional objects.

In Flatland, A Square describes his world, which also has a limited dimensionality compared to Spaceland. a Flatlander's eye can only see lines, and occasionally can see depth (when the fog cooperates). Just like Socrates' cavemen, A Square (at first) could not comprehend that there was more in the world than his two dimensional Flatland. At some point in the story, a member of Spaceland (a sphere) offers to share with A square the secrets of space. The sphere tells the square that he can move him in the "up" direction. A Square at first cannot conceive that there exists any direction aside from North, South, East, and West (similarly to how a human would have trouble of conceiving of a direction other than North, South, East, West, Up, and Down). When the Sphere finally lifts A Square off of his plane, A Square is overwhelmed. (There is a certain amount of shock accompanying these sort of forced epiphanies.) A Square eventually realizes that he can see all of Flatland's houses, people, etc... simply by virtue of being in this new, third dimension.

In Socrates' Cave Allegory, the freed cavemen eventually are forced to return to their old habits in the cave. At this point, they know that there is more to life than shadows on a cave wall, and scoff at those who believe that the shadows were "everything". Of course, none of the cavemen who had never "seen the light" would believe in any of the experiences that the freed cavemen briefly had. Analogously, A Square has trouble conveying the idea of a third dimension to other Flatlanders. He soon realizes that he will not be able to convince anybody about the truth of his travels, so he decides to keep quiet.

Of particular interest, is A Square's immediate understanding (by analogy) that there must be something Beyond the Third Dimension (now available at your local book store). At a much later point in Socrates' discussion/lecture with Glaucon, Socrates does speak about a progression of dimensions. Socrates stops, though at the third dimension. This is probably because there was very little known about spatial geometry back in his day. After all, for his purposes, three dimensions was all that was needed. He was making his way towards astronomy. To him, this only involved three dimensions.

Just like the cavemen and A Square, it would probably be silly for humans to believe that there is nothing more than three dimensions. Just as the third dimension helped A Square to understand better his two dimensional world, a comprehension of the fourth dimension would surely aid man in understanding more about our three-dimensional world.

And Jonathan Swift wrote a book called Gulliver's Travels.

Dan Margalit