Math 8, The Fourth Dimension

Plato’s Book VII of Republic portrays a conversation between the philosophers Glaucon and Socrates on the subjects of education, knowledge and wisdom. The speakers begin by describing the captivity of men in a metaphorical cave. The men are blind in the darkness, deceived by the shadows cast before them, and oblivious to their own ignorance. Only when individuals escape from the cave into the light do they perceive their illusion and understand truth. Sight itself is knowledge, but knowledge is meaningless without direction. Wisdom then is not constituted by acquired knowledge, but is rather one’s perspective relative to the light. Those who have seen the true world are infinitely more knowledgeable and powerful when they return to the cave to assist those mired in ignorance. Their frame of thinking has expanded. From the outside, they can clearly perceive the falsity of the shadows. Plato chooses an apt metaphor to describe ignorance, knowledge and wisdom. Those without knowledge cannot know their ignorance. Wisdom is attained through the shifting of perspective, moving outside of one’s former context, be this emotionally, culturally or intellectually. Those who understand the context under which a system is operating have a more powerful frame of reference. With lucidity, they can observe a lower perspective from a higher one.

Once Plato has defined the nature of wisdom, he recommends a sequence of education that will foster its pursuit. He outlines a series of schools that study the dimensions of space, one logically following the next. Arithmetic is first, for it represents the fundamental concepts on which higher dimension geometries develop. The geometry of planes follows. Plato then criticizes philosophers for neglecting the next level of understanding, solids. He urges that the gap between the geometry of planes and the geometry of solids in revolution (astronomy) be bridged by this crucial middle step. The gap signifies a hole in philosophy’s understanding of space. This point underlies a major theme of Book VII. If philosophers omit a dimension, their perception of space will be lacking. One cannot effectively build on a higher dimension without understanding lower dimensions. Philosophers lose their context and perspective and, hence, fail in achieving wisdom as defined by the metaphor of the cave. Geometers must prove and connect all successive levels of space in order to proceed towards wisdom and truth.

To connect this theme to the Abbott’s Flatland and Swift’s Gulliver’s Travels, all three works seek to teach lessons about our own dimensions by means of exploring others. They use higher and lower perspectives to gain a better understanding of our own outlook and its limitations. Furthermore, if there exists a gap in our understanding, if one perspective is neglected, then we face the risk of misunderstanding other aspects of dimensionality. Although Plato rightly extols the goodness of pure knowledge represented in the geometries of dimensions, there is a definite practical wisdom to be gained by exploring novel dimensions and then examining our own perspective from “above”.