In Godel, Escher, Bach--my current literary love interest-- Douglas R. Hofstadter is on the hunt for one of the most interesting, tantalizing, and elusive constructs of the human mind: the concept of "I", of the individual's ability to self-reference. During his own intellectual odyssey, he realizes that the works of mathematician Kurt Godel, artist and visual philosopher M.C. Escher, and musical titan J.S. Bach all demonstrate similarly self-referential properties--properties which he believes are tied to the way our consciousness works.
But what is self-reference? On the surface, it seems fairly simple: To exhibit self-reference, something has to "refer" to itself. A person can refer to trees, books, emotions, laws, wine bottles, interest rates, and, of course, other people. So what should be so strange about me referring to myself? You cook, I cook. He laughs, I laugh. The change of person seems almost trivial...but thinking deeper about the situation reveals some very evident weirdness. What am "I"? "I" changes for every person, but the concept stays the same. Or to pose another question, how do we so easily reconcile our perceptions of external stimuli with our perceptions of ourselves? We do it so casually, but in science it would be tantamount to trying to use a microscope to examine itself or trying to verify the length of a ruler by referring to it's own inches markings. When it comes to most things, in fact, the idea of self-reference and self-perception is quite strange.
Let's take a cue from another self-referential situation. A TV camera in a news studio is filming two anchors give their daily 6 o'clock report. The camera films people, blue screens, fake plants, and a giant set. When the news is done, the anchors have gone, and the cameramen are packing up, one of them swings the camera over to one of the TV monitors on the edge the set. The TV monitor now displays itself, since the camera is pointed at it. What is on the TV screen within the TV? Another set with another TV of course! And, as you've probably have seen for yourself, this effect goes on to infinity...or theoretically goes on to infinity, but in reality stops at the resolution of the camera. The resulting image is an absurd repetition of TV images buried in more TVs, on and on and on. TVs "referring" to themselves is a strange situation indeed.
This is what Hofstadter classifies as a "Strange Loop" or "Tangled Hierarchy," in which A leads to B, which leads C, which somehow magically leads back to A. Perhaps you may recall a famous optical illusion where monk-like people trudge down stairs on the top of a castle, and although all of the stairs go "down," the small people end up back where they started. That's an Escher, by the way. So is the picture of a waterfall turning a mill which lets the water out on to a stream that somehow flows back down (up?) to the top of the waterfall. So is the picture of two hands emerging from a picture, drawing each other. As you can see, Escher's world is one full of Tangled Hierarchies, put forth in the most visually jarring ways.
Although Escher's work helps to visualize the strangeness of these loops, Hofstadter is first and foremost a scientist, and will attack these logic pretzels through formal mathematics. It is then that Hofstadter introduces Kurt Godel into the mix of things. Godel used the Strange Loop concept (not by name, of course) to do something both ingenious and disruptive--he proved that there could never be a complete and consistent mathematical system. Godel's role does take some explaining: In brief, before the early 20th century, theoretical mathematicians--in all of their brainy bravado--believed that they could eventually produce a system of mathematics that was complete and internally consistent. That is to say, they thought that they could devise the perfect system, where all truths would be provable and everything provable would be true.
Godel's work simply shows the magic of simple statements like "This sentence is false," (which is another example of a Strange Loop) can be translated into mathematical jargon and disrupt logical paradise. The details are gnarled and, I'll admit a few neurons were warped in the process of the understanding them. Nevertheless, the end result is crystal clear: Godel's Incompleteness Theorem demonstrated that some statements can be known to be true, but cannot be proven within their respective system. Thus, no system can be mathematically perfect.
Now, you should know that this search for these "air tight" systems was not just some current academic fad. It was a strong belief by many mathematicians, for hundreds of years, that number theory could be perfected and completed. So it was a huge event in the history of mathematics when a 25-year-old Austrian, barely out of graduate school, destroys any hope of achieving this end. And, of course, it's all the more intriguing for our story to realize that it was a Strange Loop that was used to strangle mathematical perfection.
So, what does all of this have to do with your consciousness? Let's keep the suspense about the big question, and answer the second one first. Johann Sebastian Bach was a mathematician's musician. If you have never heard Bach's work (which is improbable...you probably just didn't know it was Bach), then download one of his Fugue's or the Goldberg Variations, and you will here something that might well be described as "the music of numbers." Bach's works were full of inversions, transformations, and expansions--terms usually left to Algebra II textbooks. Yet, Bach could devise simple melodies which could be inverted (essentially take the notes and put them upside down on the staff), reversed (play it backwards), and transposed (played in a different key) and somehow these transformations could be put together with the original melody and produce beautiful, complex harmonies.
Bach also produced an interesting tune, known as the Crab Canon, which--through subtle melodic shifts that are barely noticable--rises to higher and higher keys until...it gets to the original one. Kind of like a...yep, Tangled Hierarchy.
So, that's all the steam I have for now. Yet Hofstadter keeps going. The book is over 700 pages. I'm barely at 300. I will let you ponder how consciousness may fit into Hofstadter's theme, partially to give you something to think about, partially because I don't entirely know myself.