The allegory of the cave has got to be one of the most innovative and instructive philosphical models in all of history; truly all perception is but a shadow of reality, and it is possible to have mastery over all of banal experience without gaining any insight into the true nature of reality. One thing I often think of as I consider the cave in which we're stuck is the notion that we, as animals, cannot ever see things, but rather only their reflected light. All that we can know is what sort of patterns of light rays are reflected or refracted by an object... the many sorts of optical illusions to which the human eye may fall victim are ample evidence that two things which may seem the same can be quite different.
Optical illusions are but a simple demonstration of the manner in which the brain constructs its own reality; many other perceptions may be skewed or utterly false without our knowing it. It could be argued that all of our so-called "reality" is actually entirely a construct. Indeed, the laws and rules of even mathematics itself are pretty much arbitrary; if the language of the universe is created, rather than discovered, by humans, than perhaps the same could be said of the universe itself. If this is true, then there is no light in the cave, only shadows, and he who turns his head is only fooling himself into believing he has achieved enlightenment.
But that's a question for the disenfranchised youth and coffee-house philosophizers.
A fascinating aspect of the allegory of the cave is the manner in which it so well describes our relationship to objects in the fourth dimension; what we know of its inhabitants-A Hypercube and her sisters-is only shadows, three-dimensional hints as to their true nature. Is it possible that we might turn our heads and truly understand the construction of a 4-space object? And if we could, is there any way we could convey that understanding using 3-space communication? Or would we be called crazy? Obviously Mr. Abbott was interested in this question as he condemned A Square to prison for his knowledge, and Plato expresses the frustration of the enlightened in trying to return to mundane reality. Plato and Abbott also share an interest in the relationship between mathematics, that most pure of all the arts, and the human condition. In book VII it is written, "geometry will draw the soul toward truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down." In this, as in Abbott's use of complex geometry to illuminate the evils of society and express the arrogant ignorance of mankind, the authors suggest that understanding the most fundamental organization of the universe provides access to an understanding on a much higher level. Is it possible that a knowledge of mathematics will lift the spirit toward true knowledge? Ask me at the end of the semester...